Correspondence vs. Disquotation

In my previous two posts I have discussed the correspondence theory of truth and disquotation principles. In this post I’m going to use the former to argue against the latter. More specifically, I’m going to argue that the “if and only if” of the disquotation principles should be merely an “only if”. In other words, instead of

1. <p> is true if and only if p

I will argue that we should affirm only

2. <p> is true only if p

and reject

3. <p> is true if p.

(3) is more naturally expressed as (4):

4. If p then <p> is true.

I’m going to argue that (4) is false, and that therefore (1) is false.

Why (4) may seem to be necessarily true

My claim that (4) is false is bound to be met with some skepticism, for many philosophers regard (4) as logically necessary. Before launching into my critique of (4) it will be helpful to begin by giving this perspective its due.

Conflating use and mention. One line of thought that may lead one to endorse (4) as logically necessary is based on a misreading of it and on a conflation of the use/mention distinction. Thus, it may be thought that (4) should be read as

5. If p is true then <p> is true.

But (5) is the wrong way to read (4). In my previous post discussing disquotation principles I noted that the variable p occurs twice. When p occurs in quotes or angle brackets it is mentioned, but not used because it functions merely as a referring device allowing us to predicate truth of this truthbearer. In contrast, when p occurs without quotes or angle brackets it is used, but not mentioned because p is being used to describe a putative state of affairs. To read (4) as (5) is confused because if we’re going to predicate truth of p in the antecedent then we have to refer not to the state of affairs described by p but to p itself. So (5) should read like (6):

6. If <p> is true then <p> is true.

Now, (6) is obviously a tautology. As such it is both logically necessary and trivially true. (More exactly, it’s logically necessary that (6) is true if the proposition it expresses exists.) And so one who mistakenly reads (4) in this way may come away with the impression that (4) is logically necessary. But (6) doesn’t say the same thing as (4). In (4) the antecedent isn’t saying something about p (e.g., that p is true). Rather, it’s saying something about reality, i.e.,

7. If reality is as <p> describes, then <p> is true.

Subject to the additional need for quantification that I will address shortly, (7) is the correct way to read (4). We can easily see this by considering standard examples of the disquotation principle, such as

8. <The cat is on the mat> is true if and only if the cat is on the mat.

What (8) says is that the proposition <The cat is on the mat> is true if and only if things are as that proposition describes, that is, if and only if the cat in question really is on the mat in question.

Taking the existence of propositions for granted. Another way in which one might come to regard (4) as logically necessary is by taking the existence of propositions in general, or of specific propositions, for granted. Of course, if a proposition exists then it must have a truth value depending on whether reality is as it describes or not. And so we arrive at (4), that if reality is as a given proposition describes, then that proposition is true.

But p in (4) isn’t a proposition; it is a variable or a placeholder for propositions. To evaluate an expression that contains a variable one must ask what the domain of that variable is. Its domain is the collection of things that can be admissibly plugged into it. In (4) this is the collection of propositions. To make the meaning of (4) fully explicit we must make its domain explicit as follows:

9. For all propositions p, if p accurately describes reality then p is true.

(Since the quantifier explicitly identifies p as a proposition, and since all instances of p in (9) are cases in which p is mentioned, but not used—so that there are no possible use/mention conflations to worry about—I have dropped the angle brackets for notational simplicity.)

It must be observed, however, that (9) doesn’t commit us to the existence of propositions, any more than

10. For all hobbits h, if h lives at Bag End, then h is named Bilbo

commits us to the existence of hobbits. For all (9) tells us, the domain of p, the collection of propositions, could be empty.

Neither (9), (7), (4), nor (1) entitles us to take the existence of any given proposition, or any propositions at all, for granted. Assuming the existence of propositions may seem like an innocuous assumption. After all, we know that propositional units of meaning exist. We contemplate and express them all the time. But the thought that (4) is a logically necessary truth requires quantifying not just over actual propositions (some of which we know to exist), but also over logically possible worlds. In other words, the claim is that

11. For all logically possible worlds w and for all propositions p, if p accurately describes reality in w then p is true at w.

But what if it’s logically possible that there be an empty world, one in which nothing exists? Or what if it’s logically possible that there be a world in which there are no minds, and thus no propositions? That (4) is necessarily true requires not just the obviously correct assumption that some propositions exist, but the far from obvious assumption that it is logically necessary that some propositions exist. Perhaps that’s right. But one can’t establish it by appealing to (4) on pain of begging the question.

Why (4) is false

Having argued that (4) is not, or at least is not obviously, a logically necessary truth, I’m now going to argue that it is in fact false. It is false, I say, because it conflicts with the correspondence theory of truth, which rests on much firmer intuitive grounds.

According to the correspondence theory, truth is a relation of correspondence between a truthbearer and a truthmaker. The fundamental truthbearers are propositions. Accordingly, for a proposition to be true both the proposition and a corresponding truthmaker must exist. (In the special case of analytic propositions, the proposition is it’s own truthmaker.) But look again at what (4) says. As glossed in (7) it says that

7. If reality is as <p> describes, then <p> is true.

The antecedent “reality is as <p> describes” assumes a truthmaker for <p> but—and here’s the crucial point—it doesn’t assume the existence of <p>. It therefore gives us only half of what’s required for <p> to be true. It gives us a truthmaker for <p> but not <p> itself, the truthbearer. Consequently, (7) is false—and with it (1), (4), (9), and (11) as well. That reality is as <p> describes is necessary but not sufficient for <p>’s being true. It must also be the case that <p> exists.

We should replace (4), therefore, with

12. If p and <p> exists, then <p> is true.

Concluding remarks

Now, as I’ve said above, in the actual world we can safely take the existence of many propositions for granted. But we cannot use principles like (4) to establish the existence of propositions in all logically possible worlds. Moreover, even in the actual world we must be careful not to assume that just because we can create declarative sentences that seem to express propositions that they actually do so. Lewis Carroll’s famous nonsense poem Jabberwocky contains many statements that could be used to express propositions. We might think to plug some of these into (4), for example,

13. If the slithy toves did gyre and gimble in the wabe then <The slithy toves did gyre and gimble in the wabe> is true.

But since “the slithy toves did gyre and gimble in the wabe” is nonsense it doesn’t express a proposition. To apply (4), then, we must be able to do two things:

• Make sure that we’re entitled to assume the existence of the proposition in question.
• Make sure that we are successfully expressing a proposition and not just using a string of words without definite sense.