The following equivalence is taken by many to support the deflationary thesis that truth has no substantive nature, a nature that could justify a substantive theory along correspondentist, or coherentist, or pragmatic, or other lines. For example, someone who maintains that truth is rational acceptability at the ideal (Peircean) limit of inquiry is advancing a substantive theory of truth that purports to nail down the nature of truth. Here is the equivalence:

1) <p> is true iff p.

The angle brackets surrounding a declarative sentence make of it a name of the proposition the sentence expresses. For example, <snow is white> -- the proposition that snow is white -- is true iff snow is white. (1) suggests that the predicate ' ___ is true' does not express a substantive property. We can dispense with the predicate and say what we want without it. It suggests that there is no such legitimate metaphysical question as: What is the nature of truth? Having gotten rid of truth, can we get rid of falsity as well?

A false proposition is one that is not true. This suggests that 'false,' as a predicate applicable to propositions and truth-bearers generally, is definable in terms of 'true' and 'not.' Perhaps as follows:

2) <p> is false iff <p> is not true.

From (2) we may infer

2*) <p> is false iff ~(<p> is true)

and then, given (1),

2**) <p> is false iff ~p.

This suggests that if we are given the notions of 'proposition' and 'negation,' we can dispense with the supposed properties of truth and falsity. (1) shows us how to dispense with 'true' and (2**) show us how to dispense with 'false.'

But we hit a snag when we ask what 'not' means. Now the standard way to explain the logical constants employs truth tables. Here is the truth table for the logician's 'not' which is symbolized by the tilde, '~'.

But now we see that our explanation is circular. We set out to explain the meaning of 'false' in terms of 'not' only to find that 'not' cannot be explained except in terms of 'false.' We have moved in a circle.

The Ostrich has a response to this:

. . . we can define negation without reaching for the notions of truth and falsity. Assume that the notion of ‘all possible situations’ is coherent, and suppose it is coherent for any proposition ‘p’ to map onto a subset of that set. Then ‘not p’ maps onto the complement. The question is whether the very idea of a complement of a subset covertly appeals to the concept of negation. But then that suggests that negation is a primitive indefinable concept, rather than what you are claiming (namely that it is truth and falsity which are primitive).

So let's assume that there is a set S of possible worlds,and that every proposition (except impossible propositions) maps onto to an improper or a proper subset of S. The necessary propositions map onto the improper subset of S, namely S itself. Each contingent proposition p maps onto a proper subset of S, but a different proper subset for different propositions. If so, ~p maps onto the complement of the proper subset that p maps onto. And let's assume that negation can be understood in terms of complementation.

The most obvious problem with the Ostrich response is that it relies on the notion of a proposition. But this notion cannot be understood apart from the notions of truth and falsity. Propositions are standardly introduced as the primary vehicles of the truth-values. They alone are the items appropriately characterizable as either true or false. Therefore, to understand what a proposition is one must have an antecedent grasp of the difference between truth and falsity.

To understand the operation of negation we have to understand that upon which negation operates, namely, propositions, and to understand propositions, we need to understand truth and falsity.

A second problem is this. Suppose contingent p maps onto proper subset T of S. Why that mapping rather than some other? Because T is the set of situations or worlds in which p is true . . . . The circularity again rears its ugly head.

The Ostrich, being a nominalist, might try to dispense with propositions in favor of declarative sentences. But when we learned our grammar back in grammar school we learned that a declarative sentence is one that expresses a complete thought, and a complete thought is -- wait for it -- a proposition or what Frege calls *ein Gedanke*: not a thinking, but the accusative of a thinking.

Truth and falsity resist elimination.

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Tertium Non Datur