Matrix- based logic for avoiding paradoxes and its paraconsistent alternative

CDD: 160 MATRIX-BASED LOGIC FOR AVOIDING PARADOXES AND ITS PARACONSISTENT ALTERNATIVE PAUL WEINGARTNER Department of Philosophy University of Salzburg and Institut für Wissenschaftstheorie Salzburg, A–5020. AUSTRIA Abstract: The present article shows that there are consistent and decidable manyvalued systems of propositional logic which satisfy two or all the three criteria for non-trivial inconsistent theories by da Costa (1974). The weaker one of these paraconsistent system is also able to avoid a series of paradoxes which come up when classical logic is applied to empirical sciences. These paraconsistent systems are based on a 6-valued system of propositional logic for avoiding difficulties in several domains of empirical science (Weingartner (2009)). Keywords: Paraconsistent system. Paradox. Relevance. Many-valued logic. 1. INTRODUCTION Newton da Costa is wellknown for his contributions to Paraconsistent Logics, more correctly it can be said he is the founder of this relatively new domain of Logic. I remember that I saw Newton for the first time at the international congress for Logic, Methodology and Philosophy of Science 1987 in Moscow. I was impressed by his invited lecture “Logic and Pragmatic Truth”. Later we met several times on congresses in Europe and Southamerica. I dedicate the following article to Newton da Costa from whom I got many important stimulations for my thought. Manuscrito — Rev. Int. Fil., Campinas, v. 34, n. 1, p. 365-388, jan.-jun. 2011. 366 PAUL WEINGARTNER Before I begin let me give some general remarks on how I see the approach on Paraconsistent Logic. First of all, since I am a realist, I do not believe that reality can contain contradictions. On the other hand, since man is imperfect, both his thinking and his oral and written communications can contain contradictions. Therefore also our descriptions, hypotheses and theories about reality can contain contradictions. In this sense there is justified motivation to investigate epistemic systems which allow inconsistencies and try to handle them in a logical deductive way. It is not so well known that already Aristotle investigated deductive arguments containing contradictory premises. In his seminal article “On the theory of inconsistent formal systems” Newton da Costa formulated three principles for calculi which ‘are intended to serve as bases for non–trivial inconsistent theories’ (da Costa (1974, p. 498)). DC1 “In these calculi the principle of contradiction, ¬(A ∧ ¬A), must not be a valid schema”; DC2 “From two contradictory formulas A and non–A it will not in general be possible to deduce an arbitrary formula B”; DC3 Each calculus “must contain the most part of the schemata and rules of CL (Classical Logic) which do not interfere with the first two conditions”. A system which satisfies DC2 and DC3 will be called a weak paraconsistent system. One which satisfies DC1, DC2 and DC3 will be called a strong paraconsistent system. The purpose of this paper is to construct two models in the sense of matrix calculi of propositional logic which satisfy either DC2 and DC3 or DC1, DC2 and DC3. These models are (weak/strong) paraconsistent alternatives of a basic logic (called RMQ) for the application in empirical sciences and especially also in modern physics (WeingartManuscrito — Rev. Int. Fil., Campinas, v. 34, n. 1, p. 365-388, jan.-jun. 2011. MATRIX-BASED LOGIC FOR AVOIDING PARADOXES 367 ner (2009)). This basic logic is a finite matrix system and hence it contains its own semantics. It has the following properties: 1) RMQ is a 6–valued matrix system (3 values for truth, 3 for falsity) and so it contains its own semantics. Every well formed formula of RMQ is unambiguously determined by a particular matrix which contains 6n values for n (n = 1, 2, . . .) different propositional variables. 2) RMQ is motivated by two criteria called replacement (RC) and reduction (RD) which avoid difficulties in the application of logic (see below 6 and 7). 3) RMQ is consistent and decidable. 4) RMQ has the finite model property. 5) RMQ has two concepts of validity: a weaker one (classically valid which is identical with materially valid) and a stronger one (strictly valid). All theorems of two–valued Classical Logic (Classical Propositional Calculus CPC) are at least classically valid, that is materially valid, in RMQ. Only a restricted class of them are strictly valid in RMQ. Therefore: RMQ satisfies DC3 above. 6) The validity of a proposition is decided by calculating the highest value (cv) in its matrix. If cv = 3 the proposition (formula) is classically valid, that is materially valid. If cv = 2 the proposition (formula) is strictly valid. 7) The strictly valid theorems of RMQ avoid a great number of well– known paradoxes in the domain of scientific explanation, law statements, disposition predicates, verisimilitude, theory of human actions, deontic logic . . . etc.1 1 See below 2.1 and theorems in 3.2 and 3.3. Manuscrito — Rev. Int. Fil., Campinas, v. 34, n. 1, p. 365-388, jan.-jun. 2011. 368 PAUL WEINGARTNER 8) The strictly valid theorems of RMQ avoid the well known difficulties when logic is applied to physics; especially those with commensurability, distributivity and with Bell’s inequalities. See Weingartner (2009, Section 4.3, p. 148 - 153). 9) RMQ is closed under transitivity of implication, under modus ponens, and under equivalence substitution. 10) RMQ contains a modal system with 14 modalities which is close to the modal system T (of Feys) concerning the theorems with one modal operator (no iteration) applied to well formed formulas. The paper will be divided into the following sections: In section 2 the two criteria RC and RD will be described. The system RMQ will be defined in section 3 and 3.1. In section 3.2 those theorems of RMQ will be statet, which are only classically or materially valid, in 3.3 those which are strictly valid in RMQ and satisfy RC and RD. In section 4 the two paraconsistent alternatives to RMQ are presented. The first (weak paraconsistent system) is obtained from RMQ by weakening the negation of RMQ. The second is obtained from RMQ by weakening its conjunction. Advantages and disadvantages of these two systems are discussed. The modal logic included of RMQ will not be discussed in this paper.See Weingartner (2009). 2. THE MOTIVATING CRITERIA RC AND RD The criteria RC and RD have been introduced in order to avoid paradoxes when logic is applied to empirical sciences. Such paradoxes have been described in several papers (Weingartner, Schurz (1986), Weingartner (2001)).. For difficulties in the domain of Quantum Physics see Mittelstaedt (1978, 2004), Weingartner (2004, 2009, 2010). The criteria originate in a paper devoted to give a solution of the problem of verisimilitude in the sense of rehabilitating Popper’s original idea Manuscrito — Rev. Int. Fil., Campinas, v. 34, n. 1, p. 365-388, jan.-jun. 2011. MATRIX-BASED LOGIC FOR AVOIDING PARADOXES 369 b (...truncated)