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## Tarski's Definition of Logical ConsequenceDuring the years 1929–1933 Tarski put together a definition of the concept ' In 1935 Tarski was persuaded to attend the International Congress of Philosophers in Paris. Worrying about what he could say to impress the philosophers, he formed the idea of presenting the truth definition as a vehicle for giving formal definitions of various notions from logical metatheory, among them the notion of logical consequence. The result was a pair of papers, [17] presenting the definition of logical consequence, and [18] discussing the general idea of defining semantic notions [5, pp. 95ff]. The paper [17] on logical consequence answered a methodological question, not a question of conceptual analysis. You can't do conceptual analysis until you have a concept to analyse. But when Tarski wrote, there was no agreed concept of logical consequence to be analysed. (One should look first at what was available in the literature of his time. For example Hilbert and Ackermann [8, p. 1] have a proof-theoretic notion of To see what the methodological question was, we need to put the paper in context. Gödel had recently shown that there is no maximal proof calculus for pure logic of second or higher order. Ramsey [13] had discussed languages with infinite conjunctions, and both Bernays [1, pp. 86ff.] and Tarski himself [19, p. 288] had considered proof rules with infinitely many premises. So some very general questions about proof calculi were in the air, and some robust and well-motivated definitions were needed for handling them. Tarski seems to have clarified the central question in his own mind along the following lines: What are the weakest constraints that we can put on a rule for deriving propositions from sets of propositions in a formal language, which make it reasonable to count any rule satisfying these constraints as an inference rule? He proposed to label these constraints as saying that the conclusion of the rule is a 'logical consequence' of its premises. Now for Tarski in 1935 there were two kinds of formal language. In the first kind, which we can call 'pure' languages, all symbols are logical. In the second kind, which we can call 'applied', there are also nonlogical symbols, but these symbols are all required to be fully meaningful. For pure languages, Tarski adopted just the constraint that whenever the premises are true the conclusion must be true too. This constraint looks trivial, but in Paris in 1935 it served the purpose of advertising his recent formal definition of 'true'. For applied languages Tarski had to decide what to do about It's noticeable that Tarski's own text says almost nothing about relations between the meanings of the nonlogical constants (there is a brief parenthetical remark in the middle of P. 415 in [19]), but has at least a page on the importance of the difference between (i) changing a symbol to one with a different meaning and (ii) replacing the symbol by a variable that arbitrary objects can be assigned to. That tells me that Tarski in 1935 was really more interested in fine-tuning the notion of satisfaction than in accommodating the philosophers in Paris. The paper does use the word 'model', though not in the modern sense. The name 'model-theoretic definition of logical consequence' is not Tarski's, and I think it came into use only after the later developments that we turn to next. |

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