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.

Logical consequence

There are two main accounts of ψ being a logical consequence of ϕ:

• Inferentialist: there is a proof from ϕ to ψ

• Model theoretic: every model of ϕ is a model of ψ.

Both suffer from a related problem.

On inferentialism, the problem is that there are many different concepts of proof all of which yield an equivalent relation of between ϕ and ψ. First, we have a distinction as to how the structure of a proof is indicated: is a tree, a sequence of statements set off by subproof indentation, or something else. Second, we have a distinction as to the choice of primitive rules. Do we, for instance, have only pure rules like disjunction-introduction or do we allow mixed rules like De Morgan? Do we allow conveniences like ternary conjunction-elimination, or idempotent? Which truth-functional symbols do we take as undefined primitives and which ones do we take as abbreviations for others (e.g., maybe we just have a Sheffer stroke)?

It is tempting to say that it doesn’t matter: any reasonable answers to these questions make exactly the same ψ be logical consequence of the same ϕ.

Yes, of course! But that’s the point. All of these proof systems have something in common which makes them "reasonable"; other proof systems, like ones including the rule of arbitrary statement introduction, are not reasonable. What makes them reasonable is that the proofs they yield capture logical consequence: they have a proof from ϕ to ψ precisely when ψ logically follows from ϕ. The concept of logical consequence is thus something that goes beyond them.

None of these are the definition of proof. This is just like the point we learn from Benacerraf that none of the set-theoretic “constructions of the natural numbers” like 3 = {0, 1, 2} or 3 = {{{0}}} gives the definition of the natural numbers. The set theoretic constructions give a model of the natural numbers, but our interest is in the structure they all have in common. Likewise with proof.

The problem becomes even worse if we take a nominalist approach to proof like Goodman and Quine do, where proofs are concrete inscriptions. For then what counts as a proof depends on our latitude with regard to the choice of font!

The model theoretic approach has a similar issue. A model, on the modern understanding, is a triple (M,R,I) where M is a set of objects, R is a set of relations and I is an interpretation. We immediately have the Benacerraf problem that there are many set-theoretic ways to define triples, relations and interpretations. And, besides that, why should sets be the only allowed models?

One alternative is to take logical consequence to be primitive.

Another is not to worry, but to take the important and fundamental relation to be metaphysical consequence, and be happy with logical consequence being relative to a particular logical system rather than something absolute. We can still insist that not everything goes for logical consequence: some logical systems are good and some are bad. The good ones are the ones with the property that if ψ follows from ϕ in the system, then it is metaphysically necessary that if ϕ then ψ.