Semantic Definition of Truth
SDT was introduced by Alfred Tarski in 1933 (see Tarski 1933, 1956). I will present its modern version, that is, based on the concept of model. More specifically, we define truth-in-L-M, where L is a language and M is a model.
L is a set of sentences, M consists of U—the set of individual objects, subsets of U (set-theoretical counterparts of properties of objects, subsets of U X • • • X U (set-theoretical counterparts of n-termed relations defined on U) and, eventually, fixed individuals and functions. Thus, M is a structure defined in set-theoretical terms. L has a well-defined syntax prescribing the ways of constructing sentences from simpler expressions (constituents), that is, individual constants, predicates, variables or quantifiers and propositional connectives. L can be formalized or not. In particular, it possible to understand L as a technical scientific language or as a specialized (for instance, by adding symbols) part of natural language. To fix the scenario, any theory is a proper consistent subset of L (in symbols, T C L (see below why L is not a consistent theory).L and M are connected via a semantic interpretation V. Omitting various formal details, V is a function, which ascribes semantic values to sentences of L and their constituents. More specifically, V values sentences of L by truth-values (truth, falsehood), individual constants (if any) of L—by distinguished objects from U as references, predicate letters—by subsets of U or relations defined on U, function symbols (if any)—by functions, and quantifiers as ranging over U. In consequence truth-in-L-M is defined under an interpretation generated by V. The concept of interpretation is just semantic in its very essence, because expressions take extra-linguistic items as their values (I omit here the problem of the nature of truth and falsehood).[94] The first step toward SDT consists in defining the concept of satisfaction.
Let P be an one-place (unary) predicate letter; by definition, it refers to a subset of U. We say that the formula Px is satisfied by an object o if and only if o e V(P), that is, o is an element of the set being the value of P. The full definition of satisfaction proceeds by induction on complexity of formulas of L and can be skipped here without any loss of generality. I only mention that since formulas can be of an arbitrary finite length, it is convenient to introduce infinite sequences of objects in order to obtain a general scheme for all possible syntactic cases.Px as an open formula is neither true nor false. On the other hand, Pa (a is the individual name for the object o) has a definite logical value dependent on V. For instance, the formula ‘x is a prime number' is true for V(x) = 2, but false for V(x) = 4n. This fact shows an intuitive connection between satisfaction and true. Tarski's ingenious idea consists in considering truth as a special case of satisfaction. Let A be a sentence (closed formula, that is, not having free variables). One can prove that A is either satisfied by all (infinite) sequences of objects from U or is not satisfied by any sequences of objects (satisfied by no such sequence object). Tarski proposed to define truth as satisfaction by all sequences of objects and falsity as satisfaction by no sequence of objects object. The satisfaction by all sequences of objects is equivalent to satisfaction by at least one such sequence or to satisfaction by the empty sequence of objects. These facts motivate:
(SDT) A sentence n(A) of a language L is true in a model M under an interpretation V if and only if is satisfied by all infinite sequences of objects from U (or at least one such sequence or the empty sequence; otherwise A is false in M).[95]
The symbol n(A) refers to a name of the sentence A and belongs to ML, although the sentence A itself is an element of L. (TD) entails
(TE) A sentence n(A) is true if and only if A* (in symbols, Tn(A A*),
for any A belonging to L (A*—refers to a translation of A into ML).
Tarski claimed that every reasonable truth-definition should entail (TE) (frequently called T-scheme); this requirement is called the convention T ((CT) for brevity), which establishes the condition of the material adequacy for a truth-definition. The use of symbols n(A) and A* underlined that the construction of (SDT) proceeds in ML. On Tarski's account, this way allows to eliminate semantic paradoxes.Two important metamathematical results have an utmost relevance for my further remarks. These are (PA—Peano arithmetic; VER(T)—the set of truths of T):
(GM) A set X C L has a model if and only if X is consistent;
(TU) If T suffices for capturing PA, the set VER(T) is not definable in T.
(GM) (the Godel-Malcev completeness theorem) and (TU) (the Tarski undefinability theorem) have significant consequences. The former supplements the earlier assumption that T is consistent. If so, T has a model and, intuitively speaking, concerns something being its subject-matter. One can observe that the assumption of consistency of T is at odds with the reality of science, because inconsistency of scientific theories appears as a notorious facts. However, another fact, also notorious, is that if a theory is demonstrated as inconsistent, steps toward its repair to achieve consistency appear immediately. Other strategy used in such situations tries to isolate inconsistencies until a new, typically more general, theory is formulated (see Vickers 2013 for a historical and methodological analysis of inconsistent science).[96] Thus, we can assume that scientists tacitly consider theories as consistent, at least in majority of cases. Still one consequence of (GM) deserves attention. L in its integrity has no model, because for every sentence A, the formula —A e L. Hence, L is inconsistent. On the other hand, L, unless it is purely formal, possesses an interpretation V. Consequently, M and V should be very carefully distinguished. In fact, models are algebraic structures, but interpretations act as functions. (TU) justifies the view (see footnote 2) that semantics cannot be reduced to syntax.
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