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Intuitionistic Logic with Strong Negation Was Regarded in [5], [7], [8] and [10]

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Intuitionistic Logic with Strong Negation Was Regarded in [5], [7], [8] and [10] Intuitionistic Logic GuEwc. with Strong Negation Introduction A Kripke model of intuitionistic predicate logic can be described (see [1]) as a quadruple .;(----(M, 4, 8, v) where (M, ~) is a poser (partially ordered set), ~ is a non-decreasing function associating a set of individual constants with each X E M, and for each X e M and each formula A, ~xA is equal to either "true" or "uncertain". The details can be found in w 3 below. In particular, Vx(nA) -~ "true" iff for each Y ~ X~, Ty.A. =/= "true". In the spirit of Grzegorczyk's paper [2] 9ff may be interpreted as scheme of a scientific research. Elements of M are the stages of the reseaxch, ~ is the precedence relation, ~ (X) is the set of objects involved in the reseaxch at stage X. For an atomic formula A, vxA is a product of experiment. "The compound sentences are not a product of experi- ment"- writes Grzegorczyk- "They arise from reasoning. This con- cerns also negations: we see that the lemon is yellow, we do not see that it is not blue". This paper is a reaction for this remurk of Grzegorczyk. In many cases the falsehood of a simple scientific sentence can be ascertained as directly (or indirectly) as its truth. An example: a litmus-paper is used to verify sentence "The solution is acid". We regard a generalizations of Kripke models when ~x A can be equal to "false", "uncertain" or "true". That gives rise to a conservative extension of the intuitionistic logic which is nicer at least in one aspect: it is more symmetric, it satisfies very natural duality laws. We use the strong negation to formalize the arising logic. The pro- positional intuitionistic logic with strong negation was regarded in [5], [7], [8] and [10]. We use here Vorob'ev's ealclflus in [10]. Thomason deve- loped in [9] semantics, which is very close to ours, and the correspon- ding calculus CF. Unfortunately CF is not a conservative extension of the ordinaxy intuitionistic logic. For example formula Vx(Av C) ~ (VxAvC), where x does not occur in C, is provable in CF. It seems that even in the propositional case the duality laws of intuitionistic logic with strong negation were not mentioned before. In w 1 we introduce a Hilbert-type calettlus H formalizing intuitionistic logic with strong negation. In w 2 H is interpreted in the ordinury intuition- 4 -- Studia Logica, 1--2/77 50 Yuri Gttrevich istic calculus H and it is proved that H extends H conservatively. In w 3 Kripke models of II are defined. In w ~ the completeness theorem is proved. In w 5 duality laws are proved. In w 6 complete and independent systems of logical operators for H and for the propositional part of H axe presented. In w 7 a 3-vnlued logic associated with H is discussed. In w 8 a Gentzen-typo calculus corresponding to H is considered. The paper was written in ]r in 1972 and translated into English in 1976. Discussions with Leo Esa,kia were very usefull to the author. Note: Metalogic of this paper is classic. w 1. Predicate calculus I1,_ this section we define a calculus H formMizing intuitionistic logic with strong negation. Let H be the intuitionistie predicate calculus of [3] enri,:.hed by a denumerable list of individual constants. Recall that A ~B abbreviates (A ~ B)&(B= A). It is obtained from It by adding a new unary propositional connective "--" (called strong negation or minus) and the following axiom schemata: --(A = B)~A &--B, --(A &B)~--Av--B, 3. --(AvB)~--A &--B, -nA~.~A, -- --A~A, -- 3xA ~Vx -- A, .. -- VxA ~ 3x -- A, (for atomic A's only) --A ~ hA. tterc and below minus and other unary logical operators bind closer than any binary connective. Here and below A and B range over the H-formulae if the contrary is not said explicitly. In this section the sign F means provability and deducibility on H. Clearly H satisfies the Deduction Theorem. THEOREM ].l. ~- A ~ (A ~ B). PROOF. It is enough to deduce B from A and --A without variation of variables. If A is atomic use 5. In the other cases use i, ..., 7 respect- iYely. # Hence F--A ~ n A for all A's, not only for atomic one's. In order to prove replacement theorems fix an H-formula C and a propositional letter p. Let C~ be the result of replacing all occurrences of :p in C by A. Let V., be the set of individual variables x such that x Intuitio~istio logic with st~'ong negatio~ 5]. occurs free i~ A and some occurrence of p in C lies in the scope of 3x or ~x. T~LEORE~[ 1.2. (Replacement property of equivalence). If the mit~us does not occur in C then A--~B F C t-.~ C B where the variables of Ira w V E are varied. PROOF. See proof of the Replacement Theorem (Theorem 14) in [3]. ~: Let A ~ B abbreviate (A~B)& ( -A~--B) (the strong equivalence). LE~'IA 1.3. (i) A =--BF, --A ~ --B, and the same forT; (if) A1 -~ BI, A2 ~ B~ ~ AI ~ A2 ~ Bl ~ Be, and the same for & and v ; (iii) A :-=B FVxA ~-:VxB where x is varied, and the same for 3. Pi~ooF is clear. ~: THEORE~ 1.4. (Replacement property of the strong equivalence). A --B ~ CA = Cn where the variables of VAuV B are varied. PROOF by induction on C. The induction step uses Lemma 1.3. w 2. Reduced formulae Her~ we interpret H in H and prove that H extends H conserv,~tively. A is called reduced (cf. [10]) iff the scope of each occurrence of minus in A is an atomic formula. ~dm H-proof (A1, ..., A~) is called reduced fff the formlflas A1, ..., An are reduced. The reduction operation r is defined inductively: rA =A and r(--A) ----- --A if A is atomic; r(TA) = r(A); r(A ~ B) =r(A)~ r(B), and the same for &and v; r(VxA) = Vx(rA), a.nd the same for 3; r(--(A ~ B)) = rA &(--rB); r(--(A &B)) = (-rA)v(--rB) and r(--(AvB)) = ( --rA)& (r-B); r(--VxA) = 3x(--rA) and r(--3xA) = Vx(-rA); r(--TA) =r(----A) =rA. Theorems 2.1 and 2.2 below generalize the analoguous results in [10]. T~[EOrCE~ 2.1. rA is reduced and the formula A ~r(A) is provable in H. PROOF. Easy induction on A. @ THEORE3~ 2.2. If A is provable in H then there exists a reduced proof of rA in H. 52 Yuri G~revich PROOF. Easy induction on the length of an H-proof of A. @ THEOREM 2.3. Let P~, ..., Pk, Q~, ..., Qk be different predicate letters where Pi ~md Qi are m~.- ary; E = A Vx~ ... xmi (Qix~ ... x,~ i ~ ~ Pxx ... x,,,f) ; i A ra~tge over the H-formulae with predicate letters among P~ ..., Pk; qA be the result of replacing --P~, ..~ --Pk in rA by Q1,...~Qk. Then A is provable in, H iff E ~ qA is provable in H. PRom< Suppose that A is provable hi H. By Theorem 2.2 there exists a, reduced proof (A~, ...~ A,) of rA. Without loss of generality all predicate letters OCmlrring in this proof axe among P~,..., Pk. Then (qA~,...,qA,,) is a deduction of qA in H from hypothesises of the form Qit~ ... t,,~ ~ qP~t~ ... t,,~ which are deducible in H from E. Now let (B~, ..., B,) be an H-proof of E ~ qA. Without loss of generality all predicate letters occurring in this proof are amongPl, ..., Pk, Q~, ..., Qk. Let A~, ..., A,~ and D be the results of replacing Q1, ..., Qk by -P~, ..., --Pk in B~, ...,B,~ a.nd E respectively. Then (A~, ..., A,,) is an H-proof of D ~ rA and D is provable in H. By Theorem ,.1,'~ A is provable in H. @ Let wA be the result of repl~ming the strong neg,~tioll in A by the ordinary one. LE~La 2.4. _Let A be reduced. I.f A is provable in H then wA is pro- vable in H. PROOF. Easy induction on the length of a reduced H-proof of A. :~ THEORE~f 2.5. Let A be an H-formula. If A i.~ provable in H the~ A is provable in H. PROOF. Use Lemma 2.4. N: w 3. Kripke models We ree,~ll here Kripke modcls of H (according to [1]) and define Kripke models of H. Let C,,(A) be the set of individua.1 constants occuriing in A. A Kripke model of 11 (respectively H) is a quadruple 3C = (M, ~<, 8, r> where: (M,<<? is a poser (the poser of stnges of S); b associates a set of individual constants with each stage in such a way that X ~< Y implies bX ~_ bY; and z: M • (the set of H-sentences)-~{0, 1} (resp. ~: M • (the set of H-sentences)-+{--1, 0, 1}) satisfies the following conditions 3.1-3.7 (resp. 3.1-3.14): 3.1 If A is ~tomic, rxA =/=0 and X~ Y then Cn(A) ~_ dX and zvA = zxA; 3.2 ~v(A & B) = 1 iff min{zxA, zxB} = 1; 3.3 rx(A v B) = 1 iff Cn(AvB) ~_ 6X and max{z_u rxB}=l ; Intuitionistic logic with strong negation 53 3.4 Tx(qA) =1 iff Cn(A) ~_ (~X and for each Y~X, TyA<I; 3.5 zx(A ~ B) =1 iff Cn(A ~ B) ~ ~X and for each Y ~ X, if vzA =I then ~yB =I; 3.6 ~xVxA(x)----1 iff for every Y>~X ~nd cE~Y, vt~A(c)_--i ; 3.7 vx3A(x) ---- 1 iff there exists e~X such that ~:xA(c) = 1; 3.8 vx(A & B) = --1 iff Cn(A & B) ~_ ~X andmin{vxA, vxB } = --1; 3.9 ~x(AvB) = --1 iff max{7:xA, vxB} = --1; 3.10 Vx(TA ) = --1 if/ ~xA =1; 3.11 ~x(A D B) = --1 iff zxA = 1 and ~xB = --1; 3.12 ~:xVxA (x) = --i ill there exists c ~ ~X such that ~xA (e) = --1; 3.13 vx~xA(x) = --1 iff for every :Y~ X and c~SY, vyA(c) ---- --1; 3.14 ~x(--A) = - ~xA.
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