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Philosophy and Model Theory$
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Tim Button and Sean Walsh

Print publication date: 2018

Print ISBN-13: 9780198790396

Published to Oxford Scholarship Online: May 2018

DOI: 10.1093/oso/9780198790396.001.0001

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Sameness of structure and theory

Sameness of structure and theory

(p.107) 5 Sameness of structure and theory
Philosophy and Model Theory

Tim Button

Sean Walsh

Oxford University Press

This chapter is motivated by the following methodological question: under what circumstances should we call two structures the same? We start by surveying several formal notions of sameness of structure: definitional equivalence, interpretability, mutual interpretability, and biinterpretability. We then discuss the notions of sameness of theory which arise by considering the class of models of a theory, and some new notions of sameness of theory (such as faithful interpretability). We close the chapter by examining three ambitious claims concerning sameness of theory: that it preserves truth-value, that it preserves arithmetical consequence, and that it delivers two proofs for the price of one. We argue that the more philosophically ambitious versions of these preservation-theses fail, in general.

Keywords:   Definitional equivalence, interpretability (mutual, faithful), biinterpretability, Orey sentence, bridge principle, duality

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