De Morgan observes, “When any writer attempts to shew how the perception of convertibility ‘A is B gives B is A’ follows from the principles of identity, difference, and excluded middle, I shall be able to judge of the process; as it is, I find that others do not go beyond the simple assertion, and that I myself can detect the petitio principii in every one of my own attempts” (Syllabus of Logic, p. 47).
466 The test that I should be disposed to apply to any attempted proof of the validity of the process of conversion is to ask wherein the principle involved in the proof makes manifest the inconvertibility of an O proposition, and the illegitimacy of the simple conversion of A. It is clear that we have no right to assume that any self-evident principles that we may call to our aid[461] are equivalent to the law of identity.
[461] For example,—If one class is wholly or partially contained in a second, then the second is at least partially contained in the first; If one class is wholly excluded from a second, then the second is wholly excluded from the first.
The following attempt to establish the conversion of A and of I by means of the law of identity may be taken as an example: “Every affirmative proposition may be considered as asserting that there are certain things which possess the attributes connoted both by the subject and the predicate—the class SP. Hence the principle of identity justifies the conversion of an affirmative proposition. For if there are S’s which possess the attribute P, the principle of identity necessitates that some of the objects which possess that attribute are S’s.” The law of identity is referred to here, but we may fairly ask in what form that law really comes in. Does the argument amount to more than that as thus analysed the validity of the conversion in question is self-evident?[462] Might we not for the words “the principle of identity necessitates” substitute the words “it is self-evident”?[463]
[462] In so far as the argument is intended to amount to more than this, it contains a petitio principii.
[463] Compare, further, the discussion of the legitimacy of conversion in section [99].
No doubt if immediate inferences are no more than verbal transformations, then they can all be based on the principle of identity as interpreted by Mill, namely, on the principle that whatever is true in one form of words is true in any other form of words having the same meaning. But if conversion (or any other form of immediate inference) is more than mere verbal transformation, the equivalence of the convertend and the converse is just what we have to shew; they are not merely two different forms of words having the same meaning.
423. The Laws of Thought and Mediate Inferences.—Mansel expresses the view that syllogistic reasoning—and indeed all formal reasoning whatsoever—can be based exclusively on the laws of 467 identity, contradiction, and excluded middle. The principle of identity is, he says, immediately applicable to affirmative moods in any figure, and the principle of contradiction to negatives.[464] His proof of this position consists in quantifying the predicates of the propositions constituting the syllogism, and then making use—for affirmatives—of the axiom that “what is given as identical with the whole or a part of any concept, must be identical with the whole or a part of that which is identical with the same concept,” and—for negatives—of the axiom that “some or all S, being given as identical with all or some M, is distinct from every part of that which is distinct from all M.”
[464] Prolegomena Logica, p. 222.
These formulae, however, go distinctly beyond the laws of identity and contradiction as ordinarily stated. They may indeed be regarded as equivalent to the dictum de omni et nullo, adapted so as to be applicable to syllogisms made up of propositions with quantified predicates; and if it is assumed that the dictum is only another form of stating the laws of identity and contradiction then the question needs no further discussion. Only in this case we must no longer express the law of identity either in the form “What is true is true,” or in the form “A is A”; nor the law of contradiction either in the form “If a judgment is true, its contradictory is not true,” or in the form “A is not not-A.” The laws as thus formulated cannot be regarded as adequate expressions of the axiom upon which syllogistic reasoning proceeds. They do not bring out the function of the middle term which is the characteristic feature of the syllogism, nor could the rules of the syllogism be deduced from them.