Summer Research Project: What is Logical Consequence?

Computer Science and Mathematics double major Jesse Jenks received a Summer Research Award in the Humanities and Social Sciences (information on the Summer Research Grants in Arts, Humanities, and Social Sciences is available here). He describes his experience working on his summer research project under the supervision of Prof. Cannon in the Philosophy Department:

BY JESSE JENKS:

In the beginning of the 20th century, many prominent logicians and mathematicians, such as Frege, Russell, Hilbert, and 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 which 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” (Etchemendy 1988).

These two ways of understanding logical consequence have a long history dating back to Aristotle and Euclid. Proof-theory in particular is foundational to almost all of mathematics. For philosophers in the 19th century, the idea that was taken for granted was that a statement is logically true if and only if it can be proven. But in 1929, Gödel’s famous incompleteness theorems revealed that not all logically true statements are provable. This is now considered one of the most important results in logic and led 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 an argument of a certain form is a logically valid argument if it is impossible for the premises to be true and the conclusion false (Cannon 2016). 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. b65pddmk-kgrhqvlueyjc2kv3bmylokpdd-1_35

The modern version of model theory is derived from Tarski’s original definition, but is based in set theory. Although this is the most commonly taught version of model theory, this is problematic for foundationalists who believe that logic is the basis for mathematics. But Tarski did not originally require set-theoretic definitions. Instead, he used what Etchemendy calls “interpretational” semantics. Etchemendy’s criticism of Tarski essentially centers around the question of what “possible” means. For example, we could interpret this to mean an argument is logically valid if it is metaphysically impossible for the conclusion to be false if the premises are true. This is called “representational” semantics. The more standard approach is to say an argument is logically valid if a) we can define the “form” of an argument, and b) every argument of the same form consists of a materially (or empirically) true conclusion or a materially false premise. This is the most appealing version of model theory since it avoids both problems from metaphysics and concerns from foundationalists. However, Etchemendy points out that under an interpretational view, in almost any standard logical system we can construct sentences which are logically true, like “there are at least three objects in the universe”, which is a metaphysical claim about the size of the universe, and is not a matter of logic. This problem runs much deeper and could potentially undermine Tarski’s work. My summer research focused on what Etchemendy’s argument was and how other philosophers have responded to his claim.

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Photo courtesy of S. Harris, http://www.sciencecartoonsplus.com

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