Bulletin of the Section of LogicVolume 19/4 (1990), pp. 133137

reedition 2005 [original edition, pp. 133138]

Barbara Klunder

TOPOS BASED SEMANTICS FORCONSTRUCTIVE LOGIC WITH STRONG NEGATION

The theory of elementary toposes plays the fundamental role in thecategorial analysis of the intuitionistic logic. The main theorem of thistheory uses the fact that sets E(A,) (for any object A of a topos E) areHeyting algebras with operations defined in categorial terms. More exactly,subobject classifier true: 1 permits us define truth-morphism on and operations in E(A,) are defined by them uniformly.

The aim of this paper is to show usefulness of toposes in the categorialanalysis of the constructive logic with strong negation (CLSN , for short)too. In any topos E we distinguish an object and its truth-arrows thatthe sets E(A,) have the structure of a Nelson algebra. The object (in-ternal Nelson algebra) in E is defined as a result of an application, to theinternal Heyting algebra , the topos counterpart of the well-known clas-sical construction of the Nelson algebra N(B) for a given Heyting algebraB, (see [1], [4] for a generalization).

We denote by HA the variety of all Heyting algebras and by NA thevariety of all Nelson algebras. Explanations of definitions and notations ofused notions from topos theory are in [2]. Truth-morphisms are denotedlike respective connectives.

1. Object and its truth-arrows

Let E be an elementary topos. We shall write 4 instead of and projections from 4 to will be denoted by pr4i for i = 1, 2, 3, 4. Ofcourse 4 = ( )2 and projections defining this product we denote

134 Barbara Klunder

analogously by pr2i (i = 1, 2). It is obvious that < pr41, pr

42 > = pr

21 and

< pr43, pr44 > = pr

22.

Let (, : ) be an equalizer of a pair (, false). Ofcourse falsef = falseA for any f : A . We shall use morphism2 : 4 which is defined as follows: 2 = < pr1, pr2 >(pr1, pr2 : define product ).Lemma 1. Each of morphisms , , : 4 defined bycompositions:

:= < pr41 pr43, pr42 pr44 > 2 := < pr41 pr43, pr42 pr44 > 2 := < pr41 pr43, pr41 pr44 > 2

as well as each of , : defined by := < pr1, pr1 > := < pr2, pr1 >

equalize the pair (, false)Proof. Equation = false is a conclusion of the after-mentionedcomputation. All equations hold because E(,) is a Heyting algebra. = ((pr41 pr43) 2) ((pr42 pr44) 2)

= (pr41 2 pr43 2) (pr42 2 pr44 2)= (pr41 2 pr43 2 pr42 2) (pr41 2 pr43 2 pr44 2)= (pr41 2 pr42 2 pr43 2) (pr41 2 pr43 2 pr44 2)= (( pr1) pr43 2) (pr41 2 ( pr2))= (false(pr43 2) (pr41 2 false)

(The last equation holds because = false)= false false = false.The proof of = false is analogous to the one presented for

conjunction.Of course = (pr41 2 pr43 2) (pr41 2 pr44 2). But in

the Heyting algebra E(,) : (pr41 2 pr43 2) pr41 2 pr43 2.Thus (pr41 2 pr43 2) (pr41 2 pr44 2) pr43 2 pr44 2

and pr43 2 pr44 2 = pr2 = false and = false. Itis clear that = = false. 2

Topos Based Semantics for Constructive Logic with Strong Negation 135

Now we can define truth-morphism ,,: and ,: as liftings of the respective s along , i.e. for {,,,,} isthe unique morphism such that = . Because false true =true false = false we can distinguish two morphisms T, F : which are liftings of and respectively along.

2. Nelson algebra structure of E(A,)

Let A be an arbitrary object of a topos E. For any f, g E(A,) wedefine:

f g := < f, g >f g := < f, g >f g := < f, g >f := f f := fFA := F !ATA := T !A

Theorem 1. The algebra is a Nelson al-gebra isomorphic to the algebra N().Proof. If is sufficient to verify that the map H : E(A,) N(E(A,))defined as follows:

H(h) = (pr1 , pr2 h)is a required isomorphism. 2

Examples. (1) In topos Set we have = {0, 1} and all operations ,,, are usual operations in Boolean algebra 2 = < {0, 1},,,,, 0, 1 >.It is easy to see that = {(0, 1), (0, 0), (1, 0)} and truth-morphism establisha structure of Nelson algebra on . Of course in Set, Set(1, A) = A andN(Set(1,)) = N(2) = Set(1,).

(2) We shall consider topos SetP for any poset P . In SetP , : P Set is defined by components P = [p)+ (P is equal to set of all hereditarysets of [p) = {q P : p q}). Truth-morphisms are defined by componentstoo. For example P : P P P is a set-theoretical intersection.Now it is easy to see that : P Set is defined by components

136 Barbara Klunder

P = {< S, T > P P : S T = 0}.

3. Completeness theorem

Let 0 be a set of variables {x0, x1, . . .} and be a set of all formulas builtup using of the connectives of CLSN . We shall say that any sentence is true on a valuation V : 0 E(1,) if V = T , where V : E(1,) isan usual extension of V . Any sentence V is valid in topos E iff V = Tfor any valuation V . This fact we denote by E |= . One have noticedthat we simply define an usual semantics in a Nelson algebra E(1,).

Theorem 2. Let . The following conditions are equivalent:(i) is tautology of CLSN(ii) E |= , for any topos E(iii) SetP |= , for any poset P(iv) SetPIL |= , where PIL is the canonical frame for the intuition-

istic logic.

Proof. In the proof we shall use the following facts:

(1) [1], [3], [4]. For any the following conditions are equivalent:(i) is a tautology of CLSN(ii) is valid in every Nelson algebra(iii) is valid in every Nelson algebra of the form N(B) for every

Heyting algebra B.In particular, the least subvariety of NA containing all algebras of the

form N(B) (B HA) is equal to NA.(2) [2] Let be a sentence in the language of intuitionistic logic. Then thefollowing conditions are equivalent:

(i) is a tautology of intuitionistic logic(ii) PIL |= (iii) SetPIL |= .

(2) Let capital letters H,S, P, I denote usual operators of homomorphicimages, subalgebras, products and isomorphisms respectively. For any classK of Heyting algebras we have HSP (N(K)) = IS(N(HSP (K))). Henceif K is a variety then HSP (N(K)) = IS(N(K)).

Topos Based Semantics for Constructive Logic with Strong Negation 137

It is obvious that all implications except (iv) (i) hold.Proof (iv) (i). Let A = {PIL}. HSP (N(A)) = IS(N(HSP (A))) =

IS(N(HA)) = HSP (N(HA)) = NA. 2

References

[1] M. M. Fidel, An algebraic study of a propositional system of Nel-son, Mathematical Logic, Proc. of the First Brazilian Conference, MarcelDekker, New York 1978, pp. 99112.

[2] R. Goldblatt, Topoi. The categorial analysis of logic, Studies inLogic, vol. 98, North-Holland Publishing Co.

[3] H. Rasiowa, Algebraische Charakterisierung der intuitionistischenLogik mit starken Negation, Constructivity in Mathematics Proc. of theColl. held at Amsterdam 1957, Studies in Logic and the Foundationof Mathematics, pp. 234240.

[4] D. Vakarelov, Notes on N -lattices and constructive logic withstrong negation, Studia Logica, vol. 36 (1977), pp. 109125. North-Holland Publishing Co.

Institute of MathematicsNicolaus Copernicus UniversityTorun, Poland