Logical Nihilism and the Logic of ‘prem’
Keywordsnihilism, non-reflexive logic, non-transitive logic, dual valuations, modus ponens
AbstractAs the final component of a chain of reasoning intended to take us all the way to logical nihilism, Russell (2018) presents the atomic sentence ‘prem’ which is supposed to be true when featuring as premise in an argument and false when featuring as conclusion in an argument. Such a sentence requires a non-reflexive logic and an endnote by Russell (2018) could easily leave the reader with the impression that going non-reflexive suffices for logical nihilism. This paper shows how one can obtain non-reflexive logics in which ‘prem’ behaves as stipulated by Russell (2018) but which nonetheless has valid inferences supporting uniform substitution of any formula for propositional variables such as modus tollens and modus ponens.
Arnon Avron, 1991, “Natural 3-valued logics – characterization and proof theory”, Journal of Symbolic Logic 56 (1): 276–294. DOI: http://dx.doi.org/10.2307/2274919
N.D. Belnap, 1977, “A useful four-valued logic”, in J.M. Dunn and G. Epstein (eds.), Modern Uses of Multiple-Valued Logic, D. Reidel.
Tim Button, 2016, “Knot and tonk: Nasty connectives on many-valued truth-tables for classical sentential logic”, Analysis 76 (1): 7–19. DOI: http://dx.doi.org/10.1093/analys/anv106
Alonzo Church, 1944, “Review of formalization of logic by Rudolf Carnap”, The Philosophical Review 53 (5): 493–498. DOI: http://dx.doi.org/10.2307/2181359
R.T. Cook, 2012, “The T-schema is not a logical truth”, Analysis 72 (2): 231–239. DOI: http://dx.doi.org/10.1093/analys/ans031
Bogdan Dicher, 2020, “Requiem for logical nihilism, or: logical nihilism annihilated”, Synthese. DOI: http://dx.doi.org/10.1007/s11229-019-02510-4
Paul Égré, Lorenzo Rossi and Jan Sprenger, 2020, “De Tinettian logics of indicative conditionals, part I: Trivalent semantics and validity”, Journal of Philosophical Logic. DOI: http://dx.doi.org/10.1007/s10992-020-09549-6
Andreas Fjellstad, 2015, “How a semantics for tonk should be”, Review of Symbolic Logic 8 (3): 488–505. DOI: http://dx.doi.org/10.1017/s1755020314000513
Andreas Fjellstad, 2017, “Non-classical elegance for sequent calculus enthusiasts”, Studia Logica 105(1): 93–119.DOI: http://dx.doi.org/10.1007/s11225-016-9683-y
Andreas Fjellstad, 2020, “Structural proof theory for first-order weak Kleene logics”, Journal of Applied Non-Classical Logics 30 (3): 272–289. DOI: http://dx.doi.org/10.1080/11663081.2020.1782593
Rohan French, 2016, “Structural reflexivity and the paradoxes of self-reference”, Ergo: An Open Access Journal of Philosophy 3. DOI: http://dx.doi.org/10.3998/ergo.12405314.0003.005
Paul C. Gilmore, 1986, “Natural deduction based set theories: A new resolution of the old paradoxes”, Journal of Symbolic Logic 51 (2): 393–411. DOI: http://dx.doi.org/10.1017/S0022481200031261
Patrick Greenough, 2001, “Free assumptions and the liar paradox”, American Philosophical Quarterly 38 (2): 115–135. DOI: http://dx.doi.org/10.2307/20010029
Jan Łukasiewicz, 1970, Selected Works, L. Borkowski (ed.), North-Holland Publishing Company.
Julien Murzi and Lorenzo Rossi, 2017, “Naïve validity”, Synthese. DOI: http://dx.doi.org/10.1007/s11229-017-1541-6
Sara Negri and Jan von Plato, 2001, Structural Proof Theory, Cambridge University Press.
Carlo Nicolai and Lorenzo Rossi, 2018, “Principles for object-linguistic consequence: From logical to irreflexive”, Journal of Philosophical Logic 47: 549–577. DOI: http://dx.doi.org/10.1007/s10992-017-9438-x
Hitoshi Omori and Heinrich Wansing, 2017, “40 years of FDE: An introductory overview”, Studia Logica 105 (6): 1021–1049. DOI: http://dx.doi.org/10.1007/s11225-017-9748-6
GillianRussell, 2018, “Logical nihilism: Could there be no logic?”, Philosophical Issues 28 (1): 308–324. DOI: http://dx.doi.org/10.1111/phis.12127
Peter Schroeder-Heister, 2016, “Restricting initial sequents: The trade-offs between identity, contraction and cut”, pages 339–351 in R. Kahle, T. Strahm and T. Studer (eds.), Advances in Proof Theory, Springer International Publishing.
Geoff Sutcliffe, Francis Jeffry Pelletier and Allen Hazen, 2018, “Making Belnap’s ‘useful 4-valued logic’ useful”, in The Thirty-First International Flairs Conference.
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