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$$\in_K$$: a non-Fregean logic of explicit knowledge. (English) Zbl 1231.03016
Summary: We present a new logic-based approach to reasoning about knowledge which is independent of a possible worlds semantics. $$\in_K$$ (Epsilon-$$K$$) is a non-Fregean logic whose models consist of propositional universes with subsets for true, false and known propositions. Knowledge is, in general, not closed under rules of inference; the only valid epistemic principles are the knowledge axiom $$K_{i \varphi} \rightarrow \varphi$$ and some minimal conditions concerning common knowledge in a group. Knowledge is explicit and all forms of the logical omniscience problem are avoided. Various stronger epistemic properties such as positive and/or negative introspection, the $$K$$-axiom, closure under logical connectives, etc. can be restored by imposing additional semantic constraints. This yields corresponding sublogics for which we present sound and complete axiomatizations. As a useful tool for general model constructions we study abstract versions of some 3-valued logics in which we interpret truth as knowledge. We establish a connection between $$\in_K$$ and the well-known syntactic approach to explicit knowledge proving a result concerning equi-expressiveness. Furthermore, we discuss some self-referential epistemic statements, such as the knower paradox, as relaxations of variants of the liar paradox and show how these epistemic “paradoxes” can be solved in $$\in_K$$. Every specific $$\in_K$$-logic is defined as a certain extension of some underlying classical abstract logic.

##### MSC:
 03B42 Logics of knowledge and belief (including belief change) 03A05 Philosophical and critical aspects of logic and foundations
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##### References:
 [1] Barwise J., Etchemendy J.: The Liar. Oxford University Press, New York, Oxford (1987) · Zbl 0678.03001 [2] Bloom S.L., Brown D.J.: ’Classical Abstract Logics’. Dissertationes Mathematicae 102, 43–51 (1973) · Zbl 0317.02072 [3] Bloom S.L., Suszko R.: ’Investigation into the Sentential Calculus with Identity’. Notre Dame Journal of Formal Logic 13, 289–308 (1972) · Zbl 0238.02016 [4] Brown D.J., Suszko R.: ’Abstract Logics’. Dissertationes Mathematicae 102, 9–42 (1973) · Zbl 0317.02071 [5] Fagin R., Halpern J.Y., Moses Y., Vardi M.Y.: Reasoning about Knowledge. MIT Press, Cambridge, Mass. (2003) · Zbl 0839.68095 [6] Fitting M.: ’A Logic of Explicit Knowledge’. In: Behounek, L., Bilkova, M. (eds) The Logica Yearbook 2004, pp. 11–22. Filosofia, Prague (2005) [7] Halpern, J., and R. Pucella, ’Dealing with logical omniscience’, DOI: 10.1145/1324249.1324273 . TARK ’07 Proceedings of the 11th Conference on Theoretical aspects of rationality and knowledge, ACM New York, NY, USA, 2007. [8] Lewitzka, S., ’ $${\in_T\,(\Sigma)}$$ -Logik: Eine Erweiterung der Prädikatenlogik erster Stufe mit Selbstreferenz und totalemWahrheitsprädikat’, Diplomarbeit, Technische Universität Berlin, 1998. [9] Lewitzka S.: ’Abstract Logics, Logic Maps and Logic Homomorphisms’. Logica Universalis 1, 243–276 (2007) · Zbl 1131.03006 [10] Lewitzka, S., ’ $${\in_4}$$ : A 4-valued truth theory and meta-logic’, preprint, 2007. [11] Lewitzka S.: ’ $${\in_I}$$ : An intuitionistic logic without Fregean Axiom and with predicates for truth and falsity’. Notre Dame Journal of Formal Logic 50, 275–301 (2009) · Zbl 1190.03016 [12] Lewitzka S., Brunner A.B.M.: ’Minimally generated abstract logics’. Logica Universalis 3, 219–241 (2009) · Zbl 1255.03022 [13] Meyer, J. J. Ch., and W. van der Hoeck, Epistemic Logic for AI and Computer Science, Cambridge Tracts in Theoretical Computer Science 41, Cambridge University Press 1995. · Zbl 0868.03001 [14] Robering, K., ’Logics with Propositional Quantifiers and Propositional Identity’, in S. Bab and K. Robering (eds.), Judgements and Propositions. Logical, Linguistic, and Cognitive Issues, Logische Philosophie, Bd. 21, Logos Verlag, Berlin, 2010. [15] Sträter, W., ’ $${\in_T}$$ Eine Logik erster Stufe mit Selbstreferenz und totalem Wahrheitsprädikat’, KIT-Report 98, Technische Universität Berlin, 1992. [16] Suszko R.: ’Non-Fregean Logic and Theories’, Analele Universitatii Bucuresti. Acta Logica 11, 105–125 (1968) · Zbl 0233.02010 [17] Suszko, R., ’Abolition of the Fregean Axiom’, in R. Parikh (ed.), Logic Colloquium, Lecture Notes in Mathematics 453:169–239, Springer-Verlag, 1975. · Zbl 0308.02026 [18] Suszko R.: ’The Fregean axiom and Polish mathematical logic in the 1920s’. Studia Logica 36, 373–380 (1977) · Zbl 0404.03004 [19] Wansing H.: ’A general possible worlds framework for reasoning about knowledge and belief’. Studia Logica 49, 523–539 (1990) · Zbl 0744.03028 [20] Zeitz, P., Parametrisierte $${\in_T}$$ -Logik – eine Theorie der Erweiterung abstrakter Logiken um die Konzepte Wahrheit, Referenz und klassische Negation, Dissertation, Logos Verlag Berlin, 2000. · Zbl 0992.03013
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