What Do We Mean by Logical Consequence?

In the beginning of the 20th century, many prominent logicians and mathematicians, such as Frege, Russell, Hilbert, and many others, felt that mathematics needed a very rigorous foundation in logic. The standard approach in the early part of the 20th century was to use a syntactic or proof-theoretic definition of logical consequence. This says that “for one sentence to be a logical consequence of [a set of premises] is simply for that sentence to be derivable from [them] by means of some standard system of deduction”[3, pg. 65]. However, many famous results of the time, especially Gödel’s incompleteness theorems led to logicians such as Tarski to define logical consequence with what was eventually developed into the standard “model-theoretic” definition. This way of defining logical consequence says that a argument of a certain form is a logically valid argument if it is impossible for the premises to be true and the conclusion false[9, 1, pg. 5, pg. 12]. Many philosophers have written about the effectiveness of this definition, but in 1990 John Etchemendy offered a fundamental criticism of Tarski’s definition both as to whether it is conceptually correct, and whether it captures the right set of arguments, or interpretations. This paper explores Etchemendy’s argument and various responses from prominent philosophers.