Validity, the Squeezing Argument and Alternative Semantic Systems: the Case of Aristotelian Syllogistic [Book Review]
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Journal of Philosophical Logic 41 (2):387-418 (2012)
We investigate the philosophical significance of the existence of different semantic systems with respect to which a given deductive system is sound and complete. Our case study will be Corcoran’s deductive system D for Aristotelian syllogistic and some of the different semantic systems for syllogistic that have been proposed in the literature. We shall prove that they are not equivalent, in spite of D being sound and complete with respect to each of them. Beyond the specific case of syllogistic, the goal is to offer a general discussion of the relations between informal notions—in this case, an informal notion of deductive validity—and logical apparatuses such as deductive systems and (model-theoretic or other) semantic systems that aim at offering technical, formal accounts of informal notions. Specifically, we will be interested in Kreisel’s famous ‘squeezing argument’; we shall ask ourselves what a plurality of semantic systems (understood as classes of mathematical structures) may entail for the cogency of specific applications of the squeezing argument. More generally, the analysis brings to the fore the need for criteria of adequacy for semantic systems based on mathematical structures. Without such criteria, the idea that the gap between informal and technical accounts of validity can be bridged is put under pressure
|Keywords||Syllogistic Semantic systems for syllogistic Proofs of completeness Validity Squeezing argument|
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References found in this work BETA
Hartry H. Field (2008). Saving Truth From Paradox. Oxford University Press.
Peter Smith (2013). An Introduction to Gödel's Theorems. Cambridge University Press.
Alfred Tarski (2002). On the Concept of Following Logically. History and Philosophy of Logic 23 (3):155-196.
Georg Kreisel (1967). Informal Rigour and Completeness Proofs. In Imre Lakatos (ed.), Problems in the Philosophy of Mathematics. North-Holland 138--157.
John Corcoran (2003). Aristotle's Prior Analytics and Boole's Laws of Thought. History and Philosophy of Logic. 24 (4):261-288.
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