Logic, Inductive and Deductive

by William Minto

The justification for taking special note of this form in Logic is that we are apt by inadvertence to make an inclusive inference from it. Let it be said that None but those who work hard can reasonably expect to pass, and we are apt to take this as meaning that all who work hard may reasonably expect to pass. But what is denied of every Not-S is not necessarily affirmed of every S.

The expression of Tense or Time in the Syllogistic Forms. Seeing that the Copula in S is P or S is in P does not express time, but only a certain relation between S and P, the question arises Where are we to put time in the analytic formula? "Wheat is dear;" "All had fled;" time is expressed in these propositions, and our formula should render the whole content of what is given. Are we to include it in the Predicate term or in the Subject term? If it must not be left out altogether, and we cannot put it with the copula, we have a choice between the two terms.

It is a purely scholastic question. The common technical treatment is to view the tense as part of the predicate. "All had fled," All S is P, i.e., the whole subject is included in a class constituted on the attributes of flight at a given time. It may be that the Predicate is solely a predicate of time. "The Board met yesterday at noon." S is P, i.e., the meeting of the Board is one of the events characterised by having happened at a certain time, agreeing with other events in that respect.

But in some cases the time is more properly regarded as part of the subject. E.g., "Wheat is dear". S does not here stand for wheat collectively, but for the wheat now in the market, the wheat of the present time: it is concerning this that the attribute of dearness is predicated; it is this that is in the class of dear things.

The expression of Modality in the Syllogistic Forms. Propositions in which the predicate is qualified by an expression of necessity, contingency, possibility or impossibility [i.e., in English by must, may, can, or cannot], were called in Mediæval Logic Modal Propositions. "Two and two must make four." "Grubs may become butterflies." "Z can paint." "Y cannot fly."

There are two recognised ways of reducing such propositions to the form S is P. One is to distinguish between the Dictum and the Mode, the proposition and the qualification of its certainty, and to treat the Dictum as the Subject and the Mode as the Predicate. Thus: "That two and two make four is necessary"; "That Y can fly is impossible".

The other way is to treat the Mode as part of the predicate. The propriety of this is not obvious in the case of Necessary propositions, but it is unobjectionable in the case of the other three modes. Thus: "Grubs are things that have the potentiality of becoming butterflies"; "Z has the faculty of painting"; "Y has not the faculty of flying".

The chief risk of error is in determining the quantity of the subject about which the Contingent or Possible predicate is made. When it is said that "Victories may be gained by accident," is the predicate made concerning All victories or Some only? Here we are apt to confuse the meaning of the contingent assertion with the matter of fact on which in common belief it rests. It is true only that some victories have been gained by accident, and it is on this ground that we assert in the absence of certain knowledge concerning any victory that it may have been so gained. The latter is the effect of the contingent assertion: it is made about any victory in the absence of certain knowledge, that is to say, formally about all.

The history of Modals in Logic is a good illustration of intricate confusion arising from disregard of a clear traditional definition. The treatment of them by Aristotle was simple, and had direct reference to tricks of disputation practised in his time. He specified four "modes," the four that descended to mediæval logic, and he concerned himself chiefly with the import of contradicting these modals. What is the true contradictory of such propositions as, "It is possible to be" (δυνατὸν εἶναι), "It admits of being" (ἐνδέχεται εἶναι), "It must be" (ἀναγκαῖον εἶναι), "It is impossible to be" (ἀδύνατον εἶναι)? What is implied in saying "No" to such propositions put interrogatively? "Is it possible for Socrates to fly?" "No." Does this mean that it is not possible for Socrates to fly, or that it is possible for Socrates not to fly?

A disputant who had trapped a respondent into admitting that it is possible for Socrates not to fly, might have pushed the concession farther in some such way as this: "Is it possible for Socrates not to walk?" "Certainly." "Is it possible for him to walk?" "Yes." "When you say that it is possible for a man to do anything do you not believe that it is possible for him to do it?" "Yes." "But you have admitted that it is possible for Socrates not to fly?"