Abstract Logics and non-Fregean Extensions Department of Computer Sciences
The study of "logics" from an abstract point of view requires a general and unifying theory which is able to capture a large class of particular logics. Several general notions of ``logic" have been introduced and studied over the last decades. The theory of Abstract Logics in the sense of Brown, Suszko, Bloom [1,2] provides an easily accessible and powerful framework which relies on basic order-theoretic and algebraic notions and abstracts from the specific proof-theoretic or semantic representation of a logic. At the heart of this theory is the concept of logical consequence which goes back to Tarski and whose abstract properties can be given by a closure operator. Non-Fregean Logic is a research program started by Suszko [19, 20, 3] with the purpose of studying logic without the so-called Fregean Axiom which says that formulas of the same truth value have the same denotation. That is, the Fregean Axiom reduces the denotation (i.e., Bedeutung, reference, meaning) of a sentence to its truth value. This is actually the situation in current logics. In non-Fregean logics, however, sentences are interpreted over a model-theoretic universe of entities, called by us propositions (Suszko used the term situations), which besides a truth value may represent a more complex semantic content (in a limit case: the sense/intension expressed by the sentence). Two sentences may denote different propositions of the same truth value – the Fregean Axiom does not hold. The language of a non-Fregean logic contains an identity connective by which propositional identity and also propositional self-reference [18] can be expressed. In a series of papers it was shown that (classical and some non-classical) non-Fregean logics can be augmented with predicates for truth and falsity [18, 22, 9, 14] and by alethic modalities (independently from any possible worlds semantics) [10, 13] such that propositional self-references, including semantic antinomies such as the liar statement, can be asserted without compromising the consistency of the logical system. A given abstract logic, viewed in a sense as a parameter or object logic, can be conservatively extended to a non-Fregean logic with a truth predicate. This is shown for the classical and for some non-classical cases in [22, 10, 11, 14]. The resulting non-Fregean extension is itself an abstract logic of the same logical type. Moreover, the extension can be seen as the semantic closure of the underlying object logic in the sense of Tarski's truth theory: the extension contains its own truth predicate which satisfies an analogue to Tarski’s T-scheme. The truth predicate applies in particular to sentences of the object logic. In the first two sessions of the tutorial we give an introduction to the basic theory of Abstract Logics. We present equivalent definitions showing the duality between closure operators and closure spaces, discuss particular examples such as classical, intuitionistic and many-valued abstract logics, study several notions of compactness in abstract logics and discuss topological representations. In the third session we discuss the main ideas underlying non-Fregean Logic and present Epsilon-T-Logic [18, 22, 12, 14] as a non-Fregean logic which extends Suszko’s SCI (the Sentential Calculus with Identity [3]) by a total truth predicate and propositional quantifiers. We show how propositional self-reference can be achieved and how semantic antinomies, such as the liar paradox, are solved in this logic. In particular, we discuss Epsilon-T-Logic as a semantic closure of underlying classical and some non-classical abstract parameter logics. Finally, we shortly discuss epistemic non-Fregean logic [10, 13]. Bibliography [1] D.J. Brown and R. Suszko, Abstract Logics, Dissertationes Mathematicae, 102, 9 –42, 1973. [2] S.L. Bloom and D.J. Brown, Classical Abstract Logics, Dissertationes Mathematicae, 102, 43 – 51, 1973. [3] S. L. Bloom and R. Suszko, Investigation into the sentential calculus with identity, Notre Dame Journal of Formal Logic 13(3), 289 -- 308, 1972. [4] B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, 2. ed., Cambridge University Press, 2002. [5] J.M. Dunn, G.H. Hardegree, Algebraic Methods in Philosophical Logic, Clarendon Press, Oxford, 2001.
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[6] J. M. Font, Belnap's Four-Valued Logic and De Morgan Lattices, Logic Journal of the IGPL 5(3), 1 -- 29, 1997. [7] J.M. Font and V. Verdu, A first approach to abstract modal logics, Journal of Symbolic Logic, 54(3), 1989, 1042 -- 1062. [8] S. Lewitzka, Abstract Logics, Logic Maps and Logic Homomorphisms, Logica Universalis, 1(2), 2007, 243 -- 276. [9] S. Lewitzka, Epsilon-I: an intuitioninistic logic without Fregean axiom and with predicates for truth and falsity, Notre Dame Journal of Formal Logic 50(3), 275 -- 301, 2009. [10] S. Lewitzka, Epsilon-K: a non-Fregean logic of explicit knowledge, Studia Logica 97(2), 233 -- 264, 2011. [11] S. Lewitzka, Semantically closed intuitionistic abstract logics, J. of Logic and Computation, 22(3), 351 – 374, 2012. [12] S. Lewitzka, Construction of a canonical model for a first-order non-Fregean logic with a connective for reference and a total truth predicate, The Logic Journal of the IGPL, 2012. [13] S. Lewitzka, Necessity as justified truth, arXiv, 2012. [14] S. Lewitzka, On some many-valued abstract logics and their Epsilon-T extensions, arXiv, 2012. [15] S. Lewitzka and A.B.M. Brunner, Minimally generated abstract logics, Logica Universalis, 3(2), 2009, 219 -- 241. [16] G. Priest, An Introduction to Non-Classical Logic, Cambridge University Press, 2001. [17] K. Robering, Logics with Propositional Quantifiers and Propositional Identity, in: S. Bab, K. Robering (eds.), Judgements and Propositions, Logos Verlag, Berlin, 2010. [18] W. Sträter, Epsilon-T: Eine Logik erster Stufe mit Selbstreferenz und totalem Wahrheitspraedikat, Dissertation, KIT-Report 98, Technische Universitaet Berlin, 1992. [19] R. Suszko, Non-Fregean Logic and Theories, Analele Universitatii Bucuresti, Acta Logica, 11, 105 -- 125, 1968. [20] R. Suszko, Abolition of the Fregean Axiom, in: R. Parikh (ed.) Logic Colloquium, Lecture Notes in Mathematics 453, Springer Verlag, 169 -- 239, 1975. [21] B. C. van Fraassen, Formal Semantics and Logic, The Macmillan Company, New York, 1971. [22] Ph. Zeitz, Parametrisierte Epsilon-T-Logik: eine Theorie der Erweiterung abstrakter Logiken um die Konzepte Wahrheit, Referenz und klassische Negation, Dissertation, Logos Verlag Berlin, 2000. | ||||
