A SIMPLE AXIOMATIZATION OF LUKASIEWICZ S MODAL LOGIC

Transcription

1 Bulletin of the Section of Logic Volume 41:3/4 (2012), pp Zdzis law Dywan A SIMPLE AXIOMATIZATION OF LUKASIEWICZ S MODAL LOGIC Abstract We will propose a new axiomatization of four-valued Lukasiewicz s modal logic in the version with rejection. Moreover, two controversies connected with violation of fundamental intuitions about necessity are discussed and an argumentation against suitability of this logic for analysis of Aristotle s modal syllogistic is given. In this paper we are interested in four-valued Lukasiewicz s modal logic presented in [4] and also in [3]. This logic is determined by the following axioms: (Lk 1 ) (Lk 2 ) (Lk 3 ) (Lk 4 ) δ(p) (δ( p) δ(q)) p Mp Mp p Mp. The letters p, q denote propositional variables and, stands for connectives of negation and implication. The operation δ is the so-called functoral variable. Another classical connectives: (conjunction), (equivalence) and the connective of necessity L(= M ) are introduced by definitions in the common way. Our axiomatic system contains rules of modus ponens and substitution in their versions for acceptance of theses and rejection of non-theses, classical theses and the following modal axioms: (Ax 1 ) Lp p L(q q) (Ax 2 ) L(p p) (Ax 3 ) L(p p).

2 150 Zdzis law Dywan In this system the connective of necessity L is primitive and the connective of possibility M(= L ) is introduced by definition. The idea of this axiomatic approach was found by me by examination of J. Porte s results in [5]. From Lukasiewicz s work [4] it is well-known that a formula is a modal tautology if and only if it is a tautology in both of the following matrices simultaneously: 0 1 L L where asterisk denotes distinguished value and other classical connectives are defined in the common way. Lemma 1. Theses of Lukasiewicz s modal logic can be axiomatized by using the rule of modus ponens, the rule of substitution, classical theses and the following two formulas: (Lm 1 ) Lp p (Lm 2 ) Lp (q Lq). Proof: Lemmon in his proof of this theorem in [1], [2] used three axioms, but Tkaczyk in [6] proved that one of them is redundant and it is sufficient to use these two. Lemma 2. In every logic containing classical theses with the rule of modus ponens the following formula is a thesis (Ext) (φ ψ) (θ(p/φ) θ(p/ψ)) where φ, ψ are any formulas and variable p appears in θ only in the scope of classical connectives. Proof: Obvious. Theorem 1. The axioms (Ax 1 ),..., (Ax 3 ) (with the classical theses) are deducible equivalent to Lukasiewicz s axioms.

3 A Simple Axiomatization of Lukasiewicz s Modal Logic 151 Proof: First, we prove that axiom (Ax 1 ) (together with classical theses) is deducible equivalent to Lukasiewicz s axioms (Lk 1 ) and (Lk 2 ). It is easy to check out that axiom (Ax 1 ) is a modal tautology. Then, it is derivable from Lukasiewicz s axioms. We have to prove that all theses of Lukasiewicz s logic can be infered from (Ax 1 ). By virtue of Lemma 1 it is enough to prove that Lemmon s axioms (Lm 1 ) and (Lm 2 ) are derivable from (Ax 1 ). Using substitution from (Ax 1 ) we obtain the following theses: Lp p L(p p) Lq q L(p p). Using these theses and thesis (Ext), by virtue of Lemma 2, we obtain equivalences: Lp p p L(p p) p and Lp (q Lq) p L(p p) (q Lq) p L(p p) (q q L(p p)). Note that axioms (Lm 1 ) and (Lm 2 ) are equivalent to substitutions of some classical theses. Thus they are deducible in our logic. So the sets of theses of Lukasiewicz s logic and our logic are identical. Now we prove that axioms of rejection in both logics are mutually derivable. It can be easily verified that (Ax 2 ) and (Ax 3 ) are not modal tautologies. So rejecting axioms (Lk 3 ) and (Lk 4 ) we can also reject them. To finish our proof we have to prove inverse derivability, i.e. if we reject axioms (Ax 2 ) and (Ax 3 ), then we can reject (Lk 3 ) and (Lk 4 ) in our logic. It suffices to prove that from formulas (Lk 3 ) and (Lk 4 ) we can, in usual way, obtain formulas (Ax 2 ) and (Ax 3 ). Note that the following formulas are modal tautologies: (M (p p) (p p)) L(p p) M (p p) L(p p). Since we have already proved that the sets of the theses of our logic and Lukasiewcz s are identical, then they are also theses of our logic. By the substitution p/ (p p) in axioms (Lk 3 ) and (Lk 4 ) we obtain the following formulas:

4 152 Zdzis law Dywan M (p p) (p p) M (p p). Now using modus ponens we obtain our axioms of rejection (Ax 2 ) and (Ax 3 ). Let us look at our only axiom (Ax 1 ) of Lukasiewicz s logic Lp p L(q q). We can obtain from it the following dilemma: formula L(p p) is not a thesis of modal logic or necessity is an assertion (i.e. Lp p). Both these cases are bad. Under usual understanding of necessity modal logic should accept necessity of the principle of identity and simultaneously reject the identification of necessity and assertion. Axiom (Ax 1 ) recalls a definition and therefore raises controversies since that fact is in contradiction with intuitions of using modal notions. We can express the connection between necessity and assertion by the formula Lp p φ. We can say that necessity is assertion limited by some additional condition expressed here by the formula φ. In this formula some connection with p should appear, but in our axiom (Ax 1 ) it does not appear. In other words formulas p and φ (i.e. L(q q)) can be separated. Now let us look at the consequences of this property in applications. Suppose that we have an Aristotelian assertoric syllogism φ, ψ θ The following formula is a substitution of a classical thesis (φ A) ψ φ (ψ A) (φ A) (ψ A) where A is the abbreviation of expression L(p p). Hence using axiom (Ax 1 ) and thesis (Ext) we obtain formula Lφ ψ φ Lψ Lφ Lψ. Note that formulas: (φ A) ψ, φ (ψ A), (φ A) (ψ A) represent in fact the same expressions. Then we can say the same about formulas: Lφ ψ, φ Lψ, Lφ Lψ. Hence modal syllogisms with both necessary

5 A Simple Axiomatization of Lukasiewicz s Modal Logic 153 premises and with mixed premises cannot be distinguished while one is necessary and another assertoric. Therefore we can say that Lukasiewicz s modal logic is useless for investigating Aristotelian modal syllogistic. References [1] E. J. Lemmon, Algebraic semantics for modal logics I, J. Symb. Log. 31 (1966), pp [2] E. J. Lemmon, Algebraic semantics for modal logics II, J. Symb. Log. 31 (1966), pp [3] J. Lukasiewicz, Aristotle s Syllogistic from the Standpoint of Modern Formal Logic, Oxford University Press, Second edition, [4] J. Lukasiewicz, A system of modal logic, The Journal of Computing Systems, 1:3 (1953), pp [5] J. Porte, The Ω-system and the system of L-modal logic, Notre Dame of Formal Logic 20:4 (1979), pp [6] M. Tkaczyk, On axiomatization of Lukasiewicz s four-valued modal logic, Logic and Logical Philosophy 20:3 (2011), pp Faculty of Philosophy John Paul II Catholic University of Lublin zdzislaw.dywan@kul.pl