Ayer on Truth

Ayer wants a way to translate any sentence containing the word "true" into a sentence which does not contain that word. This is what Ayer calls a "definition in use": the idea is not to say "'true' means by definition ________", but rather to show how to systematically translate sentences containing the word 'true' into sentences that do not contain the word 'true'.

Ayer’s idea about how to do this seems ambiguous between three readings:

This handout considers the first problem. It’s surprisingly difficult! Ayer writes that the bearers of truth are propositions. So suppose we have a variable that ranges over propositions, say, p. Then we may try:

(i)     (p)(p is true p)

But this won’t work. It is ill-formed: the quantifier takes objects of a certain sort, in this case propositions, as values. So an instance of this schema would be:

the proposition that snow is white is true the proposition that snow is white

and that’s not a sentence. We need a way to get a singular term on the left and a sentence on the right. But we have to have the same variable, p, appear in both places.

The key to a solution might seem to be that there is a device for converting a sentence into a singular term: the operator 'the proposition that'. Take a sentence like 'Snow is white', add the operator 'The proposition that', and you get the singular term 'The proposition that snow is white'.

So maybe we could try something like:

(ii)     (S)(The proposition that S is true S)

But this attempt too is a failure. The quantifier (S) is an objectual quantifier which takes sentences as its values. But sentences are a kind of thing, and names of sentences are singular terms rather than sentences. An instance of (ii) would be:

The proposition that ‘Snow is white’ is true ‘Snow is white’

and this again is clearly ill-formed. Objectual variables must always be singular terms, since they always range over objects. There’s no way to make an objectual variable stand alone as a complete sentence.

Still, we were on the right track. What we need is a different sort of quantification: not objectual quantification, but substitutional quantification. In ordinary, objectual quantification the variable ranges over a set of objects which can be values of the variable. So, for instance,

'(x)Fx' is true    'F' applies to every value of x
                         the open sentence 'Fx' is satisfied by every value of x
                         every value of x is a member of the extension of F.

In substitutional quantification, the variable does not have a range. Rather, it has a set of substituends; the substitution instances for x will be linguistic expressions.

                                        '(Lx)Fx' is true all substitution instances of 'F . . .' are true
                                                                      (where a substitution instance of 'F . . .' is a sentence with
                                                                      '. . .' replaced by one of the substituends of x).

Consider a very small universe. It consists of three shapes:

And the language we are considering has three names, one for each shape:

Fred Joe Burt.

The range of the objectual variable is ,,: the objects in the universe.

The substituends of the substitutional quantifier are the names ‘Fred’, ‘Joe’, ‘Burt’.

Provided that every object has a name, and every name has an object, the two interpretations of the quantifiers will always have the same truth conditions.

'(x) x is white' is true    'is white' applies to Fred, Joe, and Burt
                                    Fred is white and Joe is white and Burt is white.

'(Lx) x is white' is true 'Fred is white' is true and 'Joe is white' is true and 'Burt is white' is true.

Now, employing this interpretation of the quantifiers, we can construct an appropriate version of the redundancy theory. In the sentences we have considered so far, the substituends were always singular terms. But they need not be: we can also have complete sentences as substituends. Thus instead of using objectual quantification over sentences, we can use substitutional quantification with sentences as substitution instances. This gives us our third attempt to formulate the redundancy theory of truth for propositions.

(iii)     (LS)(The proposition that S is true S).

I think that this is actually extremely plausible. On the other hand, I don’t think that it shows that a more substantive theory of truth, e.g. a correspondence or coherence or pragmatic theory, is unnecessary. The account described here could be true, and still not reveal the nature of truth. It seems to analyze the truth of "the proposition that S is true" in terms of the truth of "S." But it’s not clear that it tells us anything about the truth of S itself.

Last update: February 22, 2010.
Comments? Send them to cbrown@trinity.edu.

Curtis Brown | Philosophy of Language | Philosophy Department | Trinity University