Alfred Tarski (1944) The semantic conception of truth and the foundations of semantics (Reprinted as Chapter 4 of Martinich’s a

Alfred Tarski (1944) The semantic conception of truth and the foundations of semantics (Reprinted as Chapter 4 of Martinich’s anthology). This is an abridged and updated version of his 1935 long paper Der Wahrheitsbegriff in den formalisierten Sprache (The concept of truth in formalized languages), itself a translation from his book in Polish of 1933.

Background (very succinctly and approximately)

In order to solve the paradoxes of naïve set theory (and the liar’s paradox, see below), as of 1908, Russell introduced the theory of types. In a nutshell, we have a nested hierarchy of "ranges of significance", i.e. of classes of arguments for which a propositional function has values. For instance, "man" (an individual) is of type zero, "beauty" or "being beautiful" (a property) is of type I, "being an aesthetic notion" (a property of properties) is of type II, "being a kind of attributes valuable to humans" (a property of properties of properties) is of type III, and so on. The graphic devices to designate types were (guess..) multiply embedded brackets. A propositional function is only allowed to accept as arguments entities of the immediately lower type. We remove one set of brackets (the outer brackets) each time we go to a lower type. [As we will see, the lower boundaries — individuals to properties - have been drastically redefined, notably after the introduction of events by Davidson].

Sentential connectives in logic are defined via rules of inference, or, alternatively, via "truth tables". Meta-logic made it its business to analyze these procedures and pinpoint circularities and subtle problems. The status of "models" or "realizations" of classes of sentences and sentential functions (entities that satisfy them) was thoroughly investigated, notably by Tarski. The status of definitions (recursive definitions in particular) became a central issue.

The relation of logical consequence has to apply in virtue of form alone (a purely syntactic relation, of sign to sign), but this notion was inextricably linked with the notion of joint satisfiability of all (thus) connected sentences by all models that satisfy one of them (a semantic relation of signs to objects). [More on the notion of satisfaction here below].

The notion of satisfaction and that of "yielding a truth" (both quintessentially semantic) were intimately related. The risk of circularity was hard to avoid.

The notion of "truth of a universal proposition" (or of a propositional function for all possible substitutions of its variables) is taken to be primitive and indefinable (we saw it in Russell, and in Wittgenstein — tautologies — and it is also adopted by Tarski).

The strict subdivision between logical and extra-logical terms of a language proves hard to maintain on principled grounds. Tarski (and, later on, Quine) will argue in favor of a smooth boundary (the Kantian distinction between analytic and synthetic truths also becomes smooth).

The concept(ion) of truth that we want must be both materially adequate and formally correct (T’s emphasis). Sentences are "physical objects", unlike propositions, and Tarski wants to stay away from propositions. He wants to "apply the term ‘true’ to sentences". The notion of truth we want to establish is to be related to a language. The cases we saw previously

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(and many more we can construct) make the case quite vivid.

Truth is "an old notion" and we want to do justice to its everyday use (no matter how vague and imprecise), and to (some at least of) its previous philosophical developments.

Aristotle, in the Metaphysics:

"To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, or what is not that it is not, is true".

The central idea here is one of truth as "correspondence to reality". It is the kind of criterion adopted in the tribunals and in everyday life (with kids, for instance), but it is hopeless. This notion of "correspondence" cannot be cashed in precise terms. It embodies (to say the least):

a theory of accurate perception

a theory of accurate direct linguistic reports of actual experience

a theory of accurate in-direct linguistic reports of actual experience

a theory of inferences and presuppositions

a theory of pragmatics (the proper use of linguistic expressions)

a theory of what is warrantedly assertible "in the ideal limit"

And much more. The case is hopeless.

Moreover (in analogy with the contradictions to which the naive notion of set leads), the naive notion of truth leads us to the paradox of the liar (Epimenides and Eubulides, then J. Richard, B. Russell, C. Burali-Forti, K. Grelling and more).

The liar’s paradox

Ingredients of the paradox:

An object language (say, English) in which, among all sorts of predicates (tall, red etc.) we have the predicates "true" and "false";

A sentence s, in this language, unambiguously and objectively identified (via its number, or its name, or a typographical description, or the sentence itself between brackets);

The sentence s refers to itself (says something about "this very sentence");

It says that s is false (let’s say, intuitively, that it states "I am false");

We construct, in the same object language, another sentence S that contains s as a proper sub-part, and says that s is true;

We (crucially) apply the compositionality of truth conditions.

We have an insoluble contradiction. S as a whole is true if, and only if, it is false and it is false if, and only if, it is true. Aristotle’s criterion blows up.

Other versions

There are versions that generate the paradox via other predicates. For instance the Grelling-Nelson version, based on the predicates "autologous" and "heterologous". "Polysyllabic" is autologous because it applies to itself (polysyllabic is itself a polysyllabic word). So is "English", itself a word in English. But ‘long’ and ‘German’ are heterologous. In essence, we construct a sentence such as "Heterologous is itself heterologous" and we get the paradox (this is very close — and not by chance - to Russell’s paradox of the set that contains all sets that are not member of themselves). Berry’s version is "the least integer not nameable in fewer than nineteen syllables" (the expression between quotes has 18 syllables, and it is a name for that integer).

(Andy Barss has yet another version, ask Heidi).

Tarski’s solution

Raise the predicates "true" and "false" outside the object language. Make them part of a theoretical meta-language that contains the object language (English) as a sub-part. We do not want "semantically closed languages". Any language is semantically closed if it is such that a theory of truth is constructed within that language, and has the term (or concept, or notion) of true (and false) both as a regular predicate of the language, and as a theoretical term in the theory of truth. Any such language inevitably leads to the liar’s paradox. The only viable theory of truth is a theory of the meta-language, and it is a theory of truth for the object language. The predicate true is a predicate of the meta-language and the term "true" of the object language is quoted between brackets, just like any other expression of the object language.

Core ingredients:

We want an objective, formal criterion, expressed in the meta-language by a bi-conditional, such that, given any declarative tensed sentence X of the object language, we construct a formula of the object language such as:

X is true in the object language, if and only if, p.

The predicate true (in bold here, for greater clarity) is a predicate of the meta-language, not of the object language.

"if and only if’ is the standard bi-conditional of symbolic logic (its definition and rules of inference are exactly characterized).

p is an expression of the object language [We will go back to this point]

X is quoted in the meta-language. It is the name, in the meta-language, of a sentence of the object language.

A general device is established in the meta-language to create a one-to-one correspondence between expressions in the object language and their name in the meta-language.

Bracketing (inverted commas) is as good as any, and Tarski adopts it, but one may want to adopt a different device, if it is more convenient (for instance a Gödel number).

The definition of truth

An infinite, countable conjunction of formulae like the following ones:

"Snow is white" is true in English, if, and only if, snow is white.

"Grass is green" is true in English, if, and only if, grass is green.

"Napoleon lost the battle of Waterloo" is true in English, if, and only if, Napoleon lost the battle of Waterloo.

And so on, and so on.

This infinite, open, countable conjunction is the definition of the predicate true in the meta-language.

Why this definition is far from trivial

Consider some of the alternatives that are conceptually open (for simplicity, we stick to English as the object language, and to meta-English as the meta-language).

Let’s adopt the convention that expressions of English are plain, while expressions of meta-English are in bold

Solution 1: All is in English

"Snow is white" is true in English if, and only if, snow is white.

Impracticable: We have a semantically closed language, and it leads to the liar’s paradox.

Solution 2: All is in meta-English

"Snow is white" is true in meta-English if, and only if, snow is white.

Impracticable: We have again a semantically closed (higher-level) language, and it leads to the liar’s paradox.

Solution 3: A mixed formula, in which only the bi-conditional is in the meta-language

"Snow is white" is true in English if, and only if, snow is white.

Since the concept of truth is itself in English, our language is still (in all relevant respects) semantically closed.

Solution 4: Tarski’s solution: A more extensively mixed formula

(T) "Snow is white" is true in English if, and only if, p.

The formula as a whole is in the meta-language, bracketing is a device of the meta-language, the concept of truth is a concept of the meta-language, and so is the notion of English itself (as the object language, characterized in our meta-language). The rightmost part p is in English. Tarski wants his theory to be "materially adequate", doing justice to the traditional conception of truth. It would be unacceptable to come out with an expression p that only professional logicians can understand. We must, therefore, have a sentence in the object language for a sufficient and necessary criterion. Tarski says that p is an "arbitrary sentence" of English, because the identification

p = snow is white

is something that must be given as a result of the theory.

p will turn out to be the sentence in the object language of which "Snow is white" is the name in the meta-language. We have a "disquotational" theory of truth.

But he concedes that we may want to have translations into the meta-language of all the p sentences of the object language.

We may, then, consider its application to another language:

"La neve e’ bianca" is true in Italian, if, and only if, snow is white.

"Cripliskits gamanit swishii" is true in Ruritanian if, and only if, grass is green.

We name a sentence in the object language, and give its bi-conditional truth condition for that language in our meta-language, by means of a sentence translated into our meta-language. (Other solutions are examined, and rejected, by Tarski in his paper)

Solution 5: Our post-Tarskian extensively mixed formulae

We want p to express the real-world (intensional, I-Language) condition that must hold for X to be true. Notice that, as Frege and Russell happily acknowledged, Shakespeare’s thoughts about Hamlet, and our thoughts about snow being white, are real, they are part of the real world (even if Hamlet is not). Since we understand English, we can use it to describe those real-world conditions that make a sentence of English true. As Fodor has rightly stressed, it is metaphysically possible that it could have been French, or Swahili, or some other language that has the same expressive power as English, and the same compositional and truth-functional properties. More plausibly, it is the Language of Thought (LOT). (Whether a pictorial, purely iconic, mental medium might have done the job is problematic. Probably it would not have worked, but let’s leave this issue open).

(PT) "John loves Mary" is true in English if, and only if, L(J,M)

"All ravens are black" is true in English if, and only if, " x(RxÉ Bx)

or (see Heidi’s handout for last Wednesday) " x: raven x(black x)

Intuitively, we have:

A sentence of the object language, phonologically (and/or graphically) fully identified.

Its structural description, including its Logical Form.

We connect them bi-conditionally under criteria of compositionality and truth-evaluability.

That’s all we need.

Some specifications:

In our theory (in our meta-language), English is an I-Language (with certain values for the parameters of UG, and its lexicon), and we have a theory with a conceptual and graphical apparatus to specify which derivation every speaker computes from an overt sentence (in plain English) to get to its syntactic structure, all the way to LF. The whole syntactic derivation should appear on the rightmost side of a (PT) formula (this is what Tarski calls a "translation" into the meta-language). For brevity, let’s just concentrate on the LF of the sentence.

This conception of truth and meaning is "materially adequate", because we attribute to every native speaker the capacity to derive the rightmost part of the formulae. We claim that this is the conception of truth and meaning that tacitly (though perhaps very vaguely) every speaker has for her language.

Let’s not be bamboozled by the symbols on the right hand side. We do not have an "explanation" simply because we imitate the mathematicians, and use abstract symbols. These symbols are only supposed to capture, in a suitably transparent meta-language, the abstract content of the "thought" whose standard expression in the object language is reported on the left hand side. In particular, we want all the sentential constituents to be clearly individuated, and all and only the inferences that the sentence licenses (in the mind of all native speakers, solely in virtue of their tacit knowledge of language) to come out explicit and un-ambiguous.

These formulae are, therefore, sensitive to syntax, compositional, and truth-functionally evaluable. According to Fodor, we have sentences in "mentalese" (the language of thought). Translation has to stop somewhere. We must eventually come to a formula that is directly interpreted by the system. According to other authors, we have a level of syntactic representations that are semantically interpretable. In the minimalist program, we have the output of a derivation such that it satisfies the requirements (the constraints) imposed by the conceptual-intentional system. The formula on the right hand side is the product of the interface between NS (Narrow Syntax) and the conceptual apparatus.

Truth and satisfaction

Tarski wants a fully explicit definition of truth. (His infinite countable conjunction is explicit, because it is generative — in our terminology — and it is offered in the mode of an exemplification). He wants it to gravitate (so to speak) around the notion of "satisfaction". They are both semantic notions, so all is OK. In logical functional calculus the two notions are intimately inter-connected. Given a logical schema, containing propositional functions, and n variables, we single out a domain (an ordered set of individuals) that is said to constitute an interpretation of the schema. The schema is "valid" in a given domain, if in application to that domain it yields a truth for every interpretation of its schematic letters (its variables). A schema is said to be "satisfiable" in a given domain if it yields a truth for some interpretation of its schematic letters (for some values of its variables extracted from the domain). A theorem (by P. Bernays and M. Schoenfinkel, 1928) says that, if a schema is satisfiable at all, then it is satisfiable in a domain that contains 2ⁿ individuals. Tarski cannot straightforwardly adopt this characterization of satisfaction and satisfiability, because it presupposes the notion of truth. So he introduces the notion of satisfaction through a "recursive procedure". We start with the simplest sentential functions, i.e. those that contain only one free variable, and single out the objects that satisfy those functions. For instance, ‘snow’ satisfies the sentential function "x is white". Then we consider more complex sentential functions, for instance by means of conjunctions and disjunctions. Technicalities aside (see note 15) we get a notion of satisfaction that is based on ordered multiples (sequences) of objects. As a special case (notice: as a special case) we have sentences. They contain no free variables. "A sentence is either satisfied by all objects, or by no objects". In the first case, the sentence is true, in the second, it is false.

A gloss:

Akin to Russell’s strategy, we start with the general notion of a propositional function, and then land onto sentences as a special case. Tarski’s central move is to construct sentences as sentential functions with no free variables (they contain only non-logical constants connected via logical relations). They are always satisfied (or, by contrast, never satisfied). The recursive character of the definition grants (in our terminology) compositionality. The definition via the (T) schema is equivalent to a binary sentential function that has a countable infinity of admissible arguments: It is satisfied if, and only if, we substitute in it, for the value of one variable, the name of a sentence of a lower-level language, and, for the value of the other variable, that very sentence of that very language (or a translation of it into the higher-level language). This schema is the definition of truth for that lower-level language.

An interesting consideration:

Tarski says: We know how to recursively construct sentential functions, but how can we recursively construct sentences? He says that no general method is known (and he was right, at the time). Now we know how to do that. It’s a syntactic (generative) procedure, not a logical one. With this momentous improvement, we can happily adopt Tarski’s schema.