Semantic Theory of Truth—Informal Aspects

Abstract

This chapter outlines an intuitive approach to STT, very closely related to Tarski’s original approach. The essential role of the Liar antinomy and its solution by introducing the language/metalanguage distinction is stressed as well as the role of interpreted languages is pointed out. Finally, heuristics of forming the semantic truth definition via the concept of satisfaction is reported.

Appendix: Yablo Sequences and Self-Reference

Stephen Yablo (Yablo 1993) produced a version of LP in which “self reference is neither necessary nor sufficient. Consider the following sequence of sentence (Yablo uses ‘untrue’, not ‘false’):

• (A₁) for all k > 1, A_k is false;

• (A₂) for all k > 2, A_k is false;

• (A₃) for all k > 3, A_k is false;

• ……………………………

Consider (I follow Cook 2014, pp. 11–12, but my reasoning is slightly different; see also Kripke (2019a) and Cook 2014, pp. 27–28 for a metamathematical derivation of the Yablo paradox ) the sentence A_m (it says that for all n > m, A_m is false). Assume that A_m is true. Hence, it is true what A_m says. Thus, n > m, A_m is false. In particular, the sentence A_m+1 is false. However, due to the assumption that A_m true it is impossible, because falsity of A_m+1 implies that there is sentence A_n (n > m + 1) which is true. Consequently, A_m+1 must be true. Since A_m and A_m+1 are both true, they are equivalent. Using the formula A_m ⇔ (A_m+1 is false), replacing equivalents gives A_m ⇔ (A_m is false). The last formula immediately leads to LP. Assume that A_m is false. Thus, all sentences A_k (1 < k < m) are also false. Suppose that A_i and A_i+1 are such sentences. Since they are both false, they are equivalent too. Since we have the formula A_i ⇔ (A_i+1 is false), replacing equivalents gives the equivalence A_i ⇔ (A_i is false).

The above argument explicitly shows that self-reference and T-scheme are involved into the Yablo Paradox . I do not find a sufficient argument for Yablo’s conclusion that LP can be formulated without referring to self-reference (a similar view was expressed by Kripke —a personal communication), if not explicitly, then—at least implicitly. His informal reasoning is incomplete; the proof in Cook 2014 although does not appeal to self-reference plays with A₁ is false and A₁ is true without noticing that the paradox arises as a result of asserting own falsity by A₁. In fact , the reduction of A_i+1 to A_i (for any i) implicitly involves self-reference . Thus, I conclude that the Yablo paradox does not invalidate the Leśniewski -Tarski diagnosis. Further remarks about this issue will be found in Chap. 8, Sect. 8.8. Let me note that I entirely omit the problem of circularity in semantic paradoxes (see Cook 2014 for an extensive discussion of this problem).