Contraction, Infinitary Quantifiers, and Omega Paradoxes
Tóm tắt
Our main goal is to investigate whether the infinitary rules for the quantifiers endorsed by Elia Zardini in a recent paper are plausible. First, we will argue that they are problematic in several ways, especially due to their infinitary features. Secondly, we will show that even if these worries are somehow dealt with, there is another serious issue with them. They produce a truth-theoretic paradox that does not involve the structural rules of contraction.
Tài liệu tham khảo
Bacon, A. (2013). Curry’s Paradox and ω-Inconsistency. Studia Logica, 101(1), 1–9.
Barrio, E. (2010). Theories of truth without standard models and Yablo’s sequences. Studia Logica, 96(3), 375–391.
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2013). Reaching transparent truth. Mind, 122(488), 841–866.
Fjellstad, A. (2016). Omega-inconsistency without cuts and nonstandard models. Australasian Journal of Logic, 13(5), 96–122.
French, R., & Ripley, D. (2015). Contractions of noncontractive consequence relations. The Review of Symbolic Logic, 8(3), 506–528.
Girard, J.Y. (1987). Linear logic. Theoretical Computer Science, 50(1), 1–101.
Halbach, V. (2011). Axiomatic theories of truth. Cambridge: Cambridge University Press.
Leitgeb, H. (2007). What theories of truth should be like (but cannot be). Philosophy Compass, 2(2), 276–290.
Mares, E., & Paoli, F. (2014). Logical consequence and the paradoxes. Journal of Philosophical Logic, 43(2-3), 439–469.
McGee, V. (1985). How truthlike can a predicate be? A negative result. Journal of Philosophical Logic, 14, 399–410.
Montagna, F. (2004). Storage operators and multiplicative quantifiers in many-valued logics. Journal of Logic and Computation, 14, 299–322.
Negri, S., & von Plato, J. (1998). Cut elimination in the presence of axioms. Bulletin of Symbolic Logic, 4, 418–435.
Negri, S. (2003). Contraction-free sequent calculi for geometric theories with an application to Barr’ theorem. Archive for Mathematical Logic, 42, 389–401.
Paoli, F. (2002). Substructural logics: a primer. Dordrecht: Kluwer.
Paoli, F. (2005). The ambiguity of quantifiers. Philosophical Studies, 124(3), 313–330.
Ripley, D. (2015). Comparing substructural theories of truth. Ergo, 13(2), 299–328.
Shapiro, L. (2015). Naive structure, contraction and paradox. Topoi, 34(1), 75–87.
Shaw-Kwei, M. (1954). Logical paradoxes for many-valued systems. Journal of Symbolic Logic, 19(1), 37–40.
Smith, P. (2007). An introduction to Gödel’s theorems. Cambridge: Cambridge University Press.
Troelstra, A.S., & Schwichtenberg, H. (2000). Basic proof theory. Cambridge: Cambridge University Press.
Zardini, E. (2011). Truth without contra(dic)ction. The Review of Symbolic Logic, 4(4), 498–535.
Zardini, E. (2014). Naive truth and naive logical properties. The Review of Symbolic Logic, 7(2), 351–384.
Zardini, E. (2015). ∀ & ω. In Torza, A. (Ed.), Quantifiers, quantifiers and quantifiers: Springer.
Zardini, E. (2016). Restriction by non-contraction. Notre Dame Journal of Formal Logic, 57(2), 287–327.