Teoría de la verdad en Tarski

Abstract

Alfred Tarski has dedicated three essays to the semantic treatment of the concept of truth. The first essay, “The Concept of Truth in Formalized Languages”, was published in 1936, the second, “The Semantic Conception of Truth and the Foundations of Semantics” in 1944, and the last, entitled “Truth and Proof”, in 1969. In each of these works there are changes in the way of dealing with the problem, and in the last one we find the most remarkable changes. An analysis of these changes gives sufficient reason: 1) to establish the limits of the original semantic task, intended to reach a definition of “truth”, 2) to evaluate the effort in displacing the problem of truth from its traditional philosophic site to a field mathematically measurable. Tarski’s original thesis has two aspects: a) in the context of ordinary language it is not possible to reach a definition of truth that is materially adequate and formally correct; b) it is possible to reach a definition of truth in finite formal languages, but not in infinite ones. We can notice through these three essays that Tarski bans every attempt to define “truth” in the context of ordinary language, and suggests the precept that the term “truth” must be placed within a well-determined linguistic context. Later, Tarski has maintained the precept, but has attenuated the restriction with respect to ordinary language.
The first essay indicates a possible way of treating ordinary language indirectly so as to allow the use of the notion of truth within it. The strategy would consist in extending to ordinary language the results achieved in formalized languages, by translating into it the definitions of true sentences obtained for some formalized languages.
In the second essay Tarski maintains the two aspects of the thesis originally established. Nevertheless, it is possible to note changes with respect to the definitory terms used to obtain a definition of truth in a formalized language. This method appeals to another semantic notion — that of ‘satisfaction’ — to define recursively the notion of truth. Several remarks can be made concerning this procedure, for example the difficulty represented by the situation of circularity generated by the definitions of “truth” and “satisfaction”.
In the third essay we can distinguish three important changes: i) The analysis of the reductionist thesis which tends to eliminate the term “truth” from its logical linguistic usage; ii) The broadening of the concept of “formalized language” considered as the exclusive domain of a non-contradictory use of “truth”; iii) The presentation of the partial definitory mechanism of “truth” combined with the theory of proof.
The reductionist thesis of the term “truth” is the extreme possibility contained in the methodological alternative: either saving the rules of logic, or giving up the employment of the term “truth” as a real predicate that qualifies the names of sentences. Tarski rejects this thesis and offers two reasons that reveal his increasing inclination towards the domain of ordinary language. The first reason is an example: if the nihilist thesis is accepted, it would be impossible to use the expression “all the theorems of the author X have turned out true”. The second reason is more general: “it could be said that the nihilist theory of truth eliminates the notion of truth from the conceptual human stock”. It is important to note that Tarski has in mind, once more, ordinary linguistic contexts and not formalized ones.
The second important change consists in modifying the opposition between ordinary language and formalized languages. Tarski considers here that a formalized language is not essentially opposed to ordinary language. Nevertheless, this modification does not admit the possibility of establishing a general definition of “truth” in ordinary language; Tarski only accepts the possibility of establishing a formal pattern based on a fragment of ordinary language that can function as a general definition of “truth”. This pattern would estipulate three conditions for a fragment of any ordinary language: a) It must have syntactical rules precise enough to allow us to distinguish a sentence from an expression that is not a sentence; b) It must have a finite number of sentences; c) The term “true” must not appear in it. These three conditions make such a fragment scarcely representative of ordinary language, especially the second restriction. From this fragment of language Tarski suggests the tabulation of its sentences, and for each one he constructs a partial definition of truth by applying to it the following pattern: “P is true if and only if p”. Then he proceeds to form the logical sum of all definitions thus obtained. In this way we arrive at a formula that may be considered as a general definition of truth. Now, changing his original position, Tarski attempts to extend the definition of truth to scientific languages which are formed in part by ordinary language and in part by a formalized language. From a methodological point of view, this definition presents a practical problem: the definition allows us to decide the truth-value of every sentence only when it is placed in the context of the disjunction or logical sum of the language of which it is a part. This means that there is no general criterion which allows us to decide the truth-value of an isolated sentence. This fact leads Tarski to the third change: the introduction of procedures to provide “partial criteria of truth”, characteristic of each science or group of sciences. Since we are dealing in this case with logic, we appeal to the proper procedure, which is the theory of proof. We know that there is no perfect coincidence between the class of true propositions and deducible propositions, since not every true proposition is deducible. From this, Tarski postulates the infinitist notion of “truth”, understood as the limit of a series: the notion of a true proposition functions, then, as the ideal limit that cannot be reached, but to the approximation of which we tend, by means of a gradual extension of the class of deducible propositions.
Three remarks can be made on the theory of truth taken as a limit. First, it is evident that Tarski still insists on the extensionalist method of treating the notion of truth. Secondly, Tarski is introducing an alteration to the axiomatic method in a not very precise way. When he speaks of “gradual extension”, he can refer to nothing but the possibility of increasing the number of axioms of a system, with the purpose of attaining a wider deducibility of sentences. The limit of this procedure is that either the axioms are weakened when it is saturated. Finally, when Tarski postulates the notion of a true proposition as an “ideal limit that cannot be reached”, he explicitly reveals the basic ambiguity underlying his treatment of the notion of “truth” all through his essays. He refers to the general notion of truth as a common property of all propositions that are true; but some times, “truth” is reduced to the qualification of a definitive proposition, since, when we were trying to construct the disjunctive pattern of the definition of truth, we started from the partial adequation to the pattern of each proposition considered as a true and fulfilling certain conditions. So, in his last work, Tarski uses ambiguously the term “truth”: at times it applies to the set of all true propositions and at others it is limited to each sentential object. Therefore, it is necessary to distinguish between the partial criterion of truth and the search for truth, understood as a vague expression that denots the scientific trend to reach a kind of objective and secure knowledge.