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Replacement in Logic

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Abstract

We study a range of issues connected with the idea of replacing one formula by another in a fixed (linguistic) context. The replacement core of a consequence relation ⊢ is the relation holding between a set of formulas {A 1, ..., A m , ...} and a formula B when for every context C(·), we have C(A 1), ..., C(A m ), ... ⊢ C(B). Section 1 looks at some differences between which inferences are lost on passing to the replacement cores of the classical and intuitionistic consequence relations. For example, we find that while the inference from A and B to \(A \land B\), sanctioned by both these initial consequence relations, is retained on passage to the replacement core in the classical case, it is lost in the intuitionistic case. Further discussion of these two (and some other) logics occupies Sections 3 and 4. Section 2 looks at the m = 1 case, describing A as replaceable by B according to ⊢ when B is a consequence of A by the replacement core of ⊢, and inquiring as to which choices of ⊢ render this induced replaceability relation symmetric. Section 5 investigates further conceptual refinements— such as a contrast between horizontal and vertical replaceability—suggested by some work of R. B. Angell and R. Harrop (and a comment on the latter by T. J. Smiley) in the 1950s and 1960s. Appendix 1 examines a related aspect of term-for-term replacement in connection with identity in predicate logic. Appendix 2 is a repository for proofs which would otherwise clutter up Section 3.

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  1. Department of Philosophy, Monash University, Clayton, VIC, 3800, Australia

    Lloyd Humberstone

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Humberstone, L. Replacement in Logic. J Philos Logic 42, 49–89 (2013). https://doi.org/10.1007/s10992-011-9212-4

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