Fractional-Valued Modal Logic and Soft Bilateralism

Mario Piazza; Gabriele Pulcini; Matteo Tesi

doi:10.18778/0138-0680.2023.17

Authors

Mario Piazza Scuola Normale Superiore, Classe di Lettere e Filosofia Author
Gabriele Pulcini University of Rome “Tor Vergata”, Department of Literary, Philosophical and Art History Studies Author
Matteo Tesi Scuola Normale Superiore, Classe di Lettere e Filosofia Author

DOI:

https://doi.org/10.18778/0138-0680.2023.17

Keywords:

modal logic, general proof theory (including proof-theoretic semantics), many-valued logics

Abstract

In a recent paper, under the auspices of an unorthodox variety of bilateralism, we introduced a new kind of proof-theoretic semantics for the base modal logic \(\mathbf{K}\), whose values lie in the closed interval \([0,1]\) of rational numbers [14]. In this paper, after clarifying our conception of bilateralism – dubbed “soft bilateralism” – we generalize the fractional method to encompass extensions and weakenings of \(\mathbf{K}\). Specifically, we introduce well-behaved hypersequent calculi for the deontic logic \(\mathbf{D}\) and the non-normal modal logics \(\mathbf{E}\) and \(\mathbf{M}\) and thoroughly investigate their structural properties.

References

A. Avron, A constructive analysis of RM, The Journal of Symbolic Logic, vol. 52(4) (1987), pp. 939–951, DOI: https://doi.org/10.2307/2273828 DOI: https://doi.org/10.2307/2273828

A. Avron, Hypersequents, logical consequence and intermediate logics for concurrency, Annals of Mathematics and Artificial Intelligence, vol. 4(3–4) (1991), pp. 225–248, DOI: https://doi.org/10.1007/BF01531058 DOI: https://doi.org/10.1007/BF01531058

A. Avron, The method of hypersequents in the proof theory of propositional non-classical logics, [in:] Logic: From foundations to applications, Clarendon Press (1996), pp. 1–32.

N. Francez, Bilateralism in proof-theoretic semantics, Journal of Philosophical Logic, vol. 43(2–3) (2014), pp. 239–259, DOI: https://doi.org/10.1007/s10992-012-9261-3 DOI: https://doi.org/10.1007/s10992-012-9261-3

N. Francez, Proof-theoretic Semantics, College Publications (2015).

V. Goranko, G. Pulcini, T. Skura, Refutation systems: An overview and some applications to philosophical logics, [in:] F. Liu, H. Ono, J. Yu (eds.), Knowledge, Proof and Dynamics, Springer (2020), pp. 173–197, DOI: https://doi.org/10.1007/978-981-15-2221-5_9 DOI: https://doi.org/10.1007/978-981-15-2221-5_9

N. Kürbis, Proof-theoretic semantics, a problem with negation and prospects for modality, Journal of Philosophical Logic, vol. 44(6) (2015), pp. 713–727, DOI: https://doi.org/10.1007/s10992-013-9310-6 DOI: https://doi.org/10.1007/s10992-013-9310-6

N. Kürbis, Some comments on Ian Rumfitt’s bilateralism, Journal of Philosophical Logic, vol. 45(6) (2016), pp. 623–644, DOI: https://doi.org/10.1007/s10992-016-9395-9 DOI: https://doi.org/10.1007/s10992-016-9395-9

N. Kürbis, Bilateralist detours: From intuitionist to classical logic and back, [in:] Logique et Analyse, vol. 239 (2017), pp. 301–316, DOI: https://doi.org/10.2143/LEA.239.0.32371556

G. Mints, Lewis’ systems and system T (1965–1973), [in:] Selected papers in proof theory, Bibliopolis (1992), pp. 221–294.

G. Mints, A Short Introduction to Modal Logic, Center for the Study of Language (CSLI) (1992).

M. Piazza, G. Pulcini, Fractional semantics for classical logic, The Review of Symbolic Logic, vol. 13(4) (2020), pp. 810–828, DOI: https://doi.org/10.1017/S1755020319000431 DOI: https://doi.org/10.1017/S1755020319000431

M. Piazza, G. Pulcini, M. Tesi, Linear logic in a refutational setting, unpublished manuscript.

M. Piazza, G. Pulcini, M. Tesi, Fractional-valued modal logic, The Review of Symbolic Logic, (2021), p. 1–20, DOI: https://doi.org/10.1017/S1755020321000411 DOI: https://doi.org/10.1017/S1755020321000411

T. Piecha, P. Schroeder-Heister, Advances in Proof-Theoretic Semantics, Springer (2016). DOI: https://doi.org/10.1007/978-3-319-22686-6

G. Pottinger, Uniform, cut-free formulations of T, S4 and S5, Journal of Symbolic Logic, vol. 48(3) (1983), p. 900, DOI: https://doi.org/10.2307/2273495 DOI: https://doi.org/10.2307/2273495

G. Pulcini, A. Varzi, Classical logic through rejection and refutation, [in:] M. Fitting (ed.), Landscapes in logic (Vol. 2), College Publications (1992).

G. Pulcini, A. C. Varzi, Complementary Proof Nets for Classical Logic (2023), DOI: https://doi.org/10.1007/s11787-023-00337-9 DOI: https://doi.org/10.1007/s11787-023-00337-9

I. Rumfitt, ‘Yes’ and ‘No’, Mind, vol. 109(436) (2000), pp. 781–823, DOI: https://doi.org/10.1093/mind/109.436.781 DOI: https://doi.org/10.1093/mind/109.436.781

T. Skura, Refutation systems in propositional logic, [in:] Handbook of Philosophical Logic: Volume 16, Springer (2010), pp. 115–157, DOI: https://doi.org/10.1007/978-94-007-0479-4_2 DOI: https://doi.org/10.1007/978-94-007-0479-4_2

H. Wansing, The idea of a proof-theoretic semantics and the meaning of the logical operations, Studia Logica, vol. 64(1) (2000), pp. 3–20, DOI: https://doi.org/10.1023/A:1005217827758 DOI: https://doi.org/10.1023/A:1005217827758

H. Wansing, A more general general proof theory, Journal of Applied Logic, vol. 25 (2017), pp. 23–46, DOI: https://doi.org/10.1016/j.jal.2017.01.002 DOI: https://doi.org/10.1016/j.jal.2017.01.002