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,<
3.4 Tx(qA) =1 iff Cn(A) ~_ (~X and for each Y~X, TyA Suppose that A is not provable in H. By Theorem 2.3, E ~ qA is not provable in H. By Theorem 3.5 there exists a ICripke model ,~ = (M, <~, 6, a',) of H where A is defined but not true. Without loss of generality E is true in X. For, (E ~ qA) is defined but not true at some stage X of .~. Hence there exists Y >~ X such that ryE = 1 r ~y(qA). Take the sub- model of ~ with the set {ZE M: Z~> Y} of stages. For every formula P:q ... cm~ and stage X of ,~ define: if axPiq ... emi ~- l, -~~ Piel ... emi = 1 ff axOie~ ... emi = 1, otherwise The definition is emweet since E is true in ~. By Lemma 3.2 there exists a unique model .~o ~.ir, ~, d, v) of H such that T extends re. By induction on a subformula B of A it is easy to cheek that at each stage X, r,:B = 1 implies (rx(qB) = 1. Hence A is not true in s 4~ CO~mLL.tg~ 4.2. (Adequacy Theorem for H). A is provable in II if.f it is true in each model of H where it is defined. w 5. Duality Here we prove that an inessential extension of H satisfies very natural duality laws. Let calculus HI be obtained from II by adding a unary propositional connective a, binary propositional connective fl and the following axiom schemas: aA ~ -7--A, (AfiB) --~ --(--A ~ --B). HI-formulae can be regarded as abbreviations of H-formulae. Now we define duality of the lbgical opera,tors: & is dual to v and vice versa,, V is dual to 3 and vice versa, 7 is dual to a and vice versa, is dual to fl and vice versa, -- is dual to itself. Below in this section s range over the logical operators of t11, ~ is the operator dual to s, A and B range over the Ill-formulas and the sign means provability and deducibility in HI. LE3n~A 5.1. t- .... A --A; I---sA =~ ~(--A) where s is"l or a; Intuitionistic logic with st~'ong 't~,egatio~ 55 F--(AsB) =--(--A)~(--B) where s is &, v,D or fl; F-sxA ---Sx(--A) where s is V or 3. PROOF is clear. COROLLARY 5.2. There exists an algorithm A ~A ~ which associates a formula A ~ with each formula A in such a way that A -- A ~ and the scope of each occurrence of minus in A ~ is atomic, and a logical operator s occt~.rs in A ~ iff s or ~ occurs in A. Below in this section: A' is the result of repl~cing each logical oper,~tor in A by the du,~l operatoI'; .~ is the result of replacing each atomic formul~ in A by its strong negation; A* = A'. THEORES'[ 5.3. FA* ~-A. PRooF by induction on A. @ Let A-~B abbreviates (A ~ B) & ( • D --A) (the strong implication). THEOnE~ 5.4. If ~ A ~B the'J~ ~ B'~A'. PROOF. Let (A~, ..., An) be a proof of A-->B. Then (_4~, ..., A,,) is proof of A-+B. Now A-~B F --B-~--.:4 F B*-->.4* ~ B'-~A'. ~: COROLLARY 5.5. If F A ~ B then ~ A' =~ B'. z.Yote. Calculus H1 c~n be conservatively extended ia such a way that the duality stntements 5.3-5.5 remain true. For example, H1 can be enriched by the connective -~ and the connective du~l to -->. w Propositional logic Here we prove Adequacy Theorem for the propositional paa't of H and present complete and independent systems of logical oper~tors for H ~nd its propositionnl part. Let PH be the propositional p~rt of H. Formulae of PH are those of H built from propositional letters by propositional connectives. Axioms of PH are those oi H which are PH-lo~'mul~e. ~[odus ponens is the only inference r(fle of PH. In this section A, B r~nge over the PH-formtLl~e, E ~',~nge over the H-formulue. LE~L~'IA 6.1. _Let (1,~, ..., .F,,) be a,~ H-proof, p be a propositional letter and for each i = 1,...,~, A~ be obtained from E~ by (i) omitting all Vx and ]x, and (ii) replaciq~g all atomic formulae which are not propositional letters by p. Then (A~,..., A,,) is a PH-proof. PROOF is clear. @ Hence a PH-formul~ provable in H is provable in PH. 56 Yuri Gurevich A triple .)ff ----(M, 4, ~} will be c~lled a Kripke model of P/t iff (M,~} is a poser and ~: M• set of PH-sentenees)-+{--1, O,1} satisfies the releva.nt conditions among 3.1-3.14. A is true in 9ff fff for every XE M, Zx A = 1. LEM~L~ 6.2. .Get M and ~ be as above and zo: M • (the set of atomic PH-sentences)--->( --1, 0, 1} satisfies condition 3.1. Then there exists a unique extensio~, ~ of 3 ~ such that (M, ~, v) is a Kripke model of PH. PROOF is clear. From Adequacy Theorems for /~ follows TItEORESi 6.3. (Adequacy Theorem for PH). A is provable i t~ PH iff it is true in all Kripke models of PtI. Formulae F~ and F2 are called strongly equivalent iff ('~1 -~--~2) is provable in /t. According to Corollary 3.4 strongly equivalent formulae can be considered to have the same meaning. It is worth while to study formulae modnlo strong equivalence. Formula ~A --(A ~ --A) is provable in Pt-I (see [10]). This fact can be easily checked. Conjuction and disjunction are mutually expressible using minus (Lemma 5.1). So we proved THEOrEm[ 6.4. ( --, &, ~ } and ( --, v, ~ } are complete systems of con~. nectives of PiI. LE~I_~A 6.5. System (-1, &, v, ~} is not complete i,~ PH. Pn00F by induction to nbsurdity. Suppose that (-lp ~ A) is provable in PH where minus does not occur in A. Without loss of generality p is the only propositional letter of A. Let ~ be a one-stage model of P/t. Then (A~p), or (A~-lp), or (A~(p & ~p)), or (A~(pv ~p)) is true in Yd'. If p is true in ~" then formulae ( --p ~ p) ~nd ( --p -- (pv ~p)) are not true in Y. Ifp is uncertain in ~ then formulae ( --p --= -lp) and ( --p ~ (p & -Tp)) are not true in 9ft. LEM~.[A 6.6. System (--, ~ } is incomplete in PH. PROOF. Consider the Kripke model of Figtu'e i where 1 ~ 2, 1 ~ 3, riP = 0, r~p ---- 1 a,nd rap = --1. Let A be built from p using connectives- ,~nd ~ only. p -P Figure 1 Intuitionistic logic with strong negation 57 If vfA = v3A then viA = rfA. We prove this fact by induction. Cases A =p and A = --B are clear. Let A ----B~ C. If v~A ----~3A :e --A, F-+O -- B, 1"->0 F-'-, --A ; .F-> --B --(AvB), F~O -- (Av B), F-+O /~-:- - (Av B) -A (t), F~O I~ --A (y) --3xA (x), F-->O T'-> --3xA (x) (y does not occur in A(x) A, F~ F F-~A F->A A, F-> O F--> -- -- A -- A , F--> T~EOREM 8.1. If F F-E in tI with all variables held constant than ~- F~E in G, and vice versa. PROOF imitates the corresponding" proof in [3]. THEORE~I 8.2. Giren a proof in G of a sequent in which no variable occults both free a~d bound, another proof in G of the same sequent can be found which contains no cut. PROOF imitates the corresponding proof in [3]. COROLLARY 8.3. 1~7, i~ (i) 'if k A v B then k A or k B, (ii) .if k --(A & B) then k --A or ~- --B, (iii) .if k 3xA (x) then k VxA (x) or k A (c) for some indi- vidual c(mstant e, (iv) if ~ --VxA(x) then kVx--A(x) or F --A(c) for some in- divid~eal constant c. (One CUR read " k --A" ~s "A is logically f~lse". So (ii) states that ff A & G is logicully fnlse then either A or B is logie~flly fnlse). References [1] M. C. FImTING, I~tuitionistic logic, model theory and forcing, North-Holland, Amsterdam, 1969. [2] A. GRZEGORCZYK,A philosophically plausible formal interpretation of intuitionistic logic, Indagatienes :~Iathemafivae 26 (1964), pp. 596-601. Intuitionislic logic with st ro~,g ~e.qation 59 [3] S.C. KLEEN]~, Lntroduction to ~etamalhematics, Van Nostrand, New York, 1952. [4] S. K[CIrK~, ,b'ematttical a.~talysis of intuitionistic logic [, in Forn~al syste~s and rccursive ~al~ctions, ~'orth-Holland, Amsterdam, 1965, pp. 94-130. [5] A.A. 5IARKOV, Konstrutticnaja logika (in Russian), Uspekhi ~)Iatematigeskih Naak 5.3(1950), pp. 187-188. [6] J.C.C. McKt~-sl~Y, Proof of independe~zce of the primitive symbols of Heyting's calculus of proposition, s, Journal o I Sytnbolic Logic 4 (1939), pp. 155-158. [7] D. NELSON, Constr~ctible falsity, Journal o] Symbolic Logic, 14 (1949), pp. 16-21. [8] H. RASIOWA, .,V-lattices ~nd constructive logic with st~'o~g ,~zegation, F l~nda~enta ;)Iather~taticae 46 (1958), pp. 61-80. [9] R. I-I. TI~OMASO~~, A semantical study of construetible falsity, ZeitschriIt [iir ~Jtathernatische Logik ~oid GJ,undlaf]en der Mathentatik 15 (1969), pp. 247-257. [10J N. N. VoRo~'Ev, Constructive propositional calcul~ts with strong negation (in Russian), Transactio~s o[ Steklov's Institute 72 (1964), 195-227. D:EPAICTMENT OF MATIIEMAT[CS BEN GURION UNIVERSITY OF TH~ NEGEV BEER SHEVA, ISRAEL Studia Logica XXXVI, 1--2