### First-order belief and paraconsistency

Srećko Kovač

DOI: http://dx.doi.org/10.12775/LLP.2009.008

#### Abstract

A first-order logic of belief with identity is proposed, primarily to give an account of possible de re contradictory beliefs, which sometimes occur as consequences of de dicto non-contradictory beliefs. A model has two separate, though interconnected domains: the domain of objects and the domain of appearances. The satisfaction of atomic formulas is defined by a particular S-accessibility relation between worlds. Identity is non-classical, and is conceived as an equivalence relation having the classical identity relation as a subset. A tableau system with labels, signs, and suffixes is defined, extending the basic language LQB by quasiformulas (to express the denotations of predicates). The proposed logical system is paraconsistent since φ ∧ ¬φ does not “explode” with arbitrary syntactic consequences.

#### Keywords

appearance; belief; identity; labelled and signed tableau; object; paraconsistent; tableau suffix

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#### References

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Ye, R., and M. Fitting, M., “Belief, names, and modes of presentation”, pp. 389–408 in: Advances in Modal Logic, vol. 3., e.a. F. Wolter (Ed.), World Scientific, New Jersey, etc., 2002.

Béziau, J.-Y., “Paraconsistent logic from a modal viewpoint”, Journal of Applied Logic 3 (2005): 7–14.

Béziau, J.-Y., “A new four-valued approach to modal logic”, http://www.lia.ufc.br/~locia/artigos/modal4.pdf, 200X.

Bloesch, A., “A tableau style proof system for two paraconsistent logics”, Notre Dame Journal of Formal Logic 34 (1993): 295–301.

Carnielli, W.A., “Systematization of finite many-valued logics through the method of tableaux”, Journal of Symbolic Logic 52 (1987): 473–493.

Carnielli, W.A., “On sequents and tableaux for many-valued logics”, The Journal of Non-Classical Logic 8 (1991): 59–78.

Fitting, M., Proof Methods for Modal and Intuitionistic Logics, D. Reidel, Dordrecht, Boston, Lancaster, 1983.

Fitting, M., First-Order Modal Logic, Kluwer, Dordrecht, Boston, London, 1999.

Fitting, M., “First-order intensional logic”, Annals of Pure and Applied Logic 127 (2004): 171–193.

Fitting, M., “FOIL axiomatized”, Studia Logica 84 (2006): 1–22.

Frege, G., “Über Sinn und Bedeutung”, pp. 40–65 in: Funktion, Begriff, Bedeutung, G. Patzig (Ed.), 6. ed. Vandenhoeck und Ruprecht, Göttingen, 1986.

Jaśkowski, S., “Propositional calculus for contradictory deductive systems”, Studia Logica 24 (1969): 143–157. In Polish 1948.

Jaśkowski, S., “A propositional calculus for inconsistent deductive systems”, Logic and Logical Philosophy 7 (1999), 35–56. A modified version of [11].

Jaśkowski, S., “On the discussive conjunction in the propositional calculus for inconsistent deductive systems”, Logic and Logical Philosophy 7 (1999): 57–59. In Polish 1949.

Kovač, S., “Contradictions, objects, and belief”, pp. 417–434 in: Perspectives on Universal Logic, J.-Y. Béziau and A. Costa-Leite (Eds.), Polimetrica, Monza – Milano, 2007.

Kracht, M., and O. Kutz, “The semantics of modal predicate logic II. Modal individuals revisited”, in: Intensionality, R. Kahle (Ed.), A K Peters, Wellesley, Ma., 2005.

Kutz, O., “New semantics for modal predicate logics”, in: Foundations of Formal Sciences II, B. Löwe et al. (Eds.), Kluwer, Dordrecht, Boston, London, 2003.

Ye, R., Belief, Names and Modes of Presentation: A First-Order Logic Formalization, PhD thesis, City University of New York, 1999.

Ye, R., and M. Fitting, M., “Belief, names, and modes of presentation”, pp. 389–408 in: Advances in Modal Logic, vol. 3., e.a. F. Wolter (Ed.), World Scientific, New Jersey, etc., 2002.

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)