Abstract
We present a logic of evidence that reduces agents’ epistemic idealisations by combining classical propositional logic with substructural modal logic for formulas in the scope of epistemic modalities. To this aim, we provide a neighborhood semantics of evidence, which provides a modal extension of Fine’s semantics for relevant propositional logic. Possible worlds semantics for classical propositional logic is then obtained by defining the set of possible worlds as a special subset of information states in Fine’s semantics. Finally, we prove that evidence is a hyperintensional and non-prime notion in our logic, and provide a sound and complete axiomatisation of our evidence logic.
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Anderson, A. R., & Belnap, N. D. (1975). Entailment: The logic of relevance and neccessity, Vol. I. Princeton University Press
Baldi, P., & Hosni, H. (2020). Depth-bounded belief functions. International Journal of Approximate Reasoning, 123, 26–40.
Beall, J., Brady, R., Dunn, J., et al. (2012). On the ternary relation and conditionality. Journal of Philosophical Logic, 41, 1–18.
Bendana, J., & Mandelbaum, E. (2021). The fragmentation of belief, chap 3 (pp. 78–107). The Fragmented Mind: Oxford University Press, Oxford.
van Benthem, J., & Pacuit, E. (2011). Dynamic logics of evidence-based belief. Studia Logica, 99, 61–92.
van Benthem, J., Pacuit, E., & Fernández-Duque, D. (2014). Evidence and plausibility in neighborhood structures. Annals of Pure and Applied Logic 165(1):106–133. The Constructive in Logic and Applications
Berto, F., & Restall, G. (2019). Negation on the australian plan. Journal of Philosophical Logic, 48(6), 1119–1144.
Bílková, M., Majer, O., & Peliš, M. (2016). Epistemic logics for sceptical agents. Journal of Logic and Computation, 26(6), 1815–1841.
Bilkova, M., Frittella, S., Kozhemiachenko, D., & et al. (2023). Describing and quantifying contradiction between pieces of evidence via Belnap Dunn logic and Dempster-Shafer theory. In: Miranda, E., Montes, I., Quaeghebeur, E., et al (eds.) Proceedings of the thirteenth international symposium on imprecise probability: Theories and applications, proceedings of machine learning research, (vol. 215, pp. 37–47). PMLR
Bílková, M., Majer, O., Peliš, M., & et al. (2010). Relevant agents. In: Beklemishev L., Goranko V., Shehtman V. (eds.) Advances in modal logic, (vol. 8, pp. 22–38). College Publications, London
Chellas, B. F. (1980). Modal Logic: An introduction. Cambridge: Cambridge University Press.
Cresswell, M. J. (1985). Structured meanings. MIT Press.
Dunn, J. (2015). The relevance of relevance to relevance logic. Lecture Notes in Computer Science Proceedings ICLA, 2015, 11–29.
D’Agostino, M., Finger, M., & Gabbay, D. (2013). Semantics and proof-theory of depth bounded boolean logics. Theoretical Computer Science, 480, 43–68.
Edwin Mares, J. S., & Restall, G. (1997). Situations, constraints and channels. In: Johan van Benthem, AtM (ed) Handbook of logic and language, (pp. 329–344). North Holland, Cambridge
Fagin, R., & Halpern, J. Y. (1987). Belief, awareness, and limited reasoning. Artificial Intelligence, 34(1), 39–76.
Fagin, R., Halpern, J. Y., Moses, Y., et al. (1995). Reasoning about knowledge. Cambridge: MIT Press.
Ferenz, N., & Tedder, A. (2023). Neighbourhood semantics for modal relevant logics. Journal of Philosophical Logic, 52(1), 145–181.
Fine, K. (1974). Models for entailment. Journal of Philosophical Logic, 3(4), 347–372.
Fuhrmann, A. (1988). Relevant logics, modal logics and theory change. Phd thesis, Australian National University
Grzegorczyk, A. (1964). A philosophically plausible formal interpretation of intuitionistic logic. Indagationes Mathematicae, 26(5), 596–601.
Hawke P, Özgün A, & Berto, F. (2019). The fundamental problem of logical omniscience. Journal of Philosophical Logic
Jeremy Seligman, L. S. M. (1997). Situation theory. In: Johan van Benthem AtM (ed.) Handbook of logic and language, (pp. 253–328) North Holland, Cambridge
Leitgeb, H. (2019). Hype: A system of hyperintensional logic. Journal of Philosophical Logic, 48(2), 305–405.
Levesque, H. (1984). A logic of implicit and explicit belief. Proceedings of AAAI, 1984, 198–202.
Mares, E. D. (2004). Relevant logic: A philosophical interpretation. Cambridge: Cambridge University Press.
Pacuit, E. (2017). Neighborhood semantics for modal logic. Dordrecht, Springer.
Perry, J. (1989). Possible worlds and subject matter (pp. 145–160). Palo Alto, CA: CSLI Publications.
Restall, G. (1995). Information flow and relevant logics. In J. Seligman & D. Westerstahl (Eds.), Logic, language and computation: The 1994 moraga proceedings (pp. 463–477). CSLI Press.
Restall, G. (1996). Truthmakers, entailment and necessity. Australasian Journal of Philosophy, 74(2), 331–340.
Restall, G. (2000). An introduction to substructural logics. London, Routledge.
Routley, R., Meyer, R. K., Plumwood, V., amp, et al. (1982). Relevant logics and their rivals. Atascadero, Ridgeview.
Sedlár, I. (2019). Hyperintensional logics for everyone. Synthese, 198(2), 933–956.
Sedlar, I., & Vigiani, P. (2022). Relevant reasoners in a classical world. In: Duque, D. F., Palmigiano, A., Pichinat, S. (eds.) Advances in modal logic, (vol. 14, pp. 697–718). College Publications, London
Sedlár, I. (2015). Substructural epistemic logics. Journal of Applied Non-Classical Logics, 25(3), 256–285.
Seki, T. (2003). General frames for relevant modal logics. Notre Dame Journal of Formal Logic, 44(2), 93–109.
Sequoiah-Grayson, S. (2013). Epistemic closure and commutative, nonassociative residuated structures. Synthese, 190(1), 113–128.
Sequoiah-Grayson, S. (2016). Epistemic relevance and epistemic actions (pp. 133–146). Springer International Publishing, Cham
Sequoiah-Grayson, S. (2021) A logic of affordances. In: Blicha, M., Sedlár I. (eds.) The logica yearbook 2020, (pp. 219–236)
Stalnaker, R. C. (1976). Possible worlds. Noûs, 10(1), 65–75.
Tedder, A., & Ferenz, N. (2021). Neighbourhood semantics for quantified relevant logics. Journal of Philosophical Logic
Wansing, H. (2016). On split negation, strong negation, information, falsification, and verification (pp. 161–189). Springer International Publishing, Cham
Acknowledgements
We thank Shay Logan, Andrew Tedder, Riccardo Bruni and the audience of the PLEXUS Inaugural Conference for valuable comments on earlier drafts of the paper. We thank the work of two anonymous reviewers of the journal, which helped to improve the quality of the paper.
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Vigiani, P. A Hyperintensional Logic of Non-prime Evidence. J Philos Logic (2024). https://doi.org/10.1007/s10992-024-09750-x
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DOI: https://doi.org/10.1007/s10992-024-09750-x