How to Define the Concept of Logical Consequence

Hashimoto Kouji

doi:10.14989/jps_560_70

This paper consists of five sections. In the first two sections, Alfred Tarski's definition of logical consequence, which was advanced in his article "On the Concept of Logical Consequence", is examined. There, I show that his definition is originally intended to satisfy the following two conditions. The first one, which I call the condition of scientism, demands that the definition should contain only the concepts whose meanings are established scientifically and exclude metaphysical or other undefined concepts. The second one, the condition of material adequateness, requires the definition to conform to the ordinary usage of the definiendum. Then I argue that although Tarski's definition satisfies the condition of scientism, whether it also satisfies the condition of material adequateness is not clear. In the next two sections, I take John Etchemendy's arguments against Tarski's definition, which appear in his book, The Concept of Logical Consequence. In this book, Etchemendy contends that Tarski's analysis is wrong, that his definition does not capture the essential feature of the ordinary concept of logical consequence. Namely, it is claimed that Tarski's definition does not satisfy the condition of material adequateness. Analyzing his arguments, I explain the central difficulty Tarski's definition meets. It seems impossible for his definition to satisfy both the condition of scientism and the condition of material adequateness at once. Any definition of logical consequence must satisfy the condition of material adequateness as long as it aims at right one. In the last section, I outline the three possible ways to attain such a definition. The first way introduces the metaphysical theory about the essential feature of logical consequence in ordinary usage. The second way adopts a kind of syntactical definition. The third way is basically the same as Tarski's definition, but it needs to modify our understanding of the essential feature of logical consequence. The first and second ways can satisfy the condition of material adequateness because they give up satisfying the condition of scientism. On the other hand, the third way seems to be able to satisfy both of the conditions. Therefore, I think that it is worth while pursuing the third way further.

How to Define the Concept of Logical Consequence

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