The principles of science

by William Stanley Jevons

CONTENTS.

BOOK I.
FORMAL LOGIC, DEDUCTIVE AND INDUCTIVE.
CHAPTER I.
INTRODUCTION.
SECTION		PAGE
1.	Introduction	[1]
2.	The Powers of Mind concerned in the Creation of Science	[4]
3.	Laws of Identity and Difference	[5]
4.	The Nature of the Laws of Identity and Difference	[6]
5.	The Process of Inference	[9]
6.	Deduction and Induction	[11]
7.	Symbolic Expression of Logical Inference	[13]
8.	Expression of Identity and Difference	[14]
9.	General Formula of Logical Inference	[17]
10.	The Propagating Power of Similarity	[20]
11.	Anticipations of the Principle of Substitution	[21]
12.	The Logic of Relatives	[22]
CHAPTER II.
TERMS.
1.	Terms	[24]
2.	Twofold meaning of General Names	[25]
3.	Abstract Terms	[27]
4.	Substantial Terms	[28]
5.	Collective Terms	[29]
6.	Synthesis of Terms	[30]
7.	Symbolic Expression of the Law of Contradiction	[31]
8.	Certain Special Conditions of Logical Symbols	[32]
CHAPTER III.
PROPOSITIONS.
1.	Propositions	[36]
2.	Simple Identities	[37]
3.	Partial Identities	[40]
4.	Limited Identities	[42]
5.	Negative Propositions	[43]
6.	Conversion of Propositions	[46]
7.	Twofold Interpretation of Propositions	[47]
CHAPTER IV.
DEDUCTIVE REASONING.
1.	Deductive Reasoning	[49]
2.	Immediate Inference	[50]
3.	Inference with Two Simple Identities	[51]
4.	Inference with a Simple and a Partial Identity	[53]
5.	Inference of a Partial from Two Partial Identities	[55]
6.	On the Ellipsis of Terms in Partial Identities	[57]
7.	Inference of a Simple from Two Partial Identities	[58]
8.	Inference of a Limited from Two Partial Identities	[59]
9.	Miscellaneous Forms of Deductive Inference	[60]
10.	Fallacies	[62]
CHAPTER V.
DISJUNCTIVE PROPOSITIONS.
1.	Disjunctive Propositions	[66]
2.	Expression of the Alternative Relation	[67]
3.	Nature of the Alternative Relation	[68]
4.	Laws of the Disjunctive Relation	[71]
5.	Symbolic Expression of the Law of Duality	[73]
6.	Various Forms of the Disjunctive Proposition	[74]
7.	Inference by Disjunctive Propositions	[76]
CHAPTER VI.
THE INDIRECT METHOD OF INFERENCE.
1.	The Indirect Method of Inference	[81]
2.	Simple Illustrations	[83]
3.	Employment of the Contrapositive Proposition	[84]
4.	Contrapositive of a Simple Identity	[86]
5.	Miscellaneous Examples of the Method	[88]
6.	Mr. Venn’s Problem	[90]
7.	Abbreviation of the Process	[91]
8.	The Logical Alphabet	[94]
9.	The Logical Slate	[95]
10.	Abstraction of Indifferent Circumstances	[97]
11.	Illustrations of the Indirect Method	[98]
12.	Second Example	[99]
13.	Third Example	[100]
14.	Fourth Example	[101]
15.	Fifth Example	[101]
16.	Fallacies Analysed by the Indirect Method	[102]
17.	The Logical Abacus	[104]
18.	The Logical Machine	[107]
19.	The Order of Premises	[114]
20.	The Equivalence of Propositions	[115]
21.	The Nature of Inference	[118]
CHAPTER VII.
INDUCTION.
1.	Induction	[121]
2.	Induction an Inverse Operation	[122]
3.	Inductive Problems for Solution by the Reader	[126]
4.	Induction of Simple Identities	[127]
5.	Induction of Partial Identities	[130]
6.	Solution of the Inverse or Inductive Problem, involving Two Classes	[134]
7.	The Inverse Logical Problem, involving Three Classes	[137]
8.	Professor Clifford on the Types of Compound Statement involving Four Classes	[143]
9.	Distinction between Perfect and Imperfect Induction	[146]
10.	Transition from Perfect to Imperfect Induction	[149]
BOOK II.
NUMBER, VARIETY, AND PROBABILITY.
CHAPTER VIII.
PRINCIPLES OF NUMBER.
1.	Principles of Number	[153]
2.	The Nature of Numbe	[156]
3.	Of Numerical Abstraction	[158]
4.	Concrete and Abstract Number	[159]
5.	Analogy of Logical and Numerical Terms	[160]
6.	Principle of Mathematical Inference	[162]
7.	Reasoning by Inequalities	[165]
8.	Arithmetical Reasoning	[167]
9.	Numerically Definite Reasoning	[168]
10.	Numerical meaning of Logical Conditions	[171]
CHAPTER IX.
THE VARIETY OF NATURE, OR THE DOCTRINE OF COMBINATIONS AND PERMUTATIONS.
1.	The Variety of Nature	[173]
2.	Distinction of Combinations and Permutations	[177]
3.	Calculation of Number of Combinations	[180]
4.	The Arithmetical Triangle	[182]
5.	Connexion between the Arithmetical Triangle and the Logical Alphabet	[189]
6.	Possible Variety of Nature and Art	[190]
7.	Higher Orders of Variety	[192]
CHAPTER X.
THEORY OF PROBABILITY.
1.	Theory of Probability	[197]
2.	Fundamental Principles of the Theory	[200]
3.	Rules for the Calculation of Probabilities	[203]
4.	The Logical Alphabet in questions of Probability	[205]
5.	Comparison of the Theory with Experience	[206]
6.	Probable Deductive Arguments	[209]
7.	Difficulties of the Theory	[213]
CHAPTER XI.
PHILOSOPHY OF INDUCTIVE INFERENCE.
1.	Philosophy of Inductive Inference	[218]
2.	Various Classes of Inductive Truths	[219]
3.	The Relation of Cause and Effect	[220]
4.	Fallacious Use of the Term Cause	[221]
5.	Confusion of Two Questions	[222]
6.	Definition of the Term Cause	[224]
7.	Distinction of Inductive and Deductive Results	[226]
8.	The Grounds of Inductive Inference	[228]
9.	Illustrations of the Inductive Process	[229]
10.	Geometrical Reasoning	[233]
11.	Discrimination of Certainty and Probability	[235]
CHAPTER XII.
THE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY OF PROBABILITY.
1.	The Inductive or Inverse Application of the Theory	[240]
2.	Principle of the Inverse Method	[242]
3.	Simple Applications of the Inverse Method	[244]
4.	The Theory of Probability in Astronomy	[247]
5.	The General Inverse Problem	[250]
6.	Simple Illustration of the Inverse Problem	[253]
7.	General Solution of the Inverse Problem	[255]
8.	Rules of the Inverse Method	[257]
9.	Fortuitous Coincidences	[261]
10.	Summary of the Theory of Inductive Inference	[265]
BOOK III.
METHODS OF MEASUREMENT.
CHAPTER XIII.
THE EXACT MEASUREMENT OF PHENOMENA.
1.	The Exact Measurement of Phenomena	[270]
2.	Division of the Subject	[274]
3.	Continuous quantity	[274]
4.	The Fallacious Indications of the Senses	[276]
5.	Complexity of Quantitative Questions	[278]
6.	The Methods of Accurate Measurement	[282]
7.	Conditions of Accurate Measurement	[282]
8.	Measuring Instruments	[284]
9.	The Method of Repetition	[288]
10.	Measurements by Natural Coincidence	[292]
11.	Modes of Indirect Measurement	[296]
12.	Comparative Use of Measuring Instruments	[299]
13.	Systematic Performance of Measurements	[300]
14.	The Pendulum	[302]
15.	Attainable Accuracy of Measurement	[303]
CHAPTER XIV.
UNITS AND STANDARDS OF MEASUREMENT.
1.	Units and Standards of Measurement	[305]
2.	Standard Unit of Time	[307]
3.	The Unit of Space and the Bar Standard	[312]
4.	The Terrestrial Standard	[314]
5.	The Pendulum Standard	[315]
6.	Unit of Density	[316]
7.	Unit of Mass	[317]
8.	Natural System of Standards	[319]
9.	Subsidiary Units	[320]
10.	Derived Units	[321]
11.	Provisional Units	[323]
12.	Theory of Dimensions	[325]
13.	Natural Constants	[328]
14.	Mathematical Constants	[330]
15.	Physical Constants	[331]
16.	Astronomical Constants	[332]
17.	Terrestrial Numbers	[333]
18.	Organic Numbers	[333]
19.	Social Numbers	[334]
CHAPTER XV.
ANALYSIS OF QUANTITATIVE PHENOMENA.
1.	Analysis of Quantitative Phenomena	[335]
2.	Illustrations of the Complication of Effects	[336]
3.	Methods of Eliminating Error	[339]
4.	Method of Avoidance of Error	[340]
5.	Differential Method	[344]
6.	Method of Correction	[346]
7.	Method of Compensation	[350]
8.	Method of Reversal	[354]
CHAPTER XVI.
THE METHOD OF MEANS.
1.	The Method of Means	[357]
2.	Several Uses of the Mean Result	[359]
3.	The Mean and the Average	[360]
4.	On the Average or Fictitious Mean	[363]
5.	The Precise Mean Result	[365]
6.	Determination of the Zero Point	[368]
7.	Determination of Maximum Points	[371]
CHAPTER XVII.
THE LAW OF ERROR.
1.	The Law of Error	[374]
2.	Establishment of the Law of Error	[375]
3.	Herschel’s Geometrical Proof	[377]
4.	Laplace’s and Quetelet’s Proof of the Law	[378]
5.	Logical Origin of the Law of Error	[383]
6.	Verification of the Law of Error	[383]
7.	The Probable Mean Result	[385]
8.	The Probable Error of Results	[386]
9.	Rejection of the Mean Result	[389]
10.	Method of Least Squares	[393]
11.	Works upon the Theory of Probability	[394]
12.	Detection of Constant Errors	[396]
BOOK IV.
INDUCTIVE INVESTIGATION.
CHAPTER XVIII.
OBSERVATION.
1.	Observation	[399]
2.	Distinction of Observation and Experiment	[400]
3.	Mental Conditions of Correct Observation	[402]
4.	Instrumental and Sensual Conditions of Correct Observation	[404]
5.	External Conditions of Correct Observation	[407]
6.	Apparent Sequence of Events	[409]
7.	Negative Arguments from Non-Observation	[411]
CHAPTER XIX.
EXPERIMENT.
1.	Experiment	[416]
2.	Exclusion of Indifferent Circumstances	[419]
3.	Simplification of Experiments	[422]
4.	Failure in the Simplification of Experiments	[424]
5.	Removal of Usual Conditions	[426]
6.	Interference of Unsuspected Conditions	[428]
7.	Blind or Test Experiments	[433]
8.	Negative Results of Experiment	[434]
9.	Limits of Experiment	[437]
CHAPTER XX.
METHOD OF VARIATIONS.
1.	Method of Variations	[439]
2.	The Variable and the Variant	[440]
3.	Measurement of the Variable	[441]
4.	Maintenance of Similar Conditions	[443]
5.	Collective Experiments	[445]
6.	Periodic Variations	[447]
7.	Combined Periodic Changes	[450]
8.	Principle of Forced Vibrations	[451]
9.	Integrated Variations	[452]
CHAPTER XXI.
THEORY OF APPROXIMATION.
1.	Theory of Approximation	[456]
2.	Substitution of Simple Hypotheses	[458]
3.	Approximation to Exact Laws	[462]
4.	Successive Approximations to Natural Conditions	[465]
5.	Discovery of Hypothetically Simple Laws	[470]
6.	Mathematical Principles of Approximation	[471]
7.	Approximate Independence of Small Effects	[475]
8.	Four Meanings of Equality	[479]
9.	Arithmetic of Approximate Quantities	[481]
CHAPTER XXII.
QUANTITATIVE INDUCTION.
1.	Quantitative Induction	[483]
2.	Probable Connexion of Varying Quantities	[484]
3.	Empirical Mathematical Laws	[487]
4.	Discovery of Rational Formulæ	[489]
5.	The Graphical Method	[492]
6.	Interpolation and Extrapolation	[495]
7.	Illustrations of Empirical Quantitative Laws	[499]
8.	Simple Proportional Variation	[501]
CHAPTER XXIII.
THE USE OF HYPOTHESIS.
1.	The Use of Hypothesis	[504]
2.	Requisites of a good Hypothesis	[510]
3.	Possibility of Deductive Reasoning	[511]
4.	Consistency with the Laws of Nature	[514]
5.	Conformity with Facts	[516]
6.	Experimentum Crucis	[518]
7.	Descriptive Hypotheses	[522]
CHAPTER XXIV.
EMPIRICAL KNOWLEDGE, EXPLANATION AND PREDICTION.
1.	Empirical Knowledge, Explanation and Prediction	[525]
2.	Empirical Knowledge	[526]
3.	Accidental Discovery	[529]
4.	Empirical Observations subsequently Explained	[532]
5.	Overlooked Results of Theory	[534]
6.	Predicted Discoveries	[536]
7.	Predictions in the Science of Light	[538]
8.	Predictions from the Theory of Undulations	[540]
9.	Prediction in other Sciences	[542]
10.	Prediction by Inversion of Cause and Effect	[545]
11.	Facts known only by Theory	[547]
CHAPTER XXV.
ACCORDANCE OF QUANTITATIVE THEORIES.
1.	Accordance of Quantitative Theories	[551]
2.	Empirical Measurements	[552]
3.	Quantities indicated by Theory, but Empirically Measured	[553]
4.	Explained Results of Measurement	[554]
5.	Quantities determined by Theory and verified by Measurement	[555]
6.	Quantities determined by Theory and not verified	[556]
7.	Discordance of Theory and Experiment	[558]
8.	Accordance of Measurements of Astronomical Distances	[560]
9.	Selection of the best Mode of Measurement	[563]
10.	Agreement of Distinct Modes of Measurement	[564]
11.	Residual Phenomena	[569]
CHAPTER XXVI.
CHARACTER OF THE EXPERIMENTALIST.
1.	Character of the Experimentalist	[574]
2.	Error of the Baconian Method	[576]
3.	Freedom of Theorising	[577]
4.	The Newtonian Method, the True Organum	[581]
5.	Candour and Courage of the Philosophic Mind	[586]
6.	The Philosophic Character of Faraday	[587]
7.	Reservation of Judgment	[592]
BOOK V.
GENERALISATION, ANALOGY, AND CLASSIFICATION.
CHAPTER XXVII.
GENERALISATION.
1.	Generalisation	[594]
2.	Distinction of Generalisation and Analogy	[596]
3.	Two Meanings of Generalisation	[597]
4.	Value of Generalisation	[599]
5.	Comparative Generality of Properties	[600]
6.	Uniform Properties of all Matter	[603]
7.	Variable Properties of Matter	[606]
8.	Extreme Instances of Properties	[607]
9.	The Detection of Continuity	[610]
10.	The Law of Continuity	[615]
11.	Failure of the Law of Continuity	[619]
12.	Negative Arguments on the Principle of Continuity	[621]
13.	Tendency to Hasty Generalisation	[623]
CHAPTER XXVIII.
ANALOGY.
1.	Analogy	[627]
2.	Analogy as a Guide in Discovery	[629]
3.	Analogy in the Mathematical Sciences	[631]
4.	Analogy in the Theory of Undulations	[635]
5.	Analogy in Astronomy	[638]
6.	Failures of Analogy	[641]
CHAPTER XXIX.
EXCEPTIONAL PHENOMENA.
1.	Exceptional Phenomena	[644]
2.	Imaginary or False Exceptions	[647]
3.	Apparent but Congruent Exceptions	[649]
4.	Singular Exceptions	[652]
5.	Divergent Exceptions	[655]
6.	Accidental Exceptions	[658]
7.	Novel and Unexplained Exceptions	[661]
8.	Limiting Exceptions	[663]
9.	Real Exceptions to Supposed Laws	[666]
10.	Unclassed Exceptions	[668]
CHAPTER XXX.
CLASSIFICATION.
1.	Classification	[673]
2.	Classification involving Induction	[675]
3.	Multiplicity of Modes of Classification	[677]
4.	Natural and Artificial Systems of Classification	[679]
5.	Correlation of Properties	[681]
6.	Classification in Crystallography	[685]
7.	Classification an Inverse and Tentative Operation	[689]
8.	Symbolic Statement of the Theory of Classification	[692]
9.	Bifurcate Classification	[694]
10.	The Five Predicates	[698]
11.	Summum Genus and Infima Species	[701]
12.	The Tree of Porphyry	[702]
13.	Does Abstraction imply Generalisation?	[704]
14.	Discovery of Marks or Characteristics	[708]
15.	Diagnostic Systems of Classification	[710]
16.	Index Classifications	[714]
17.	Classification in the Biological Sciences	[718]
18.	Classification by Types	[722]
19.	Natural Genera and Species	[724]
20.	Unique or Exceptional Objects	[728]
21.	Limits of Classification	[730]
BOOK VI.
CHAPTER XXXI.
REFLECTIONS ON THE RESULTS AND LIMITS OF SCIENTIFIC METHOD.
1.	Reflections on the Results and Limits of Scientific Method	[735]
2.	The Meaning of Natural Law	[737]
3.	Infiniteness of the Universe	[738]
4.	The Indeterminate Problem of Creation	[740]
5.	Hierarchy of Natural Laws	[742]
6.	The Ambiguous Expression—“Uniformity of Nature”	[745]
7.	Possible States of the Universe	[749]
8.	Speculations on the Reconcentration of Energy	[751]
9.	The Divergent Scope for New Discovery	[752]
10.	Infinite Incompleteness of the Mathematical Sciences	[754]
11.	The Reign of Law in Mental and Social Phenomena	[759]
12.	The Theory of Evolution	[761]
13.	Possibility of Divine Interference	[765]
14.	Conclusion	[766]
INDEX		[773]

THE PRINCIPLES OF SCIENCE.

CHAPTER I.
INTRODUCTION.

Science arises from the discovery of Identity amidst Diversity. The process may be described in different words, but our language must always imply the presence of one common and necessary element. In every act of inference or scientific method we are engaged about a certain identity, sameness, similarity, likeness, resemblance, analogy, equivalence or equality apparent between two objects. It is doubtful whether an entirely isolated phenomenon could present itself to our notice, since there must always be some points of similarity between object and object. But in any case an isolated phenomenon could be studied to no useful purpose. The whole value of science consists in the power which it confers upon us of applying to one object the knowledge acquired from like objects; and it is only so far, therefore, as we can discover and register resemblances that we can turn our observations to account.

Nature is a spectacle continually exhibited to our senses, in which phenomena are mingled in combinations of endless variety and novelty. Wonder fixes the mind’s attention; memory stores up a record of each distinct impression; the powers of association bring forth the record when the like is felt again. By the higher faculties of judgment and reasoning the mind compares the new with the old, recognises essential identity, even when disguised by diverse circumstances, and expects to find again what was before experienced. It must be the ground of all reasoning and inference that what is true of one thing will be true of its equivalent, and that under carefully ascertained conditions Nature repeats herself.

Were this indeed a Chaotic Universe, the powers of mind employed in science would be useless to us. Did Chance wholly take the place of order, and did all phenomena come out of an Infinite Lottery, to use Condorcet’s expression, there could be no reason to expect the like result in like circumstances. It is possible to conceive a world in which no two things should be associated more often, in the long run, than any other two things. The frequent conjunction of any two events would then be purely fortuitous, and if we expected conjunctions to recur continually, we should be disappointed. In such a world we might recognise the same kind of phenomenon as it appeared from time to time, just as we might recognise a marked ball as it was occasionally drawn and re-drawn from a ballot-box; but the approach of any phenomenon would be in no way indicated by what had gone before, nor would it be a sign of what was to come after. In such a world knowledge would be no more than the memory of past coincidences, and the reasoning powers, if they existed at all, would give no clue to the nature of the present, and no presage of the future.

Happily the Universe in which we dwell is not the result of chance, and where chance seems to work it is our own deficient faculties which prevent us from recognising the operation of Law and of Design. In the material framework of this world, substances and forces present themselves in definite and stable combinations. Things are not in perpetual flux, as ancient philosophers held. Element remains element; iron changes not into gold. With suitable precautions we can calculate upon finding the same thing again endowed with the same properties. The constituents of the globe, indeed, appear in almost endless combinations; but each combination bears its fixed character, and when resolved is found to be the compound of definite substances. Misapprehensions must continually occur, owing to the limited extent of our experience. We can never have examined and registered possible existences so thoroughly as to be sure that no new ones will occur and frustrate our calculations. The same outward appearances may cover any amount of hidden differences which we have not yet suspected. To the variety of substances and powers diffused through nature at its creation, we should not suppose that our brief experience can assign a limit, and the necessary imperfection of our knowledge must be ever borne in mind.

Yet there is much to give us confidence in Science. The wider our experience, the more minute our examination of the globe, the greater the accumulation of well-reasoned knowledge,—the fewer in all probability will be the failures of inference compared with the successes. Exceptions to the prevalence of Law are gradually reduced to Law themselves. Certain deep similarities have been detected among the objects around us, and have never yet been found wanting. As the means of examining distant parts of the universe have been acquired, those similarities have been traced there as here. Other worlds and stellar systems may be almost incomprehensively different from ours in magnitude, condition and disposition of parts, and yet we detect there the same elements of which our own limbs are composed. The same natural laws can be detected in operation in every part of the universe within the scope of our instruments; and doubtless these laws are obeyed irrespective of distance, time, and circumstance.

It is the prerogative of Intellect to discover what is uniform and unchanging in the phenomena around us. So far as object is different from object, knowledge is useless and inference impossible. But so far as object resembles object, we can pass from one to the other. In proportion as resemblance is deeper and more general, the commanding powers of knowledge become more wonderful. Identity in one or other of its phases is thus always the bridge by which we pass in inference from case to case; and it is my purpose in this treatise to trace out the various forms in which the one same process of reasoning presents itself in the ever-growing achievements of Scientific Method.

The Powers of Mind concerned in the Creation of Science.

It is no part of the purpose of this work to investigate the nature of mind. People not uncommonly suppose that logic is a branch of psychology, because reasoning is a mental operation. On the same ground, however, we might argue that all the sciences are branches of psychology. As will be further explained, I adopt the opinion of Mr. Herbert Spencer, that logic is really an objective science, like mathematics or mechanics. Only in an incidental manner, then, need I point out that the mental powers employed in the acquisition of knowledge are probably three in number. They are substantially as Professor Bain has stated them‍[24]:‍—

1. The Power of Discrimination.

2. The Power of Detecting Identity.

3. The Power of Retention.

We exert the first power in every act of perception. Hardly can we have a sensation or feeling unless we discriminate it from something else which preceded. Consciousness would almost seem to consist in the break between one state of mind and the next, just as an induced current of electricity arises from the beginning or the ending of the primary current. We are always engaged in discrimination; and the rudiment of thought which exists in the lower animals probably consists in their power of feeling difference and being agitated by it.

Yet had we the power of discrimination only, Science could not be created. To know that one feeling differs from another gives purely negative information. It cannot teach us what will happen. In such a state of intellect each sensation would stand out distinct from every other; there would be no tie, no bridge of affinity between them. We want a unifying power by which the present and the future may be linked to the past; and this seems to be accomplished by a different power of mind. Lord Bacon has pointed out that different men possess in very different degrees the powers of discrimination and identification. It may be said indeed that discrimination necessarily implies the action of the opposite process of identification; and so it doubtless does in negative points. But there is a rare property of mind which consists in penetrating the disguise of variety and seizing the common elements of sameness; and it is this property which furnishes the true measure of intellect. The name of “intellect” expresses the interlacing of the general and the single, which is the peculiar province of mind.‍[25] To cogitate is the Latin coagitare, resting on a like metaphor. Logic, also, is but another name for the same process, the peculiar work of reason; for λογος is derived from λεγειν, which like the Latin legere meant originally to gather. Plato said of this unifying power, that if he met the man who could detect the one in the many, he would follow him as a god.

Laws of Identity and Difference.

At the base of all thought and science must lie the laws which express the very nature and conditions of the discriminating and identifying powers of mind. These are the so-called Fundamental Laws of Thought, usually stated as follows:‍—

1. The Law of Identity. Whatever is, is.

2. The Law of Contradiction. A thing cannot both be and not be.

3. The Law of Duality. A thing must either be or not be.

The first of these statements may perhaps be regarded as a description of identity itself, if so fundamental a notion can admit of description. A thing at any moment is perfectly identical with itself, and, if any person were unaware of the meaning of the word “identity,” we could not better describe it than by such an example.

The second law points out that contradictory attributes can never be joined together. The same object may vary in its different parts; here it may be black, and there white; at one time it may be hard and at another time soft; but at the same time and place an attribute cannot be both present and absent. Aristotle truly described this law as the first of all axioms—one of which we need not seek for any demonstration. All truths cannot be proved, otherwise there would be an endless chain of demonstration; and it is in self-evident truths like this that we find the simplest foundations.

The third of these laws completes the other two. It asserts that at every step there are two possible alternatives—presence or absence, affirmation or negation. Hence I propose to name this law the Law of Duality, for it gives to all the formulæ of reasoning a dual character. It asserts also that between presence and absence, existence and non-existence, affirmation and negation, there is no third alternative. As Aristotle said, there can be no mean between opposite assertions: we must either affirm or deny. Hence the inconvenient name by which it has been known—The Law of Excluded Middle.

It may be allowed that these laws are not three independent and distinct laws; they rather express three different aspects of the same truth, and each law doubtless presupposes and implies the other two. But it has not hitherto been found possible to state these characters of identity and difference in less than the threefold formula. The reader may perhaps desire some information as to the mode in which these laws have been stated, or the way in which they have been regarded, by philosophers in different ages of the world. Abundant information on this and many other points of logical history will be found in Ueberweg’s System of Logic, of which an excellent translation has been published by Professor T. M. Lindsay (see pp. 228–281).

The Nature of the Laws of Identity and Difference.

I must at least allude to the profoundly difficult question concerning the nature and authority of these Laws of Identity and Difference. Are they Laws of Thought or Laws of Things? Do they belong to mind or to material nature? On the one hand it may be said that science is a purely mental existence, and must therefore conform to the laws of that which formed it. Science is in the mind and not in the things, and the properties of mind are therefore all important. It is true that these laws are verified in the observation of the exterior world; and it would seem that they might have been gathered and proved by generalisation, had they not already been in our possession. But on the other hand, it may well be urged that we cannot prove these laws by any process of reasoning or observation, because the laws themselves are presupposed, as Leibnitz acutely remarked, in the very notion of a proof. They are the prior conditions of all thought and all knowledge, and even to question their truth is to allow them true. Hartley ingeniously refined upon this argument, remarking that if the fundamental laws of logic be not certain, there must exist a logic of a second order whereby we may determine the degree of uncertainty: if the second logic be not certain, there must be a third; and so on ad infinitum. Thus we must suppose either that absolutely certain laws of thought exist, or that there is no such thing as certainty whatever.‍[26]

Logicians, indeed, appear to me to have paid insufficient attention to the fact that mistakes in reasoning are always possible, and of not unfrequent occurrence. The Laws of Thought are often called necessary laws, that is, laws which cannot but be obeyed. Yet as a matter of fact, who is there that does not often fail to obey them? They are the laws which the mind ought to obey rather than what it always does obey. Our thoughts cannot be the criterion of truth, for we often have to acknowledge mistakes in arguments of moderate complexity, and we sometimes only discover our mistakes by collision between our expectations and the events of objective nature.

Mr. Herbert Spencer holds that the laws of logic are objective laws,‍[27] and he regards the mind as being in a state of constant education, each act of false reasoning or miscalculation leading to results which are likely to prevent similar mistakes from being again committed. I am quite inclined to accept such ingenious views; but at the same time it is necessary to distinguish between the accumulation of knowledge, and the constitution of the mind which allows of the acquisition of knowledge. Before the mind can perceive or reason at all it must have the conditions of thought impressed upon it. Before a mistake can be committed, the mind must clearly distinguish the mistaken conclusion from all other assertions. Are not the Laws of Identity and Difference the prior conditions of all consciousness and all existence? Must they not hold true, alike of things material and immaterial? and if so, can we say that they are only subjectively true or objectively true? I am inclined, in short, to regard them as true both “in the nature of thought and things,” as I expressed it in my first logical essay;‍[28] and I hold that they belong to the common basis of all existence. But this is one of the most difficult questions of psychology and metaphysics which can be raised, and it is hardly one for the logician to decide. As the mathematician does not inquire into the nature of unity and plurality, but develops the formal laws of plurality, so the logician, as I conceive, must assume the truth of the Laws of Identity and Difference, and occupy himself in developing the variety of forms of reasoning in which their truth may be manifested.

Again, I need hardly dwell upon the question whether logic treats of language, notions, or things. As reasonably might we debate whether a mathematician treats of symbols, quantities, or things. A mathematician certainly does treat of symbols, but only as the instruments whereby to facilitate his reasoning concerning quantities; and as the axioms and rules of mathematical science must be verified in concrete objects in order that the calculations founded upon them may have any validity or utility, it follows that the ultimate objects of mathematical science are the things themselves. In like manner I conceive that the logician treats of language so far as it is essential for the embodiment and exhibition of thought. Even if reasoning can take place in the inner consciousness of man without the use of any signs, which is doubtful, at any rate it cannot become the subject of discussion until by some system of material signs it is manifested to other persons. The logician then uses words and symbols as instruments of reasoning, and leaves the nature and peculiarities of language to the grammarian. But signs again must correspond to the thoughts and things expressed, in order that they shall serve their intended purpose. We may therefore say that logic treats ultimately of thoughts and things, and immediately of the signs which stand for them. Signs, thoughts, and exterior objects may be regarded as parallel and analogous series of phenomena, and to treat any one of the three series is equivalent to treating either of the other series.

The Process of Inference.

The fundamental action of our reasoning faculties consists in inferring or carrying to a new instance of a phenomenon whatever we have previously known of its like, analogue, equivalent or equal. Sameness or identity presents itself in all degrees, and is known under various names; but the great rule of inference embraces all degrees, and affirms that so far as there exists sameness, identity or likeness, what is true of one thing will be true of the other. The great difficulty doubtless consists in ascertaining that there does exist a sufficient degree of likeness or sameness to warrant an intended inference; and it will be our main task to investigate the conditions under which reasoning is valid. In this place I wish to point out that there is something common to all acts of inference, however different their apparent forms. The one same rule lends itself to the most diverse applications.

The simplest possible case of inference, perhaps, occurs in the use of a pattern, example, or, as it is commonly called, a sample. To prove the exact similarity of two portions of commodity, we need not bring one portion beside the other. It is sufficient that we take a sample which exactly represents the texture, appearance, and general nature of one portion, and according as this sample agrees or not with the other, so will the two portions of commodity agree or differ. Whatever is true as regards the colour, texture, density, material of the sample will be true of the goods themselves. In such cases likeness of quality is the condition of inference.

Exactly the same mode of reasoning holds true of magnitude and figure. To compare the sizes of two objects, we need not lay them beside each other. A staff, string, or other kind of measure may be employed to represent the length of one object, and according as it agrees or not with the other, so must the two objects agree or differ. In this case the proxy or sample represents length; but the fact that lengths can be added and multiplied renders it unnecessary that the proxy should always be as large as the object. Any standard of convenient size, such as a common foot-rule, may be made the medium of comparison. The height of a church in one town may be carried to that in another, and objects existing immovably at opposite sides of the earth may be vicariously measured against each other. We obviously employ the axiom that whatever is true of a thing as regards its length, is true of its equal.

To every other simple phenomenon in nature the same principle of substitution is applicable. We may compare weights, densities, degrees of hardness, and degrees of all other qualities, in like manner. To ascertain whether two sounds are in unison we need not compare them directly, but a third sound may be the go-between. If a tuning-fork is in unison with the middle C of York Minster organ, and we afterwards find it to be in unison with the same note of the organ in Westminster Abbey, then it follows that the two organs are tuned in unison. The rule of inference now is, that what is true of the tuning-fork as regards the tone or pitch of its sound, is true of any sound in unison with it.

The skilful employment of this substitutive process enables us to make measurements beyond the powers of our senses. No one can count the vibrations, for instance, of an organ-pipe. But we can construct an instrument called the siren, so that, while producing a sound of any pitch, it shall register the number of vibrations constituting the sound. Adjusting the sound of the siren in unison with an organ-pipe, we measure indirectly the number of vibrations belonging to a sound of that pitch. To measure a sound of the same pitch is as good as to measure the sound itself.

Sir David Brewster, in a somewhat similar manner, succeeded in measuring the refractive indices of irregular fragments of transparent minerals. It was a troublesome, and sometimes impracticable work to grind the minerals into prisms, so that the power of refracting light could be directly observed; but he fell upon the ingenious device of compounding a liquid possessing the same refractive power as the transparent fragment under examination. The moment when this equality was attained could be known by the fragments ceasing to reflect or refract light when immersed in the liquid, so that they became almost invisible in it. The refractive power of the liquid being then measured gave that of the solid. A more beautiful instance of representative measurement, depending immediately upon the principle of inference, could not be found.‍[29]

Throughout the various logical processes which we are about to consider—Deduction, Induction, Generalisation, Analogy, Classification, Quantitative Reasoning—we shall find the one same principle operating in a more or less disguised form.

Deduction and Induction.

The processes of inference always depend on the one same principle of substitution; but they may nevertheless be distinguished according as the results are inductive or deductive. As generally stated, deduction consists in passing from more general to less general truths; induction is the contrary process from less to more general truths. We may however describe the difference in another manner. In deduction we are engaged in developing the consequences of a law. We learn the meaning, contents, results or inferences, which attach to any given proposition. Induction is the exactly inverse process. Given certain results or consequences, we are required to discover the general law from which they flow.

In a certain sense all knowledge is inductive. We can only learn the laws and relations of things in nature by observing those things. But the knowledge gained from the senses is knowledge only of particular facts, and we require some process of reasoning by which we may collect out of the facts the laws obeyed by them. Experience gives us the materials of knowledge: induction digests those materials, and yields us general knowledge. When we possess such knowledge, in the form of general propositions and natural laws, we can usefully apply the reverse process of deduction to ascertain the exact information required at any moment. In its ultimate foundation, then, all knowledge is inductive—in the sense that it is derived by a certain inductive reasoning from the facts of experience.

It is nevertheless true,—and this is a point to which insufficient attention has been paid, that all reasoning is founded on the principles of deduction. I call in question the existence of any method of reasoning which can be carried on without a knowledge of deductive processes. I shall endeavour to show that induction is really the inverse process of deduction. There is no mode of ascertaining the laws which are obeyed in certain phenomena, unless we have the power of determining what results would follow from a given law. Just as the process of division necessitates a prior knowledge of multiplication, or the integral calculus rests upon the observation and remembrance of the results of the differential calculus, so induction requires a prior knowledge of deduction. An inverse process is the undoing of the direct process. A person who enters a maze must either trust to chance to lead him out again, or he must carefully notice the road by which he entered. The facts furnished to us by experience are a maze of particular results; we might by chance observe in them the fulfilment of a law, but this is scarcely possible, unless we thoroughly learn the effects which would attach to any particular law.

Accordingly, the importance of deductive reasoning is doubly supreme. Even when we gain the results of induction they would be of no use unless we could deductively apply them. But before we can gain them at all we must understand deduction, since it is the inversion of deduction which constitutes induction. Our first task in this work, then, must be to trace out fully the nature of identity in all its forms of occurrence. Having given any series of propositions we must be prepared to develop deductively the whole meaning embodied in them, and the whole of the consequences which flow from them.

Symbolic Expression of Logical Inference.

In developing the results of the Principle of Inference we require to use an appropriate language of signs. It would indeed be quite possible to explain the processes of reasoning by the use of words found in the dictionary. Special examples of reasoning, too, may seem to be more readily apprehended than general symbolic forms. But it has been shown in the mathematical sciences that the attainment of truth depends greatly upon the invention of a clear, brief, and appropriate system of symbols. Not only is such a language convenient, but it is almost essential to the expression of those general truths which are the very soul of science. To apprehend the truth of special cases of inference does not constitute logic; we must apprehend them as cases of more general truths. The object of all science is the separation of what is common and general from what is accidental and different. In a system of logic, if anywhere, we should esteem this generality, and strive to exhibit clearly what is similar in very diverse cases. Hence the great value of general symbols by which we can represent the form of a reasoning process, disentangled from any consideration of the special subject to which it is applied.

The signs required in logic are of a very simple kind. As sameness or difference must exist between two things or notions, we need signs to indicate the things or notions compared, and other signs to denote the relations between them. We need, then, (1) symbols for terms, (2) a symbol for sameness, (3) a symbol for difference, and (4) one or two symbols to take the place of conjunctions.

Ordinary nouns substantive, such as Iron, Metal, Electricity, Undulation, might serve as terms, but, for the reasons explained above, it is better to adopt blank letters, devoid of special signification, such as A, B, C, &c. Each letter must be understood to represent a noun, and, so far as the conditions of the argument allow, any noun. Just as in Algebra, x, y, z, p, q, &c. are used for any quantities, undetermined or unknown, except when the special conditions of the problem are taken into account, so will our letters stand for undetermined or unknown things.

These letter-terms will be used indifferently for nouns substantive and adjective. Between these two kinds of nouns there may perhaps be differences in a metaphysical or grammatical point of view. But grammatical usage sanctions the conversion of adjectives into substantives, and vice versâ; we may avail ourselves of this latitude without in any way prejudging the metaphysical difficulties which may be involved. Here, as throughout this work, I shall devote my attention to truths which I can exhibit in a clear and formal manner, believing that in the present condition of logical science, this course will lead to greater advantage than discussion upon the metaphysical questions which may underlie any part of the subject.

Every noun or term denotes an object, and usually implies the possession by that object of certain qualities or circumstances common to all the objects denoted. There are certain terms, however, which imply the absence of qualities or circumstances attaching to other objects. It will be convenient to employ a special mode of indicating these negative terms, as they are called. If the general name A denotes an object or class of objects possessing certain defined qualities, then the term Not A will denote any object which does not possess the whole of those qualities; in short, Not A is the sign for anything which differs from A in regard to any one or more of the assigned qualities. If A denote “transparent object,” Not A will denote “not transparent object.” Brevity and facility of expression are of no slight importance in a system of notation, and it will therefore be desirable to substitute for the negative term Not A a briefer symbol. De Morgan represented negative terms by small Roman letters, or sometimes by small italic letters;‍[30] as the latter seem to be highly convenient, I shall use a, b, c, . . . p, q, &c., as the negative terms corresponding to A, B, C, . . . P, Q, &c. Thus if A means “fluid,” a will mean “not fluid.”

Expression of Identity and Difference.

To denote the relation of sameness or identity I unhesitatingly adopt the sign =, so long used by mathematicians to denote equality. This symbol was originally appropriated by Robert Recorde in his Whetstone of Wit, to avoid the tedious repetition of the words “is equal to;” and he chose a pair of parallel lines, because no two things can be more equal.‍[31] The meaning of the sign has however been gradually extended beyond that of equality of quantities; mathematicians have themselves used it to indicate equivalence of operations. The force of analogy has been so great that writers in most other branches of science have employed the same sign. The philologist uses it to indicate the equivalence of meaning of words: chemists adopt it to signify identity in kind and equality in weight of the elements which form two different compounds. Not a few logicians, for instance Lambert, Drobitsch, George Bentham,‍[32] Boole,‍[33] have employed it as the copula of propositions. De Morgan declined to use it for this purpose, but still further extended its meaning so as to include the equivalence of a proposition with the premises from which it can be inferred;‍[34] and Herbert Spencer has applied it in a like manner.‍[35]

Many persons may think that the choice of a symbol is a matter of slight importance or of mere convenience; but I hold that the common use of this sign = in so many different meanings is really founded upon a generalisation of the widest character and of the greatest importance—one indeed which it is a principal purpose of this work to explain. The employment of the same sign in different cases would be unphilosophical unless there were some real analogy between its diverse meanings. If such analogy exists, it is not only allowable, but highly desirable and even imperative, to use the symbol of equivalence with a generality of meaning corresponding to the generality of the principles involved. Accordingly De Morgan’s refusal to use the symbol in logical propositions indicated his opinion that there was a want of analogy between logical propositions and mathematical equations. I use the sign because I hold the contrary opinion.

I conceive that the sign = as commonly employed, always denotes some form or degree of sameness, and the particular form is usually indicated by the nature of the terms joined by it. Thus “6,720 pounds = 3 tons” is evidently an equation of quantities. The formula — × — = + expresses the equivalence of operations. “Exogens = Dicotyledons” is a logical identity expressing a profound truth concerning the character and origin of a most important group of plants.

We have great need in logic of a distinct sign for the copula, because the little verb is (or are), hitherto used both in logic and ordinary discourse, is thoroughly ambiguous. It sometimes denotes identity, as in “St. Paul’s is the chef-d’œuvre of Sir Christopher Wren;” but it more commonly indicates inclusion of class within class, or partial identity, as in “Bishops are members of the House of Lords.” This latter relation involves identity, but requires careful discrimination from simple identity, as will be shown further on.

When with this sign of equality we join two nouns or logical terms, as in

Hydrogen = The least dense element,

we signify that the object or group of objects denoted by one term is identical with that denoted by the other, in everything except the names. The general formula

A = B

must be taken to mean that A and B are symbols for the same object or group of objects. This identity may sometimes arise from the mere imposition of names, but it may also arise from the deepest laws of the constitution of nature; as when we say

Gravitating matter = Matter possessing inertia,

Exogenous plants = Dicotyledonous plants,

Plagihedral quartz crystals = Quartz crystals causing the plane of polarisation of light to rotate.

We shall need carefully to distinguish between relations of terms which can be modified at our own will and those which are fixed as expressing the laws of nature; but at present we are considering only the mode of expression which may be the same in either case.

Sometimes, but much less frequently, we require a symbol to indicate difference or the absence of complete sameness. For this purpose we may generalise in like manner the symbol ~, which was introduced by Wallis to signify difference between quantities. The general formula

B ~ C

denotes that B and C are the names of two objects or groups which are not identical with each other. Thus we may say

Acrogens ~ Flowering plants.

Snowdon ~ The highest mountain in Great Britain.

I shall also occasionally use the sign ᔕ to signify in the most general manner the existence of any relation between the two terms connected by it. Thus ᔕ might mean not only the relations of equality or inequality, sameness or difference, but any special relation of time, place, size, causation, &c. in which one thing may stand to another. By A ᔕ B I mean, then, any two objects of thought related to each other in any conceivable manner.

General Formula of Logical Inference.

The one supreme rule of inference consists, as I have said, in the direction to affirm of anything whatever is known of its like, equal or equivalent. The Substitution of Similars is a phrase which seems aptly to express the capacity of mutual replacement existing in any two objects which are like or equivalent to a sufficient degree. It is matter for further investigation to ascertain when and for what purposes a degree of similarity less than complete identity is sufficient to warrant substitution. For the present we think only of the exact sameness expressed in the form

A = B.

Now if we take the letter C to denote any third conceivable object, and use the sign ᔕ in its stated meaning of indefinite relation, then the general formula of all inference may be thus exhibited:‍—

From     A = B ᔕ C

we may infer   A ᔕ C

or, in words—In whatever relation a thing stands to a second thing, in the same relation it stands to the like or equivalent of that second thing. The identity between A and B allows us indifferently to place A where B was, or B where A was; and there is no limit to the variety of special meanings which we can bestow upon the signs used in this formula consistently with its truth. Thus if we first specify only the meaning of the sign ᔕ, we may say that if C is the weight of B, then C is also the weight of A. Similarly

If C is the father of B, C is the father of A;

If C is a fragment of B, C is a fragment of A;

If C is a quality of B, C is a quality of A;

If C is a species of B, C is a species of A;

If C is the equal of B, C is the equal of A;

and so on ad infinitum.

We may also endow with special meanings the letter-terms A, B, and C, and the process of inference will never be false. Thus let the sign ᔕ mean “is height of,” and let

A = Snowdon,

B = Highest mountain in England or Wales,

C = 3,590 feet;

then it obviously follows since “3,590 feet is the height of Snowdon,” and “Snowdon = the highest mountain in England or Wales,” that, “3,590 feet is the height of the highest mountain in England or Wales.”

One result of this general process of inference is that we may in any aggregate or complex whole replace any part by its equivalent without altering the whole. To alter is to make a difference; but if in replacing a part I make no difference, there is no alteration of the whole. Many inferences which have been very imperfectly included in logical formulas at once follow. I remember the late Prof. De Morgan remarking that all Aristotle’s logic could not prove that “Because a horse is an animal, the head of a horse is the head of an animal.” I conceive that this amounts merely to replacing in the complete notion head of a horse, the term “horse,” by its equivalent some animal or an animal. Similarly, since

The Lord Chancellor = The Speaker of the House of Lords,

it follows that

The death of the Lord Chancellor = The death of the Speaker of the House of Lords;

and any event, circumstance or thing, which stands in a certain relation to the one will stand in like relation to the other. Milton reasons in this way when he says, in his Areopagitica, “Who kills a man, kills a reasonable creature, God’s image.” If we may suppose him to mean

God’s image = man = some reasonable creature,

it follows that “The killer of a man is the killer of some reasonable creature,” and also “The killer of God’s image.”

This replacement of equivalents may be repeated over and over again to any extent. Thus if person is identical in meaning with individual, it follows that

Meeting of persons = meeting of individuals;

and if assemblage = meeting, we may make a new replacement and show that

Meeting of persons = assemblage of individuals.

We may in fact found upon this principle of substitution a most general axiom in the following terms‍[36]:‍—

Same parts samely related make same wholes.

If, for instance, exactly similar bricks and other materials be used to build two houses, and they be similarly placed in each house, the two houses must be similar. There are millions of cells in a human body, but if each cell of one person were represented by an exactly similar cell similarly placed in another body, the two persons would be undistinguishable, and would be only numerically different. It is upon this principle, as we shall see, that all accurate processes of measurement depend. If for a weight in a scale of a balance we substitute another weight, and the equilibrium remains entirely unchanged, then the weights must be exactly equal. The general test of equality is substitution. Objects are equally bright when on replacing one by the other the eye perceives no difference. Objects are equal in dimensions when tested by the same gauge they fit in the same manner. Generally speaking, two objects are alike so far as when substituted one for another no alteration is produced, and vice versâ when alike no alteration is produced by the substitution.

The Propagating Power of Similarity.

The relation of similarity in all its degrees is reciprocal. So far as things are alike, either may be substituted for the other; and this may perhaps be considered the very meaning of the relation. But it is well worth notice that there is in similarity a peculiar power of extending itself among all the things which are similar. To render a number of things similar to each other we need only render them similar to one standard object. Each coin struck from a pair of dies not only resembles the matrix or original pattern from which the dies were struck, but resembles every other coin manufactured from the same original pattern. Among a million such coins there are not less than 499,999,500,000 pairs of coins resembling each other. Similars to the same are similars to all. It is one great advantage of printing that all copies of a document struck from the same type are necessarily identical each with each, and whatever is true of one copy will be true of every copy. Similarly, if fifty rows of pipes in an organ be tuned in perfect unison with one row, usually the Principal, they must be in unison with each other. Similarity can also reproduce or propagate itself ad infinitum: for if a number of tuning-forks be adjusted in perfect unison with one standard fork, all instruments tuned to any one fork will agree with any instrument tuned to any other fork. Standard measures of length, capacity, weight, or any other measurable quality, are propagated in the same manner. So far as copies of the original standard, or copies of copies, or copies again of those copies, are accurately executed, they must all agree each with every other.

It is the capability of mutual substitution which gives such great value to the modern methods of mechanical construction, according to which all the parts of a machine are exact facsimiles of a fixed pattern. The rifles used in the British army are constructed on the American interchangeable system, so that any part of any rifle can be substituted for the same part of another. A bullet fitting one rifle will fit all others of the same bore. Sir J. Whitworth has extended the same system to the screws and screw-bolts used in connecting together the parts of machines, by establishing a series of standard screws.

Anticipations of the Principle of Substitution.

In such a subject as logic it is hardly possible to put forth any opinions which have not been in some degree previously entertained. The germ at least of every doctrine will be found in earlier writers, and novelty must arise chiefly in the mode of harmonising and developing ideas. When I first employed the process and name of substitution in logic,‍[37] I was led to do so from analogy with the familiar mathematical process of substituting for a symbol its value as given in an equation. In writing my first logical essay I had a most imperfect conception of the importance and generality of the process, and I described, as if they were of equal importance, a number of other laws which now seem to be but particular cases of the one general rule of substitution.

My second essay, “The Substitution of Similars,” was written shortly after I had become aware of the great simplification which may be effected by a proper application of the principle of substitution. I was not then acquainted with the fact that the German logician Beneke had employed the principle of substitution, and had used the word itself in forming a theory of the syllogism. My imperfect acquaintance with the German language had prevented me from acquiring a complete knowledge of Beneke’s views; but there is no doubt that Professor Lindsay is right in saying that he, and probably other logicians, were in some degree familiar with the principle.‍[38] Even Aristotle’s dictum may be regarded as an imperfect statement of the principle of substitution; and, as I have pointed out, we have only to modify that dictum in accordance with the quantification of the predicate in order to arrive at the complete process of substitution.‍[39] The Port-Royal logicians appear to have entertained nearly equivalent views, for they considered that all moods of the syllogism might be reduced under one general principle.‍[40] Of two premises they regard one as the containing proposition (propositio continens), and the other as the applicative proposition. The latter proposition must always be affirmative, and represents that by which a substitution is made; the former may or may not be negative, and is that in which a substitution is effected. They also show that this method will embrace certain cases of complex reasoning which had no place in the Aristotelian syllogism. Their views probably constitute the greatest improvement in logical doctrine made up to that time since the days of Aristotle. But a true reform in logic must consist, not in explaining the syllogism in one way or another, but in doing away with all the narrow restrictions of the Aristotelian system, and in showing that there exists an infinite variety of logical arguments immediately deducible from the principle of substitution of which the ancient syllogism forms but a small and not even the most important part.

The Logic of Relatives.

There is a difficult and important branch of logic which may be called the Logic of Relatives. If I argue, for instance, that because Daniel Bernoulli was the son of John, and John the brother of James, therefore Daniel was the nephew of James, it is not possible to prove this conclusion by any simple logical process. We require at any rate to assume that the son of a brother is a nephew. A simple logical relation is that which exists between properties and circumstances of the same object or class. But objects and classes of objects may also be related according to all the properties of time and space. I believe it may be shown, indeed, that where an inference concerning such relations is drawn, a process of substitution is really employed and an identity must exist; but I will not undertake to prove the assertion in this work. The relations of time and space are logical relations of a complicated character demanding much abstract and difficult investigation. The subject has been treated with such great ability by Peirce,‍[41] De Morgan,‍[42] Ellis,‍[43] and Harley, that I will not in the present work attempt any review of their writings, but merely refer the reader to the publications in which they are to be found.

CHAPTER II.
TERMS.

Every proposition expresses the resemblance or difference of the things denoted by its terms. As inference treats of the relation between two or more propositions, so a proposition expresses a relation between two or more terms. In the portion of this work which treats of deduction it will be convenient to follow the usual order of exposition. We will consider in succession the various kinds of terms, propositions, and arguments, and we commence in this chapter with terms.

The simplest and most palpable meaning which can belong to a term consists of some single material object, such as Westminster Abbey, Stonehenge, the Sun, Sirius, &c. It is probable that in early stages of intellect only concrete and palpable things are the objects of thought. The youngest child knows the difference between a hot and a cold body. The dog can recognise his master among a hundred other persons, and animals of much lower intelligence know and discriminate their haunts. In all such acts there is judgment concerning the likeness of physical objects, but there is little or no power of analysing each object and regarding it as a group of qualities.

The dignity of intellect begins with the power of separating points of agreement from those of difference. Comparison of two objects may lead us to perceive that they are at once like and unlike. Two fragments of rock may differ entirely in outward form, yet they may have the same colour, hardness, and texture. Flowers which agree in colour may differ in odour. The mind learns to regard each object as an aggregate of qualities, and acquires the power of dwelling at will upon one or other of those qualities to the exclusion of the rest. Logical abstraction, in short, comes into play, and the mind becomes capable of reasoning, not merely about objects which are physically complete and concrete, but about things which may be thought of separately in the mind though they exist not separately in nature. We can think of the hardness of a rock, or the colour of a flower, and thus produce abstract notions, denoted by abstract terms, which will form a subject for further consideration.

At the same time arise general notions and classes of objects. We cannot fail to observe that the quality hardness exists in many objects, for instance in many fragments of rock; mentally joining these together, we create the class hard object, which will include, not only the actual objects examined, but all others which may happen to agree with them, as they agree with each other. As our senses cannot possibly report to us all the contents of space, we cannot usually set any limits to the number of objects which may fall into any such class. At this point we begin to perceive the power and generality of thought, which enables us in a single act to treat of indefinitely or even infinitely numerous objects. We can safely assert that whatever is true of any one object coming under a class is true of any of the other objects so far as they possess the common qualities implied in their belonging to the class. We must not place a thing in a class unless we are prepared to believe of it all that is believed of the class in general; but it remains a matter of important consideration to decide how far and in what manner we can safely undertake thus to assign the place of objects in that general system of classification which constitutes the body of science.

Twofold Meaning of General Names.

Etymologically the meaning of a name is that which we are caused to think of when the name is used. Now every general name causes us to think of some one or more of the objects belonging to a class; it may also cause us to think of the common qualities possessed by those objects. A name is said to denote the object of thought to which it may be applied; it implies at the same time the possession of certain qualities or circumstances. The objects denoted form the extent of meaning of the term; the qualities implied form the intent of meaning. Crystal is the name of any substance of which the molecules are arranged in a regular geometrical manner. The substances or objects in question form the extent of meaning; the circumstance of having the molecules so arranged forms the intent of meaning.

When we compare general terms together, it may often be found that the meaning of one is included in the meaning of another. Thus all crystals are included among material substances, and all opaque crystals are included among crystals; here the inclusion is in extension. We may also have inclusion of meaning in regard to intension. For, as all crystals are material substances, the qualities implied by the term material substance must be among those implied by crystal. Again, it is obvious that while in extension of meaning opaque crystals are but a part of crystals, in intension of meaning crystal is but part of opaque crystal. We increase the intent of meaning of a term by joining to it adjectives, or phrases equivalent to adjectives, and the removal of such adjectives of course decreases the intensive meaning. Now, concerning such changes of meaning, the following all-important law holds universally true:—When the intent of meaning of a term is increased the extent is decreased; and vice versâ, when the extent is increased the intent is decreased. In short, as one is increased the other is decreased.

This law refers only to logical changes. The number of steam-engines in the world may be undergoing a rapid increase without the intensive meaning of the name being altered. The law will only be verified, again, when there is a real change in the intensive meaning, and an adjective may often be joined to a noun without making a change. Elementary metal is identical with metal; mortal man with man; it being a property of all metals to be elements, and of all men to be mortals.

There is no limit to the amount of meaning which a term may have. A term may denote one object, or many, or an infinite number; it may imply a single quality, if such there be, or a group of any number of qualities, and yet the law connecting the extension and intension will infallibly apply. Taking the general name planet, we increase its intension and decrease its extension by prefixing the adjective exterior; and if we further add nearest to the earth, there remains but one planet, Mars, to which the name can then be applied. Singular terms, which denote a single individual only, come under the same law of meaning as general names. They may be regarded as general names of which the meaning in extension is reduced to a minimum. Logicians have erroneously asserted, as it seems to me, that singular terms are devoid of meaning in intension, the fact being that they exceed all other terms in that kind of meaning, as I have elsewhere tried to show.‍[44]

Abstract Terms.

Comparison of objects, and analysis of the complex resemblances and differences which they present, lead us to the conception of abstract qualities. We learn to think of one object as not only different from another, but as differing in some particular point, such as colour, or weight, or size. We may then convert points of agreement or difference into separate objects of thought which we call qualities and denote by abstract terms. Thus the term redness means something in which a number of objects agree as to colour, and in virtue of which they are called red. Redness forms, in fact, the intensive meaning of the term red.

Abstract terms are strongly distinguished from general terms by possessing only one kind of meaning; for as they denote qualities there is nothing which they cannot in addition imply. The adjective “red” is the name of red objects, but it implies the possession by them of the quality redness; but this latter term has one single meaning—the quality alone. Thus it arises that abstract terms are incapable of plurality. Red objects are numerically distinct each from each, and there are multitudes of such objects; but redness is a single quality which runs through all those objects, and is the same in one as it is in another. It is true that we may speak of rednesses, meaning different kinds or tints of redness, just as we may speak of colours, meaning different kinds of colours. But in distinguishing kinds, degrees, or other differences, we render the terms so far concrete. In that they are merely red there is but a single nature in red objects, and so far as things are merely coloured, colour is a single indivisible quality. Redness, so far as it is redness merely, is one and the same everywhere, and possesses absolute oneness. In virtue of this unity we acquire the power of treating all instances of such quality as we may treat any one. We possess, in short, general knowledge.

Substantial Terms.

Logicians appear to have taken little notice of a class of terms which partake in certain respects of the character of abstract terms and yet are undoubtedly the names of concrete existing things. These terms are the names of substances, such as gold, carbonate of lime, nitrogen, &c. We cannot speak of two golds, twenty carbonates of lime, or a hundred nitrogens. There is no such distinction between the parts of a uniform substance as will allow of a discrimination of numerous individuals. The qualities of colour, lustre, malleability, density, &c., by which we recognise gold, extend through its substance irrespective of particular size or shape. So far as a substance is gold, it is one and the same everywhere; so that terms of this kind, which I propose to call substantial terms, possess the peculiar unity of abstract terms. Yet they are not abstract; for gold is of course a tangible visible body, entirely concrete, and existing independently of other bodies.

It is only when, by actual mechanical division, we break up the uniform whole which forms the meaning of a substantial term, that we introduce number. Piece of gold is a term capable of plurality; for there may be a great many pieces discriminated either by their various shapes and sizes, or, in the absence of such marks, by simultaneously occupying different parts of space. In substance they are one; as regards the properties of space they are many.‍[45] We need not further pursue this question, which involves the distinction between unity and plurality, until we consider the principles of number in a subsequent chapter.

Collective Terms.

We must clearly distinguish between the collective and the general meanings of terms. The same name may be used to denote the whole body of existing objects of a certain kind, or any one of those objects taken separately. “Man” may mean the aggregate of existing men, which we sometimes describe as mankind; it is also the general name applying to any man. The vegetable kingdom is the name of the whole aggregate of plants, but “plant” itself is a general name applying to any one or other plant. Every material object may be conceived as divisible into parts, and is therefore collective as regards those parts. The animal body is made up of cells and fibres, a crystal of molecules; wherever physical division, or as it has been called partition, is possible, there we deal in reality with a collective whole. Thus the greater number of general terms are at the same time collective as regards each individual whole which they denote.

It need hardly be pointed out that we must not infer of a collective whole what we know only of the parts, nor of the parts what we know only of the whole. The relation of whole and part is not one of identity, and does not allow of substitution. There may nevertheless be qualities which are true alike of the whole and of its parts. A number of organ-pipes tuned in unison produce an aggregate of sound which is of exactly the same pitch as each separate sound. In the case of substantial terms, certain qualities may be present equally in each minutest part as in the whole. The chemical nature of the largest mass of pure carbonate of lime is the same as the nature of the smallest particle. In the case of abstract terms, again, we cannot draw a distinction between whole and part; what is true of redness in any case is always true of redness, so far as it is merely red.

Synthesis of Terms.

We continually combine simple terms together so as to form new terms of more complex meaning. Thus, to increase the intension of meaning of a term we write it with an adjective or a phrase of adjectival nature. By joining “brittle” to “metal,” we obtain a combined term, “brittle metal,” which denotes a certain portion of the metals, namely, such as are selected on account of possessing the quality of brittleness. As we have already seen, “brittle metal” possesses less extension and greater intension than metal. Nouns, prepositional phrases, participial phrases and subordinate propositions may also be added to terms so as to increase their intension and decrease their extension.

In our symbolic language we need some mode of indicating this junction of terms, and the most convenient device will be the juxtaposition of the letter-terms. Thus if A mean brittle, and B mean metal, then AB will mean brittle metal. Nor need there be any limit to the number of letters thus joined together, or the complexity of the notions which they may represent.

Thus if we take the letters

P = metal,
Q = white,
R = monovalent,
S = of specific gravity 10·5,
T = melting above 1000° C.,
V = good conductor of heat and electricity,

then we can form a combined term PQRSTV, which will denote “a white monovalent metal, of specific gravity 10·5, melting above 1000° C., and a good conductor of heat and electricity.”

There are many grammatical usages concerning the junction of words and phrases to which we need pay no attention in logic. We can never say in ordinary language “of wood table,” meaning “table of wood;” but we may consider “of wood” as logically an exact equivalent of “wooden”; so that if

X = of wood,
Y = table,

there is no reason why, in our symbols, XY should not be just as correct an expression for “table of wood ” as YX. In this case indeed we might substitute for “of wood ” the corresponding adjective “wooden,” but we should often fail to find any adjective answering exactly to a phrase. There is no single word by which we could express the notion “of specific gravity 10·5:” but logically we may consider these words as forming an adjective; and denoting this by S and metal by P, we may say that SP means “metal of specific gravity 10·5.” It is one of many advantages in these blank letter-symbols that they enable us completely to neglect all grammatical peculiarities and to fix our attention solely on the purely logical relations involved. Investigation will probably show that the rules of grammar are mainly founded upon traditional usage and have little logical signification. This indeed is sufficiently proved by the wide grammatical differences which exist between languages, though the logical foundation must be the same.

Symbolic Expression of the Law of Contradiction.

The synthesis of terms is subject to the all-important Law of Thought, described in a previous section (p. [5]) and called the Law of Contradiction, It is self-evident that no quality can be both present and absent at the same time and place. This fundamental condition of all thought and of all existence is expressed symbolically by a rule that a term and its negative shall never be allowed to come into combination. Such combined terms as Aa, Bb, Cc, &c., are self-contradictory and devoid of all intelligible meaning. If they could represent anything, it would be what cannot exist, and cannot even be imagined in the mind. They can therefore only enter into our consideration to suffer immediate exclusion. The criterion of false reasoning, as we shall find, is that it involves self-contradiction, the affirming and denying of the same statement. We might represent the object of all reasoning as the separation of the consistent and possible from the inconsistent and impossible; and we cannot make any statement except a truism without implying that certain combinations of terms are contradictory and excluded from thought. To assert that “all A’s are B’s” is equivalent to the assertion that “A’s which are not B’s cannot exist.”

It will be convenient to have the means of indicating the exclusion of the self-contradictory, and we may use the familiar sign for nothing, the cipher 0. Thus the second law of thought may be symbolised in the forms

Aa = 0  ABb = 0  ABCa = 0

We may variously describe the meaning of 0 in logic as the non-existent, the impossible, the self-inconsistent, the inconceivable. Close analogy exists between this meaning and its mathematical signification.

Certain Special Conditions of Logical Symbols.

In order that we may argue and infer truly we must treat our logical symbols according to the fundamental laws of Identity and Difference. But in thus using our symbols we shall frequently meet with combinations of which the meaning will not at first sight be apparent. If in one case we learn that an object is “yellow and round,” and in another case that it is “round and yellow,” there arises the question whether these two descriptions are identical in meaning or not. Again, if we proved that an object was “round round,” the meaning of such an expression would be open to doubt. Accordingly we must take notice, before proceeding further, of certain special laws which govern the combination of logical terms.

In the first place the combination of a logical term with itself is without effect, just as the repetition of a statement does not alter the meaning of the statement; “a round round object” is simply “a round object.” What is yellow yellow is merely yellow; metallic metals cannot differ from metals, nor circular circles from circles. In our symbolic language we may similarly hold that AA is identical with A, or

A = AA = AAA = &c.

The late Professor Boole is the only logician in modern times who has drawn attention to this remarkable property of logical terms;‍[46] but in place of the name which he gave to the law, I have proposed to call it The Law of Simplicity.‍[47] Its high importance will only become apparent when we attempt to determine the relations of logical and mathematical science. Two symbols of quantity, and only two, seem to obey this law; we may say that 1 × 1 = 1, and 0 × 0 = 0 (taking 0 to mean absolute zero or 1 – 1); there is apparently no other number which combined with itself gives an unchanged result. I shall point out, however, in the chapter upon Number, that in reality all numerical symbols obey this logical principle.

It is curious that this Law of Simplicity, though almost unnoticed in modern times, was known to Boëthius, who makes a singular remark in his treatise De Trinitate et Unitate Dei (p. 959). He says: “If I should say sun, sun, sun, I should not have made three suns, but I should have named one sun so many times.”‍[48] Ancient discussions about the doctrine of the Trinity drew more attention to subtle questions concerning the nature of unity and plurality than has ever since been given to them.

It is a second law of logical symbols that order of combination is a matter of indifference. “Rich and rare gems” are the same as “rare and rich gems,” or even as “gems, rich and rare.” Grammatical, rhetorical, or poetic usage may give considerable significance to order of expression. The limited power of our minds prevents our grasping many ideas at once, and thus the order of statement may produce some effect, but not in a simply logical manner. All life proceeds in the succession of time, and we are obliged to write, speak, or even think of things and their qualities one after the other; but between the things and their qualities there need be no such relation of order in time or space. The sweetness of sugar is neither before nor after its weight and solubility. The hardness of a metal, its colour, weight, opacity, malleability, electric and chemical properties, are all coexistent and coextensive, pervading the metal and every part of it in perfect community, none before nor after the others. In our words and symbols we cannot observe this natural condition; we must name one quality first and another second, just as some one must be the first to sign a petition, or to walk foremost in a procession. In nature there is no such precedence.

I find that the opinion here stated, to the effect that relations of space and time do not apply to many of our ideas, is clearly adopted by Hume in his celebrated Treatise on Human Nature (vol. i. p. 410). He says:‍[49]—“An object may be said to be no where, when its parts are not so situated with respect to each other, as to form any figure or quantity; nor the whole with respect to other bodies so as to answer to our notions of contiguity or distance. Now this is evidently the case with all our perceptions and objects, except those of sight and feeling. A moral reflection cannot be placed on the right hand or on the left hand of a passion, nor can a smell or sound be either of a circular or a square figure. These objects and perceptions, so far from requiring any particular place, are absolutely incompatible with it, and even the imagination cannot attribute it to them.”

A little reflection will show that knowledge in the highest perfection would consist in the simultaneous possession of a multitude of facts. To comprehend a science perfectly we should have every fact present with every other fact. We must write a book and we must read it successively word by word, but how infinitely higher would be our powers of thought if we could grasp the whole in one collective act of consciousness! Compared with the brutes we do possess some slight approximation to such power, and it is conceivable that in the indefinite future mind may acquire an increase of capacity, and be less restricted to the piecemeal examination of a subject. But I wish here to make plain that there is no logical foundation for the successive character of thought and reasoning unavoidable under our present mental conditions. We are logically weak and imperfect in respect of the fact that we are obliged to think of one thing after another. We must describe metal as “hard and opaque,” or “opaque and hard,” but in the metal itself there is no such difference of order; the properties are simultaneous and coextensive in existence.

Setting aside all grammatical peculiarities which render a substantive less moveable than an adjective, and disregarding any meaning indicated by emphasis or marked order of words, we may state, as a general law of logic, that AB is identical with BA, or AB = BA. Similarly, ABC = ACB = BCA = &c.

Boole first drew attention in recent years to this property of logical terms, and he called it the property of Commutativeness.‍[50] He not only stated the law with the utmost clearness, but pointed out that it is a Law of Thought rather than a Law of Things. I shall have in various parts of this work to show how the necessary imperfection of our symbols expressed in this law clings to our modes of expression, and introduces complication into the whole body of mathematical formulæ, which are really founded on a logical basis.

It is of course apparent that the power of commutation belongs only to terms related in the simple logical mode of synthesis. No one can confuse “a house of bricks” with “bricks of a house,” “twelve square feet” with “twelve feet square,” “the water of crystallization” with “the crystallization of water.” All relations which involve differences of time and space are inconvertible; the higher must not be made to change places with the lower, nor the first with the last. For the parties concerned there is all the difference in the world between A killing B and B killing A. The law of commutativeness simply asserts that difference of order does not attach to the connection between the properties and circumstances of a thing—to what I call simple logical relation.

CHAPTER III.
PROPOSITIONS.

We now proceed to consider the variety of forms of propositions in which the truths of science must be expressed. I shall endeavour to show that, however diverse these forms may be, they all admit the application of the one same principle of inference that what is true of a thing is true of the like or same. This principle holds true whatever be the kind or manner of the likeness, provided proper regard be had to its nature. Propositions may assert an identity of time, space, manner, quantity, degree, or any other circumstance in which things may agree or differ.

We find an instance of a proposition concerning time in the following:—“The year in which Newton was born, was the year in which Galileo died.” This proposition expresses an approximate identity of time between two events; hence whatever is true of the year in which Galileo died is true of that in which Newton was born, and vice versâ. “Tower Hill is the place where Raleigh was executed” expresses an identity of place; and whatever is true of the one spot is true of the spot otherwise defined, but in reality the same. In ordinary language we have many propositions obscurely expressing identities of number, quantity, or degree. “So many men, so many minds,” is a proposition concerning number, that is to say, an equation; whatever is true of the number of men is true of the number of minds, and vice versâ. “The density of Mars is (nearly) the same as that of the Earth,” “The force of gravity is directly as the product of the masses, and inversely as the square of the distance,” are propositions concerning magnitude or degree. Logicians have not paid adequate attention to the great variety of propositions which can be stated by the use of the little conjunction as, together with so. “As the home so the people,” is a proposition expressing identity of manner; and a great number of similar propositions all indicating some kind of resemblance might be quoted. Whatever be the special kind of identity, all such expressions are subject to the great principle of inference; but as we shall in later parts of this work treat more particularly of inference in cases of number and magnitude, we will here confine our attention to logical propositions which involve only notions of quality.

Simple Identities.

The most important class of propositions consists of those which fall under the formula

A = B,

and may be called simple identities. I may instance, in the first place, those most elementary propositions which express the exact similarity of a quality encountered in two or more objects. I may compare the colour of the Pacific Ocean with that of the Atlantic, and declare them identical. I may assert that “the smell of a rotten egg is like that of hydrogen sulphide;” “the taste of silver hyposulphite is like that of cane sugar;” “the sound of an earthquake resembles that of distant artillery.” Such are propositions stating, accurately or otherwise, the identity of simple physical sensations. Judgments of this kind are necessarily pre-supposed in more complex judgments. If I declare that “this coin is made of gold,” I must base the judgment upon the exact likeness of the substance in several qualities to other pieces of substance which are undoubtedly gold. I must make judgments of the colour, the specific gravity, the hardness, and of other mechanical and chemical properties; each of these judgments is expressed in an elementary proposition, “the colour of this coin is the colour of gold,” and so on. Even when we establish the identity of a thing with itself under a different name or aspect, it is by distinct judgments concerning single circumstances. To prove that the Homeric χαλκός is copper we must show the identity of each quality recorded of χαλκός with a quality of copper. To establish Deal as the landing-place of Cæsar all material circumstances must be shown to agree. If the modern Wroxeter is the ancient Uriconium, there must be the like agreement of all features of the country not subject to alteration by time.

Such identities must be expressed in the form A = B. We may say

Colour of Pacific Ocean = Colour of Atlantic Ocean.

Smell of rotten egg = Smell of hydrogen sulphide.

In these and similar propositions we assert identity of single qualities or causes of sensation. In the same form we may also express identity of any group of qualities, as in

χαλκός = Copper.
Deal = Landing-place of Cæsar.

A multitude of propositions involving singular terms fall into the same form, as in

The Pole star = The slowest-moving star.

Jupiter = The greatest of the planets.

The ringed planet = The planet having seven satellites.

The Queen of England = The Empress of India.

The number two = The even prime number.

In mathematical and scientific theories we often meet with simple identities capable of expression in the same form. Thus in mechanical science “The process for finding the resultant of forces = the process for finding the resultant of simultaneous velocities.” Theorems in geometry often give results in this form, as

Equilateral triangles = Equiangular triangles.

Circle = Finite plane curve of constant curvature.

Circle = Curve of least perimeter.

The more profound and important laws of nature are often expressible in the form of simple identities; in addition to some instances which have already been given, I may suggest,

Crystals of cubical system = Crystals not possessing the power of double refraction.

All definitions are necessarily of this form, whether the objects defined be many, few, or singular. Thus we may say,

Common salt = Sodium chloride.

Chlorophyl = Green colouring matter of leaves.

Square = Equal-sided rectangle.

It is an extraordinary fact that propositions of this elementary form, all-important and very numerous as they are, had no recognised place in Aristotle’s system of Logic. Accordingly their importance was overlooked until very recent times, and logic was the most deformed of sciences. But it is impossible that Aristotle or any other person should avoid constantly using them; not a term could be defined without their use. In one place at least Aristotle actually notices a proposition of the kind. He observes: “We sometimes say that that white thing is Socrates, or that the object approaching is Callias.”‍[51] Here we certainly have simple identity of terms; but he considered such propositions purely accidental, and came to the unfortunate conclusion, that “Singulars cannot be predicated of other terms.”

Propositions may also express the identity of extensive groups of objects taken collectively or in one connected whole; as when we say,

The Queen, Lords, and Commons = The Legislature of the United Kingdom.

When Blackstone asserts that “The only true and natural foundation of society are the wants and fears of individuals,” we must interpret him as meaning that the whole of the wants and fears of individuals in the aggregate form the foundation of society. But many propositions which might seem to be collective are but groups of singular propositions or identities. When we say “Potassium and sodium are the metallic bases of potash and soda,” we obviously mean,

Potassium = Metallic base of potash;
Sodium = Metallic base of soda.

It is the work of grammatical analysis to separate the various propositions often combined into a single sentence. Logic cannot be properly required to interpret the forms and devices of language, but only to treat the meaning when clearly exhibited.

Partial Identities.

A second highly important kind of proposition is that which I propose to call a partial identity. When we say that “All mammalia are vertebrata,” we do not mean that mammalian animals are identical with vertebrate animals, but only that the mammalia form a part of the class vertebrata. Such a proposition was regarded in the old logic as asserting the inclusion of one class in another, or of an object in a class. It was called a universal affirmative proposition, because the attribute vertebrate was affirmed of the whole subject mammalia; but the attribute was said to be undistributed, because not all vertebrata were of necessity involved in the proposition. Aristotle, overlooking the importance of simple identities, and indeed almost denying their existence, unfortunately founded his system upon the notion of inclusion in a class, instead of adopting the basis of identity. He regarded inference as resting upon the rule that what is true of the containing class is true of the contained, in place of the vastly more general rule that what is true of a class or thing is true of the like. Thus he not only reduced logic to a fragment of its proper self, but destroyed the deep analogies which bind together logical and mathematical reasoning. Hence a crowd of defects, difficulties and errors which will long disfigure the first and simplest of the sciences.

It is surely evident that the relation of inclusion rests upon the relation of identity. Mammalian animals cannot be included among vertebrates unless they be identical with part of the vertebrates. Cabinet Ministers are included almost always in the class Members of Parliament, because they are identical with some who sit in Parliament. We may indicate this identity with a part of the larger class in various ways; as for instance,

Mammalia = part of the vertebrata.
Diatomaceæ = a class of plants.

Cabinet Ministers = some members of Parliament.

In ordinary language the verbs is and are express mere inclusion more often than not. Men are mortals, means that men form a part of the class mortal; but great confusion exists between this sense of the verb and that in which it expresses identity, as in “The sun is the centre of the planetary system.” The introduction of the indefinite article a often expresses partiality; when we say “Iron is a metal” we clearly mean that iron is one only of several metals.

Certain recent logicians have proposed to avoid the indefiniteness in question by what is called the Quantification of the Predicate, and they have generally used the little word some to show that only a part of the predicate is identical with the subject. Some is an indeterminate adjective; it implies unknown qualities by which we might select the part in question if the qualities were known, but it gives no hint as to their nature. I might make use of such an indeterminate sign to express partial identities in this work. Thus, taking the special symbol V = Some, the general form of a partial identity would be A = VB, and in Boole’s Logic expressions of the kind were much used. But I believe that indeterminate symbols only introduce complexity, and destroy the beauty and simple universality of the system which may be created without their use. A vague word like some is only used in ordinary language by ellipsis, and to avoid the trouble of attaining accuracy. We can always employ more definite expressions if we like; but when once the indefinite some is introduced we cannot replace it by the special description. We do not know whether some colour is red, yellow, blue, or what it is; but on the other hand red colour is certainly some colour.

Throughout this system of logic I shall dispense with such indefinite expressions; and this can readily be done by substituting one of the other terms. To express the proposition “All A’s are some B’s” I shall not use the form A = VB, but

A = AB.

This formula states that the class A is identical with the class AB; and as the latter must be a part at least of the class B, it implies the inclusion of the class A in that of B. We might represent our former example thus,

Mammalia = Mammalian vertebrata.

This proposition asserts identity between a part (or it may be the whole) of the vertebrata and the mammalia. If it is asked What part? the proposition affords no answer, except that it is the part which is mammalian; but the assertion “mammalia = some vertebrata” tells us no more.

It is quite likely that some readers will think this mode of representing the universal affirmative proposition artificial and complicated. I will not undertake to convince them of the opposite at this point of my exposition. Justification for it will be found, not so much in the immediate treatment of this proposition, as in the general harmony which it will enable us to disclose between all parts of reasoning. I have no doubt that this is the critical difficulty in the relation of logical to other forms of reasoning. Grant this mode of denoting that “all A’s are B’s,” and I fear no further difficulties; refuse it, and we find want of analogy and endless anomaly in every direction. It is on general grounds that I hope to show overwhelming reasons for seeking to reduce every kind of proposition to the form of an identity.

I may add that not a few logicians have accepted this view of the universal affirmative proposition. Leibnitz, in his Difficultates Quædam Logicæ, adopts it, saying, “Omne A est B; id est æquivalent AB et A, seu A non B est nonens.” Boole employed the logical equation x = xy concurrently with x = vy; and Spalding‍[52] distinctly says that the proposition “all metals are minerals” might be described as an assertion of partial identity between the two classes. Hence the name which I have adopted for the proposition.

Limited Identities.

An important class of propositions have the form

AB = AC,

expressing the identity of the class AB with the class AC. In other words, “Within the sphere of the class A, all the B’s are all the C’s;” or again, “The B’s and C’s, which are A’s, are identical.” But it will be observed that nothing is asserted concerning things which are outside of the class A; and thus the identity is of limited extent. It is the proposition B = C limited to the sphere of things called A. Thus we may say, with some approximation to truth, that “Large plants are plants devoid of locomotive power.”

A barrister may make numbers of most general statements concerning the relations of persons and things in the course of an argument, but it is of course to be understood that he speaks only of persons and things under the English Law. Even mathematicians make statements which are not true with absolute generality. They say that imaginary roots enter into equations by pairs; but this is only true under the tacit condition that the equations in question shall not have imaginary coefficients.‍[53] The universe, in short, within which they habitually discourse is that of equations with real coefficients. These implied limitations form part of that great mass of tacit knowledge which accompanies all special arguments.

To De Morgan is due the remark, that we do usually think and argue in a limited universe or sphere of notions, even when it is not expressly stated.‍[54]

It is worthy of inquiry whether all identities are not really limited to an implied sphere of meaning. When we make such a plain statement as “Gold is malleable” we obviously speak of gold only in its solid state; when we say that “Mercury is a liquid metal” we must be understood to exclude the frozen condition to which it may be reduced in the Arctic regions. Even when we take such a fundamental law of nature as “All substances gravitate,” we must mean by substance, material substance, not including that basis of heat, light, and electrical undulations which occupies space and possesses many wonderful mechanical properties, but not gravity. The proposition then is really of the form

Material substance = Material gravitating substance.

Negative Propositions.

In every act of intellect we are engaged with a certain identity or difference between things or sensations compared together. Hitherto I have treated only of identities; and yet it might seem that the relation of difference must be infinitely more common than that of likeness. One thing may resemble a great many other things, but then it differs from all remaining things in the world. Diversity may almost be said to constitute life, being to thought what motion is to a river. The perception of an object involves its discrimination from all other objects. But we may nevertheless be said to detect resemblance as often as we detect difference. We cannot, in fact, assert the existence of a difference, without at the same time implying the existence of an agreement.

If I compare mercury, for instance, with other metals, and decide that it is not solid, here is a difference between mercury and solid things, expressed in a negative proposition; but there must be implied, at the same time, an agreement between mercury and the other substances which are not solid. As it is impossible to separate the vowels of the alphabet from the consonants without at the same time separating the consonants from the vowels, so I cannot select as the object of thought solid things, without thereby throwing together into another class all things which are not solid. The very fact of not possessing a quality, constitutes a new quality which may be the ground of judgment and classification. In this point of view, agreement and difference are ever the two sides of the same act of intellect, and it becomes equally possible to express the same judgment in the one or other aspect.

Between affirmation and negation there is accordingly a perfect equilibrium. Every affirmative proposition implies a negative one, and vice versâ. It is even a matter of indifference, in a logical point of view, whether a positive or negative term be used to denote a given quality and the class of things possessing it. If the ordinary state of a man’s body be called good health, then in other circumstances he is said not to be in good health; but we might equally describe him in the latter state as sickly, and in his normal condition he would be not sickly. Animal and vegetable substances are now called organic, so that the other substances, forming an immensely greater part of the globe, are described negatively as inorganic. But we might, with at least equal logical correctness, have described the preponderating class of substances as mineral, and then vegetable and animal substances would have been non-mineral.

It is plain that any positive term and its corresponding negative divide between them the whole universe of thought: whatever does not fall into one must fall into the other, by the third fundamental Law of Thought, the Law of Duality. It follows at once that there are two modes of representing a difference. Supposing that the things represented by A and B are found to differ, we may indicate (see p. [17]) the result of the judgment by the notation

A ~ B.

We may now represent the same judgment by the assertion that A agrees with those things which differ from B, or that A agrees with the not-B’s. Using our notation for negative terms (see p. [14]), we obtain

A = Ab

as the expression of the ordinary negative proposition. Thus if we take A to mean quicksilver, and B solid, then we have the following proposition:‍—

Quicksilver = Quicksilver not-solid.

There may also be several other classes of negative propositions, of which no notice was taken in the old logic. We may have cases where all A’s are not-B’s, and at the same time all not-B’s are A’s; there may, in short, be a simple identity between A and not-B, which may be expressed in the form

A = b.

An example of this form would be

Conductors of electricity = non-electrics.

We shall also frequently have to deal as results of deduction, with simple, partial, or limited identities between negative terms, as in the forms

a = b,  a = ab,  aC = bC, etc.

It would be possible to represent affirmative propositions in the negative form. Thus “Iron is solid,” might be expressed as “Iron is not not-solid,” or “Iron is not fluid;” or, taking A and b for the terms “iron,” and “not-solid,” the form would be A ~ b.

But there are very strong reasons why we should employ all propositions in their affirmative form. All inference proceeds by the substitution of equivalents, and a proposition expressed in the form of an identity is ready to yield all its consequences in the most direct manner. As will be more fully shown, we can infer in a negative proposition, but not by it. Difference is incapable of becoming the ground of inference; it is only the implied agreement with other differing objects which admits of deductive reasoning; and it will always be found advantageous to employ propositions in the form which exhibits clearly the implied agreements.

Conversion of Propositions.

The old books of logic contain many rules concerning the conversion of propositions, that is, the transposition of the subject and predicate in such a way as to obtain a new proposition which will be true when the original proposition is true. The reduction of every proposition to the form of an identity renders all such rules and processes needless. Identity is essentially reciprocal. If the colour of the Atlantic Ocean is the same as that of the Pacific Ocean, that of the Pacific must be the same as that of the Atlantic. Sodium chloride being identical with common salt, common salt must be identical with sodium chloride. If the number of windows in Salisbury Cathedral equals the number of days in the year, the number of days in the year must equal the number of the windows. Lord Chesterfield was not wrong when he said, “I will give anybody their choice of these two truths, which amount to the same thing; He who loves himself best is the honestest man; or, The honestest man loves himself best.” Scotus Erigena exactly expresses this reciprocal character of identity in saying, “There are not two studies, one of philosophy and the other of religion; true philosophy is true religion, and true religion is true philosophy.”

A mathematician would not think it worth while to mention that if x = y then also y = x. He would not consider these to be two equations at all, but one equation accidentally written in two different manners. In written symbols one of two names must come first, and the other second, and a like succession must perhaps be observed in our thoughts: but in the relation of identity there is no need for succession in order (see p. [33]), each is simultaneously equal and identical to the other. These remarks will hold true both of logical and mathematical identity; so that I shall consider the two forms

A = B and B = A

to express exactly the same identity differently written. All need for rules of conversion disappears, and there will be no single proposition in the system which may not be written with either end foremost. Thus A = AB is the same as AB = A, aC = bC is the same as bC = aC, and so forth.

The same remarks are partially true of differences and inequalities, which are also reciprocal to the extent that one thing cannot differ from a second without the second differing from the first. Mars differs in colour from Venus, and Venus must differ from Mars. The Earth differs from Jupiter in density; therefore Jupiter must differ from the Earth. Speaking generally, if A ~ B we shall also have B ~ A, and these two forms may be considered expressions of the same difference. But the relation of differing things is not wholly reciprocal. The density of Jupiter does not differ from that of the Earth in the same way that that of the Earth differs from that of Jupiter. The change of sensation which we experience in passing from Venus to Mars is not the same as what we experience in passing back to Venus, but just the opposite in nature. The colour of the sky is lighter than that of the ocean; therefore that of the ocean cannot be lighter than that of the sky, but darker. In these and all similar cases we gain a notion of direction or character of change, and results of immense importance may be shown to rest on this notion. For the present we shall be concerned with the mere fact of identity existing or not existing.

Twofold Interpretation of Propositions.

Terms, as we have seen (p. [25]), may have a meaning either in extension or intension; and according as one or the other meaning is attributed to the terms of a proposition, so may a different interpretation be assigned to the proposition itself. When the terms are abstract we must read them in intension, and a proposition connecting such terms must denote the identity or non-identity of the qualities respectively denoted by the terms. Thus if we say

Equality = Identity of magnitude,

the assertion means that the circumstance of being equal exactly corresponds with the circumstance of being identical in magnitude. Similarly in

Opacity = Incapability of transmitting light,

the quality of being incapable of transmitting light is declared to be the same as the intended meaning of the word opacity.

When general names form the terms of a proposition we may apply a double interpretation. Thus

Exogens = Dicotyledons

means either that the qualities which belong to all exogens are the same as those which belong to all dicotyledons, or else that every individual falling under one name falls equally under the other. Hence it may be said that there are two distinct fields of logical thought. We may argue either by the qualitative meaning of names or by the quantitative, that is, the extensive meaning. Every argument involving concrete plural terms might be converted into one involving only abstract singular terms, and vice versâ. But there are reasons for believing that the intensive or qualitative form of reasoning is the primary and fundamental one. It is sufficient to point out that the extensive meaning of a name is a changeable and fleeting thing, while the intensive meaning may nevertheless remain fixed. Very numerous additions have been lately made to the extensive meanings both of planet and element. Every iron steam-ship which is made or destroyed adds to or subtracts from the extensive meaning of the name steam-ship, without necessarily affecting the intensive meaning. Stage coach means as much as ever in one way, but in extension the class is nearly extinct. Chinese railway, on the other hand, is a term represented only by a single instance; in twenty years it may be the name of a large class.

CHAPTER IV.
DEDUCTIVE REASONING.

The general principle of inference having been explained in the previous chapters, and a suitable system of symbols provided, we have now before us the comparatively easy task of tracing out the most common and important forms of deductive reasoning. The general problem of deduction is as follows:—From one or more propositions called premises to draw such other propositions as will necessarily be true when the premises are true. By deduction we investigate and unfold the information contained in the premises; and this we can do by one single rule—For any term occurring in any proposition substitute the term which is asserted in any premise to be identical with it. To obtain certain deductions, especially those involving negative conclusions, we shall require to bring into use the second and third Laws of Thought, and the process of reasoning will then be called Indirect Deduction. In the present chapter, however, I shall confine my attention to those results which can be obtained by the process of Direct Deduction, that is, by applying to the premises themselves the rule of substitution. It will be found that we can combine into one harmonious system, not only the various moods of the ancient syllogism but a great number of equally important forms of reasoning, which had no recognised place in the old logic. We can at the same time dispense entirely with the elaborate apparatus of logical rules and mnemonic lines, which were requisite so long as the vital principle of reasoning was not clearly expressed.

Immediate Inference.

Probably the simplest of all forms of inference is that which has been called Immediate Inference, because it can be performed upon a single proposition. It consists in joining an adjective, or other qualifying clause of the same nature, to both sides of an identity, and asserting the equivalence of the terms thus produced. For instance, since

Conductors of electricity = Non-electrics,

it follows that

Liquid conductors of electricity = Liquid non-electrics.

If we suppose that

Plants = Bodies decomposing carbonic acid,

it follows that

Microscopic plants = Microscopic bodies decomposing carbonic acid.

In general terms, from the identity

A = B

we can infer the identity

AC = BC.

This is but a case of plain substitution; for by the first Law of Thought it must be admitted that

AC = AC,

and if, in the second side of this identity, we substitute for A its equivalent B, we obtain

AC = BC.

In like manner from the partial identity

A = AB

we may obtain

AC = ABC

by an exactly similar act of substitution; and in every other case the rule will be found capable of verification by the principle of inference. The process when performed as here described will be quite free from the liability to error which I have shown‍[55] to exist in “Immediate Inference by added Determinants,” as described by Dr. Thomson.‍[56]

Inference with Two Simple Identities.

One of the most common forms of inference, and one to which I shall especially direct attention, is practised with two simple identities. From the two statements that “London is the capital of England” and “London is the most populous city in the world,” we instantaneously draw the conclusion that “The capital of England is the most populous city in the world.” Similarly, from the identities

Hydrogen = Substance of least density,

Hydrogen = Substance of least atomic weight,

we infer

Substance of least density = Substance of least atomic weight.

The general form of the argument is exhibited in the symbols

	B = A	(1)
	B = C	(2)
hence	A = C.	(3)

We may describe the result by saying that terms identical with the same term are identical with each other; and it is impossible to overlook the analogy to the first axiom of Euclid that “things equal to the same thing are equal to each other.” It has been very commonly supposed that this is a fundamental principle of thought, incapable of reduction to anything simpler. But I entertain no doubt that this form of reasoning is only one case of the general rule of inference. We have two propositions, A = B and B = C, and we may for a moment consider the second one as affirming a truth concerning B, while the former one informs us that B is identical with A; hence by substitution we may affirm the same truth of A. It happens in this particular case that the truth affirmed is identity to C, and we might, if we preferred it, have considered the substitution as made by means of the second identity in the first. Having two identities we have a choice of the mode in which we will make the substitution, though the result is exactly the same in either case.

Now compare the three following formulæ,

(1)	A = B = C, hence A = C
(2)	A = B ~ C, hence A ~ C
(3)	A ~ B ~ C, no inference.

In the second formula we have an identity and a difference, and we are able to infer a difference; in the third we have two differences and are unable to make any inference at all. Because A and C both differ from B, we cannot tell whether they will or will not differ from each other. The flowers and leaves of a plant may both differ in colour from the earth in which the plant grows, and yet they may differ from each other; in other cases the leaves and stem may both differ from the soil and yet agree with each other. Where we have difference only we can make no inference; where we have identity we can infer. This fact gives great countenance to my assertion that inference proceeds always through identity, but may be equally well effected in propositions asserting difference or identity.

Deferring a more complete discussion of this point, I will only mention now that arguments from double identity occur very frequently, and are usually taken for granted, owing to their extreme simplicity. In regard to the equivalence of words this form of inference must be constantly employed. If the ancient Greek χαλκός is our copper, then it must be the French cuivre, the German kupfer, the Latin cuprum, because these are words, in one sense at least, equivalent to copper. Whenever we can give two definitions or expressions for the same term, the formula applies; thus Senior defined wealth as “All those things, and those things only, which are transferable, are limited in supply, and are directly or indirectly productive of pleasure or preventive of pain.” Wealth is also equivalent to “things which have value in exchange;” hence obviously, “things which have value in exchange = all those things, and those things only, which are transferable, &c.” Two expressions for the same term are often given in the same sentence, and their equivalence implied. Thus Thomson and Tait say,‍[57] “The naturalist may be content to know matter as that which can be perceived by the senses, or as that which can be acted upon by or can exert force.” I take this to mean—

Matter = what can be perceived by the senses;

Matter = what can be acted upon by or can exert force.

For the term “matter” in either of these identities we may substitute its equivalent given in the other definition. Elsewhere they often employ sentences of the form exemplified in the following:‍[58] “The integral curvature, or whole change of direction of an arc of a plane curve, is the angle through which the tangent has turned as we pass from one extremity to the other.” This sentence is certainly of the form‍—

The integral curvature = the whole change of direction, &c. = the angle through which the tangent has turned, &c.

Disguised cases of the same kind of inference occur throughout all sciences, and a remarkable instance is found in algebraic geometry. Mathematicians readily show that every equation of the form y = mx + c corresponds to or represents a straight line; it is also easily proved that the same equation is equivalent to one of the general form Ax + By + C = 0, and vice versâ. Hence it follows that every equation of the form in question, that is to say, every equation of the first degree, corresponds to or represents a straight line.‍[59]

Inference with a Simple and a Partial Identity.

A form of reasoning somewhat different from that last considered consists in inference-between a simple and a partial identity. If we have two propositions of the forms

A = B,
B = BC,

we may then substitute for B in either proposition its equivalent in the other, getting in both cases A = BC; in this we may if we like make a second substitution for B, getting

A = AC.

Thus, since “The Mont Blanc is the highest mountain in Europe, and the Mont Blanc is deeply covered with snow,” we infer by an obvious substitution that “The highest mountain in Europe is deeply covered with snow.” These propositions when rigorously stated fall into the forms above exhibited.

This mode of inference is constantly employed when for a term we substitute its definition, or vice versâ. The very purpose of a definition is to allow a single noun to be employed in place of a long descriptive phrase. Thus, when we say “A circle is a curve of the second degree,” we may substitute a definition of the circle, getting “A curve, all points of which are at equal distances from one point, is a curve of the second degree.” The real forms of the propositions here given are exactly those shown in the symbolic statement, but in this and many other cases it will be sufficient to state them in ordinary elliptical language for sake of brevity. In scientific treatises a term and its definition are often both given in the same sentence, as in “The weight of a body in any given locality, or the force with which the earth attracts it, is proportional to its mass.” The conjunction or in this statement gives the force of equivalence to the parenthetic phrase, so that the propositions really are

Weight of a body = force with which the earth attracts it.

Weight of a body = weight, &c. proportional to its mass.

A slightly different case of inference consists in substituting in a proposition of the form A = AB, a definition of the term B. Thus from A = AB and B = C we get A = AC. For instance, we may say that “Metals are elements” and “Elements are incapable of decomposition.”

Metal = metal element.

Element = what is incapable of decomposition.

Hence

Metal = metal incapable of decomposition.

It is almost needless to point out that the form of these arguments does not suffer any real modification if some of the terms happen to be negative; indeed in the last example “incapable of decomposition” may be treated as a negative term. Taking

A = metal	C = capable of decomposition
B = element	c = incapable of decomposition;

the propositions are of the forms

A = AB
B = c

whence, by substitution,

A = Ac.

Inference of a Partial from Two Partial Identities.

However common be the cases of inference already noticed, there is a form occurring almost more frequently, and which deserves much attention, because it occupied a prominent place in the ancient syllogistic system. That system strangely overlooked all the kinds of argument we have as yet considered, and selected, as the type of all reasoning, one which employs two partial identities as premises. Thus from the propositions

Sodium is a metal	(1)
Metals conduct electricity,	(2)

we may conclude that

Sodium conducts electricity. (3)

Taking A, B, C to represent the three terms respectively, the premises are of the forms

A = AB   (1)
B = BC.  (2)

Now for B in (1) we can substitute its expression as given in (2), obtaining

A = ABC,  (3)

or, in words, from

Sodium = sodium metal,	(1)
Metal = metal conducting electricity,	(2)

we infer

Sodium = sodium metal conducting electricity,  (3)

which, in the elliptical language of common life, becomes

“Sodium conducts electricity.”

The above is a syllogism in the mood called Barbara‍[60] in the truly barbarous language of ancient logicians; and the first figure of the syllogism contained Barbara and three other moods which were esteemed distinct forms of argument. But it is worthy of notice that, without any real change in our form of inference, we readily include these three other moods under Barbara. The negative mood Celarent will be represented by the example

	Neptune is a planet,	(1)
	No planet has retrograde motion;	(2)
Hence	Neptune has not retrograde motion.	(3)

If we put A for Neptune, B for planet, and C for “having retrograde motion,” then by the corresponding negative term c, we denote “not having retrograde motion.” The premises now fall into the forms

A = AB	(1)
B = Bc,	(2)

and by substitution for B, exactly as before, we obtain

A = ABc.  (3)

What is called in the old logic a particular conclusion may be deduced without any real variation in the symbols. Particular quantity is indicated, as before mentioned (p. [41]), by joining to the term an indefinite adjective of quantity, such as some, a part of, certain, &c., meaning that an unknown part of the term enters into the proposition as subject. Considerable doubt and ambiguity arise out of the question whether the part may not in some cases be the whole, and in the syllogism at least it must be understood in this sense.‍[61] Now, if we take a letter to represent this indefinite part, we need make no change in our formulæ to express the syllogisms Darii and Ferio. Consider the example—

Some metals are of less density than water,	(1)
All bodies of less density than water will float upon the surface of water; hence	(2)
Some metals will float upon the surface of water.	(3)

Let	A = some metals,
	B = body of less density than water,
	C = floating on the surface of water

then the propositions are evidently as before,

	A = AB,	(1)
	B = BC;	(2)
hence	A = ABC,	(3)

Thus the syllogism Darii does not really differ from Barbara. If the reader prefer it, we can readily employ a distinct symbol for the indefinite sign of quantity.

Let	P = some,
	Q = metal,

B and C having the same meanings as before. Then the premises become

PQ = PQB,	(1)
B = BC;	(2)

hence, by substitution, as before,

PQ = PQBC.  (3)

Except that the formulæ look a little more complicated there is no difference whatever.

The mood Ferio is of exactly the same character as Darii or Barbara, except that it involves the use of a negative term. Take the example,

Bodies which are equally elastic in all directions do not doubly refract light;

Some crystals are bodies equally elastic in all directions; therefore, some crystals do not doubly refract light.

Assigning the letters as follows:‍—

A = some crystals,
B = bodies equally elastic in all directions,
C = doubly refracting light,
c = not doubly refracting light.

Our argument is of the same form as before, and may be concisely stated in one line,

A = AB = ABc.

If it is preferred to put PQ for the indefinite some crystals, we have

PQ = PQB = PQBc.

The only difference is that the negative term c takes the place of C in the mood Darii.

Ellipsis of Terms in Partial Identities.

The reader will probably have noticed that the conclusion which we obtain from premises is often more full than that drawn by the old Aristotelian processes. Thus from “Sodium is a metal,” and “Metals conduct electricity,” we inferred (p. [55]) that “Sodium = sodium, metal, conducting electricity,” whereas the old logic simply concludes that “Sodium conducts electricity.” Symbolically, from A = AB, and B = BC, we get A = ABC, whereas the old logic gets at the most A = AC. It is therefore well to show that without employing any other principles of inference than those already described, we may infer A = AC from A = ABC, though we cannot infer the latter more full and accurate result from the former. We may show this most simply as follows:‍—

By the first Law of Thought it is evident that

AA = AA;

and if we have given the proposition A = ABC, we may substitute for both the A’s in the second side of the above, obtaining

AA = ABC . ABC.

But from the property of logical symbols expressed in the Law of Simplicity (p. [33]) some of the repeated letters may be made to coalesce, and we have

A = ABC . C.

Substituting again for ABC its equivalent A, we obtain

A = AC,

the desired result.

By a similar process of reasoning it may be shown that we can always drop out any term appearing in one member of a proposition, provided that we substitute for it the whole of the other member. This process was described in my first logical Essay,‍[62] as Intrinsic Elimination, but it might perhaps be better entitled the Ellipsis of Terms. It enables us to get rid of needless terms by strict substitutive reasoning.

Inference of a Simple from Two Partial Identities.

Two terms may be connected together by two partial identities in yet another manner, and a case of inference then arises which is of the highest importance. In the two premises

A = AB (1)
B = AB (2)

the second member of each is the same; so that we can by obvious substitution obtain

A = B.

Thus, in plain geometry we readily prove that “Every equilateral triangle is also an equiangular triangle,” and we can with equal ease prove that “Every equiangular triangle is an equilateral triangle.” Thence by substitution, as explained above, we pass to the simple identity,

Equilateral triangle = equiangular triangle.

We thus prove that one class of triangles is entirely identical with another class; that is to say, they differ only in our way of naming and regarding them.

The great importance of this process of inference arises from the fact that the conclusion is more simple and general than either of the premises, and contains as much information as both of them put together. It is on this account constantly employed in inductive investigation, as will afterwards be more fully explained, and it is the natural mode by which we arrive at a conviction of the truth of simple identities as existing between classes of numerous objects.

Inference of a Limited from Two Partial Identities.

We have considered some arguments which are of the type treated by Aristotle in the first figure of the syllogism. But there exist two other types of argument which employ a pair of partial identities. If our premises are as shown in these symbols,

B = AB    (1)
B = CB,   (2)

we may substitute for B either by (1) in (2) or by (2) in (1), and by both modes we obtain the conclusion

AB = CB,   (3)

a proposition of the kind which we have called a limited identity (p. [42]). Thus, for example,

Potassium = potassium metal	(1)
Potassium = potassium capable of floating on water;	(2)

hence

Potassium metal = potassium capable of floating on water.

(3)

This is really a syllogism of the mood Darapti in the third figure, except that we obtain a conclusion of a more exact character than the old syllogism gives. From the premises “Potassium is a metal” and “Potassium floats on water,” Aristotle would have inferred that “Some metals float on water.” But if inquiry were made what the “some metals” are, the answer would certainly be “Metal which is potassium.” Hence Aristotle’s conclusion simply leaves out some of the information afforded in the premises. It even leaves us open to interpret the some metals in a wider sense than we are warranted in doing. From these distinct defects of the old syllogism the process of substitution is free, and the new process only incurs the possible objection of being tediously minute and accurate.

Miscellaneous Forms of Deductive Inference.

The more common forms of deductive reasoning having been exhibited and demonstrated on the principle of substitution, there still remain many, in fact an indefinite number, which may be explained with nearly equal ease. Such as involve the use of disjunctive propositions will be described in a later chapter, and several of the syllogistic moods which include negative terms will be more conveniently treated after we have introduced the symbolic use of the second and third laws of thought.

We sometimes meet with a chain of propositions which allow of repeated substitution, and form an argument called in the old logic a Sorites. Take, for instance, the premises

Iron is a metal,	(1)
Metals are good conductors of elec­tri­city,	(2)
Good conductors ofelectricity are useful for tele­graph­ic purposes.	(3)

It obviously follows that

Iron is useful for telegraphic purposes.

(4)

Now if we take our letters thus,

A = Iron,   B = metal,   C = good conductor of electricity,   D = useful for telegraphic purposes,

the premises will assume the forms

A = AB,	(1)
B = BC,	(2)
C = CD.	(3)

For B in (1) we can substitute its equivalent in (2) obtaining, as before,

A = ABC.

Substituting for C in this intermediate result its equivalent as given in (3), we obtain the complete conclusion

A = ABCD.

(4)

The full interpretation is that Iron is iron, metal, good conductor of electricity, useful for telegraphic purposes, which is abridged in common language by the ellipsis of the circumstances which are not of immediate importance.

Instead of all the propositions being exactly of the same kind as in the last example, we may have a series of premises of various character; for instance,

Common salt is sodium chloride,	(1)
Sodium chloride crystallizes in a cubical form,	(2)
What crystallizes in a cubical form does not possess the power of double refraction;	(3)

it will follow that

Common salt does not possess the power of double refraction.

(4)

Taking our letter-terms thus,

A = Common salt,
B = Sodium chloride,
C = Crystallizing in a cubical form,
D = Possessing the power of double refraction,

we may state the premises in the forms

A = B,	(1)
B = BC,	(2)
C = Cd.	(3)

Substituting by (3) in (2) and then by (2) as thus altered in (1) we obtain

A = BCd,

(4)

which is a more precise version of the common conclusion.

We often meet with a series of propositions describing the qualities or circumstances of the one same thing, and we may combine them all into one proposition by the process of substitution. This case is, in fact, that which Dr. Thomson has called “Immediate Inference by the sum of several predicates,” and his example will serve my purpose well.‍[63] He describes copper as “A metal—of a red colour—and disagreeable smell—and taste—all the preparations of which are poisonous—which is highly malleable—ductile—and tenacious—with a specific gravity of about 8.83.” If we assign the letter A to copper, and the succeeding letters of the alphabet in succession to the series of predicates, we have nine distinct statements, of the form A = AB (1) A = AC (2) A = AD (3) . . . A = AK (9). We can readily combine these propositions into one by substituting for A in the second side of (1) its expression in (2). We thus get

A = ABC,

and by repeating the process over and over again we obviously get the single proposition

A = ABCD . . . JK.

But Dr. Thomson is mistaken in supposing that we can obtain in this manner a definition of copper. Strictly speaking, the above proposition is only a description of copper, and all the ordinary descriptions of substances in scientific works may be summed up in this form. Thus we may assert of the organic substances called Paraffins that they are all saturated hydrocarbons, incapable of uniting with other substances, produced by heating the alcoholic iodides with zinc, and so on. It may be shown that no amount of ordinary description can be equivalent to a definition of any substance.

Fallacies.

I have hitherto been engaged in showing that all the forms of reasoning of the old syllogistic logic, and an indefinite number of other forms in addition, may be readily and clearly explained on the single principle of substitution. It is now desirable to show that the same principle will prevent us falling into fallacies. So long as we exactly observe the one rule of substitution of equivalents it will be impossible to commit a paralogism, that is to break any one of the elaborate rules of the ancient system. The one new rule is thus proved to be as powerful as the six, eight, or more rules by which the correctness of syllogistic reasoning was guarded.

It was a fundamental rule, for instance, that two negative premises could give no conclusion. If we take the propositions

Granite is not a sedimentary rock,	(1)
Basalt is not a sedimentary rock,	(2)

we ought not to be able to draw any inference concerning the relation between granite and basalt. Taking our letter-terms thus:

A = granite,   B = sedimentary rock,   C = basalt,

the premises may be expressed in the forms

A ~ B,	(1)
C ~ B.	(2)

We have in this form two statements of difference; but the principle of inference can only work with a statement of agreement or identity (p. [63]). Thus our rule gives us no power whatever of drawing any inference; this is exactly in accordance with the fifth rule of the syllogism.

It is to be remembered, indeed, that we claim the power of always turning a negative proposition into an affirmative one (p. [45]); and it might seem that the old rule against negative premises would thus be circumvented. Let us try. The premises (1) and (2) when affirmatively stated take the forms

A = Ab	(1)
C = Cb.	(2)

The reader will find it impossible by the rule of substitution to discover a relation between A and C. Three terms occur in the above premises, namely A, b, and C; but they are so combined that no term occurring in one has its exact equivalent stated in the other. No substitution can therefore be made, and the principle of the fifth rule of the syllogism holds true. Fallacy is impossible.

It would be a mistake, however, to suppose that the mere occurrence of negative terms in both premises of a syllogism renders them incapable of yielding a conclusion. The old rule informed us that from two negative premises no conclusion could be drawn, but it is a fact that the rule in this bare form does not hold universally true; and I am not aware that any precise explanation has been given of the conditions under which it is or is not imperative. Consider the follow­ing example:

Whatever is not metallic is not capable of powerful magnetic influence,	(1)
Carbon is not metallic,	(2)
Therefore, carbon is not capable of powerful magnetic influence.	(3)

Here we have two distinctly negative premises (1) and (2), and yet they yield a perfectly valid negative conclusion (3). The syllogistic rule is actually falsified in its bare and general statement. In this and many other cases we can convert the propositions into affirmative ones which will yield a conclusion by substitution without any difficulty.

To show this let

A = carbon,
B = metallic,

C = capable of powerful magnetic influence.

The premises readily take the forms

b = bc,	(1)
A = Ab,	(2)

and substitution for b in (2) by means of (1) gives the conclusion

A = Abc.

(3)

Our principle of inference then includes the rule of negative premises whenever it is true, and discriminates correctly between the cases where it does and does not hold true.

The paralogism, anciently called the Fallacy of Undistributed Middle, is also easily exhibited and infallibly avoided by our system. Let the premises be

Hydrogen is an element,	(1)
All metals are elements.	(2)

According to the syllogistic rules the middle term “element” is here undistributed, and no conclusion can be obtained; we cannot tell then whether hydrogen is or is not a metal. Represent the terms as follows

A = hydrogen,
B = element,
C = metal.

The premises then become

A = AB,	(1)
C = CB.	(2)

The reader will here, as in a former page (p. [62]), find it impossible to make any substitution. The only term which occurs in both premises is B, but it is differently combined in the two premises. For B we must not substitute A, which is equivalent to AB, not to B. Nor must we confuse together CB and AB, which, though they contain one common letter, are different aggregate terms. The rule of substitution gives us no right to decompose combinations; and if we adhere rigidly to the rule, that if two terms are stated to be equivalent we may substitute one for the other, we cannot commit the fallacy. It is apparent that the form of premises stated above is the same as that which we obtained by translating two negative premises into the affirmative form.

The old fallacy, technically called the Illicit Process of the Major Term, is more easy to commit and more difficult to detect than any other breach of the syllogistic rules. In our system it could hardly occur. From the premises

All planets are subject to gravity,	(1)
Fixed stars are not planets,	(2)

we might inadvertently but fallaciously infer that, “Fixed stars are not subject to gravity.” To reduce the premises to symbolic form, let

A = planet
B = fixed star
C = subject to gravity;

then we have the propositions

A = AC	(1)
B = Ba.	(2)

The reader will try in vain to produce from these premises by legitimate substitution any relation between B and C; he could not then commit the fallacy of asserting that B is not C.

There remain two other kinds of paralogism, commonly known as the fallacy of Four Terms and the Illicit Process of the Minor Term. They are so evidently impossible while we obey the rule of the substitution of equivalents, that it is not necessary to give any illustrations. When there are four distinct terms in two propositions as in A = B and C = D, there could evidently be no opening for substitution. As to the Illicit Process of the Minor Term it consists in a flagrant substitution for a term of another wider term which is not known to be equivalent to it, and which is therefore not allowed by our rule to be substituted for it.

CHAPTER V.
DISJUNCTIVE PROPOSITIONS.

In the previous chapter I have exhibited various cases of deductive reasoning by the process of substitution, avoiding the introduction of disjunctive propositions; but we cannot long defer the consideration of this more complex class of identities. General terms arise, as we have seen (p. [24]), from classifying or mentally uniting together all objects which agree in certain qualities, the value of this union consisting in the fact that the power of knowledge is multiplied thereby. In forming such classes or general notions, we overlook or abstract the points of difference which exist between the objects joined together, and fix our attention only on the points of agreement. But every process of thought may be said to have its inverse process, which consists in undoing the effects of the direct process. Just as division undoes multiplication, and evolution undoes involution, so we must have a process which undoes generalization, or the operation of forming general notions. This inverse process will consist in distinguishing the separate objects or minor classes which are the constituent parts of any wider class. If we mentally unite together certain objects visible in the sky and call them planets, we shall afterwards need to distinguish the contents of this general notion, which we do in the disjunctive proposition—

A planet is either Mercury or Venus or the Earth or . . . or Neptune.

Having formed the very wide class “vertebrate animal,” we may specify its subordinate classes thus:—“A vertebrate animal is either a mammal, bird, reptile, or fish.” Nor is there any limit to the number of possible alternatives. “An exogenous plant is either a ranunculus, a poppy, a crucifer, a rose, or it belongs to some one of the other seventy natural orders of exogens at present recognized by botanists.” A cathedral church in England must be either that of London, Canterbury, Winchester, Salisbury, Manchester, or of one of about twenty-four cities possessing such churches. And if we were to attempt to specify the meaning of the term “star,” we should require to enumerate as alternatives, not only the many thousands of stars recorded in catalogues, but the many millions unnamed.

Whenever we thus distinguish the parts of a general notion we employ a disjunctive proposition, in at least one side of which are several alternatives joined by the so-called disjunctive conjunction or, a contracted form of other. There must be some relation between the parts thus connected in one proposition; we may call it the disjunctive or alternative relation, and we must carefully inquire into its nature. This relation is that of ignorance and doubt, giving rise to choice. Whenever we classify and abstract we must open the way to such uncertainty. By fixing our attention on certain attributes to the exclusion of others, we necessarily leave it doubtful what those other attributes are. The term “molar tooth” bears upon the face of it that it is a part of the wider term “tooth.” But if we meet with the simple term “tooth” there is nothing to indicate whether it is an incisor, a canine, or a molar tooth. This doubt, however, may be resolved by further information, and we have to consider what are the appropriate logical processes for treating disjunctive propositions in connection with other propositions disjunctive or otherwise.

Expression of the Alternative Relation.

In order to represent disjunctive propositions with convenience we require a sign of the alternative relation, equivalent to one meaning at least of the little conjunction or so frequently used in common language. I propose to use for this purpose the symbol ꖌ. In my first logical essay I followed the practice of Boole and adopted the sign +; but this sign should not be employed unless there exists exact analogy between mathematical addition and logical alternation. We shall find that the analogy is imperfect, and that there is such profound difference between logical and mathematical terms as should prevent our uniting them by the same symbol. Accordingly I have chosen a sign ꖌ, which seems aptly to suggest whatever degree of analogy may exist without implying more. The exact meaning of the symbol we will now proceed to investigate.

Nature of the Alternative Relation.

Before treating disjunctive propositions it is indispensable to decide whether the alternatives must be considered exclusive or unexclusive. By exclusive alternatives we mean those which cannot contain the same things. If we say “Arches are circular or pointed,” it is certainly to be understood that the same arch cannot be described as both circular and pointed. Many examples, on the other hand, can readily be suggested in which two or more alternatives may hold true of the same object. Thus

Luminous bodies are self-luminous or luminous by reflection.

It is undoubtedly possible, by the laws of optics, that the same surface may at one and the same moment give off light of its own and reflect light from other bodies. We speak familiarly of deaf or dumb persons, knowing that the majority of those who are deaf from birth are also dumb.

There can be no doubt that in a great many cases, perhaps the greater number of cases, alternatives are exclusive as a matter of fact. Any one number is incompatible with any other; one point of time or place is exclusive of all others. Roger Bacon died either in 1284 or 1292; it is certain that he could not die in both years. Henry Fielding was born either in Dublin or Somersetshire; he could not be born in both places. There is so much more precision and clearness in the use of exclusive alternatives that we ought doubtless to select them when possible. Old works on logic accordingly contained a rule directing that the Membra dividentia, the parts of a division or the constituent species of a genus, should be exclusive of each other.

It is no doubt owing to the great prevalence and convenience of exclusive divisions that the majority of logicians have held it necessary to make every alternative in a disjunctive proposition exclusive of every other one. Aquinas considered that when this was not the case the proposition was actually false, and Kant adopted the same opinion.‍[64] A multitude of statements to the same effect might readily be quoted, and if the question were to be determined by the weight of historical evidence, it would certainly go against my view. Among recent logicians Hamilton, as well as Boole, took the exclusive side. But there are authorities to the opposite effect. Whately, Mansel, and J. S. Mill have all pointed out that we may often treat alternatives as Compossible, or true at the same time. Whately gives us an example,‍[65] “Virtue tends to procure us either the esteem of mankind, or the favour of God,” and he adds—“Here both members are true, and consequently from one being affirmed we are not authorized to deny the other. Of course we are left to conjecture in each case, from the context, whether it is meant to be implied that the members are or are not exclusive.” Mansel says,‍[66] “We may happen to know that two alternatives cannot be true together, so that the affirmation of the second necessitates the denial of the first; but this, as Boethius observes, is a material, not a formal consequence.” Mill has also pointed out the absurdities which would arise from always interpreting alternatives as exclusive. “If we assert,” he says,‍[67] “that a man who has acted in some particular way must be either a knave or a fool, we by no means assert, or intend to assert, that he cannot be both.” Again, “to make an entirely unselfish use of despotic power, a man must be either a saint or a philosopher.... Does the disjunctive premise necessarily imply, or must it be construed as supposing, that the same person cannot be both a saint and a philosopher? Such a construction would be ridiculous.”

I discuss this subject fully because it is really the point which separates my logical system from that of Boole. In his Laws of Thought (p. 32) he expressly says, “In strictness, the words ‘and,’ ‘or,’ interposed between the terms descriptive of two or more classes of objects, imply that those classes are quite distinct, so that no member of one is found in another.” This I altogether dispute. In the ordinary use of these conjunctions we do not join distinct terms only; and when terms so joined do prove to be logically distinct, it is by virtue of a tacit premise, something in the meaning of the names and our knowledge of them, which teaches us that they are distinct. If our knowledge of the meanings of the words joined is defective it will often be impossible to decide whether terms joined by conjunctions are exclusive or not.

In the sentence “Repentance is not a single act, but a habit or virtue,” it cannot be implied that a virtue is not a habit; by Aristotle’s definition it is. Milton has the expression in one of his sonnets, “Unstain’d by gold or fee,” where it is obvious that if the fee is not always gold, the gold is meant to be a fee or bribe. Tennyson has the expression “wreath or anadem.” Most readers would be quite uncertain whether a wreath may be an anadem, or an anadem a wreath, or whether they are quite distinct or quite the same. From Darwin’s Origin of Species, I take the expression, “When we see any part or organ developed in a remarkable degree or manner.” In this, or is used twice, and neither time exclusively. For if part and organ are not synonymous, at any rate an organ is a part. And it is obvious that a part may be developed at the same time both in an extraordinary degree and an extraordinary manner, although such cases may be comparatively rare.

From a careful examination of ordinary writings, it will thus be found that the meanings of terms joined by “and,” “or” vary from absolute identity up to absolute contrariety. There is no logical condition of distinctness at all, and when we do choose exclusive alternatives, it is because our subject demands it. The matter, not the form of an expression, points out whether terms are exclusive or not.‍[68] In bills, policies, and other kinds of legal documents, it is sometimes necessary to express very distinctly that alternatives are not exclusive. The form and/or is then used, and, as Mr. J. J. Murphy has remarked, this form coincides exactly in meaning with the symbol ꖌ.

In the first edition of this work (vol. i., p. 81), I took the disjunctive proposition “Matter is solid, or liquid, or gaseous,” and treated it as an instance of exclusive alternatives, remarking that the same portion of matter cannot be at once solid and liquid, properly speaking, and that still less can we suppose it to be solid and gaseous, or solid, liquid, and gaseous all at the same time. But the experiments of Professor Andrews show that, under certain conditions of temperature and pressure, there is no abrupt change from the liquid to the gaseous state. The same substance may be in such a state as to be indifferently described as liquid and gaseous. In many cases, too, the transition from solid to liquid is gradual, so that the properties of solidity are at least partially joined with those of liquidity. The proposition then, instead of being an instance of exclusive alternatives, seems to afford an excellent instance to the opposite effect. When such doubts can arise, it is evidently impossible to treat alternatives as absolutely exclusive by the logical nature of the relation. It becomes purely a question of the matter of the proposition.

The question, as we shall afterwards see more fully, is one of the greatest theoretical importance, because it concerns the true distinction between the sciences of Logic and Mathematics. It is the foundation of number that every unit shall be distinct from every other unit; but Boole imported the conditions of number into the science of Logic, and produced a system which, though wonderful in its results, was not a system of logic at all.

Laws of the Disjunctive Relation.

In considering the combination or synthesis of terms (p. [30]), we found that certain laws, those of Simplicity and Commutativeness, must be observed. In uniting terms by the disjunctive symbol we shall find that the same or closely similar laws hold true. The alternatives of either member of a disjunctive proposition are certainly commutative. Just as we cannot properly distinguish between rich and rare gems and rare and rich gems, so we must consider as identical the expression rich or rare gems, and rare or rich gems. In our symbolic language we may say

A ꖌ B = B ꖌ A.

The order of statement, in short, has no effect upon the meaning of an aggregate of alternatives, so that the Law of Commutativeness holds true of the disjunctive symbol.

As we have admitted the possibility of joining as alternatives terms which are not really different, the question arises, How shall we treat two or more alternatives when they are clearly shown to be the same? If we have it asserted that P is Q or R, and it is afterwards proved that Q is but another name for R, the result is that P is either R or R. How shall we interpret such a statement? What would be the meaning, for instance, of “wreath or anadem” if, on referring to a dictionary, we found anadem described as a wreath? I take it to be self-evident that the meaning would then become simply “wreath.” Accordingly we may affirm the general law

A ꖌ A = A.

Any number of identical alternatives may always be reduced to, and are logically equivalent to, any one of those alternatives. This is a law which distinguishes mathematical terms from logical terms, because it obviously does not apply to the former. I propose to call it the Law of Unity, because it must really be involved in any definition of a mathematical unit. This law is closely analogous to the Law of Simplicity, AA = A; and the nature of the connection is worthy of attention.

Few or no logicians except De Morgan have adequately noticed the close relation between combined and disjunctive terms, namely, that every disjunctive term is the negative of a corresponding combined term, and vice versâ. Consider the term

Malleable dense metal.

How shall we describe the class of things which are not malleable-dense-metals? Whatever is included under that term must have all the qualities of malleability, denseness, and metallicity. Wherever any one or more of the qualities is wanting, the combined term will not apply. Hence the negative of the whole term is

Not-malleable or not-dense or not-metallic.

In the above the conjunction or must clearly be interpreted as unexclusive; for there may readily be objects which are both not-malleable, and not-dense, and perhaps not-metallic at the same time. If in fact we were required to use or in a strictly exclusive manner, it would be requisite to specify seven distinct alternatives in order to describe the negative of a combination of three terms. The negatives of four or five terms would consist of fifteen or thirty-one alternatives. This consideration alone is sufficient to prove that the meaning of or cannot be always exclusive in common language.

Expressed symbolically, we may say that the negative of

	ABC
is	not-A or not-B or not-C;
that is,	a ꖌ b ꖌ c.

Reciprocally the negative of

	P ꖌ Q ꖌ R
is	pqr.

Every disjunctive term, then, is the negative of a combined term, and vice versâ.

Apply this result to the combined term AAA, and its negative is

a ꖌ a ꖌ a.

Since AAA is by the Law of Simplicity equivalent to A, so a ꖌ a ꖌ a must be equivalent to a, and the Law of Unity holds true. Each law thus necessarily presupposes the other.

Symbolic expression of the Law of Duality.

We may now employ our symbol of alternation to express in a clear and formal manner the third Fundamental Law of Thought, which I have called the Law of Duality (p. [6]). Taking A to represent any class or object or quality, and B any other class, object or quality, we may always assert that A either agrees with B, or does not agree. Thus we may say

A = AB ꖌ Ab.

This is a formula which will henceforth be constantly employed, and it lies at the basis of reasoning.

The reader may perhaps wish to know why A is inserted in both alternatives of the second member of the identity, and why the law is not stated in the form

A = B ꖌ b.

But if he will consider the contents of the last section (p. [73]), he will see that the latter expression cannot be correct, otherwise no term could have a corresponding negative term. For the negative of B ꖌ b is bB, or a self-contradictory term; thus if A were identical with B ꖌ b, its negative a would be non-existent. To say the least, this result would in most cases be an absurd one, and I see much reason to think that in a strictly logical point of view it would always be absurd. In all probability we ought to assume as a fundamental logical axiom that every term has its negative in thought. We cannot think at all without separating what we think about from other things, and these things necessarily form the negative notion.‍[69] It follows that any proposition of the form A = B ꖌ b is just as self-contradictory as one of the form A = Bb.

It is convenient to recapitulate in this place the three Laws of Thought in their symbolic form, thus

Law of Identity	A = A.
Law of Contradiction	Aa = 0.
Law of Duality	A = AB ꖌ Ab.

Various Forms of the Disjunctive Proposition.

Disjunctive propositions may occur in a great variety of forms, of which the old logicians took insufficient notice. There may be any number of alternatives, each of which may be a combination of any number of simple terms. A proposition, again, may be disjunctive in one or both members. The proposition

Solids or liquids or gases are electrics or conductors of electricity

is an example of the doubly disjunctive form. The meaning of such a proposition is that whatever falls under any one or more alternatives on one side must fall under one or more alternatives on the other side. From what has been said before, it is apparent that the proposition

A ꖌ B = C ꖌ D

will correspond to

ab = cd,

each member of the latter being the negative of a member of the former proposition.

As an instance of a complex disjunctive proposition I may give Senior’s definition of wealth, which, briefly stated, amounts to the proposition “Wealth is what is transferable, limited in supply, and either productive of pleasure or preventive of pain.”‍[70]

Let	A = wealth
	B = transferable
	C = limited in supply
	D = productive of pleasure
	E = preventive of pain.

The definition takes the form

A = BC(D ꖌ E);

but if we develop the alternatives by a method to be afterwards more fully considered, it becomes

A = BCDE ꖌ BCDe ꖌ BCdE.

An example of a still more complex proposition is found in De Morgan’s writings,‍[71] as follows:—“He must have been rich, and if not absolutely mad was weakness itself, subjected either to bad advice or to most unfavourable circumstances.”

If we assign the letters of the alphabet in succession, thus,

A = he
B = rich
C = absolutely mad
D = weakness itself
E = subjected to bad advice
F = subjected to most unfavourable circumstances,

the proposition will take the form

A = AB{C ꖌ D (E ꖌ F)},

and if we develop the alternatives, expressing some of the different cases which may happen, we obtain

A = ABC ꖌ ABcDEF ꖌ ABcDEf ꖌ ABcDeF.

The above gives the strict logical interpretation of the sentence, and the first alternative ABC is capable of development into eight cases, according as D, E and F are or are not present. Although from our knowledge of the matter, we may infer that weakness of character cannot be asserted of a person absolutely mad, there is no explicit statement to this effect.

Inference by Disjunctive Propositions.

Before we can make a free use of disjunctive propositions in the processes of inference we must consider how disjunctive terms can be combined together or with simple terms. In the first place, to combine a simple term with a disjunctive one, we must combine it with every alternative of the disjunctive term. A vegetable, for instance, is either a herb, a shrub, or a tree. Hence an exogenous vegetable is either an exogenous herb, or an exogenous shrub, or an exogenous tree. Symbolically stated, this process of combination is as follows,

A(B ꖌ C) = AB ꖌ AC.

Secondly, to combine two disjunctive terms with each other, combine each alternative of one with each alternative of the other. Since flowering plants are either exogens or endogens, and are at the same time either herbs, shrubs or trees, it follows that there are altogether six alternatives—namely, exogenous herbs, exogenous shrubs, exogenous trees, endogenous herbs, endogenous shrubs, endogenous trees. This process of combination is shown in the general form

(A ꖌ B) (C ꖌ D ꖌ E) = AC ꖌ AD ꖌ AE ꖌ BC ꖌ BD ꖌ BE.

It is hardly necessary to point out that, however numerous the terms combined, or the alternatives in those terms, we may effect the combination, provided each alternative is combined with each alternative of the other terms, as in the algebraic process of multiplication.

Some processes of deduction may be at once exhibited. We may always, for instance, unite the same qualifying term to each side of an identity even though one or both members of the identity be disjunctive. Thus let

A = B ꖌ C.

Now it is self-evident that

AD = AD,

and in one side of this identity we may for A substitute its equivalent B ꖌ C, obtaining

AD = BD ꖌ CD.

Since “a gaseous element is either hydrogen, or oxygen, or nitrogen, or chlorine, or fluorine,” it follows that “a free gaseous element is either free hydrogen, or free oxygen, or free nitrogen, or free chlorine, or free fluorine.”

This process of combination will lead to most useful inferences when the qualifying adjective combined with both sides of the proposition is a negative of one or more alternatives. Since chlorine is a coloured gas, we may infer that “a colourless gaseous element is either (colourless) hydrogen, oxygen, nitrogen, or fluorine.” The alternative chlorine disappears because colourless chlorine does not exist. Again, since “a tooth is either an incisor, canine, bicuspid, or molar,” it follows that “a not-incisor tooth is either canine, bicuspid, or molar.” The general rule is that from the denial of any of the alternatives the affirmation of the remainder can be inferred. Now this result clearly follows from our process of substitution; for if we have the proposition

A = B ꖌ C ꖌ D,

and we insert this expression for A on one side of the self-evident identity

Ab = Ab,

we obtain Ab = ABb ꖌ AbC ꖌ AbD;

and, as the first of the three alternatives is self-contradictory, we strike it out according to the law of contradiction: there remains

Ab = AbC ꖌ AbD.

Thus our system fully includes and explains that mood of the Disjunctive Syllogism technically called the modus tollendo ponens.

But the reader must carefully observe that the Disjunctive Syllogism of the mood ponendo tollens, which affirms one alternative, and thence infers the denial of the rest, cannot be held true in this system. If I say, indeed, that

Water is either salt or fresh water,

it seems evident that “water which is salt is not fresh.” But this inference really proceeds from our knowledge that water cannot be at once salt and fresh. This inconsistency of the alternatives, as I have fully shown, will not always hold. Thus, if I say

Gems are either rare stones or beautiful stones,

(1)

it will obviously not follow that

A rare gem is not a beautiful stone,

(2)

nor that

A beautiful gem is not a rare stone.

(3)

Our symbolic method gives only true conclusions; for if we take

A = gem
B = rare stone
C = beautiful stone,

the proposition (1) is of the form

	A	= B ꖌ C
hence	AB	= B ꖌ BC
and	AC	= BC ꖌ C;

but these inferences are not equivalent to the false ones (2) and (3).

We can readily represent disjunctive reasoning by the modus ponendo tollens, when it is valid, by expressing the inconsistency of the alternatives explicitly. Thus if we resort to our instance of

Water is either salt or fresh,

and take

A = Water B = salt C = fresh,

then the premise is apparently of the form

A = AB ꖌ AC;

but in reality there is an unexpressed condition that “what is salt is not fresh,” from which follows, by a process of inference to be afterwards described, that “what is fresh is not salt.” We have then, in letter-terms, the two propositions

B = Bc
C = bC.

If we substitute these descriptions in the original proposition, we obtain

A = ABc ꖌ AbC;

uniting B to each side we infer

	AB	= ABc ꖌ ABbC
or	AB	= ABc;

that is,

Water which is salt is water salt and not fresh.

I should weary the reader if I attempted to illustrate the multitude of forms which disjunctive reasoning may take; and as in the next chapter we shall be constantly treating the subject, I must here restrict myself to a single instance. A very common process of reasoning consists in the determination of the name of a thing by the successive exclusion of alternatives, a process called by the old name abscissio infiniti. Take the case:

Red-coloured metal is either copper or gold	(1)
Copper is dissolved by nitric acid	(2)
This specimen is red-coloured metal	(3)
This specimen is not dissolved by nitric acid	(4)
Therefore, this specimen consists of gold	(5)

Let us assign the letter-symbols thus—

A = this specimen
B = red-coloured metal
C = copper
D = gold
E = dissolved by nitric acid.

Assuming that the alternatives copper or gold are intended to be exclusive, as just explained in the case of fresh and salt water, the premises may be stated in the forms

B = BCd ꖌ BcD	(1)
C = CE	(2)
A = AB	(3)
A = Ae	(4)

Substituting for C in (1) by means of (2) we get

B = BCdE ꖌ BcD

From (3) and (4) we may infer likewise

A = ABe

and if in this we substitute for B its equivalent just stated, it follows that

A = ABCdEe ꖌ ABcDe

The first of the alternatives being contradictory the result is

A = ABcDe

which contains a full description of “this specimen,” as furnished in the premises, but by ellipsis asserts that it is gold. It will be observed that in the symbolic expression (1) I have explicitly stated what is certainly implied, that copper is not gold, and gold not copper, without which condition the inference would not hold good.

CHAPTER VI.
THE INDIRECT METHOD OF INFERENCE.

The forms of deductive reasoning as yet considered, are mostly cases of Direct Deduction as distinguished from those which we are now about to treat. The method of Indirect Deduction may be described as that which points out what a thing is, by showing that it cannot be anything else. We can define a certain space upon a map, either by colouring that space, or by colouring all except the space; the first mode is positive, the second negative. The difference, it will be readily seen, is exactly analogous to that between the direct and indirect modes of proof in geometry. Euclid often shows that two lines are equal, by showing that they cannot be unequal, and the proof rests upon the known number of alternatives, greater, equal or less, which are alone conceivable. In other cases, as for instance in the seventh proposition of the first book, he shows that two lines must meet in a particular point, by showing that they cannot meet elsewhere.

In logic we can always define with certainty the utmost number of alternatives which are conceivable. The Law of Duality (pp. [6], [74]) enables us always to assert that any quality or circumstance whatsoever is either present or absent. Whatever may be the meaning of the terms A and B it is certainly true that

A = AB ꖌ Ab
B = AB ꖌ aB.

These are universal tacit premises which may be employed in the solution of every problem, and which are such invariable and necessary conditions of all thought, that they need not be specially laid down. The Law of Contradiction is a further condition of all thought and of all logical symbols; it enables, and in fact obliges, us to reject from further consideration all terms which imply the presence and absence of the same quality. Now, whenever we bring both these Laws of Thought into explicit action by the method of substitution, we employ the Indirect Method of Inference. It will be found that we can treat not only those arguments already exhibited according to the direct method, but we can include an infinite multitude of other arguments which are incapable of solution by any other means.

Some philosophers, especially those of France, have held that the Indirect Method of Proof has a certain inferiority to the direct method, which should prevent our using it except when obliged. But there are many truths which we can prove only indirectly. We can prove that a number is a prime only by the purely indirect method of showing that it is not any of the numbers which have divisors, and the remarkable process known as Eratosthenes’ Sieve is the only mode by which we can select the prime numbers.‍[72] It bears a strong analogy to the indirect method here to be described. We can prove that the side and diameter of a square are incommensurable, but only in the negative or indirect manner, by showing that the contrary supposition inevitably leads to contradiction.‍[73] Many other demonstrations in various branches of the mathematical sciences proceed upon a like method. Now, if there is only one important truth which must be, and can only be, proved indirectly, we may say that the process is a necessary and sufficient one, and the question of its comparative excellence or usefulness is not worth discussion. As a matter of fact I believe that nearly half our logical conclusions rest upon its employment.

Simple Illustrations.

In tracing out the powers and results of this method, we will begin with the simplest possible instance. Let us take a proposition of the common form, A = AB, say,

A Metal is an Element,

and let us investigate its full meaning. Any person who has had the least logical training, is aware that we can draw from the above proposition an apparently different one, namely,

A Not-element is a Not-metal.

While some logicians, as for instance De Morgan,‍[74] have considered the relation of these two propositions to be purely self-evident, and neither needing nor allowing analysis, a great many more persons, as I have observed while teaching logic, are at first unable to perceive the close connection between them. I believe that a true and complete system of logic will furnish a clear analysis of this process, which has been called Contrapositive Conversion; the full process is as follows:‍—

Firstly, by the Law of Duality we know that

Not-element is either Metal or Not-metal.

If it be metal, we know that it is by the premise an element; we should thus be supposing that the same thing is an element and a not-element, which is in opposition to the Law of Contradiction. According to the only other alternative, then, the not-element must be a not-metal.

To represent this process of inference symbolically we take the premise in the form

A = AB.

(1)

We observe that by the Law of Duality the term not-B is thus described

b = Ab ꖌ ab.

(2)

For A in this proposition we substitute its description as given in (1), obtaining

b = ABb ꖌ ab.

But according to the Law of Contradiction the term ABb must be excluded from thought, or

ABb = 0.

Hence it results that b is either nothing at all, or it is ab; and the conclusion is

b = ab.

As it will often be necessary to refer to a conclusion of this kind I shall call it, as is usual, the Contrapositive Proposition of the original. The reader need hardly be cautioned to observe that from all A’s are B’s it does not follow that all not-A’s are not-B’s. For by the Law of Duality we have

a = aB ꖌ ab,

and it will not be found possible to make any substitution in this by our original premise A = AB. It still remains doubtful, therefore, whether not-metal is element or not-element.

The proof of the Contrapositive Proposition given above is exactly the same as that which Euclid applies in the case of geometrical notions. De Morgan describes Euclid’s process as follows‍[75]:—“From every not-B is not-A he produces Every A is B, thus: If it be possible, let this A be not-B, but every not-B is not-A, therefore this A is not-A, which is absurd: whence every A is B.” Now De Morgan thinks that this proof is entirely needless, because common logic gives the inference without the use of any geometrical reasoning. I conceive however that logic gives the inference only by an indirect process. De Morgan claims “to see identity in Every A is B and every not-B is not-A, by a process of thought prior to syllogism.” Whether prior to syllogism or not, I claim that it is not prior to the laws of thought and the process of substitutive inference, by which it may be undoubtedly demonstrated.

Employment of the Contrapositive Proposition.

We can frequently employ the contrapositive form of a proposition by the method of substitution; and certain moods of the ancient syllogism, which we have hitherto passed over, may thus be satisfactorily comprehended in our system. Take for instance the following syllogism in the mood Camestres:‍—

“Whales are not true fish; for they do not respire water, whereas true fish do respire water.”

Let us take

A = whale
B = true fish
C = respiring water

The premises are of the forms

A = Ac	(1)
B = BC	(2)

Now, by the process of contraposition we obtain from the second premise

c = bc

and we can substitute this expression for c in (1), obtaining

A = Abc

or “Whales are not true fish, not respiring water.”

The mood Cesare does not really differ from Camestres except in the order of the premises, and it could be exhibited in an exactly similar manner.

The mood Baroko gave much trouble to the old logicians, who could not reduce it to the first figure in the same manner as the other moods, and were obliged to invent, specially for it and for Bokardo, a method of Indirect Reduction closely analogous to the indirect proof of Euclid. Now these moods require no exceptional treatment in this system. Let us take as an instance of Baroko, the argument

All heated solids give continuous spectra	(1)
Some nebulæ do not give continuous spectra	(2)
Therefore, some nebulæ are not heated solids	(3)

Treating the little word some as an indeterminate adjective of selection, to which we assign a symbol like any other adjective, let

A = some
B = nebulæ
C = giving continuous spectra
D = heated solids

The premises then become

D	= DC	(1)
AB	= ABc	(2)

Now from (1) we obtain by the indirect method the contrapositive proposition

c = cd

and if we substitute this expression for c in (2) we have

AB = ABcd

the full meaning of which is that “some nebulæ do not give continuous spectra and are not heated solids.”

We might similarly apply the contrapositive in many other instances. Take the argument, “All fixed stars are self-luminous; but some of the heavenly bodies are not self-luminous, and are therefore not fixed stars.” Taking our terms

A = fixed stars
B = self-luminous
C = some
D = heavenly bodies

we have the premises

A	= AB,	(1)
CD	= bCD	(2)

Now from (1) we can draw the contrapositive

b = ab

and substituting this expression for b in (2) we obtain

CD = abCD

which expresses the conclusion of the argument that some heavenly bodies are not fixed stars.

Contrapositive of a Simple Identity.

The reader should carefully note that when we apply the process of Indirect Inference to a simple identity of the form

A = B

we may obtain further results. If we wish to know what is the term not-B, we have as before, by the Law of Duality,

b = Ab ꖌ ab

and substituting for A we obtain

b = Bb ꖌ ab = ab.

But we may now also draw a second contrapositive; for we have

a = aB ꖌ ab,

and substituting for B its equivalent A we have

a = aA ꖌ ab = ab.

Hence from the single identity A = B we can draw the two propositions

a = ab
b = ab,

and observing that these propositions have a common term ab we can make a new substitution, getting

a = b.

This result is in strict accordance with the fundamental principles of inference, and it may be a question whether it is not a self-evident result, independent of the steps of deduction by which we have reached it. For where two classes are coincident like A and B, whatever is true of the one is true of the other; what is excluded from the one must be excluded from the other similarly. Now as a bears to A exactly the same relation that b bears to B, the identity of either pair follows from the identity of the other pair. In every identity, equality, or similarity, we may argue from the negative of the one side to the negative of the other. Thus at ordinary temperatures

Mercury = liquid-metal,

hence obviously

Not-mercury = not liquid-metal;

or since

Sirius = brightest fixed star,

it follows that whatever star is not the brightest is not Sirius, and vice versâ. Every correct definition is of the form A = B, and may often require to be applied in the equivalent negative form.

Let us take as an illustration of the mode of using this result the argument following:

Vowels are letters which can be sounded alone,	(1)
The letter w cannot be sounded alone;	(2)
Therefore the letter w is not a vowel.	(3)

Here we have a definition (1), and a comparison of a thing with that definition (2), leading to exclusion of the thing from the class defined.

Taking the terms

A = vowel,
B = letter which can be sounded alone,
C = letter w,

the premises are plainly of the forms

A = B,	(1)
C = bC.	(2)

Now by the Indirect method we obtain from (1) the Contrapositive

b = a,

and inserting in (2) the equivalent for b we have

C = aC,

(3)

or “the letter w is not a vowel.”

Miscellaneous Examples of the Method.

We can apply the Indirect Method of Inference however many may be the terms involved or the premises containing those terms. As the working of the method is best learnt from examples, I will take a case of two premises forming the syllogism Barbara: thus

Iron is metal	(1)
Metal is element.	(2)

If we want to ascertain what inference is possible concerning the term Iron, we develop the term by the Law of Duality. Iron must be either metal or not-metal; iron which is metal must be either element or not-element; and similarly iron which is not-metal must be either element or not-element. There are then altogether four alternatives among which the description of iron must be contained; thus

Iron, metal, element,	(α)
Iron, metal, not-element,	(β)
Iron, not-metal, element,	(γ)
Iron, not-metal, not-element.	(δ)

Our first premise informs us that iron is a metal, and if we substitute this description in (γ) and (δ) we shall have self-contradictory combinations. Our second premise likewise informs us that metal is element, and applying this description to (β) we again have self-contradiction, so that there remains only (α) as a description of iron—our inference is

Iron = iron, metal, element.

To represent this process of reasoning in general symbols, let

A = iron
B = metal
C = element,

The premises of the problem take the forms

A = AB	(1)
B = BC.	(2)

By the Law of Duality we have

A = AB ꖌ Ab	(3)
A = AC ꖌ Ac.	(4)

Now, if we insert for A in the second side of (3) its description in (4), we obtain what I shall call the development of A with respect to B and C, namely

A = ABC ꖌ ABc ꖌ AbC ꖌ Abc.

(5)

Wherever the letters A or B appear in the second side of (5) substitute their equivalents given in (1) and (2), and the results stated at full length are

A = ABC ꖌ ABCc ꖌ ABbC ꖌ ABbCc.

The last three alternatives break the Law of Contradiction, so that

A = ABC ꖌ 0 ꖌ 0 ꖌ 0 = ABC.

This conclusion is, indeed, no more than we could obtain by the direct process of substitution, that is by substituting for B in (1), its description in (2) as in p. [55]; it is the characteristic of the Indirect process that it gives all possible logical conclusions, both those which we have previously obtained, and an immense number of others or which the ancient logic took little or no account. From the same premises, for instance, we can obtain a description of the class not-element or c. By the Law of Duality we can develop c into four alternatives, thus

c = ABc ꖌ Abc ꖌ aBc ꖌ abc.

If we substitute for A and B as before, we get

c = ABCc ꖌ ABbc ꖌ aBCc ꖌ abc,

and, striking out the terms which break the Law of Contradiction, there remains

c = abc,

or what is not element is also not iron and not metal. This Indirect Method of Inference thus furnishes a complete solution of the following problem—Given any number of logical premises or conditions, required the description of any class of objects, or of any term, as governed by those conditions.

The steps of the process of inference may thus be concisely stated—

1. By the Law of Duality develop the utmost number of alternatives which may exist in the description of the required class or term as regards the terms involved in the premises.

2. For each term in these alternatives substitute its description as given in the premises.

3. Strike out every alternative which is then found to break the Law of Contradiction.

4. The remaining terms may be equated to the term in question as the desired description.

Mr. Venn’s Problem.

The need of some logical method more powerful and comprehensive than the old logic of Aristotle is strikingly illustrated by Mr. Venn in his most interesting and able article on Boole’s logic.‍[76] An easy example, originally got, as he says, by the aid of my method as simply described in the Elementary Lessons in Logic, was proposed in examination and lecture-rooms to some hundred and fifty students as a problem in ordinary logic. It was answered by, at most, five or six of them. It was afterwards set, as an example on Boole’s method, to a small class who had attended a few lectures on the nature of these symbolic methods. It was readily answered by half or more of their number.

The problem was as follows:—“The members of a board were all of them either bondholders, or shareholders, but not both; and the bondholders as it happened, were all on the board. What conclusion can be drawn?” The conclusion wanted is, “No shareholders are bondholders.” Now, as Mr. Venn says, nothing can look simpler than the following reasoning, when stated:—“There can be no bondholders who are shareholders; for if there were they must be either on the board, or off it. But they are not on it, by the first of the given statements; nor off it, by the second.” Yet from the want of any systematic mode of treating such a question only five or six of some hundred and fifty students could succeed in so simple a problem.

By symbolic statement the problem is instantly solved. Taking

A = member of board
B = bondholder
C = shareholder

the premises are evidently

A = ABc ꖌ AbC B = AB.

The class C or shareholders may in respect of A and B be developed into four alternatives,

C = ABC ꖌ AbC ꖌ aBC ꖌ abC.

But substituting for A in the first and for B in the third alternative we get

C = ABCc ꖌ ABbC ꖌ AbC ꖌ aABC ꖌ abC.

The first, second, and fourth alternatives in the above are self-contradictory combinations, and only these; striking them out there remain

C = AbC ꖌ abC = bC,

the required answer. This symbolic reasoning is, I believe, the exact equivalent of Mr. Venn’s reasoning, and I do not believe that the result can be attained in a simpler manner. Mr. Venn adds that he could adduce other similar instances, that is, instances showing the necessity of a better logical method.

Abbreviation of the Process.

Before proceeding to further illustrations of the use of this method, I must point out how much its practical employment can be simplified, and how much more easy it is than would appear from the description. When we want to effect at all a thorough solution of a logical problem it is best to form, in the first place, a complete series of all the combinations of terms involved in it. If there be two terms A and B, the utmost variety of combinations in which they can appear are

AB	aB
Ab	ab.

The term A appears in the first and second; B in the first and third; a in the third and fourth; and b in the second and fourth. Now if we have any premise, say

A = B,

we must ascertain which of these combinations will be rendered self-contradictory by substitution; the second and third will have to be struck out, and there will remain only

AB
ba.

Hence we draw the following inferences

A = AB, B = AB, a = ab, b = ab.

Exactly the same method must be followed when a question involves a greater number of terms. Thus by the Law of Duality the three terms A, B, C, give rise to eight conceivable combinations, namely

ABC	(α)	aBC	(ε)
ABc	(β)	aBc	(ζ)
AbC	(γ)	abC	(η)
Abc	(δ)	abc.	(θ)

The development of the term A is formed by the first four of these; for B we must select (α), (β), (ε), (ζ); C consists of (α), (γ), (ε), (η); b of (γ), (δ), (η), (θ), and so on.

Now if we want to investigate completely the meaning of the premises

A = AB	(1)
B = BC	(2)

we examine each of the eight combinations as regards each premise; (γ) and (δ) are contradicted by (1), and (β) and (ζ) by (2), so that there remain only

ABC	(α)
aBC	(ε)
abC	(η)
abc.	(θ)

To describe any term under the conditions of the premises (1) and (2), we have simply to draw out the proper combinations from this list; thus, A is represented only by ABC, that is to say

	A	= ABC,
similarly	c	= abc.

For B we have two alternatives thus stated,

B = ABC ꖌ aBC;

and for b we have

b = abC ꖌ abc.

When we have a problem involving four distinct terms we need to double the number of combinations, and as we add each new term the combinations become twice as numerous. Thus

A, B	produce	four  combinations
A, B, C,	"	eight	"
A, B, C, D	"	sixteen	"
A, B, C, D, E	"	thirty-two	"
A, B, C, D, E, F	"	sixty-four	"

and so on.

I propose to call any such series of combinations the Logical Alphabet. It holds in logical science a position the importance of which cannot be exaggerated, and as we proceed from logical to mathematical considerations, it will become apparent that there is a close connection between these combinations and the fundamental theorems of mathematical science. For the convenience of the reader who may wish to employ the Alphabet in logical questions, I have had printed on the next page a complete series of the combinations up to those of six terms. At the very commencement, in the first column, is placed a single letter X, which might seem to be superfluous. This letter serves to denote that it is always some higher class which is divided up. Thus the combination AB really means ABX, or that part of some larger class, say X, which has the qualities of A and B present. The letter X is omitted in the greater part of the table merely for the sake of brevity and clearness. In a later chapter on Combinations it will become apparent that the introduction of this unit class is requisite in order to complete the analogy with the Arithmetical Triangle there described.

The reader ought to bear in mind that though the Logical Alphabet seems to give mere lists of combinations, these combinations are intended in every case to constitute the development of a term of a proposition. Thus the four combinations AB, Ab, aB, ab really mean that any class X is described by the following proposition,

X = XAB ꖌ XAb ꖌ XaB ꖌ Xab.

If we select the A’s, we obtain the following proposition

AX = XAB ꖌ XAb.

Thus whatever group of combinations we treat must be conceived as part of a higher class, summum genus or universe symbolised in the term X; but, bearing this in mind, it is needless to complicate our formulæ by always introducing the letter. All inference consists in passing from propositions to propositions, and combinations per se have no meaning. They are consequently to be regarded in all cases as forming parts of propositions.