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A New Algebraic Version of Monteiro's Four-Valued Propositional Calculus

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A New Algebraic Version of Monteiro's Four-Valued Propositional Calculus Open Journal of Philosophy, 2014, 4, 319-331 Published Online August 2014 in SciRes. http://www.scirp.org/journal/ojpp http://dx.doi.org/10.4236/ojpp.2014.43036 A New Algebraic Version of Monteiro’s Four-Valued Propositional Calculus Aldo Victorio Figallo1, Estela Bianco2, Alicia Ziliani2 1Instituto de Ciencias Básicas, Universidad Nacional de San Juan, San Juan, Argentina 2Departamento de Matemática, Universidad Nacional del Sur, Bahía Blanca, Argentina Email: [email protected], [email protected], [email protected] Received 1 May 2014; revised 20 July 2014; accepted 2 August 2014 Copyright © 2014 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract In the XII Latin American Symposium on Mathematical Logic we presented a work introducing a Hilbert-style propositional calculus called four-valued Monteiro propositional calculus. This cal- culus, denoted by 4 , is introduced in terms of the binary connectives ⇒ (implication), → (weak implication), ∧ (conjunction) and the unary ones (negation) and ∇ (modal opera- tor). In this paper, it is proved that 4 belongs to the class of standard systems of implicative extensional propositional calculi as defined by Rasiowa (1974). Furthermore, we show that the definitions of four-valued modal algebra and 4 -algebra are equivalent and, in addition, obtain the completeness theorem for 4 . We also introduce the notion of modal distributive lattices with implication and show that these algebras are more convenient than four-valued modal alge- bras for the study of four-valued Monteiro propositional calculus from an algebraic point of view. This follows from the fact that the implication → is one of its basic binary operations. Keywords Mathematical Logic, Hilbert-Style Propositional Calculus, Non-Classical Logics, Four-Valued Monteiro Propositional Calculus 1. Preliminaries In 1978, A. Monteiro introduced four-valued modal algebras as a generalization of 3-valued Łukasiewicz alge- bras. These algebras raise a genuine interest both from the points of view of algebra and logic, and especially from that of Algebraic Logic. It is worth mentioning that Monteiro expressed his view that in near future these algebras would give rise to a four-valued modal logic with significant applications in computer science. His in- How to cite this paper: Figallo, A. V., Bianco, E., & Ziliani, A. (2014). A New Algebraic Version of Monteiro’s Four-Valued Propositional Calculus. Open Journal of Philosophy, 4, 319-331. http://dx.doi.org/10.4236/ojpp.2014.43036 A. V. Figallo et al. sight was essentially right, although we don’t know if such applications have yet been developed. On the other hand, it is well known that Font and Rius (1990, 2000) have done an exhaustive research into four-valued modal logics. In particular, in (Font & Rius, 1990) they studied the class of abstract logics projectively generated by the class of logics defined on four-valued modal algebras by the family of their filters and they obtained an axioma- tization of them by means of a Gentzen calculus. Furthermore, in (Font & Rius, 2000) they defined two senten- tial logics. One of them is algebrizable in the sense of (Blok & Pigozzi, 1989) and the corresponding algebras are four-valued modal algebras. The other is not algebrizable in the above sense, but its algebraic counterpart is also the class of four-valued modal algebras. These results gave a positive answer to Monteiro’s conjecture. Be- sides, in (Bianco, 2004, 2008) we gave a Hilbert-style propositional calculus, called four-valued Monteiro prop- ositional calculus which we would describe below. An algebraic study of four-valued modal algebras can be found in (Figallo et al., 1991, 1992, 1994, 1995) and (Loureiro, 1980, 1983a, 1983b, 1983c, 1984, 1985). Recall that: A four-valued modal algebra is an algebra A,,,,,1∧∨∼∇ of type (2,2,1,1,0) such that the reduct A,,,1∨∧ is a distributive lattice with greatest element 1 satisfying these conditions: (L1) ∼∼xx = , (L2) ∼( xy ∧) =∼ x ∨∼ y, (L3) ∼( xx ∧∼∇) =1, (L4) xxxx∧∼ =∇ ∧∼ . From the definition, it follows that A is a De Morgan algebra (Monteiro, 1960; Birkhoff, 1967). Besides, Loureiro (1982, 1983a) proved que that the variety of these algebras is generated by the algebra 44=T ,,,,,1 ∨∧∇∼ , where 1) T4 = {0, ab , ,1} is the De Morgan algebra such that ∼=aa and ∼=bb, being a,b not comparable; 2) The operation ∇ is defined by ∇=00 and ∇=x 1 for all x =0/ . In what follows, we will denote by TM the category of four-valued modal algebras and their corresponding homomorphisms. For the notions of universal algebra including distributive lattices, De Morgan algebras and category theory used in this paper we refer the reader to (Kalman, 1958; Birkhoff, 1967; MacLane, 1971; Burris & Sankappana- var, 1981). The structure of the paper is as follows: In Section 1, we develop a Hilbert-style propositional calculus which we call four-valued Monteiro propositional calculus and denote by 4 . Besides, we prove that 4 belongs to the class of standard systems of implicative extensional propositional calculi (Rasiowa, 1974). Furthermore, we show that the notions of four-valued modal algebra and 4 -algebra are equivalent. Finally, we demon- strate that 4 is consistent and that the completeness theorem holds. From the equivalence established above, we conclude that from any algebra L,∧∨∼∇ , , ,1 ∈ we have that L,,→⇒∧ ,,,1 ∼∇ is an 4 -algebra where xy→ = ∇∼ xy ∨ and x⇒ y =( xy →) ∧( ∇ x →∇( xy ∧ )) and conversely, from any 4 -algebra we obtain a four-valued modal algebra by defining xy∨ =∼( ∼ x ∧∼ y) . These statements and the fact that ⇒ can be defined from → allow us to assert that and the variety i generated by 4 = 〈{0,a,b,1},→,∧,,∇,1〉 are polynomially equivalent. In Section 2, we introduce the variety i of modal distributive lattices with implication and we show that the category MDLi of these algebras and their corresponding homomorphisms is equivalent to TM . Thus, the category theory notably simplify the proof that should be performed to obtain this result by direct calculations. Hence, we determine a new equational descrip- tion of four-valued modal algebras which is more convenient than the latter to study 4 from an algebraic point of view, since the implication → is one of its basic binary operations. 2. Four-Valued Monteiro Propositional Calculus The main aim of this section is to describe the propositional calculus 4 and show that it has four-valued modal algebras as the algebraic counterpart. The terminology and symbols used here coincide in general with those used in (Rasiowa, 1974). 0 0 Let L= ( AF, ) be a formalized language of zero order such that in the alphabet A= ( VL,,,,012 LLU) the set 320 A. V. Figallo et al. 1) V of propositional variables which they will be denoted by αβγ,,,, is enumerable; 2) L0 is empty; 3) L1 contains two elements denoted by ∼ and ∇ called negation sign and modal operator sign, respectively; 4) L2 contains three elements denoted by ∧ , → and ⇒ called conjunction sign, weak implication sign and implication sign, respectively; 5) U contains two elements denoted by (,) . Let F be the set of all formulas over A0. For any α, β in F , we shall write for short αβ↔ instead of (αβ⇒∧⇒) ( βα) . We assume that the set Al of logical axioms consists of all formulas of the following form, where α, β,γ are any formulas in F : (A1) α→→( βα) , (A2) (α→→( βγ)) →(( αβ →→→) ( αγ)) , (A3) (αβ∧→) α, (A4) (αβ∧→) β, (A5) α→→∧( β( αβ)) , (A6) ∇α →∇( αα ∧ ) , (A7) ∇(αβ ∧) →∇ β, (A8) (α→ β) →∇( ( αβ ∧) →∇( ( γα ∧) →∇(( αγ ∧) ∧( βγ ∧ )))) , (A9) ∇∇αα → ∇ , (A10) αα→∇ , (A11) ∇(αβ →) →( α →∇ β) , (A12) (α→∇ β) →∇( αβ → ) , (A13) ∇→∧(α( βγ)) →∇→∧→(( α β) ( α γ)) , (A14) (βγααβββγααγ→) →(( ∇ →∇( ∧)) →(( ∇ →∇( ∧)) →( ∇ →∇( ∧ )))) , (A15) ∇((αγ ∧∧) ( βγ ∧)) →∇∧∧∧(( γα) ( γβ)) , (A16) (∇α → ∇( αβ ∧)) →( ∼( αβ ∧) →∼ α) , (A17) ∼β →∼( αβ ∧ ) , (A18) (∇α →∇( αβ ∧)) →(( β → α) →( ∇∼ α →∇( ∼ α ∧∼ β))) , (A19) ∇α →∇( αα ∧( ∧∼( ∼ α ∧∼ β))) , (A20) α→∼( ∼ αβ ∧ ∼ ) , (A21) (αβ⇒) ↔(( αβ →) ∧( ∇ α →∇( αβ ∧ ))) , (A22) (αβ⇒↔→) ( αβ) , (A23) ∼∼( (γα ∧) ∧∼( βα ∧)) ↔( α ∧∼∼∧∼( β γ)) , 321 A. V. Figallo et al. (A24) ∼(αα ∧∼∇ ) , (A25) ∼α →∇( αα → ) , (A26) ∇α →∇( αα ∧∇ ) , (A27) ∇ ∼α →( ∇∇ α → ∇( ∼ αα ∧ )) , (A28) (αβ⇒→) (( βα ⇒→∇⇒→∇⇒∧⇒) ( ( γα) (( γα) ( γβ)))) , (A29) (αβ⇒) →(( βα ⇒) →∇( ( αγ ⇒) →∇(( αγ ⇒) ∧( βγ ⇒ )))) , (A30) α→( ∇ β →∇( βα ∧ )) . 0 The consequence operation C in = ( AF, ) is determined by the set l and by the following rule of inference: αα, → β (R1) , (Modus Ponens) β The system 4 = ( ,C ) thus obtained, will be called four-valued Monteiro propositional calculus. It is worth mentioning that the above connectives are not independent, however, we consider them for simplicity. We shall denote by the set of all formulas derivable in 4 . As usual, if α ∈ we shall say that α is a theorem of 4 and we shall write α or α . In Lemma 1.1 we summarize the most important rules and theorems necessary for further development. Lemma 1.1. In 4 the following rules and theorems hold: α (R2) , βα→ α→→( βγ) (R3) , (αβ→→→) ( αγ) (T1) αα→ , (T2) (αβ→) →(( γα →→→) ( γβ)) , αβ→ (R4) , (γα→→→) ( γβ) (αβ→→→) ( αγ) (R5) , β→→( αγ) α→→( βγ) (R6) , β→→( αγ) (T3) (α→→( αβ)) →→( αβ) , (T4) (αβ→) →(( βγ →→→) ( αγ)) , α→→ ββ, γ (R7) , αγ→ αβ→ (R8) , (βγ→→→) ( αγ) αβ, (R9) , αβ∧ 322 A. V. Figallo et al. αα, ⇒ β (R10) , β αβ⇒ (R11) , (α→ β) ∧( ∇ α →∇( αβ ∧ )) (α→ β) ∧( ∇ α →∇( αβ ∧ )) (R12) , αβ⇒ (T5) αα⇒ , α⇒⇒ ββ, γ (R13) , αγ⇒ α (R14) , βα⇒ (T6) (α→ β) →∇( ( αβ ∧) →∇( ( γα ∧) →∇( βγ ∧ ))) , (T7) ∇(αβ ∧) →∇( βα ∧ ) , (T8) ∇(αβ ∧) →∇ α, (T9) (γ→ α) →(( γ →→ β) ( γ →( αβ ∧))) , (T10) ∇(αβ ∧) →( ∇ α ∧∇ β) , (T11) ∇(αβ ∧) →∇( ∇ α ∧∇ β) , αβ⇒ (R15) , ∇αβ ⇒∇ (T12) (αβ∧→∧) ( βα) , (T13) (∼(αβ ∧) →∼ α) →( ∼ β →∼ α) , α⇒⇒ ββ, α (R16) , ∼αβ ⇒∼ (T13) (βα→) →(( α →∇ γ) →( β →∇ γ)) , β⇒⇒ αγ, δ (R17) , (αγ→⇒) ( βδ →) α⇒⇒ βγ, δ (R18) , (αγ∧⇒) ( βδ ∧) α⇒ ββ, ⇒⇒⇒ αγ ,, δδ γ (R19) .
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