## Tuesday, September 12, 2023

### Subalternation

In logic, subalternation is the operation in which from a universal proposition you can infer a particular proposition with the same terms and quality -- e.g., from "All X is Y" you can always get "Some X is Y". As is well known, this plays a role in  the traditional Square of Opposition, but is not found in the modern Square of Opposition.  It's worth thinking through the implications of the difference.

(1) The characteristic principle of subalternation is "Some S is S". Whenever "Some S is S", you always have subalternation, assuming that the rest of your logic is not modified. This is easily shown. For instance, in TFL, in which "All X is Y" is represented as -X+Y and "Some X is Y" is represented as +X+Y, subalternation uses the principle:

-A+B
+A+A
---------
+A+B

The premises add to the conclusion and the particular conclusion derives from one premise that is particular, so it is valid. Likewise in SYLL, in which "All X is Y" is represented as X -> Y and "Some X is Y" is represented as X <- * -> Y:

X -> Y   [premise]
X <- * -> X [subalternation principle]
X <- * -> X -> Y [concatenation]
X <- * -> Y [simplification]

Likewise, Lukasiewicz showed that you can derive the whole of Aristotle's syllogistic just from

All A is A
Some A is A

as well as Barbara and Datisi syllogisms.

Modern predicate logic, then, is a form of logic in which it is assumed that "Some A is A" need not be true -- that, for instance, "Some dogs are dogs" is not necessarily right. The traditional logic assumes that "Some A is A" is a necessary truth.

(2) Besides subalternation, the other relations of opposition are contradiction, contrariety. and subcontrariety. Any Square of Opposition that has both contradiction and contrariety will have subalternation. Suppose that A ("All S is P" is contradictory to O ("Some S is not P") and contrary to E ("No S is P") and that E is contradictory to I ("Some S is P") and contrary to A. Then from the truth of A we can include that E is false, since contraries cannot both be true; from E's being false we can conclude that I is true, since contradictories cannot both be false. The same reasoning follows for subalternation of O from E, if we start with E.

Thus in the modern Square of Opposition, which still has the same contradictories but lacks subalternation, contrariety must be lacking as well, which it is. In the modern logic it is assumed that "All S is P" and "No S is P" can both be true; in the traditional logic this is assumed to be impossible.

(3) The fundamental reason for the modern Square's difference is a very curious and quite arbitrary choice that was made to treat particular propositions as existential propositions (they imply that the subject term exists) and universal propositions as conditional propositions (with "All A is B" being translated as "If something is A, it is B"). Conditional propositions can be true even if their antecedent is false, so universal affirmative and negative propositions are not contraries, and you can't get "Something exists that is A and is B" from "If something is A, it is B".

This weird division between universal and particular is logically consistent (although you have to tweak the rules of inference slightly), but it is far from being the most natural way to build the Square -- the most natural way to build the Square of Opposition is to assume either that they all imply the existence of their subject terms or that none of them do. Both of these assumptions give us something consistent (Aristotle, using mereological analogies, seems to have assumed that the subject term always in some sense exists, although he is fairly generous about what counts as existing, and free logic assumes that none of the categorical propositions are existential), and in both we can have subalternation. It's only when our account of universal propositions is not unified with our account of particular propositions that subalternation becomes impossible to accept as a logical rule.