Semantic Definition of Truthand Empirical Theories
Tarski himself explicitly emphasized STT applies to empirical theories (see Tarski 1944). In particular, he proposed to explain conditions of acceptability of theories by constraining that they do not imply false sentences.
On the other hand, Tarski did not discuss the problems pertained to realism, because he considered them as too metaphysical.[97] In what follows I will comment various objections advanced in order to demonstrate that STT has no application to scientific theories and, hence, it has nothing to do with scientific realism. Consequently, it cannot justify this view or its opposition, that is, anti-realism. The main points can be classified in the following way:(1) STT blurs the difference between logical and empirical truths;
(2) STT can be applied to formal languages, but empirical are not such;
(3) STT can be applied to theories understood as set of sentences, but so-called non-statement view is more proper;
(4) Even if we agree that a kind of truth in the semantic sense can be attributed to empirical theories, we should apply the concept of partial (approximate) truth;
(5) The world as the subject matter of empirical theories cannot be identified with semantic model.
I will argue that these objections can be met from the point of view of SDT, eventually enriched by further constraints.
Ad (1) This objection is to be find in O'Connor (1975, p. 109) and (Haack 1978, p. 113). Both authors maintain that if truth of a sentence consists in its satisfaction by all sequences, this conditions holds for logical as well as empirical sentences. However, this argument rests on entirely mistaken (see above) views on sequences as such. Once again, sequences of objects are not facts, states of affairs, etc. We can define logical truths as true in every model (or under any interpretation) and empirical truths as holding in some models only.
If someone, as, for example, Tarski himself did, maintains that the borderline between logical and empirical truths is vague or fuzzy, still employs the definition of logical truth as valis (true in all models). If so, the division of truths into logical and anti-logical has nothing to do with (SDT) as such.Ad (2) Tarski never said that (SDT) applies to formal languages. It is clear, if we consult the following proclamation (Tarski 1933, pp. 166-167; page reference to Tarski 1956):
It remains perhaps to add that we are not interested here in ‘formal’ languages in sciences in one special sense of the word ‘formal’, namely sciences to the signs and expressions of which no material sense is attached. For such sciences the problem here discussed [the problem of truth] has no relevance, it is not even meaningful. We shall always ascribe quite concrete and, for us, intelligible meanings to the signs which occur in the language we shall consider. The expressions which we call sentences still remain sentences after the signs which occur in have been translated into colloquial language.
Thus, a language L for which (SDT) applies is always interpreted, even if it is formalized. Consequently, interpretation of L always precedes definitions of semantic concepts including the notion of truth. Thus, we arrived at the problem of how formal is related to formalized. The answer is that formal language do not need to be equipped with meaning, contrary to formalized languages.
A common misunderstanding of Tarski's views consists in attributing to him the opinion that STT applies to formal (formalized) languages only (this objection goes back to Black 1948). This mistake neglects that Tarski explicitly explained that truth-bearers are correct syntactic units of the propositional category and having meaning. Yet it does not mean that Tarski's view about language and meaning have no weak points. In particular, he did not define the concept of meaning. In fact, he intentionally avoided this question and deliberately preferred to speak about interpreted languages as semantic items.[98] Yet such languages (see above) can be formalized or not.
Now the question arise whether formal semantics can be applied to non-formalized languages or which amount of formalization suffices for the exact semantic analysis. A particularly interesting questions concern natural language as a subject of logical semantic constructions. Tarski pointed out (see Tarski 1933, Sect. 1) that natural languages are closed in the sense that they mix L and ML, and thereby generate semantic paradoxes. This feature of ordinary parlance motivated Tarski's skepticism concerning applicability formal semantic constructions to natural language. In Tarski (1944), this skepticism is weaker, because Tarski explicitly says that considerable parts of natural language can be analyzed by tools of logical semantics. He even introduced the category of languages having a specified structure, that is, not formalized but syntactically well-defined. Although Tarski illustrated (SDT) by an example from formalized theory of classes and exposed mathematically his definition of truth, there is nothing in its construction which would have to preclude its application to the language of physics. In particular, there is nothing in the definition of satisfaction which restricts this concept to formalized formulas. If we way that Warsaw satisfies the predicate ‘is a city in Poland”, it would be difficult to find a difference with saying that the number iiii(0) satisfies the formula ‘4 e N”, where N is a set o natural numbers and is an odd number' and iiii(0) has the meaning ‘the successor of the successor of the successor of the successor of 0').To conclude this point, SDT is fully consistent with the view that the language of empirical theories is semantically interpreted (see also Popper 1963, pp. 398399). It can be formalized or not, but cannot be purely formal.
Ad (3) Some authors (see Morrison and Morgan 1999, p. 3) contrast the syntactic and semantic understanding of empirical theories. Roughly speaking, the former view sees theories as ordered, for instance, axiomatized, sets of sentences, whereas the latter approach identifies theories with classes of models (see Suppes 2002 for the most comprehensive elaboration of this position).[99] I consider this distinction as very misleading.
The simplest argument for that qualification stems from (GM). If being consistent (a syntactic property) has its exact counterpart in having a model, both ways of speaking, syntactic and semantic, are equivalent.[100] Hence, every consistent theory has a model, but also every model can be described by a set of sentences belonging to L. The characterization (see above) of T as CnX, although syntactic, due to the status of Cn, concerns interpreted languages, that is, involves semantics by definition. In fact, Patrick Suppes, the main proponent of the semantic view, had nothing against (SDT), although focuses not on it but on model-theoretic constructions as usable in analysis of empirical theories. Taking into account the history of contemporary methodology of science, it is easily to show that the purely syntactic point of view on theories can be attributed to the early Vienna Circle, but not to philosophers a la Tarski. If one agrees that scientists use interpreted languages, (SDT) and defining theories by Cn are perfectly compatible (see also Przel^cki 1969, 1977; Ruttkamp 2002).[101]Ad (4) Many authors maintain that scientific truth is approximate or partial (see Wojcicki 1979; Wojcicki 1995/96; Psillos 1999; Costa and French 2003). This account does not need to be at odds with STT. In fact, all mentioned authors propose some modifications of this theory in order to capture ‘the nearly true' (the phrase of Stathis Psillos) as a semantic property. Personally, I am very skeptical about conceptions of degrees of truth. On Tarskian semantics, sentences are either true or false under a given interpretation V ((SDT) implies the principle of bivalence; see Wolenski 2003 for details) and there are no other or partial logical values. Of course, one can propose many-valued logic, fuzzy logic, probabilistic logic or account via the concept of verisimilitude, but they require a fairly fundamental revision of logic and metalogic. I prefer a more opportunistic approach consisting in keeping classical logic as defining formal properties of logical values with simultaneous admitting that we have various degrees of justification.
To conclude, the is no need to revise (SDT) in its traditional model-theoretic setting.Ad (5) At first, the following account of SR via STT could be attractive. Take an empirical theory T. By the assumption of consistency and (GM) it has model, say M(T). Call this model the fragment of the real world being the subject-matter of T. By (UT), M(T) is not definable within T, provided that T contains PA; this assumption is legitimate for most empirical theories. Using a philosophical jargon, we can say that M(T) transcends T. And it is the fundamental thesis of any realism, including SR (see Wolenski 2004). An additional advantage of this picture concerns theoretical and observational parts of theories, independently of criteria of their particular delimitation. Since, as the history of science points out, observational data are consistent with many alternative theories, information coming from observational procedures, can be captured by several different theories and their models. Contemporary cosmology provides a good example of this situation.
Unfortunately, the above picture is too simplified (see Grobler 2001). First of all, the real world is not an algebraic structure. This statement is true even on Platonic view on the reality that abstract objects belong to its ontological equipment. Independently of the chosen ontological view on the reality, (SDT) does not qualify per se any model as proper (intended, standard, etc.). Using the traditional (in fact, going back to the Schoolmen) way of speaking, we might distinguish formal and material objects of knowledge or cognition. According to this distinction, every consistent cognitive result refers to a formal objects.[102] It concerns, for example, completely fictional stories. This implies that some cognitive results have no material objects, that is, considered as the parts of the real world. Consequently, a formal object of cognition becomes a material object, if some additional conditions are fulfilled. Employing this idea, (GM) says that every consistent set of sentences has a model as a formal object of semantic interpretation, but this statement does not solve the problem of how identify the material object how it is related to formal one.
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