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dc.contributor.authorMilne, Peter-
dc.contributor.editorArazim, P-
dc.contributor.editorDancak, M-
dc.description.abstractIn Lewis Carroll’s Through the Looking Glass and What Alice Found There, Alice enters through a mirror into the realm reflected. It is, of course, left-right reversed but this is only the start of the fun and games when Alice explores the world on the other side of the mirror. Borrowing, if only in part, Carroll’s theme of inversion, my aim is to take a look at classical logic in something of an inverted way, or, to be more exact, in three somewhat inverted ways. Firstly, I come at proof of the completeness of classical logic in the Lindenbaum-Henkin style backwards: I take for granted the existence of a set Σ for which it holds, for some formula φ, that ψ !in Σ if, and only if, Σu{ψ} |- φ  then read off the rules of inference governing connectives and quantifiers that most directly yield the desired (classical) semantic properties. We thus obtain general elimination rules and what I have elsewhere called general introduction rules. Secondly, the same approach lets us read off a different set of rules: those of the cut-free sequent calculus S' of (Smullyan, 1968). Smullyan uses this calculus in proving the Craig-Lyndon interpolation theorem for first-order logic (without identity and function symbols). By attending very carefully to the steps in Smullyan’s proof, we obtain a strengthening: if φ |- ψ,  /|- ¬φ and /|- ψ then there is an interpolant χ, a formula employing only the non-logical vocabulary common to φ and ψ, such that φ entails χ in the first-order version of Kleene’s 3-valued logic and χ entails ψ in the first-order version of Graham Priest’s Logic of Paradox. The result, which is hidden from view in natural deduction formulations of classical logic, extends, I believe, to firstorder logic with identity. Thirdly, we look at a contraction-free “approximation” to classical propositional logic. Adding the general introduction rules for negation or the conditional leads to Contraction being a derived rule, apparently blurring the distinction between structural and operational rules.en_UK
dc.publisherCollege Publications-
dc.relationMilne P (2016) Classical Logic through the Looking-Glass In: Arazim P, Dancak M (ed.) Logica Yearbook 2015, London: College Publications. Logica 2015, 15.6.2015 - 19.6.2015, Hejnice, Czech Republic.-
dc.rightsAuthor retains copyright. Published in Logica Yearbook 2015 edited by Pavel Arazim and Michal Dancak:
dc.subjectCompleteness proof for classical first-order logicen_UK
dc.subjectLindenbaum-Henkin constructionen_UK
dc.subjectgeneral elimination rulesen_UK
dc.subjectgeneral introduction rulesen_UK
dc.subjectCraig interpolation lemmaen_UK
dc.subjectKleene’s strong three-valued logicen_UK
dc.subjectLogic of Paradoxen_UK
dc.subjectŁukasiewicz’s infinite-valued logicen_UK
dc.titleClassical Logic through the Looking-Glassen_UK
dc.typeConference Paperen_UK
dc.type.statusBook Chapter: author post-print (pre-copy editing)-
dc.citation.btitleLogica Yearbook 2015-
dc.citation.conferencelocationHejnice, Czech Republic-
dc.citation.conferencenameLogica 2015-
Appears in Collections:Law and Philosophy Conference Papers and Proceedings

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