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26Логические исследования 2017. Т. 23. № 2. С. 76-95 УДК 510.64
Logical Investigations 2017, Vol. 23, No. 2, pp. 76-95 DOI: 10.21146/2074-1472-2017-23-2-76-95
v.l. vasyukoy1
Potoses: Categorical Paraconsistent Universum for Paraconsistent Logic and Mathematic2
Vladimir Leonidovich Vasyukov
Department of the History and Philosophy of Science Institute of Philosophy of Russian Academy of Sciences 12/1 Goncharnaya Str., Moscow, 109240, Russian Federation E-mail: [email protected]
It is well-known that the concept of da Costa algebra [3] reflects most of the logical properties of paraconsistent propositional calculi Cn, 1 < n < u introduced by N.C.A. da Costa. In [10] the construction of topos of functors from a small category to the category of sets was proposed which allows to yield the categorical semantics for da Costa's paraconsistent logic. Another categorical semantics for Cn would be obtained by introducing the concept of potos - the categorical counterpart of da Costa algebra (the name "potos" is borrowed from W.Carnielli's story of the idea of such kind of categories)
Keywords: paraconsistent logic, categorical semantics, potos, paraconsistent set theory, da Costa algebra
1. Introduction
It is well-known that the conception of da Costa algebra [3] renders the majority of the logical properties of paraconsistent propositional calculus Cn, 1 < n < w introduced by N.C.A. da Costa. In [10] the construction of the topos of functors from a small cateory to Set was proposed which allows to obtain the categorical semantics of da Costa's paraconsitent logics. Another categorical semantics of Cn would be introduced considering the construction of a potos or da Costa topos — a categorical equivalent of da Costa algebra.
xThis resarch is supported by RFH grant № 16-03-00364.
2The paper is an improved, extended and modified English version of early published in "Logical Investigations" (vol. 17) paper "Paraconsistent Categories for Paraconsistent Logic" (in Russian).
© Vasyukov V.L.
A potos is a paraconsistent universe in which paraconsistent mathematics would be developed the way it was done in case of intuitionistic mathematics in topos. But while in case of [10] all paraconsistency instances appear only as partial constructions in the intuitionistic universe (as some local artefacts) in a potos this paraconsistency is absolutely immanent and, moreover, it underlies all the constructions, it is global and fundamental. Here the classical mathematics emerges now as an artefact in paraconsistent universe, as some local deviation from paraconsistent regularity. Thus e.g. interpreting Cra-systems one would implement non-truth-functional valuation while truth-functional valuation becomes featuring just the case of Boolean toposes which are now only the particular case of potoses.
In [11] the construction of a potos as the Cartesian closed category (with the initial object 0 and the terminal object 1) along with distinguished object Q which is an implicit da Costa algebra was proposed, i.e. there are arrows true : 1 — Q, false : 1 — Q, — : Q — Q, n : Q x Q — Q, U : Q x Q — Q, D: Q x Q — Q which satisfy da Costa algebra conditions from [1, p.81] But the shortcoming of such a definition of potos is that the arrow of negation instead of other arrows is introduced only "locally" and has no connections with any categorical constructions.
In order to overcome this shortcoming we introduce the notion of so-called complementary closedness of Cartesian closed category. This allows to yield the arrows of negation "globally" following the recipe of the definition of other truth-arrows.
As the consequence of the new construction introduced we need to consider a new category of paraconsistent sets PSet where as the objects the ZFi-sets of paraconsistent set theory are exploited. The system ZF1 is correlates with Zermelo-Fraenkel set theory ZF0 the same way the paraconsistent first-order predicate calculus with identity correlates with the classical one. And at the same time category PSet turns out to be not a topos but a potos of sets.
Such type of considerations is based on J. Benabou's proposal (cf. [1]) to accept as minimal set-theoretical basis of category theory any set theory in ZFo-language exploiting only the extensionality axiom and the comprehension scheme. For any model of such a theory its ele-
ments will be the sets while the "meta-sets" of the universe of discourse (where the model is chosen) will represent the classes. Following Ben-abou we can define the set theory required as any theory formulated in ZF1 and choose any model of such a theory from the universe of discourse (in fact, extending our universe to the universe of all non-classical "meta-sets"). Then so-called Yoneda's map will assign to every set its representative class relative to our "paraconsistent" model while our sets will be exactly the sets we need for considering PSet.
In section 2 the minimal information concerning the theory of da Costa algebras is adduced which is required for the further considerations since the last are essentially exploited and substantially determine the potos construction itself.
In section 3 the notion of a potos is introduced, its properties and set-theoretical foundations are analyzed and the potos PSet is considered.
In section 4 an algebraic interpretation of da Costa systems Cnin terms of da Costa algebra is yielded along with potos-theoretic interpretation of those essentially based on this algebra. The completeness of the systems is proved in respect to the interpretation given. Besides, non-truth-functional valuation of C1 is considered and the completeness of this system is proved exploiting this valuation.
Finally in section 5 the interpretation in potos PSetA is considered and the completeness of C1 in respect to such semantics is proved.
2. Da Costa Algebras
W.A. Carnielli and L.P. Alcantara in 1984 [3] formulated the notion of da Costa algebra reflected the most of logical properties of logic Cn. It was shown that da Costa algebra is isomorphic to a paraconsistent set algebra which would be counted as an counterpart of Stone representation theorem for Boolean algebra. However, such an analogy works only if we takes a non-classical point of view: some operations in paraconsistent set algebra are formulated not in usual set-theoretical terms.
Since our theoretic-categorical constructions are essentially based on da Costa algebra then for the further proceedings the complete definitions are adduced.
Definition 1. [3, p. 81] By a da Costa algebra we mean a structure
A = (S, 0, 1, <, А, V, D,' ) such that for every a, b,c in S the following conditions hold:
1. < is a quasi-order;
2. a А b < a, a А b < b;
3. if c < a and c < b then c < a А b;
4. a А a = a, a V a = a;
5. a А (b V c) = (a А b) V (a А c);
6. a < a V b,b < a V b;
7. if a < c and b < c then a V b < c;
8. a А (a D b) < b;
9. if a А c < b then c < (a D b);
10. 0 < a, a < 1;
11. x° < (x')°, where x° = (x А x')';
12. x V x' = 1, where a = b iff a < b and b < a;
13. x'' < x, where x'' abbreviates (x')';
14. a° < (b D a) D ((b D a') D b');
15. x° А (x°)' = 0.
If there exists x € S such that it is not true that x А x' = 0 the algebra A is said to be a proper da Costa algebra.
Let us note that it would be much more natural to consider congruences instead of equations in items 4 and 5 in definition 1. This is the choice made e.g. by Carlos Caleiro and Ricardo Goncalves developed so-called Behavioral algebraization of da Costa's C-systems (cf. [2]).
Proposition 1. [3, p. 82] If A = (S, 0,1, <, А, V, d,' ) is a da Costa algebra then the following properties are verified: (C1) y < x iff x А y = y; (C2) x А 0 = 0, x V 1 = 1; (C3) x V 0 = x,x А 1 = x; (C4) x V y = y V x,x А y = y А x; (C5) if x = y then x = y; (C6) if a = b and x = y then x А a = y А b; (C7) if a = b and x = y then x V a = y V b; (C8) y < x iff y V x = x;
(C9) if x V y = 0 then x = 0 and y = 0;
(C10) if x < yo and x < (yo)' then x = 0;
(C11) p V (p' A po) = 1, p' V (p' A po) = 1;
(c12) x A y = 0 iff x < y' A yo;
(C13) x A (y A yo) = 0 iff x < y';
(C14) x A x' A xo = 0;
(c 15) if x A (x')o = 0 then x = x'';
(C16) if x' A (x')o = 0 then x = x'';
(C17) if x A y = 0 then x < y';
(C18) if y = yo then x A y = 0 iff x < y';
(C19) x A y' A yo = 0 iff x < y.
Proof. Obvious. □
Hereafter □ means the end of the proof.
Theorem 1. [3, p. 83] Every proper da Costa algebra has at least three elements.
Definition 2. [3, p. 83] A paraconsistent algebra of sets is a structure A = (S, 0,1, <n, u, )
where
1. n and U are the set operations of intersection and union;
2. < is a preorder;
3. S C p(I);
4. S is closed with respect to the binary operations n, u, and the unary operation ;
5. a n b < a, a n b < b;
6. if c < a and c < b then c < a n b;
7. a < a u b, b < a u b;
8. a n (a ^ b) < b;
9. if a n c < b then c < (a ^ b);
10. 0 < a, a < I;
11.x U x' ^ I, where a ^ b iff a < b and b < a;
12. x < x;
13. xo < (y ^ x) ^ ((y ^ x') ^ y'), where xo = (x n x')';
14. xo n (xo)' & 0;
15. xo < (x')o;
Let S0 = {x : x € S and x П x' ^0}. If S0 = 0, we have a proper paraconsistent algebra of sets. Every paraconsistent algebra of sets is a proper da Costa algebra while every Boolean algebra of sets is a non-proper paraconsistent algebra of sets.
If one consider a natural notion of congruence on da Costa algebra and then to define the notion of homomorphic image of a da Costa algebra (= isomorphism (projection on quotient by a congruence)) then the following result obviously will take place:
Theorem 2. Every proper da Costa algebra is isomorphic to a quotient to a proper paraconsistent algebra of sets.
In [3, p. 84], in fact, more weaker notion of isomorphism was considered which is not symmetric indeed. Given an equivalence relation ~ two da Costa algebras A and B are said to be ^-isomorphic if there exists a function f from A onto B preserving the operations and being ~ -injective that is, if x ф y then f (x) = f (y). And the following result take place
Theorem 3. [3, p. 84] Every proper da Costa algebra is = —isomorphic to a proper paraconsistent algebra of sets.
In [9, p. 273] the following theorem was proved:
Theorem 4. A set of principal filters of the proper da Costa algebra is = —isomorphic to a proper da Costa algebra.
Taking into account that every principal filter is determined by the single element of a da Costa algebra then this = —isomomorphism will be symmetric one.
3. Potoses
A potos is, in fact, a topos with some additional structure. In essence, we would equally well use the name "paraconsistent topos" or "da Costa topos". The name "potos" was borrowed from W.Carnielli's story of the idea of such kind of categories originated from some Brasilian mathematician.
Definition 3. A potos C is a Cartesian closed category which is also complementary closed and has a subobject classifier. That is:
(i) Chas finite products (—, -),[-, —] and Cis distributive relative to those, i.e. ([a, b], [a, c]) = [a, (b, c)] for any objects a, b, c in C;
(ii) C allows an exponentiation;
(iii) C has a terminal object 1 and an initial object 0;
(iv) a — b is an arrow in C iff a ^ b = 1, for any two objects a, b in C where a ^ b is an exponential;
(v) for any object a of C there is an object a' with the respective operations (functions) in C :
()' : obj(C) — obj(C) such that a — a',
dn : obj(C) — Hom(C) such that a — dn(a) : a'' — a,
hdn : Hom(C) — Hom(C) such that d — a — hdn(a, d) :
d—a ,
()° : obj(C) — obj(C) such that a — ao = (a, a')', cert : obj(C) — Hom(C) such that a — cert(a) : ao — (a')o, hcert : Hom(C) — Hom(C) such that d — ao — hcert(a,d) : d — (a' )o,
where dn(a) and cert(a) are monic and we have a fixed choice of co-products [—, —] and of products (—, —) for each pair of objects in the respective operations in C;
(vi) there is an operation triv : obj(C) — Hom(C) in C such that a — triv(a) : ao — (b ^ a) ^ ((b ^ a') ^ b') and triv(a) is monic;
(vii) 1 = [a, a'], 0 = (ao,ao')(with, possibly, binary coproducts and products);
(viii) a subobject classifier for C is a C-object Q together with an arrow true : 1 — Q that satisfies the following axiom: for each monic f : a ^ d there is one and only one arrow xf : d — Q such that
1
true
Q
is a pullback square.
Here complementary closedness of C is given by (v) — (vii). To put this another way, complementary closedness is given by:
(v) for any object a of C there is an object a', such that:
for any arrow f : d — a we have monos f' : d — a', dn(f) : a'' — a and cert(f) : ao — (a')o where ao = (a, a')',
(vi) for any two objects a,b in C there is a mono triv(a,b) : ao — (b ^ a) ^ ((b ^ a') ^ b');
(vii) 1 = [a, a'], 0 = (ao,ao')(with, possibly, binary coproducts and products).
Proposition 2. In a potos C the set Sub(d) of subobjects of d (= set of equivalence classes of monos with codomain d) is a da Costa algebra.
Proof. Since any potos C is a Cartesian closed category then for any object d in C the collection Sub(d) of all C-arrows that are monic with d as codomain will be preordered bounded distributive lattice. This gives us that the conditions 1-7 and 10 of the definition of da Costa algebra are fulfiled in Sub(d). The conditions 8-9 are the consequences of the exponentiation diagram. The conditions (v)-(vii) of the definition of potos provides us the conditions 11-15 are to be held. □
It is easy to see that in potos we have Sub(d) = Hom(d, Q) and thus Hom(d, Q) will be a da Costa algebra. But in this case the problem arises concerning the category Set of sets. The matter of fact is that in Set we have Sub(D) = p(D) where p(D) = {x : x is a subset of the set D} Since p(D) is a Boolean algebra of subsets and not the paraconsistent algebra of sets as we can expect from the theorem 5, then we come to the conclusion that Set cannot be a potos. But according the definition of a paraconsistent algebra of sets there are some sets which form such an algebra. So either such sets generates the subcategory PSet of Set or Set is, in a sense, a subcategory of PSet.
It is known (cf. [5]) that there is a system ZF1 of paraconsistent set theory that related to Church's version of Zermelo-Fraenkel set theory ZF0 with a universal set as a da Costa paraconsistent first-order logic C= is related to the clasical first-order predicate calculus C=. In essence, "ZFi should be 'partially' included in ZFq, though the latter is is also
to be contained, in a certain sense, in the former" [5, p. 170]. The basic set-theoretic concepts of ZF1 are analogous to those of ZF0, although the concepts involving negation give rise to two notions: one involving the weak negation (-) and the other the strong negation (-*). As a result we have, for instance, two empty sets: 0 = {x : x = x} and 0* = {x : - * (x = x)}.
The collection of all sets, plus 0, V, n, U,C (where V = {x : x = x} and xC = {y : y € x}) form in ZF0 a complete Boolean algebra. For ZFi we obtain the following result:
Proposition 3. In ZF1 the collection of all sets, plus 0* ,V, n, U,C* form a paraconsistent algebra of sets.
Proof. By immediate checking (putting x ^ y = {z : z € x — z € y}).
□
Each axiom scheme of ZFo generates two corresponding axiom schemes of ZF1, one with the strong negation and another with the weak one. Thus, we can say that ZF1 includes ZF0 and hence, Set is actually, in a sense, a subcategory of PSet. But what does it means to be a category of sets other than Set?
Shepherdson in 1952 [9] introduced a notion of an "inner model" of a logic L. He means a model whose universe is a subset of the universe of L, and in which the "true" statements are those statements of the model which are provable in L. Shepherdson assumes that the universe of L contains classes, sets, and possibly additional objects. The models have classes and sets and a membership relation €m. For every nonempty class A of L, there is a member y € A such that z €m y for every member z of A.
Later J. Benabou in [1, p. 18] trying to find the minimal set theoretical foundations for category theory defines a set theory as any theory T written in the language of Zermelo-Fraenkel set theory ZF and satisfying at least the extensional axiom E and the comprehension scheme CS.
Let us for any model M of such a theory the elements of M will be called sets and denoted by S,T,... and the formal membership and equality of sets will be denoted by S €* T and S = T. Then the "metasets" of the universe of discourse U where the model M is taken will
be called classes and denoted by S,C,... while the membership and equality in U is denoted by the usual notations € and =. A subclass S of M will be representable if there is a set S, called a representative of S, such that for all T € M we have T € S iff T €* S. The Yoneda map assign to each set S the representative class S = {T € M : T €* S}
The extensionality axiom, for example, read thus:
(E) For all sets S and T, S = T iff S = T.
Unfortunately, it is known that such naive set theory is inconsistent and the source of it is the exploitation of classical logic underlying such theory. To overcome this troublesome case we can e.g. use in the role of T Grishin's LST theory (cf. [8]) which is the set theory with the unlimited comprehension scheme based on the modified linear Girard's logic and which is free of the mentioned shortcomings. But in our case it does not matter since we have to consider paraconsistent set theory and we are concerned just with the universe of discourse U which is hypothetically contains any kind of sets.
Following Benabou's course we can now define a set theory as any theory T written in the language of ZF1 and choose any model M of such a theory from the universe of discource U (in fact, extending it to the universe of all non-classical "meta-sets"). Then Yoneda map will assign to each set S the representative class S = {T € M : T €* S} relative to our "paraconsistent" model M and our sets will be exactly the sets we need for considering the category PSet.
Proposition 4. PSet is a potos.
Proof. According to proposition 10 for any set I we always have [3, p. 84] a paraconsistent algebra of sets (S, 0,I, < П, U, ^,') where П and U are the set operations of intersection and union, < is a preorder, S С p(I),S is closed with respect to the П, U, and the unary operation '. If we consider inclusion functions as arrows then we can define x < y iff x — y U {b}, where b € S. We define x' = xc if x € S0 and x' = xc U т if x € So, taking So = {x € S: there exists т = {a,b} such that x П т = 0,xc П т = 0 and -i(x С {a,b})} = 0, xc being the set-theoretical complement of x. Finally, we define x ^ y is x' U y.
It is easy to see that in our algebra xo = I if x / S0 and xo = I— {b} if x € So, for b € x n t. Also < is a proper preorder for if b € x then we have x U {b} < x and x < x U {b} but x U {b} = x. Hence, defining x ^ y iff x < y and y < x we get an equivalence relation other than equality. Moreover, if x C y (and thus there is an inclusion arrow x ^ y) then x < y and x = y imply x ^ y. Our equivalence relation is ^ is not compatible with ', since if we take x such that —(t C x) then x U t ^ x U {a} where t = {a, b}. But (x U t)' = (x U t)c = xc — t and (x U {a})' = (x U {a})c U t = xc = t. Thus, xc — t & xc U t.
So, we can conclude that in PSet we have Sub(d) = p(d) and Sub(d) will be a paraconsistent set algebra and so do p(d). But in this case we cannot take 2 as the classifying object exploiting the fact thatp(d) = 2d because this gives rise to the Boolean algebra of characteristic arrows as in Set. Actually, if we will try to define
XA (x) = {1, if x€A0, if x/A
then we need to take into account that in PSet we have two negations and hence the right definition will be
XA (x) =
1, if x A
2, if x/A
0, if — * (x € A)
This means that in the role of classifying object in PSet we should take not the two-element Boolean algebra but the three-element da Costa algebra (according theorem 3 every proper da Costa algebra has at least three elements). An example of such an algebra would be found in [1, p. 83] where the operations are defined by the following tables:
A 0 12 V 0 1 2 D 0 1 2
0 0 00 0 0 1 2 0 1 1 1
1 0 12 1 1 1 1 1 0 1 0
2 0 22 2 212 2 0 1 1
0' = 1, 1' = 0, 2' = 1; 0 < 2 < 1.
So, we have p(d) = 3d and the set 3 = {0*, 0, {0*}} together with the function true : 1 ^ 3 such that true(0*) = 1 (where 1 := {0*})
playing the role of the subobject classifier in PSet. Also here we have arrows, false : 1 — 3 (such that false(0*) = 0) and false* : 1 — 3 (such that false*(0*) = 0*). □
We define now truth-arrows in potos in general case. Let us C will be a potos with the subobject classifier true : 1 — Q. Then the negation — : Q — Q will be the unique arrow for which the diagram
1
falsee
true
Q
Q
will be the pullback in C. Thus, — = Xfalse. The negation will be the unique arrow for which the diagram
Q ->• Q
false*
true
Q
Q
will be the pullback in C. In this case we have —* = Xfalse*
1. Since potos is a Cartesian closed category then other truth-arrows will be defined standardly:
n : Q x Q — Qisa character of the product of arrows (true, true) : 1 — Q x Q in a potos C;
U : Q x Q — Q is by definition a character of the image of C—arrow [(truen, 1n), (1n,truen)} : Q + Q — Q x Q ;
Q x Q — Q is a character of the monic c : © ^ Q x Q, which is an equalizer of the pair
Q x Q ^ Q
pri
where pr\ is a projection on the first member of the product Q x Q.
*
4. An Interpretation of Paraconsistent Logic in a Potos
Let us give an interpretation in terms of potoses of the following list of axioms and rule of inference [4, p. 3790]:
A1. a D (ft D a)
A2. (a D ft) D ((a D (ft D 7)) D (a D 7))
A3. a A ft D a
A4. a A ft D ft
A5. a D (ft D a A ft)
A6. a D a V ft
A7. ft D a V ft
A8. (a D 7) D ((ft D y) D (a V ft D 7)) A9. a V —a A10. ——a D a
A11. fto D ((a D ft) D ((a D —ft) D —a)) A12. ao A fto D (a A ft)o A13. ao A fto D (a V ft)o A14. ao A fto D (a D ft)° A15. ao D (—af a a D ft
ft
Here ao is an abbreviation for —(a A —a).This axiomatic describes, in fact, the system C1 of da Costa paraconsistent logic.
We can define a valuation v : $0 — A of the system Ci in da Costa algebra A assigning to an every propositional letter ni some truth-value V(ni) € A. It uniquely would be extended in a following way:
(1) v(—a) = v(a)';
(2) v(a A ft) = v(a) A v(ft);
(3) v(a V ft) = v(a) V v(ft);
(4) v(a D ft) = v(a) D v(ft).
to the function v : $ — A. The sentence a such that v(a) = 1 for every A-valuation v is called A-valid and this is denoted as A |= a.
Theorem 5. For any da Costa algebra A, A\= a iff a.
Proof. From left to right we check immediately C1-validity of all C1-axioms and detachement rule. For obtaining the proof of the claim from
right to left we will use the theorem 6. Putting into the correspondence to each element x of an algebra A the principal filter [x) = {q : x < q} where q € A we come to the conclusion that an algebra A+ of principal filters will be = -isomorphic da Costa algebra. Let us define A+-valuation as a function vc : $0 — A+ by means of the formula vc(ni) = [v(n,)). The rest is obvious.
There is one more way to prove this theorem if we will use the non-truth-functional valuation of C\. Following [6} we can introduce a valuation V' : $0 — A where $0 is a set of propositional letters and extend this to the set $ of all formulas in the following way:
(5) V'(a) = 0 ^ V'(—a) = 1;
(6) V'(——a) = 1 ^ V'(a) = 1;
(7) V'(f°) = V'(a D f) = V'(a D —f) = 1 ^ V'(a) = 0;
(8) V'(a D f) = 1 & V'(a) = 0 or V'(f) = 1;
(9) V' (a A f) = 1 & V'(a) = V' (f) = 1;
(10) V'(a V f) = 1 & V'(a) = 1 or V'(f) = 1;
(11) V'(a°) = V'(f°) = 1 ^ V'((a D f)°) = V'((a A f)°) = V'((a V f)°) = 1.
According to [6, p. 623} A |= a iff ^c-t a i.e. a is valid for every valuation V . □
Let us define now an interpretation of the system considered in an arbitrary potos C. The truth-value in potos we will call an arrow of the type 1 — Q and the collection of all such C-arrows will be the set C(1, Q).
C-valuation will be a function V : $0 — C(1, Q) assigning to an every propositional variable ni some truth-value V(nj) : 1 — Q. This function might be extended to the set $ of all formulas in the following way:
(12) V(a) = false ^ V(—a) = true :
(13) V(——a) = true ^ V(a) = true;
(14) V(f°) = V(a D f) = V(a D —f) = true ^ V'(f°) = V'(a D f) = V'(a D —f) = 1;
(15) V(a D f) = true & V(a) = false or V(f) = true;
(16) V(a A f)= true & V(a) = V(f) = true;
(17) V(a V ft) = true & V(a) = 1 or V(ft) = true;
(18) V(a°) = V(ft°) = true ^ V'(a°) = V'(ft°) = 1.
Thus, we extend the valuation V in such a way that to each sentence a corresponds some C-arrow V(a):1 — Q. C-validity of a (which is denoted C == a) means that V(a) = true : 1 — Q for all V.
Since a potos is a particular kind of topos, we have Sub(d) = C(1, Qd) then Sub(d) = C(d, Q) (Sub(d) and not its quotient is =-isomorphic to C(d, Q), using the definition of =-isomorphism given above), i.e. taking into correspondence to some subobject f its character xf we transfering the structure of da Costa algebra from Sub(d) on C(d, Q). The connection between potos semantics and considered theory as in case of Heyting algebra (cf. [7]) consists in that for any potos
C\= a iff C(1, Q) = a iff Sub(1) = a Hence, the validity in any potos C is equal to the validity in da Costa algebras C(1, Q) and Sub(1). This implies the following theorem:
Theorem 6. If hCl a then for any potos C we have C == a.
Proof. Let a be some C1-theorem. Then a is valid in da Costa algebra by theorem 12. In particular, C(1, Q) == a from which C\= a according to the previous claim.
We would define the way V relates to V' from above while putting V(ni) = true if V'(ni) = 1, and V(ni) = false otherwise. Then we extend this to the set $ of all formulas in the following way:
(19) V(a) = false & V'(—a) = 1 :
(20) V(——a) = true & V'(——a) = 1;
(21) V(ft°) = V(a D ft) = V(a D —ft) = true & V'(ft°) = V'(a D ft) = V (a D —ft) = 0;
(22) V(a D ft) = true & V'(a D ft) = 1;
(23) V(a A ft) = true & V'(a A ft) = 1;
(24) V(a V ft) = true & V'(a V ft);
(25) V(a°) = V(ft°) = true & V'((a D ft)°) = V'((a A ft)°) = V'((a V ft)°) = 1.
It is easy to prove that V(a) = true iff V'(a) = 1 that allows us to obtain the proof of
Lemma 1. V(a) = true iff V'(a) = 1.
Proof. In case a = ni lemma is true by the definition.
Let a = —f and V(f) = false. Then V'(—f) = 1 and V'(a) = 1 and the other way round.
In case of a = ——f and V (——f) = true we have V'(——f) = 1 and V' (a) = 1.
For a = f° and V(f°) = V(a D f) = V(a D —f) = true we have V'(f °) = V'(a D f) = V'(a D — f) = 0 and thus V'(a) = 0.
For a = y D f we have V(a D f) = true & V'(a D f) = 1. In case of a = y A f we have V (y A f) = true & V' (y A f) = 1. In case of a = y V f we have V (y V f) = true & V' (y V f) = 1. Finally, taking V(a°) = V(f °) = true we obtain V'((a D f)°) = V' ((a A f)°) = V' ((a V f)°) = 1.
The rest we obtain in a similar way. □
□
Theorem 7. For any potos C and propositional fromula a the following statement is true: C == a iff he- a.
Proof. Suppose FCl a then, by the completeness result in [6}, there is a valuation V' such that V'(a) = 1 and, by Lemma 4.3, the associated V is such that V(a) = true, and this means C¥ a. □
5. An Interpretation of Paraconsistent Logic in a Potos
PSetA
For obtaining an interpretation of Ci in a topos SetA in [10} as the categorical counterpart of da Costa algebra so-called CN-categories have been implemented. But since x < y ^ y' < x' is not a valid property concerning the paraconsistent negation in Ci then we need to reformulate the definition of CN-categories.
Definition 4. A CN-category C is a preorder category such that
(i) C has finite products (—, —), coproducts [—, —} and C is distributive relative to those, i.e. ([a,b}, [a, c}) = [a, (b,c)] for any objects a, b, c in C;
(ii) C allows an exponentiation;
(iii) a — b is an arrow in C iff a ^ b = 1, for any two objects a, b in C where a ^ b is an exponential;
(iv) C has a terminal object 1 and an initial object 0;
(v) for any object a of C there is an object Na such that we have arrows NNa — a and ao — (Nay in C where ao = N (a, Na) and for any arrow d — a there is an arrow d — Na in C;
(vi) for any two objects a,b in C there is an arrow ao — (b ^ a) ^ ((b ^ Na) ^ Nb);
(vii) 1 = [a, Na] and 0 ^ (ao,Nao).
It is easy to check that any CN-category has the following properties:a an exponential a ^ b in C will be a residual, C is cartesian closed,
y — x is an arrow in C iff (x, y) = y and [x, y] = x, ((Na, ao),a) = 0, [(Na, ao), a] = 1, every CN-category has at least three objects. In order to build the category PSetA as a potos we will use the theorem 7. According to this theorem a set A+ of all principal filters i.e. of sets [p) = {q : p < q} is a da Costa algebra =-isomorphic to A and this will be true for [p)+ where [p)+ is the relativization of A+.
Now we consider the functor O : A — PSet which will represent the classifying object in potos PSetA. Hereafter we will use A both as an algebra and the category. For any functor F : A — PSet we denote by Fp the value F(p) of functor F for object p from A. For any q and p such that p < q a functor F defines the function from Fp to Fq which we denote Fpq. A functor F will be treated as the collection {Fp : p € A} of sets indexed by elements of the set A from an algebra A and endowed with the transition mapping Fpq : Fp — Fq under p < q (in particular, Fpp will an identity function on Fp).
We continue in this fashion putting Op = [p)+ and for p and q such that p < q the function Opq : Op — Oq maps every S € [p)+ into S n [q) € [q)+, i.e. Opq(S) = Sq.
A constant functor 1 : A — PSet which is a terminal object of the category PSetA might be defined with a help of conditions 1p = {0} for p € A and 1pq = id{0} under p < q. A subobject classifier true : 1 — O
is a natural transformation whose p-th component truep : {0} — Op will be determined by the equality truep(0) = [p). Thus, the function true chooses the greatest element from every da Costa algebra of [p)+ type. Let t : F ^ G be an arbitrary subobject of PSetA-object G.
An every component tp is injective and can be treated as the inclusion function Fp — Gp. The p-th component (xT)p : Gp — [p)+ of a characteristic arrow xT : G ^ O will be defined by the equality
(Xt)p(x) = {q : p < q and Gpq(x) € Fq} for every x € Gp.
Now we construct truth arrows in a potos PSetA. Let us start with an arrow false.
An initial object 0 : A — PSet of category PSetA is the constant functor such that 0p = 0* and 0pq = id0* for p < q. Components of a natural transformation 0 ^ 1 are the inclusions 0* — {0} (the same
component for any p). According to the usual definition an arrow false is the characteristic arrow of subobject ! : 0 ^ 1. For its component
falsep : {0} ^ Op we have falsep(0) = {q : p < q and 1pq(0) € 0q} = {q : p < q and 0 € 0*} = 0* and hence a natural transformation chooses the null element from an every da Costa algebra.
Conjunction and disjunction can be handled in same way as in case of topos SetP (cf. [7]), i.e. we, in fact, need for n : O x O ^ O and U : O x O O the definitions of their p-th components in a form of np({S,T)) = S n T; Up((S,T)) = s U t . The negation is — : O ^ O whose p-th component —p : Op — Op
in case of indentifying falsep with the inclusion {0*} — Op (and since — : O ^ O is a characteristic arrow of subobject false) is as follows:
—p(S) = {q : p < q and Opq(S) € {0*}} = {q : p < q and S n [q)= 0*} = [p) n—S = (—S )p.
A negation —* : O ^ O is obtained by deducing that the p-th component —** : Op — Op of negation satisfies equality
p • a"p ^ *"p
= (-S)p Hp (-so)p = np({-p(s), -p(s°))) = (S')p.
An implication O x O ^ O we have by defining the p-th component as
^p ((S,T)) = (S ^ T)p. Finally, we will call PSetA-valuation a function V : $0 — PSetA(1, O) assigning to every propositional variable ni some truth-value V(ni) : 1 ^ O. This function might be extended to the set $ of all formulas in the following way:
(12) V(a) = false ^ V(—a) = true :
(13) V(——a) = true ^ V(a) = true;
(14) V(ft°) = V(a D ft) = V(a D —ft) = true ^ V'(ft°) = V'(a D ft) = V'(a D —ft) = 1;
(15) V(a D ft) = true & V(a) = false or V(ft) = true;
(16) V(a A ft) = true & V(a) = V(ft) = true;
(17) V(a V ft) = true & V(a) = 1 or V(ft) = true;
(18) V(a°) = V(ft°) = true ^ V'(a°) = V'(ft°) = 1. We say that the formula a be PSetA-valid (we write
PSetA \= a) if
V(a) = true : 1 — O for all PSetA-valuations V.
Using da Costa-Alves valuation V' : $0 — {0,1} from above it is easy to prove at the same way the following theorem:
Theorem 8. For any potos
PSetA, PSetA
== a iff hc1 a (i.e. a is
provable in C1). References
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[10] Vasyukov, V.L. "Paraconsistency in Categories", in: Frontiers of Paraconsistent Logic, D. Batens, C. Mortensen, G. Priest and J.-P. van Bendegem (eds.). Research Studies Press Ltd., Baldock, Hartfordshire, England, 2000, pp. 263-278.
[11] Vasyukov, V.L. Potosy dlya paraneprotivorechivoi logiki [Potoses for Paraconsistent Logics], Modern Logics: Issues of Theory, History and Implementation in Science. Papers from the X-th all-Russian Scientific Conference, SPb, 2008. P. 105-107. (In Russian)
