Quantifying into Sentence Position

23 02 2006

In his “Concept of Truth in Formalized Languages”, Tarski considers an alternative truth-definition that involves sentence-position variables inside quotes. In what he calls a “formally correct” truth definition, we would have a condition of the form “forall x (x is true iff …)”. “x” here is a variable that ranges over mentioned sentences, and “…” should be filled in with our definition. The attempt under consideration is
forall x (x is true iff exists σ (x=”σ” and σ)).
Here, “σ” is a variable that is supposed to range over used sentences, rather than mentioned sentences. We will say that x is true if it is a sentence that can be used to mean σ, so we need x to be an expression for σ, which is why we need to say “x=”σ””. However weird sentence-position quantification might be, the worse problem here is that we have to quantify into quotation marks. Note that the quoted letter sigma appears inside the truth-definition in a position where we have to quantify into it, but in my first sentence after that definition, I used that same expression to name the variable, not to give an expression with a free variable inside it naming a sentence. That usage is what we would expect given ordinary rules for using quotation marks, but Tarski considers what would happen if we allowed for this unusual usage (which would make it tough to talk about the language) and shows that we can get versions of the liar paradox, which would undermine the whole goal of trying to define truth.

However, I’ve got another reason to think that we shouldn’t have quantification into quotes that behaves the way we want it to in the attempted truth-definition – instead it should behave the way it does in my first sentence after the definition. The reason is mainly going to be because we often want to have distinct object-language sentences with the same truth-conditions (or perhaps more generally, possibly the same meanings). As a result, the range of values for the sentence-position variable will have to come with both intensional and extensional information. That is, to be used in sentence-position, it will have to have at least the extensional information of the truth-conditions, but in order to get different values for quote sigma given extensionally equivalent values of sigma, it will need to somehow have intensional information carried with it. Now, this is possible, but somewhat unwieldy.

In addition, if sigma is a variable that appears in the object language as well as in the metalanguage, then we’ll have to have a different procedure to indicate that we mean to refer to that variable, rather than have a free sentence-position variable inside quotation marks. This is also possible – in the LaTeX markup language, one can do this for special characters by putting a backslash in front of them; in some other languages one doubles up the special character, or uses some other way to “escape” it. Of course, if one wants to have that expression in quotes, rather than just the letter sigma, then one needs a further set of commands to escape the relevant characters. It’s possible, but it involves replacing a lot of the standard names for certain symbols in the language inside quotes.

So we can reconsider why we wanted to be able to quantify into quotes to begin with. The reason was so that we can have one position that names a sentence while another position uses the same sentence, with the sentence being quantified over. Since every sentence has exactly one meaning (or set of truth-conditions), while truth-conditions are in general shared by multiple sentences, it seems most natural for our metalanguage function to go the other way. Instead of going from use to mention, it should go from mention to use, because that function should be well-defined – multiple intensions correspond to the same extension, but not vice versa. Thus, we should be able to express our truth-definition roughly as the following:
forall x (x is true iff exists S (x=S and F(S))),
where “F” is the function that gets us from use to mention, or intension to extension. That is, “F(S)” corresponds to “σ” and “S” corresponds to “”σ””. We don’t have to worry here about any collision between object language and metalanguage variables, so I think this proposal is overall more natural.

But we can see that this definition is equivalent to
forall x (x is true iff F(x)),
which we see means that “F” just is the truth-predicate. I think this is why natural language has a truth-predicate rather than a quote-quantifying sentence-place variable. They can express all the same things, but one is more convenient than the other. Semantic descent is easier than semantic ascent, so that’s why it’s the function that we have built into our language.

As a result, we have to go to more work to define truth, but Tarski has showed us that this is generally possible, as long as we don’t mind the problems Field points out of the definition being non-systematic and non-explanatory.




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