Inflationism and deflationism

We have been looking at attempts to give a theory of propositional truth. That is, a true claim of one of the following forms:
 For a proposition to be true is for ...
 Necessarily: for all propositions p: p is true iff ...
(where what goes in the gap does not itself appeal to truth, on pain of circularity.)

We have considered the following proposals:

The correspondence theory of propositional truth
(For a proposition to be true is for it to correspond with a fact)
(Necessarily: for all propositions p: p is true iff p corresponds with a fact)

The coherence theory of propositional truth
(For a proposition to be true is for it to cohere with propositions P)
(Necessarily: for all propositions p: p is true iff p coheres with propositions P)

The pragmatic theory of propositional truth
(For a proposition to be true is for it to be useful to believe)
(Necessarily: for all propositions p: p is true iff p is useful to believe)

The identity theory of propositional truth
(For a proposition to be true is for it to be a fact)
(Necessarily: for all propositions p: p is true iff p is a fact)

The view that there is something as substantial as this to say about propositional truth is sometimes called inflationism about propositional truth. Such theories are sometimes called robust or substantive.
There is also a view that nothing as substantial as this can be said about propositional truth. This is typically called deflationism about propositional truth. Deflationism has been defended by Gottlob Frege, Frank Ramsey, A. J. Ayer, W. V. O. Quine, Hartry Field, and Paul Horwich.

Deflationists typically say that there are, however, interesting things that can be said about truth and the word 'true'. We will look at:
 The redundancy theory of 'true'
 Strawson's performative theory of 'true'
 The prosentential theory of 'true'
 Minimalism about propositional truth
 Tarski’s theory of sentential truth
Some agreement

There are some things that inflationists and deflationists tend to agree on. They tend to agree, for example, that:
The proposition that grass is green is true iff grass is green.

They also tend to agree on the following stronger claim:
Necessarily: the proposition that grass is green is true iff grass is green.
(i.e. It is not possible that: the proposition that grass is green is true and grass is not green, or the proposition that grass is green is not true and grass is green.)

In general, deflationists and inflationists alike tend to accept the claim expressed by any instance of the following schemas:
The proposition that # is true if and only if #
Necessarily: the proposition that # is true if and only if #
(Where instances of these schemas are obtained by replacing the symbol ‘#’ by a sentence of English that expresses a proposition.)

So they accept:
 The proposition that snow is white is true if and only if snow is white
 Necessarily: the proposition that snow is white is true if and only if snow is white
 The proposition that Tallahassee is the capital of Florida is true if and only if Tallahassee is the capital of Florida
 Necessarily: the proposition that Tallahassee is the capital of Florida is true if and only if Tallahassee is the capital of Florida
And so on ...

So all tend to agree that for any given proposition, p, we can say something substantial about the conditions under which p is true. If p is the proposition that grass is green then, necessarily, p is true iff grass is green; if p is the proposition that snow is white then, necessarily, p is true iff snow is white; and so on. But note that these are quite different conditions in each case  the conditions differ from proposition to proposition.

What inflationists think is that there is also some single condition which applies in every case:
Necessarily: for all propositions p: p is true iff C(p)
Deflationists deny this.
The redundancy theory of 'true'

The most radical version of deflationism about propositional truth is the redundancy theory of 'true', also called nihilism about propositional truth.

According to the redundancy theory of 'true', the predicate 'true' does not express a property. It fact, it adds no content at all to the sentences in which it occurs (thus the name "redundancy theory of 'true'"). The sentence 'It is true that grass is green' expresses the very same proposition as the simpler sentence 'Grass is green'  namely, the proposition that grass is green.
This amounts to saying that there is no such property as being true (thus the name "nihilism about propositional truth").

Gottlob Frege (1918) seems to have thought something like this:
“It is worthy of notice that the sentence ‘I smell the scent of violets’ has the same content as the sentence ‘It is true that I smell the scent of violets’. So it seems, then, that nothing is added to the thought by my ascribing to it the property of truth.”
So too Frank Ramsey (1927), ‘Facts and Propositions’, and Peter Strawson (1949, 1950).

The redundancy theory explains why we are happy to accept every instance of the schema: Necessarily: the proposition that # is true iff #. Why? Because "the proposition that # is true" and "#" express the very same proposition.

But can it explain why we have the predicate ‘true’ at all?
Why have the predicate 'true'?

Redundancy theorists tend to agree that some cases in which we use 'true' are cases in which we don't really need to:

Suppose I assert, 'It is true that Boise is the capital of Idaho'. I could have just asserted, 'Boise is the capital of Idaho'.

Suppose you assert, 'Women are mortal', and then I assert, 'That is true'. I could have just asserted, 'Women are mortal'.

Suppose John asserts that π is not 22/7 and I then give him $1; when asked why I gave John $1 I explain, 'Because it is true that π is not 22/7', or 'Because what John said is true'; I could have explained this way instead: 'Because π is not 22/7'.
These are all cases in which I am in a position to express the proposition under consideration. In the second case, rather than asserting, 'That is true', I can assert that women are mortal, because I am in a position to directly express the proposition that women are mortal.

Redundancy theorists argue, however, that there are cases in which 'true' is very useful. Consider the third case above. Suppose that I don't hear what John said, so I have no idea which proposition he asserted. Still, I am confident that John never makes mistakes, and so I give him $1. When asked why I cannot explain by directly asserting the proposition that John asserted, because I am not in a position to do so (I don't even know which proposition it was). But what I can do is explain, 'Because what John said is true'. According to the redundancy theory, what I am doing is indirectly asserting the proposition that John asserted  I am asserting that π is not 22/7. Having the predicate 'true' allows me to do this.
Two problems

Cases of the second kind are what redundancy theorists tend to appeal to when explaining why we have the predicate 'true'. But they also pose a problem for the theory.

Here is one problem. Suppose you assert some proposition, say the proposition that grass is green, and I assert, ‘That is true’. According to the redundancy theory, I have also expressed the proposition that grass is green. But have I? Note that we could change the example so that I have no idea which proposition you have expressed, and even make it a proposition that I cannot even express (e.g. I do not have all the required concepts).

There is a second problem (presented in Soames). To see what it is, we first need some background on variables and quantification.
Variables and quantification
Name position

Suppose I want to make all of the following claims, one for each person in the world:
 John is happy
 Mary is happy
 Tony is happy
and so on...

Rather than asserting several billion propositions using several billion sentences, I can assert a single proposition using a single sentence:
S1: For all persons x: x is happy
In S1, the symbol 'x' is called a variable, and the operator 'For all persons x' is called a quantifier. We might call S1 a quantified sentence.
In S1, 'x' is said to be in name position. That's because in the subsentence 'x is happy' it is occupying a position that would normally be occupied by a name.

There are at least two different ways of understanding the conditions under which S1 is true: the objectual interpretation of S1, and the substitutional interpretation of S1.

On its objectual interpretation, S1 is true just in case the subsentence 'x is happy' is true no matter which person is assigned as the referent of the variable 'x'.

On its substitutional interpretation, S1 is true just in case the subsentence 'x is happy' is true no matter which name for a person is substituted for the varibale 'x' (i.e. replace 'x' by).

If there are people in the world for which we have no names, then S1 might have different truth values on the two interpretations. Suppose that everyone in the world that we have a name for in our language is happy. Then S1 is true on its substitutional interpretation. But suppose that there is also someone who is not happy (we have no name for him or her). Then S1 is false on its objectual interpretation.
On its objectual interpretation, whether or not S1 is true does not depend on how many names for people we have in our language, but on its substitutional interpretation it does.
Because of this, it is only on its objectual interpretation that S1 expresses the proposition that I am trying to express.

Here are some other examples to illustrate how the truth value of a quantified sentence can differ on the two interpretations:

For some x: x is a real number and x has no name.
On the objectual interpretation this is true, on the substitutional interpretation it is false.

For some x: I dreamed about x and x does not exist.
On the objectual interpretation this is false, on the substitutional interpretation it is true.
Predicate position

Now suppose that I want to make all the following claims, one for each property of people:
 John is mortal
 John is hairy
 John is rational
and so on...

Rather than asserting an infinite number of propositions using an infinite number of sentences, I can assert a single proposition using a single sentence:
S2: For all properties F: John is F.
Here, the variable 'F' is said to be in predicate position. That's because in the subsentence 'John is F' it is occupying a position that would normally be occupied by a predicate.

Again, there are at least two different ways of understanding the conditions under which S2 is true: the objectual interpretation, and the substitutional interpretation.

On its objectual interpretation, S2 is true just in case the subsentence 'John is F' is true no matter which property is assigned as the referent of 'F'.

On its substitutional interpretation, S2 is true just in case the subsentence 'John is F' is true no matter which predicate is substituted for 'F'.

Again, if there are properties of people for which we have no predicate in our language, then S2 might have different truth values on the two interpretations. Suppose that John has every property of people that we have a name for in our language. Then S2 is true on its substitutional interpretation. But suppose that there is also some property of people that John does not have (we have no predicate for it). Then S2 is false on its objectual interpretation.
On its objectual interpretation, whether or not S2 is true does not depend on how many predicates for properties of people we have in our language, but on its substitutional interpretation it does.
Because of this, it is only on its objectual interpretation that S2 expresses the proposition that I am trying to express.
Sentence position

Finally, suppose I want to make all the following claims, one for each proposition:
 John knows that grass is green
 John knows that snow is white
 John knows that Providence is the capital of Rhode Island
and so on...

Rather than asserting a very large number of propositions using a very large number of sentences, I can assert a single proposition using a single sentence:
S3: For all propositions S: John knows that S.
Here, the variable 'S' is said to be in sentence position. That's because in the subsentence 'John knows that S' it is occupying a position that would normally be occupied by a sentence.

Again, there are at least two different ways of understanding the conditions under S3 is true: the objectual interpretation, and the substitutional interpretation.

On its objectual interpretation, S3 is true just in case 'John knows that S' is true no matter which proposition is assigned as the content of 'S'.

On its substitutional interpretation, S3 is true just in case 'John knows that S' is true no matter which sentence in our language that expresses a proposition is substituted for 'S'.

Again, if there are propositions for which we have no sentence in our language, then S3 might have different truth values on the two interpretations. Suppose that John knows every proposition that we have a sentence for in our language. Then S3 is true on its substitutional interpretation. But suppose that there is also some proposition that John does not know (we have no sentence for it). Then S3 is false on its objectual interpretation.
On its objectual interpretation, whether or not S3 is true does not depend on how many sentences that express propositions we have in our language, but on its substitutional interpretation it does.
Because of this, it is only on its objectual interpretation that S3 expresses the proposition that I am trying to express.
A second problem for the redundancy theory

Suppose I have no idea what or how much Maria asserted but, trusting her, I say ‘Everything Maria asserted is true’. What have I said  what proposition have I asserted?

A nonredundancy theorist has the property of being true to appeal to, so she can say that I have expressed the following proposition:
For all propositions p: if Maria asserted p then p is true.
Here the variable 'p' is in name position, and the quantification is to be interpreted as objectual.
But a redundancy theorist does not have the property of being true to appeal to, and so cannot say this.

She might try just wiping out the ‘is true’ part:
For all propositions p: if Maria asserted p then p.
But since 'p' is name position, this expression is illformed  it is not grammatical to follow 'then' with a name (or any other referring expression). The following, for example, is not grammatical:
If Maria asserted the proposition that Baton Rouge is in Louisiana then the proposition that Baton Rouge is in Louisiana.

She can avoid this problem by trying the following suggestion, where the variable 'P' is in sentence position:
For all propositions P: if Maria asserted that P then P.
This does not lead to any illformed instances. Here is the instance obtained by replacing 'P' by the sentence ‘grass is green’:
If Maria asserted that grass is green then grass is green.

Now, is the quantification in this sentence to be understood as objectual or substitutional?

Suppose it is to be understood as substitutional. Then the sentence is true iff the subsentence 'If Maria asserted that P then P' is true no matter which of our sentences we substitute for 'P'.
But whether or not this gives the right result depends upon whether or not we have sufficiently many sentences in our language. Suppose that Maria asserted ten propositions, nine true ones and one false one. So what I said is false. Suppose that Maria speaks a different language from ours  a richer language with which they can express more propositions than we can. Suppose that we have sentences to express each of the nine true propositions that Maria expressed, but no sentence to express the one false one. Then, on the substitutional interpretation, what I said is true, which is the wrong result.

So we must understand the quantification in this sentence as objectual. Then the sentence is true iff the subsentence 'If Maria asserted that P then P' is true no matter which proposition is assigned as the content of 'P'. But if we allow this interpretation then we seem to have the resources to give an inflationary theory of truth after all:
Necessarily: for all propositions p: p is true iff for some proposition P: x is the proposition that P, and P.

So it seems that for a redundancy theorist to give a true account of what it is that I am saying with ‘Everything Maria asserted is true’ she has to admit that we can after all give an inflationary theory of truth.
Read Quine, W. V. Q. (1986), Philosophy of Logic (Cambridge, MA: Harvard University Press), sec. 'Truth and semantic ascent'. Suppose I want to make all the following claims, one for each proposition: John knows that grass is green, John knows that snow is white, John knows that Providence is the capital of Rhode Island, and so on, but I want to do so by making just a single claim. Quine thinks that I can do so by ascending a level and using the predicate 'true'. How, according to Quine, do I do this? Can you think of any problems for Quine's approach?