18Notre Dame Journal of Formal LogicVolume 29, Number 1, Winter 1988

Constructive Predicate Logic with StrongNegation and Model Theory

SEIKI AKAMA*

In this paper, we attempt to investigate the so-called constructive predicatelogic with strong negation from a model-theoretical point of view.

Strong negation was first introduced by Nelson [8] in connection withKleene's recursive realizability. At the same time, Markov [7] showed indepen-dently that intuitionistic (Heyting) negation can be defined by strong negationand implication. Then Vorob'ev [14] formulated constructive propositional logicwith strong negation. Polish logicians such as Rasiowa [9],[10] studied in termsof lattice theory. Additionally, we can find a Gentzen-type formulation byAlmukdad and Nelson [1] and Ishimoto [5],[6], and a model-theoretic study byThomason [13] and Routley [11], to cite only a few.

Strong negation is a constructive negation different from Heyting negation.For example, in intuitionistic logic, -^(A r\B) is not equivalent to the deriva-bility of at least one formula of -*A or ~^B. And we cannot prove the equiva-lence between -^VxA(x) and -^A(t). But these are equivalent in constructivelogic with strong negation.

In this paper, we research this system on the basis of Kripke's many worldssemantics (the so-called Kripke model), and try to provide a Henkin-type proofof the completeness theorem for the system. We will also inquire into the rela-tionships among constructive predicate logic with strong negation, classical logic,and intuitionistic logic.

1 Constructive predicate logic with strong negation Constructive predicatelogic with strong negation, instead of Heyting negation, is designated as S. Asstated above, Heyting negation can be defined in S by way of strong negationand implication, as S does not have it as one of its primitives. We call the sys-

*I would like to thank the referee and Dr. Ishimoto for their valuable suggestions onthis paper.

Received November 20, 1985; revised February 24, 1986

CONSTRUCTIVE LOGIC 19

tern with Heyting negation S+. The well-formed formulas (w.f.f.) in S and S+are defined in terms of parameters (free variables), bound variables, predicatevariables including equality (=) as a binary predicate and function symbols withany finite number of arguments as well as seven logical symbols, namely, (conjunction), v (disjunction), D (implication), -i (Heyting negation), ~ (strongnegation), V (universal quantifier), and 3 (existential quantifier). Here we oughtto pay attention to the fact that implication is a logical symbol in the sense ofintuitionistic logic. In fact Rasiowa [10] introduced a symbol => defined in termsof ->(D) as follows:

A => B= (A-+B) (~B-+ -A). (1.1)Markov [7], on the other hand, used complete equality =;

A = B (A S B) (~A = ~B). (1.2)Then, S is the system in which strong negation is embedded in intuitionistic logic.

We can define the Hilbert-type version of S as follows:

YAD.BDA (1.3)\-(A D.BDC) D. (ADB)D (AD C) (1.4)\-A D (B D.A B) (1.5)\-AB.D.A (1.6)\-A/\B.D.B (1.7)\-(A DC) D. (BDC)D (A v B .D C) (1.8)YAD.AvB (1.9)h D . , 4 v (1.10)h~ D./1D5 (1.11)}-~(A DB) =.A ~B (1.12)h~( B ) =. - v ~ (1.13)\-~(A vB) s . ^^ 4 ~B (1.14)M = A (1.15)hv4(x) Dy4(0 (1.16)\-A(t) D 3xA(x) (1.17)IVX4(A) S 3X ~A(X) (1.18)I3JC4U) s VJC - (x). (1.19)

This version of S is closed under detachment and the rules of restriction on vari-ables in quantification.

\-ADB(t)^\-AD VxB(x) (1.20)\-A(t) DB=> \-3xA(x) D B. (1.21)A s 5 is an abbreviation for (AD B) A (BDA). Deleting Axiom (1.12)

from the list of axioms, Fitch's system [3] is obtained. It should be noted herethat the formula Vx(A v B(x)) D (A v VxB(x)) (x not free in A) is not prov-able in S. But it is provable in Fitch's system (see Fitch [3] and Thomason [13]for details). Fitch's system has a Kripke semantics with constant domains,whereas S requires variable domains, as discussed later.

As stated above, Heyting negation can be defined in terms of strongnegation;

-iA=AD~A. (1.22)

20 SEIKI AKAMA

In S+ involving Heyting negation, the following formulas are provable:

Y(A D B) D. (A D -^B) D -A (1.23)t-iAD.ADB (1.24)YA**~^A. (1.25)Naturally we can find that S+ is equal to Vorob'ev's system. Moreover,

the following two axioms for equality can be added to S:

Ya = a (1.26)\-a = b D.A(a) DA(b). (1.27)

2 Kripke's many-worlds semantics The Kripke model for 5 is a quintuple * G VP(A))c. vr (evN(A)**r*evN(A)).

(2.4) P is a function which maps from G to a nonempty set of parame-ters which holds P() c P(*) for any satisfying TRY*.

Here 4^ denotes a propositional variable, and Greek capitals such as , ,etc. denote meta-variables ranging over elements of G. And v* and 3* arequantifications over the elements * G G such that TRY* (cf. Fitting [4]).Namely, the relation R* says that * is better than because of partial orderrelation of R.

We can define two forcings, |=p and tN, with respect to any element of Gand propositional variable in the Kripke model (=p ^ 1 and NN 5 (2.16)

CONSTRUCTIVE LOGIC 21

N ^B e> tpB (2.17) |=N ~B tpB (2.18) \zN A *> v** t=N A for any atomic A (2.19) N v vxA(x) *> N M ( O for some t E P() (2.20) tNlxA(x) *NP^),V*(\=NA^T*\=NA)

for any e G and A.Proof: We can prove this lemma by induction on the length of the formulas.The induction steps are as follows:

a. Np A B & \=P A and \=P B=> * \=P A and * \=P B & * (=P A B.

b. \=pADBt* V*(* \=PA ^ * NP B)= V**(** \=P A =* ** Np ) ^ * Np >1 D 5.

c. f=PVj&4(jt) & V** N P ^ 4 ( O for a l l / E P ( * )=> v**** tpA(t) for all / e P(**) * NpVx4(x).

d. Np -^1 ^ N^ ^ 4 => * |=N yl ^ * |=P -^4.e. (=p -A ^ V** Np>l => v**** PA & * NP - In the case of ]=N, the induction steps are similar to those in the case of

\=P. Lemma 1 tells us that Np and ]=N have a monotonicity property.Lemma 2 Given a model

22 SEIKI AKAMA

Proof: We can prove the theorem by induction on the length of the proof. Theproof of axioms without strong negation is similar to that of classical logic. Theinduction steps for other axioms are exemplified as follows:

NP (~A v ~B) D ~(AB)& V*(* |=p ~A v ~B => * (=p ~ (A B))#> V*(* Np ~A or* t=p ~ => * YN A 5 ) * (= ^ or * \=N B).

\=PA~BD ~(ADB)& V*(* Np A A ~B => * Np - (^ D /?))* V*(* Np^land* NP ^ B => * | = N ^ 4 D 5 )^V*(* YpA and* N^ B = * NP yl and * ^J).

tp~vxA(x) D 3x~A(x)*> V*(* Np ~VJC4(X) => * Np 3JC ~ yl(jc))^ V*(* |=N VX^I(JC) => * (=p ~ ^ ( 0 for some G P(*))^ V*(* N N i4(0 =>* NA(t)) for some G P ( ) .

Likewise we can prove the consistency theorem for S+:

P~-^A DA & V*(* tp~-iA=** VPA) V*(* tN -^A => * \=PA)* V*(* Np^=** YpA).

T Yp-^AD .AD -A. ^ V*(* Np pA =* * YP A D -A).** \=PA=>** NA).

T Yp-iAD .ADB. ^ V*(* Np ^A=>T* YpAD B)*

V**(** pA =*. ** Npv4 =>** | = P )^

V**(** YNA =>. ** N N ^=>** N ^ ) .

Next we show that the converse of Theorem 1, namely, the completenesstheorem, is provable. The proof is carried out by means of the Henkin-typeproof construction. Thus we need to introduce several concepts for the proof.According to Shoenfield [12] and van Dalen [2], the intuitionistic counterpartsof the Henkin theory can be used in our proof.

Definition A set of sentences satisfying the following conditions is a primetheory with respect to L:

(3.1) is closed under \-(3.2) AvBE =>AeT or BG(3.3) 3xA(x) G => A(c) G for some constant c in L.

The constant c in (3.3) is an analog of the witness constant in Henkin the-ory. Then we extend language L to L by adding a suitable set of witness con-stants. Thus the following lemma is provable.

Lemma 3 Let and be closed. IfT\f, then there exists a prime theory' such that V \f .

CONSTRUCTIVE LOGIC 23

Proof: The prime theory of can be defined as:L

n=Lc

n

Cn+{ = Cn U [c\c is a witness constant}.

Then ' is obtained by series of extensions o g That is, ' is a union

of and the set of witness constants satisfying (3.3). Now put * to be givensuch that Tk\f and contains a lot of witnesses. Then two cases can be consid-ered. First, suppose that k is even. Take the first existential sentence 3xA(x) inL that has not been treated, such that * h 3xA(x)9 and assume that d is a wit-ness constant not appearing in Tk. Then Tk+\ = ^ U {A(d)} holds.

Suppose, on the other hand, that k is odd. Take the first disjunctive sen-tence AvB such that Tk\- Aw B was not treated before:

Tk+ = Tk U {A} if Tk, A \fkU{B}ifk,A\-.

From (3.2), it is not needed that both l 5 A \- and 2, B (- hold simulta-neously. " consists of all IVs satisfying the above-mentioned properties, ' =( J IV As a result, it suffices to show the existence of the prime theory inwhich T' \f holds.Case L " \f . First, we prove , \f by induction on /. If / is even, thenassuming ,

+ 1 \- , , , A(d) \- holds. Though we obtain , |- since , hlxA(x), this contradicts the assumption. Thus /+1 \f holds. For any /, , \f holds by induction on /. If " \- ,

z \- for some /. But by contraposition and

7 \f , ' \f holds.Case 2. ' is a prime theory. We may check three conditions on a prime theory:

(3.2): Let A v B G ' and k is the smallest number such that k\-AvB.By assumption, for any h such that k < h,Th\- A v B. Then A E Th+ or B G

+ 1 . Hence A^' or Be '.(3.3): Let 3xA(x) G V and k, h are described as before. By assumption,

since

\- 3xA(x), A(d) G + 1 c ' holds. Also it is trivial that (3.1) holds.

Then, we get Lemma 3.Next, we have to prove that there exists a model for unprovable formulas:

Theorem 2 (Model Existence Lemma) If \f then there is a Kripkemodel such that 0/?

and o . [Here Rn is an -time application of R

defined in (2.2).]Proof: Let C = {Q| / > 0} be the set of constants not appearing in L. In the firstplace, we shall extend to the prime theory , such that , \f . Consider thelanguage U such that for any k belonging to the set of natural numbers IN it is

A r - la union of L and the set of constants C = ( J c, . Then we can define a Kripkemodel satisfying Theorem 2. According to the definition in Section 2, if G is N,(2.3) is sufficient since N is a partially ordered set. Also VP and VN are definedas follows:

VP\ IN -> [n\n G IN, n is even}VN: IN-* {A?|A2GIN, n is odd).

24 SEIKI AKAMA

Moreover we shall consider a mapping P from IN to the set of constantsand mapping from N to the set of atomic formulas containing a constant overP(N). As 0Rkk9 it is clear that P(0) is a set of constants in o and (0) is a setof atomic formulas in

o.

Now, let N correspond to ordered pairs (o>o)> (i>0i)> in L' satis-fying (n), at \f j8/. And for each i, apply Lemma 3 to (n) U (,) and ft . Asa result the prime theory

z is obtained. Now {n) is the constants and atomic

formulas belonging to , . This model must be satisfied by the following state-ments since it is a prime theory:

ntP) n (=P A v B & n |=P A or n hp B=> (n) \-A or (n) V B & (n) \-A v B.

=) () h i4 v B & (n) h A or E(/) h 5=> /i |=p v4 or n VP B 1 (/ B holds. Andthere exists some n' >n such that (AZ) U {^ 4} g (') and (nf) \f B. Fromthe inductive basis, n' YP A and from n' >n and n |=P A D B, n' |=P B but itis a contradiction since (n') \- B by hypothesis. Therefore () \- A D B.

=) () M 3 ^ ^ (n) h v4 => (/2) h 5vm(m f=p ^ 4 =* w |=p ^ ) for m > /2 (m) hv4(0

CONSTRUCTIVE LOGIC 25

Theorem 5 A is valid in all models iff A is valid in all models with the setof denumerable parameters.

Proof: Necessity is trivial. Sufficiency can be proved as follows: Consider themodel A) = TA D -TA (3.8)T(VxA(x)) = VxT(A(x)) (3.9)T(lxA(x)) = 3xT(A(x)). (3.10)We provide a Kripke model for H before we consider the relationships

between S and H. The Kripke model for H is a quadruple [=P TA and NP TB. NPTA TB.

& r =p M #) .T YHADB^ V*(* N H ^ => * \=H B) =* V*(* hp 7M = * NP TB)

** tpTADTB & p T(A D B).YYH-^A V** fep TA *

P~TA) V** H A{t) for all G P(*)=* V** Np 7M(0 for all G P(*) NP VJC7M(X) ^ (=P (VJO4(X)).

Theorem 6 (Embedding Theorem) 4^ w v/// m H iff TA is valid in S.

26 SEIKI AKAMA

Proof: First we should prove the necessity condition. If we assume that A is notvalid in H then there exists a model

CONSTRUCTIVE LOGIC 27

ollary of Theorem 7, the completeness theorem for C is obtained. Thus if A isvalid in C then -A D A is a theorem of S. Though -A D A is a theorem of Csince 5 is a subsystem of C, this is equivalent to A. Then A is a theorem of CThe following corollary holds by means of -A = A D -A.

Corollary 3 (Embedding Theorem) A is valid in C iff ~^~A is valid in S+where A is a formula not containing D.

REFERENCES

[1] Almukdad, A. and D. Nelson, "Constructible falsity and inexact predicates," TheJournal of Symbolic Logic, vol. 49 (1984), pp. 231-233.

[2] van Dalen, D., Logic and Structure, 2nd Ed., Springer-Verlag, Berlin, 1983.[3] Fitch, F. B., Symbolic Logic, Ronald Press, New York, 1952.[4] Fitting, M. C , Intuitionistic Logic, Model Theory and Forcing, North-Holland,

Amsterdam, 1969.

[5] Ishimoto, A., "A Schtte-type formulation of intuitionistic functional calculus withstrong negation," Bulletin of the Tokyo Institute of Technology, vol. 100 (1970),pp. 161-189.

[6] Ishimoto, A., "The intuitionistic functional calculus with strong negation and someof its versions," Bulletin of the Tokyo Institute of Technology, vol. 111 (1972), pp.67-94.

[7] Markov, A. A., "A constructive logic" (in Russian), Uspehi MathematiceskihNauk, vol. 5 (1950), pp. 187-188.

[8] Nelson, D., "Constructible falsity," The Journal of Symbolic Logic, vol. 14 (1949),pp. 16-26.

[9] Rasiowa, H., "N-lattice and constructive logic with strong negation," FundamentaMathematicae, vol. 46 (1958), pp. 61-80.

[10] Rasiowa, H., An Algebraic Approach to Non-Classical Logics, North-Holland,Amsterdam, 1974.

[11] Routley, R., "Semantical analyses of propositional systems of Fitch and Nelson,"Studia Logica, vol. 33 (1974), pp. 283-298.

[12] Shoenfield, J. R., Mathematical Logic, Addison-Wesley, Reading, Massachusetts,1967.

[13] Thomason, R. H., "A semantical study of constructive falsity," Zeitshrift furmathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp.247-257.

[14] Vorob'ev, N. N., "A constructive propositional calculus with strong negation" (inRussian), Doklady Akademii Nauk SSSR, vol. 85 (1952), pp. 465-468.

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