Truth and Falsity from a Deflationary Point of View

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.

Comments

[The Bad Ostrich]

>>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 brief résumé of the Aristotelian theory. A sentence (logos, oratio) is a meaningful set of meaningful signs. Not merely a string of letters or noises, but a set of things which possess meaning. Propositions (enuntiatio, propositio) are a subset of sentences which, unlike prayers, commands, questions wishes etc) are capable of truth and falsity.

But how do we ‘understand truth and falsity’? According to the Aristotelian schema:

TRUE: N says that p and p
FALSE: N says that p and not p

Do we define ‘not’ in terms of truth and falsity, or do we define truth and falsity in terms of negation? A difficult question, but start with truth. This does not require the negation operator. Imagine a language where there is no word ‘true’. Is Jake telling the truth when he says that grass is green? Answer, I say that Jake says that grass is green and I say that grass is green. I have said that Jake telling the truth without any need for the word ‘true’. This does not mean that truth is nothing, for the truth of Jake’s statement relies on two facts (1) that he said that grass is green and (2) that grass is green.

Truth is something, but we can do without the word 'true'.

As for falsity, if truth is definable, why not falsity? We just need to introduce the operator ‘not’, which we apply to sentences, by means of which the sentence has a different meaning, which (as you set out above) can be defined in terms of complementation.

You will no doubt object that the same proposition can be expressed in two different languages. The sentences are not the same, ergo the proposition, if the same, must be different from the sentence. I reply: the same argument must show that if two different flowers are yellow, hence have ‘the same’ colour, then the colour must be a single individual entity somehow existing in the two different flowers but duplicated but not existing in them because single and not duplicated. But here we approach a wider question, which you will no doubt recognise, as to whether meaning and yellow ‘subsist in the nature of things or in mere conceptions only, whether also if subsistent, they are bodies or incorporeal, and whether they are separate from, or in, sensibles, and subsist about these, for such a treatise is most profound, and requires another more extensive investigation’. Indeed.

[BV]

I am afraid we don't share enough common ground to have a fruitful discussion. (But that's nothing new.) In any case, my time is consumed by other projects one of which is a book I want to complete by the end of the year. So I leave you the last word.