Article

Logical Consequence, Proof Theory, and Model Theory

Stewart Shapiro

in The Oxford Handbook of Philosophy of Mathematics and Logic

Published in print June 2007 | ISBN: 9780195325928
Published online September 2009 | | DOI: http://dx.doi.org/10.1093/oxfordhb/9780195325928.003.0021

Series: Oxford Handbooks

Logical Consequence, Proof Theory, and Model Theory

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This article's main concern is the notion of model-theoretic consequence. What does it have to do with correct reasoning? The article takes on deductive consequence only by way of contrast. Do these two notions answer to different intuitive notions of consequence? Is one of them primary, and the other secondary? Or perhaps they are autonomous and independent. Maybe there are two distinct notions of correct reasoning, valid thought, and/or inference. For what it is worth, treatments of mathematical logic usually presuppose that the model-theoretic notion is the primary one. For example, one says that a deductive system is sound or complete (or not) for the semantics—not the other way around. If a deductive system is not sound for a given semantics, then that alone disqualifies the deductive system. It is because the deductive system allows us to deduce a falsehood from truths in some interpretation of the language.

Keywords: logical consequence; proof theory; model theory; mathematical reasoning; intuitive notions; model-theoretic notion

Article.  9186 words. 

Subjects: Philosophy ; Philosophy of Mathematics and Logic

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