Definitions

In the previous post, I offered a criticism of defining logical consequence by means of proofs. A more precise way to put my criticism would be:

1. Logical consequence is equally well defined by (i) tree-proofs or by (ii) Fitch-proofs.

2. If (1), then logical consequence is either correctly defined by (i) and correctly defined by (ii) or it is not correctly defined by either.

3. If logical consequence is correctly defined by one of (i) and (ii), it is not correctly defined by the other.

4. Logical consequence is not both correctly defined by (i) and and correctly defined by (ii). (By 3)

5. Logical consequence is neither correctly defined by (i) nor by (ii). (By 1, 2, and 4)

When writing the post I had a disquiet about the argument, which I think amounts to a worry that there are parallel arguments that are bad. Consider the parallel argument against the standard definition of a bachelor:

6. A bachelor is equally well defined as (iii) an unmarried individual that is a man or as (iv) a man that is unmarried.

7. If (6), then a bachelor is either correctly defined by (iii) and correctly defined by (iv) or it is not correctly defined by either.

8. If logical consequence is correctly defined by one of (iii) and (iv), it is not correctly defined by the other.

9. A bachelor is not both correctly defined by (iii) and correctly defined by (iv). (By 9)

10. A bachelor is neither correctly defined by (iii) nor by (iv). (By 6, 7, and 10)

Whatever the problems of the standard definition of a bachelor (is a pope or a widower a bachelor?), this argument is not a problem. Premise (9) is false: there is no problem with saying that both (iii) and (iv) are good definitions, given that they are equivalent as definitions.

But now can’t the inferentialist say the same thing about premise (3) of my original argument?

No. Here’s why. That ψ has a tree-proof from ϕ is a different fact from the fact that ψ has a Fitch-proof from ϕ. It’s a different fact because it depends on the existence of a different entity—a tree-proof versus a Fitch-proof. We can put the point here in terms of grounding or truth-making: the grounds of one involve one entity and the grounds of the other involve a different entity. On the other hand, that Bob is an unmarried individual who is a bachelor and that Bob is a bachelor who is unmarried are the same fact, and have the same grounds: Bob’s being unmarried and Bob’s being a man.

Suppose one polytheist believes in two necessarily existing and essentially omniscient gods, A and B, and defines truth as what A believes, while her coreligionist defines truth as what B believes. The two thinkers genuinely disagree as to what truth is, since for the first thinker the grounds of a proposition’s being true are beliefs by A while for the second the grounds are beliefs by B. That necessarily each definition picks out the same truth facts does not save the definition. A good definition has to be hyperintensionally correct.