Questions Regarding Tarski's Semantical Formalization of the Colloquial Usage of Truth

Question

My question is in regard to a problem (albeit a simple one) that I ran into reading Tarski's paper "Concept of Truth in Formalized Languages".

On page 159 Tarski states:

(5) for all p, ‘p' is a true sentence if and only if p.

From what I understand, this is meant to be a generalization of a prior example he gave earlier as case (3) which was:

(3) ‘it is snowing' is a true sentence if and only if it is snowing

Which in turn was meant to be a specification of case (2), which stated:

(2) x is a true sentence if and only if p.

In which x is the 'name' of the sentence p. Here I have what one might regard as a trivial question, that being, is case (2) not already a generalization of case (3)? If that is the case, what is the point of case (5)? To make the generalization of (3) more rigorous?

A second question I have is in regards to case (5), Tarski states about case (5):

(5) for all p, ‘p' is a true sentence if and only if p.

But the above sentence could not serve as a general definition of the expression 'x is a true sentence’ because the totality of possible substitutions for the symbol x is here restricted to quotation-mark names. In order to remove this restriction we must have recourse to the well-known fact that to every true sentence (and generally speaking to every sentence) there corresponds a quotation-mark name which denotes just that sentence. With this fact in mind we could try to generalize the formulation (5), for example, in the following way:

(6) for all x, x is a true sentence if and only if, for a certain p, x is identical with 'p' and p.

If I am understanding his dilemma correctly, he is pointing out the fact that if the statement 'x is a true sentence' is presented in (5), the problem that arises is that you'll have a non-sensical end statement since x can only symbolize a quotation mark name, to illustrate:

[''it is snowing'' is a true sentence iff 'it is snowing' is a true sentence]

whereas what we want is:

['it is snowing' is a true sentence iff it is snowing is a true sentence]

The thing I do not understand here is how case (6) solves the problem he mentioned about case (5)? What is case (6) saying?

Furthermore, why can't Tarski simply apply functions to p in order to avoid the problems he later mentions about using the quotation marks in the manner it is used in case (5) and to remedy the problem he mentioned above about nested quotation mark names? For example, why can he not simply formulate (5) as:

If p is not a quotation mark name: For all p, p is a true sentence if and only if X(p). Else: For all p, Q(p) is a true sentence if and only if X(p).

In which Q(p) is a function that adds a quotation string before p and at the end of p, and X(p) would remove all quotation strings from p (as the quotation mark name would in essence simply be a variable in place of the actual sentence).

An easy way to represent this would be with a small snippet of python code that goes as follows:

p = "'it is snowing'"

Xp = p.replace("'", "")

Qp = "'" + p + "'"

if "'" in p:

print (p, "is a true sentence iff", Xp)

else:

print (Qp, "is a true sentence iff", Xp)

Resulting in the statement ['it is snowing' is a true sentence iff it is snowing] regardless of if you put in ['it is snowing'] or [its snowing] as p.

This seems to avoid the problem of nested quotation mark names and the problem of transforming the quotation marks (themselves) into functions. Is there something I am missing, or am I completely misunderstanding Tarski? Is there something I need to familiarize myself with before diving into Tarksi? He seems almost impossible to comprehend (so many of his sentences can have double meanings and it seems as if there are a lot of inferences he expects you to make) compared to other writers of his school (such as Lukasiewicz).

Anyways, thank you for taking the time to read or answer this question, any help or guidance would be appreciated - Thank you

Answer 1

Long comment

We have to start from (2) x is a true sentence if and only if p, where the symbol "p" is a sentence and the symbol "x" is a name for that sentence.

The issue is with quotation marks: we use them to transform an expression, e.g. a name referring to an object, into a name for itself.

See page 159: "Quotation-mark names may be treated like single words of a language, and thus like syntactically simple expressions. [...] Every quotation-mark name is then a constant individual name of a definite expression."

Thus, if we use the variable p in a semi-formal context to stay for a generic sentence, with "p" we have a name for a letter of the alphabet, which is not a variable any more.

Thus, using formula (5) for all p, ‘p' is a true sentence if and only if p, and we instantiate it with a sentence like "it is snowing", what we get will be the non-sensical:

‘p' is a true sentence if and only if it is snowing.

The same applies to (6); if we try to to instantiate the variable x, we get:

"it is snowing" is a true sentence if and only if, for a certain p, "it is snowing" is identical with 'p' and p.

Thus, also (6) is flawed; see page 160: "We see at once that the sentences (5) and (6) are not formulations of the thought we wish to express."

A formal device to solve this issue has been later introduce by Quine with Quasi-quotation: quasi-quotation is introduced as shorthand to capture the fact that what the formula expresses isn't precisely quotation, but instead something about the concatenation of symbols, like e.g.

If φ is a well-formed formula, then ⌜~φ⌝ is a well-formed formula.

Quine's device is works exactly like a function, implementing Tarski's consideration [page 161]: "we must seek a different interpretation of the quotation-mark names. We must treat these names as syntactically composite expressions, of which both the quotation marks and the expressions within them are parts."

Very useful: Monika Gruber, Alfred Tarski and The Concept of Truth in Formalized Languages (Springer, 2016).