Abstract
This paper is dedicated to developing a formalism that takes rejection seriously. Bilateral notation of signed formulas with force indicators is adopted to define signed consequences which can be viewed as the bilateral counterpart of Tarskian consequence relations. Its relation to some other bilateral approaches is discussed. It is shown how David Nelson’s logic N4 can be characterized bilaterally and the corresponding completeness result is proved. Further, bilateral variants of three familiar notions are considered and investigated: that of a fragment, of definitional equivalence, and of a conservative extension.
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Notes
Throughout the paper I will sometimes use “bilaterally” to loosely mean “while taking rejection seriously” and “unilaterally” for the converse.
Observe that this way we have built compactness into our definition, so our Scott (Tarskian) consequence relation is actually a compact Scott (Tarskian) consequence relation.
Here one can read \(v(\bar{\varphi })=t\) as “\(\bar{\varphi }\) holds according to v” and \(v(\bar{\varphi })=f\) as “\(\bar{\varphi }\) does not holds according to v”.
Let me remind the reader that the language of \(\mathsf {PN4}\) is denoted \(\mathcal {L}^p\).
Which is also sometimes called the positive logic.
That is, in terms of the assertion components.
In fact, strong implication is an important connective for logics with toggle negation, for instance, it is used to define weak replacement rule, under which systems like \(\mathsf {N4}\) and the bilattice logic are closed.
References
Almukdad, A., & Nelson, D. (1984). Constructible falsity and inexact predicates. The Journal of Symbolic Logic, 49(01), 231–233. https://doi.org/10.2307/2274105.
Arieli, O., & Avron, A. (1996). Reasoning with logical bilattices. Journal of Logic, Language and Information, 5(1), 25–63. https://doi.org/10.1007/bf00215626.
Arieli, O., & Avron, A. (1998). The value of the four values. Artificial Intelligence, 102(1), 97–141. https://doi.org/10.1016/s0004-3702(98)00032-0.
Belnap, N. D. (1977). How a computer should think. In G. Ryle (Ed.), Contemporary Aspects of Philosophy (pp. 30–56). Boston: Oriel Press.
Belnap, N. D. (1977). A useful four-valued logic. In Modern uses of multiple-valued logic (pp. 5–37). Dordrecht: Springer. https://doi.org/10.1007/978-94-010-1161-7_2
Bendall, K. (1978). Natural deduction, separation, and the meaning of logical operators. Journal of Philosophical Logic, 7(1), 245–276. https://doi.org/10.1007/bf00245930.
Bendall, K. (1979). Negation as a sign of negative judgment. Notre Dame Journal of Formal Logic, 20(1), 68–76. https://doi.org/10.1305/ndjfl/1093882402.
Blasio, C., Marcos, J., & Wansing, H. (2017). An inferentially many-valued two-dimensional notion of entailment. Bulletin of the Section of Logic, 46(3/4), 233–262. https://doi.org/10.18778/0138-0680.46.3.4.05.
Bochman, A. (1998). Biconsequence relations: A four-valued formalism of reasoning with inconsistency and incompleteness. Notre Dame Journal of Formal Logic, 39(1), 47–73.
Chagrov, A., & Zakharyaschev, M. (1997). Modal logic (Vol. 35). Oxford: Clarendon Press, Oxford logic guides.
Drobyshevich, S. A bilateral Hilbert-style investigation of 2-intuitionistic logic. Journal of Logic and Computation (to appear)
Dummett, M. (1996). The seas of language. Oxford: Oxford University Press. https://doi.org/10.1093/0198236212.001.0001.
Frege, G. (1960). Translations from the philosophical writings of Gottlob Frege. In P. Geach & M. Black (Eds.), Negation. Oxford: Oxford University Press.
Gabbay, D. M., & Wansing, H. (Eds.). (1999). What is negation?. Dordrecht: Springer. https://doi.org/10.1007/978-94-015-9309-0.
Gibbard, P. (2002). Price and Rumfitt on rejective negation and classical logic. Mind, 111(442), 297–304. https://doi.org/10.1093/mind/111.442.297.
Gyuris, V. (1999). Variations of algebraizability. Ph.D. thesis, The University of Illinois at Chicago
Humberstone, L. (2000). The revival of rejective negation. Journal of Philosophical Logic, 29(4), 331–381. https://doi.org/10.1023/a:1004747920321.
Kapsner, A. (2014). Logics and falsifications. In Trends in logic (Vol. 40). Berlin: Springer. https://doi.org/10.1007/978-3-319-05206-9.
Nelson, D. (1949). Constructible falsity. The Journal of Symbolic Logic, 14(01), 16–26. https://doi.org/10.2307/2268973.
Odintsov, S. P., & Wansing, H. (2017). Disentangling FDE-based paraconsistent modal logics. Studia Logica, 105(6), 1221–1254. https://doi.org/10.1007/s11225-017-9753-9.
Price, H. (1990). Why ‘not’? Mind, 99(394), 221–238.
Price, H. (1994). Semantic minimalism and the Frege point. In S. Tsohatzidis (Ed.), Foundations of Speech Act Theory (pp. 132–155). London: Routledge.
Rumfitt, I. (2000). “Yes” and “no”. Mind, 109(436), 781–823. https://doi.org/10.1093/mind/109.436.781.
Schroeder-Heister, P. (2018). Proof-theoretic semantics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy, spring 2018. Stanford: Metaphysics Research Lab, Stanford University.
Scott, D. (1974). Rules and derived rules. In Logical theory and semantic analysis (pp. 147–161). Netherlands: Springer. https://doi.org/10.1007/978-94-010-2191-3_13.
Smiley, T. (1996). Rejection. Analysis, 56(1), 1–9. https://doi.org/10.1093/analys/56.1.1.
Wansing, H. (2010). Proofs, disproofs and thier duals. In L. Beklemishev, V. Goranko, & V. Shekhtman (Eds.), Advances in modal logic (pp. 483–505). Singapore: World Scientific.
Wansing, H. (2013). Falsification, natural deduction and bi-intuitionistic logic. Journal of Logic and Computation, 26(1), 425–450. https://doi.org/10.1093/logcom/ext035.
Wansing, H. (2016a). On split negation, strong negation, information, falsification, and verification. In K. Bimbó (Ed.), J. Michael Dunn on information based logics. Outstanding contributions to logic (Vol. 8, pp. 161–189). Berlin: Springer.
Wansing, H. (2016b). Natural deduction for bi-connexive logic and a two-sorted typed \(\lambda \)-calculus. IfCoLog Journal of Logics and their Applications, 3(3), 413–439.
Wansing, H. (2017). A more general general proof theory. Journal of Applied Logic, 25, 23–46. https://doi.org/10.1016/j.jal.2017.01.002.
Wansing, H. (2017). Reprint of: A more general general proof theory. Journal of Applied Logic, 25, 23–46. https://doi.org/10.1016/j.jal.2017.12.003.
Acknowledgements
This work was supported by the Alexander von Humboldt Foundation. I would like to thank two anonymous referees for their useful comments.
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Drobyshevich, S. Tarskian consequence relations bilaterally: some familiar notions. Synthese 198 (Suppl 22), 5213–5240 (2021). https://doi.org/10.1007/s11229-019-02267-w
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DOI: https://doi.org/10.1007/s11229-019-02267-w