[Math Lair] Distribution of Terms

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For the concept of distributions in statistics, see Distributions.

In syllogistic logic, distribution (or distribution of terms) refers to whether a term of a proposition is applied to the entire class that the term denotes. The significance of distribution is that two of the rules for determining whether a syllogism is valid is that the middle term must be distributed at least once, and no term may be distributed in the conclusion that is not distributed in one of the premises.

While the concept of distribution is important in Aristotelian logic, this concept was not known to Aristotle; there is no Greek word for "distributed" or "distribution" in this sense, and Aristotle uses different tests to determine if a syllogism is valid.

In the classic work Studies and Exercises in Formal Logic by John Neville Keynes (the father of famous economist John Maynard Keynes), Keynes defines distribution as follows:

63. The Distribution of Terms in a Proposition.—A term is said to be distributed when reference is made to all the individuals denoted by it; it is said to be undistributed when they are only referred to partially, that is, when information is given with regard to a portion of the class denoted by the term, but we are left in ignorance with regard to the remainder of the class. It follows immediately from this definition that the subject is distributed in a universal, and undistributed in a particular, proposition. It can further be shewn that the predicate is distributed in a negative, and undistributed in an affirmative proposition. Thus, if I say All S is P, I identify every member of the class S with some member of the class P, and I therefore imply that at any rate some P is S, but I make no implication with regard to the whole of P. It is left an open question whether there is or is not any P outside the class S. Similarly if I say Some S is P. But if I say No S is P, in excluding the whole of S from P, I am also excluding the whole of P from S, and therefore P as well as S is distributed. Again, if I say Some S is not P, although I make an assertion with regard to a part only of S, I exclude this part from the whole of P, and therefore the whole of P from it. In this case, then, the predicate is distributed, although the subject is not.

In other words, a term is said to be distributed with respect to a certain proposition when it refers in its context to all the members of the class of objects that it denotes. With a quantifier of "All", the subject is distributed and the predicate is not. With a quantifier of "Some", neither the subject nor the predicate is distributed. With a quantifier of "No", both the subject and the predicate are distributed. With a quantifier of "Some ... is not", the subject is not distributed and the predicate is distributed.

If you think that it's hard to see how, in Some S is not P, "reference is made to all the individuals denoted by" P, you're not alone. The theory requires that the predicate be distributed in this case, but atttempting to explain it in the framework of the theory can end up not making much sense. In Reference and Generality, Peter Thomas Geach writes, with regard to Keynes' explanation above:

As before, Keynes's class terminology obscures the matter: let us amend "with regard to a part only . . . the whole of P from it" to "about only some of the Ss, I exclude these from among all the Ps, and therefore exclude all the Ps from among them." We may now ask: From among which Ss are all the Ps being excluded? Clearly no definite answer is possible; so Keynes has simply failed to exhibit "Some S is not P", as an assertion about all the Ps, in which the term "P" is distributed.

To be sure, if "Some S is not a man" is true, then of every man we can truly say: "Not he alone is an S". But of course such a form of predication as "Not —— alone is an S" falls right outside the traditional scheme; and the admission of such forms would wreck the doctrine of distribution anyhow. If we say that in "Some S is not a man" "man" is distributed, on the score that this sort of statement about every man is inferable, then we must also allow that "dog" in "Some dog is white" is distributed, on the score that it entails that we can say as regards every dog "Either he is white, or not he alone is a dog".

Sources used (see bibliography page for titles corresponding to numbers): 65, 66.