an algebraic approach to intuitionistic connectives xavier caicedo ... · thejournal of s~~bolic...

An Algebraic Approach to Intuitionistic Connectives

Xavier Caicedo; Roberto Cignoli

The Journal of Symbolic Logic, Vol. 66, No. 4. (Dec., 2001), pp. 1620-1636.

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THEJOURNAL LOGICOF S ~ ~ B O L I C Volume 66 Number4 Dec 2001

AN ALGEBRAIC APPROACH TO INTUITIONISTIC CONNECTIVES

XAVIER CAICEDO AND ROBERTO CIGNOLI

Abstract. It is show11 that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives. including those proposed by Gabbay. are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases. the double negatio~l of such a connective is equivalent to a formula of intnitionistic calculus. Thus, under the excluded third law it collapses to a classical formula. showing that this conditio~l in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting algebras. unless they are already equivalent to a formula of intnitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be "completed conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives.

$1. Introduction. If we consider intuitionistic and intermediate propositional cal- culi as logics with truth values in Heyting algebras, it is natural to consider new connectives for theses logics as operations in the algebras, univocally determined by their axioms, approach that we explore in this paper. Most of the proposed extensions of intuitionism by connectives have been introduced prima facie as deductive systems, before looking for a semantics for them. In particular, the proposal by Gabbay [6 ,7]of a general definition of intuitionistic connective is given in deductive terms, and attempts to maintain the "intuitionistic character" of the corresponding axiomatic systems by asking that they be conservative over pure intuitionistic calculus, have the disjunction property, and collapse to classical calculus under the excluded third law. Among other results, we show that Gabbay's systems are strongly complete for their natural algebraic semantics.

There is a natural notion of intuitionistic connective for Kripke models, preserving the fundamental properties of Kripke semantics for intuitionism and yielding conservative extensions with the disjunction property (cf. [2]). These connectives correspond to certain operations in Heyting algebras of increasing sets of partial orders. More generally, we may consider the connectives of the logic of sheaves over a topological space X . That is, the morphisms of the subobject classijiev Q of the topos S h ( X ) ,which acts as the "object of truth values" for the inner logic of this topos in the same sense that (0.1) is the set of truth values for classical logic (cf. [5, 81). As noticed in [3], these connectives are in correspondence with the operations f on the Heyting algebra of open subsets of X which satisfy the

Received November 9, 1999: revised April 18. 2000.

@ 2001 Association for Symbolic Logic 0022-48 lZ/O1/6604-0007/$2.70

AN ALGEBRAIC APPROACH TO INTUITIONISTIC CONNECTIVES 1621

equation:

observation which extends to the logic of sheaves over any complete Heyting algebra. We will see that this equation is related in arbitrary Heyting algebras to the notion

of afine completeness studied in Universal Algebra (see [l I], [12]). Moreover, in axiomatic extensions of intuitionistic calculus by a connective V, it corresponds to the validity of the axiom schema:

in turn, equivalent to strong con~pleteness of the extensions with respect to their associated varieties of enriched Heyting algebras. This allows us to study intuitionistic connectives by algebraic means, since all axiomatic extensions determining univocally a connective, including those proposed by Gabbay, will be shown to contain the latter schema.'

In sections 2 and 3, we explore the algebraic meaning of equation (1. l ) , obtaining as a by-product simple proofs of the known affine completeness of boolean and finite Heyting algebras, and we consider the properties of operations implicitly defined by equations over a variety of Heyting algebras. In Section 4, we study axiomatically defined connectives for intuitionistic calculus and intermediate logics and show that they always satisfy schema (1.2). The double negation of any such connective must be equivalent to a formula of Heyting propositional calculus. Thus, under the excluded third law it collapses to a classical propositional formula, showing that this condition in Gabbay's definition is redundant. In the last two sections, we consider some examples, and show that the intermediate logic 5?n with values in the chain of length n may be "completed" conservatively by adding a single unary connective S , so that any implicit connective of 9,+ S is equivalent to a formula of this calculus.

52. Compatible functions in Heyting algebras. We assume that the reader is fa- miliar with the theory of Heyting algebras, also called pseudo-boolean algebras, and their relation with intuitionistic propositional calculus [4, 16, 171. For more on Heyting algebras see also [I]. We will utilize +, A, V, 1 , 0 , 1 for relative pseudo- complement, meet, join, pseudo-complement, minimum, and maximum, respectively; x H y will be used as an abbreviation for (x -+ y) A (y + x).

In general, H will denote a Heyting algebra. A term over the vocabulary z = (7,A, V, -., 0 , l ) will be called a Heyting term, and the function determined in H by a Heyting term t will be denoted tH .

A n-arypolynomial of H is a function obtained by evaluating m - n variables of tH by fixed elements of H, for some m-ary term t (m 2 n).

A function f : Hn-+ H is compatible with a congruence relation 0 of H if:

(xi, yj) E 0 for i = 1 , .. . , n implies (f ( x l , . . . , x,), f (yl, . . . , y,)) E 0.

' ~ c c o r d i n ~to [21]. this axiom appears in a definition of connective proposed by Novikov in the fifties.

1622 XAVIER CAICEDO AND ROBERTO CIGNOLI

f is a compatible function of H provided it is compatible with all congruence relations of H. This is equivalent to saying that the algebras H and ( H , f ) have the same congruences.

The simplest examples of compatible functions in a Heyting algebra H are the polynomial functions; in particular, all constant functions.

An algebra H is afine complete if any compatible function of H is given by a polynomial of H . It is locally afine complete provided that any compatible function is given by a polynomial on each finite subset of H . It is known that boolean algebras and finite Heyting algebras are affine complete [9] , [15], [12, Cor. 3.6.11. These facts appear as corollaries below.

Recall that the following relations hold in any Heyting algebra, due to the ad- junction between A and -+:

LEMMA : H n -+ H in a Heyting 2.1. The following ave equivalent for any map f algebra H .

a) f ( x l , . . . ,x , ) A a = f ( x l A a , . . . , x , A a ) A a , f o v a l l x i , a E H . b) ArZl(x i ~ y i ) E HL f ( x l , . . . , x n ) *f(~l>...>yn),fovallxi,yi c) f is a compatible function of H .

PROOF.It is enough to consider unary functions. Assuming (a), then by (2.2): f ( x ) A ( x y ) = f ( x A ( x H y ) ) A ( x H Y ) = f ( y A ( X H y ) ) A ( X - y ) = f ( y ) A ( x o y ) . Therefore, from (2.1) follows (b): x H y < f ( x ) o f ( y ) . Since any congruence O on H is given by a filter F of H in the form: x O y iff x H y E F , the last inequality implies that f is compatible with O . Reciprocally, assuming (c), f must be compatible with the congruence associated to the principal filter ( ( x A a ) H x ) . Hence, a < ( x A a ) H x < f ( X A a ) H f ( x ) , and so f ( x A a ) A a = f ( x ) A a , by (2.1). -1

Condition (a) of the lemma is equivalent to the apparently stronger equation:

A compatible function which is not a polynomial is given in the next example (cf. [12,5 4.21):

EXAMPLE2.1. Let H be the totally-ordered Heyting algebra obtained by adding two new elements a,p to the set co of natural numbers in such a way that n < a < /3 for each n E co. Since the filter {a,p } is contained in every filter of H different from { P } ,the following prescription

a if x is even or x = a,f ( x )=

/3 i f x is oddov x = /3,

defines a compatible function f : H -+ H . On the other hand, given a k + 1 -variable Heyting term t and k elements a l , . . . , a,, of H , it is not hard to see that for sufficiently large n k w , t H ( n , a l , . . . ,a k ) = a: implies t H ( n+ 1, a l , . . . , a k ) = a. Hence, f can not coincide with a polynomial of H .

1623 AN ALGEBRAIC APPROACH TO INTUITIONISTIC CONNECTIVES

If {f j ) i E I is a family of compatible functions of the same arity in H for which the join f ( x l , . . . ,x,) = Vi f j(x1,. . . ,x,) exists for all x l , . . . ,x , E H , then f satisfies condition (a) of the previous lemma because in a Heyting algebra A distributes over all existing joins. Hence, all existing joins of polynomials are compatible. The next theorem, which generalizes the disjunctive normal form of boolean algebras, shows that all compatible functions are joins of polynomials.

THEOREM2.2. Let f : H n + H be a compatible function. Then for any subset S s H a n d x l ,. . . ,x , E S :

f ( x l , . . . , x n )= V - U ~ ) A . . . A ( ~ ,f (a1 , . . . ,a,!n(xl o n , , ) . (ul . . . . , a , ) ~ S n

PROOF.Fix ( x I , .. . ,x,) E S n . Then by (2.3)and (2.2),for all al , . . . , a, E S :

. f ( a l ,. . . ,a , ) ~ ( a l - x ~ ) A + . . A ( ~ , o x , )

= f ( x 1 , . . . , x , ) A ( a l o x l ) ~ . . . A ( a , o x , ) L f ( x l , . . . , x,).

Therefore f ( X I , . . . ,x,) is an upper bound of the set

T = { f (a l , . . . ,a , ) A (a1 - X I ) A . . . A (a, o x,) I ( a ] ,. . . ,a , ) E S n ) .

But,

Hence, f ( x l , . . . , x , ) = maxT. -1

COROLLARY2.3. Any Heyting algebra is locally afine complete. Anyjnite Heyting algebra is a$ne complete.

If, for a given a function f : H" -+ H , the join

exists for all xl .. . . .x , in H , then f is the minimum compatible function above f . For example, let H3 be the three-element chain 0 < a < 1 endowed with its natural Heyting algebra structure and f : H; -+ H3 be the tukasiewicz three- valued implication. which is not compatible, then f ( x . y ) = ( x -+ y ) V a. The functions f and f are shown in the following tables:

f o a 1 f 0 a 1

a l O a l

A compatible function is always dense in a polynomial, as shown in the next theorem.

THEOREM2.4. For any n-ary compatible function f in a Heyting algebra H , there is a Heytingpolynomial p ( x l , . . . ,x,) of H such that


More precisely, one may take:

where x 1 = x : x 0 = ~ x .

PROOF. By Theorem 2.2, p(x1, . . . ,x,) <_ f ( x l , . . . . x,) , since x 1 = ( x H I ) , x0 = ( x H 0 ) . By (2.2)and (2.3),

P ( X I , .. . > x n )= V f ( ~ 1, . . . ,x , ) A x ; ' A . . . A x ~ ' ~ " ( S I . . . . , s , , ) E { O , I ) "

= f ( x l , . . . ,X , ) A V x S 1A . . . A x ; : "I

(51.... . S n ) ~ { 0 . 1 ) "

Thus,

( x . x ) f ( x . . x ) 7 V x ; A . . . A x ' " ( s,.....S , ) € { O , l ) "

L f ( x l , . . . , x n ) ,

considering that

- 3 V x;' A . . . A x : ) = 1 ( S I ,...,sn)€{O.l)"

in any Heyting algebra. COROLLARY2.5. [9] Any boolean algebra is afine complete. The next two observations will be useful. COROLLARY ,2.6. For any compatiblejilnction f

PROOF. From Theorem 2.4, 77f = 7 7 p , and the statement holds for any Heyt- ing polynomial, as may be shown by induction on complexity. -1

COROLLARY2.7. Let f : H -+ H be a compatible function such that f ( 1 ) , f ( 0 ) E

{ 1 : 0 ) . Theneither f ( x ) = I X , f ( x ) = 0 , f ( x ) x V l x , o r x 5 f ( x ) 5 l l x . PROOF. By Theorem 2.4 one of the next situations must hold:

f ( 1 ) f ( 0 ) 1 1 x v 1 x 5 f ( x ) 5 1 - ( x v 1 x ) = 1

1 0 x 5 f ( x )< 1 1 x 0 1 1 X < f ( x )5 7 - 7 1 ~= 1 x

0 0 0 < f ( x ) <_ 7-0 = 0. -I

EXAMPLE2.2. Recall that an operator - on a Heyting algebra is a De Morgan negation if it satisfies: -- x = x and - ( x v y ) =- x A -- y (cf. [ l , 161). It follows that - 1 = 0 and - 0 = 1 . Thus, in case -- is compatible, Corollary 2.7 implies

x = l x . Hence, A Heyting algebra admitting a compatible De Morgan negation -- must be a boolean algebra, where - x = - x .


An operator is a necessitation if it satisfies Ux 5 x and 1 = 1. If is compatible, the last equation and Corollary 2.7 imply x 5 Ox. Hence,

The only compatible necessitation operator in a Heyting algebra is the identity. (cf. [IS, Prop. 4.11).

$3. Equationally defined compatible operations on Heyting algebras. A set E ( f ) of equations in the signature of Heyting algebras augmented with the n-ary function symbol f will be said to define an implicit operation of Heyting algebras if for any Heyting algebra H there is at most one function f : H n -+ H such that ( H ,f H )

satisfies the universal closure of the equations in E (f ). f will be an implicit compatible operation provided all f are compatible. Beth's definability theorem guarantees that an implicit operation must be explicitly definable by a first order formula in the vocabulary of Heyting algebras. That does not mean that it has to be given by a Heyting term, even if it is compatible, as the following example illustrates.

EXAMPLE3.1. The system E(y) consisting of the following three equations de-fines an implicit compatible operation y (x) of Heyting algebras, the smallest dense element above x .

Cl . 7 y (0) = 0, c2.y(0) -+ (x v 1 x ) = 1, c3. y(x) = X v y(0).

Recall that an element x of a Heyting algebra H is dense if l i x = 1, and the dense elements form a filter of H . It should be clear that H has an element y (0) satisfying C1 and C2 if and only if the filter of dense elements of H is principal with generator y (0). This element exists in all finite Heyting algebras and, more generally, in atomic Heyting algebras where the supremum of the atoms exists. C3 determines y (x) univocally as the smallest dense element above x , showing also that y is a compatible function (being a polynomial). This operation is not expressible by a Heyting term, not even by an infinite combination of Heyting terms, because in the three-element chain H3 = (0, a, 1) we have y (0) = a , while t (0) E {0,1) for any Heyting term t .

The axioms of the operation * on Heyting algebras introduced by Touraille in [20] imply compatibility and are satisfied by the operation y. But Touraille's equations do not determine * univocally since they are satisfied by the identity also.

EXAMPLE3.2. The following set of equations E(p) defines an implicit non com-patible operation p(x), the dual pseudo-complement (see [I , VIII.31).

P I . x v p(x v y ) = x v p(y), P2. p(1) = 0, P3. p ( O ) = l .

Indeed, it is easy to check that any p(x) satisfying E (p) must be the least element of the set { y E H : y V x = 1). If p were compatible, we would have p(x) = 7 x for each x by P2,P3 and Corollary 2.7. Then, by P1,the equation 7 x V x = 1 would hold in any H where p exists, contradicting the fact that p is defined in all finite Heyting algebras. Not being compatible, p can not be expressible by a Heyting term.

- -


The constant y (0)of Example 3.1 may be expressed as the infimum of the elements of the form x V T X .The next example shows that this kind of constant is not always determined by equations.

EXAMPLE3.3. Let H be a Heyting algebra where d H = A{TXV T T X : x E H ) exists. Then the stipulation A ( x ) := x +6 uniquely defines a polynomial function AH: H i H . Despite its natural definition, A and 6 can not be characterized by equations, because 6 = A(1) is not preserved by A-subalgebras. Indeed, if H is the finite Heyting algebra whose Hasse diagram is depicted in Figure 1 , then dH = i . Thus, AH ( x )= 1 for x < i and ( e )= AH( 1 ) = i . Hence, the Heyting subalgebra S = but in this subalgebra 1 > i .(0 ,i, 1) is closed under A ~ , =

A compatible implicit operation does not need to exist in all Heyting algebras. The operation y of Example 3.1 does not exists in the ordered real interval [0, I] . This is a particular case of the next theorem, which shows that an implicit compatible operation can not exist in all Heyting algebras, unless it is explicitly definable by a Heyting term.

THEOREM3.1. I fa system ofequations E ( f ) is satisfied by a unique n-ary comnpat- ible function f in each algebra H of a variety V ofHeyting algebras, then there is a n-ary Heyting term t(x1, . . . , x,) such that f H = t H in any H E V.

PROOF. Let F be the free algebra of V on countable many generators X,, n E w, and let ~ F ( x ~ ,. . . , X , ) = qF ( X I , . . . ,x,, . . . , x,+I,), where q(x1 , .. . ,x,, . . . ,X n + k )

is a Heyting term. Now, given a finite or countable Heyting algebra H E V and al , . . . ,a, E H , let h : F + H be an onto homomorphism such that h ( x , ) = a, for i = 1, . . . , n , and h ( x , )= a,, for i = n, . . . , k. Since f is compatible with the kernel of h , the function f :H n i H given by

is well defined in H and, by construction, h is an homomorphism from ( F , f F ) onto

(a7).

- - -


Therefore, all positive universal sentences satisfied by the first algebra are satisfied by the second. In particular (H ,f)k E ( f ). This shows that f = f H does-not depend on a l , . . . , a, or h. Moreover, f ( a l ,. . . , a,) = f (h (%l) , . . . ,h (x,))=

h ( f ~ ( % l = . . . ,x I , , . . . , x k ) ) = qH(h (%l ), . . . , A(x,,), . . . ,, . . . ,x,)) h ( q F ( x l , h ( ~ , + ~ ) ) , . . . , a, , . . . , a,). Hence, f~ = t H , where t (x l, . . . ,x,) == q H ( a l q ( x l , . . . ,xI,,. . . ,xI , ) .Therefore t satisfies E ( f ) in all finite or countable H E V . As E ( f ) is at most countable this holds in any H E V by the downward Lowenheim- Skolem theorem, and the identity f = t follows by uniqueness. i

If E ( f ) defines implicitly a compatible operation, let V ( E ( f )) be the variety of enriched Heyting algebras (H , f H ) satisfying E ( f ). For each A E V ( E ( f )), let H ( A )be the Heyting algebra reduct of A , and set

Clearly, this class is closed under products and, by compatibility o f f , it is closed under homomorphic images. Moreover, Theorem 3.1 shows that iff is not given by a Heyting term, then Red(V(E ( f ))) can not be a variety and, by Birkhoffs theorem, it can not be closed under subalgebras. On the other hand, if E ( f ) k f = t for some Heyting term t then Red(V(E ( f ))) =V ( E ( f I t ) ) . Thus:

COROLLARY3.2. An equationally dejined implicit compatible operation of Heyting algebras is explicitly dejinable by a Heyting term if and only i f the class of Heyting algebras where it exists is a variety (equivalently, it is closed under subalgebras).

An algebra is nun trivial if it has more than one element. A set of equations E ( f ) is nun trivial if V ( E ( f )) contains non trivial algebras.

THEOREM3.3. I f E ( f ) is nun trivial and defines an implicit compatible operation f of Heyting algebras, then there is a Heyting term t such that 11f = t in any algebra where f is dejined. Moreover, f is defined in all boolean algebras.

PROOF. If (H , f ) E V ( E ( f )) is non trivial, with f compatible, a maximal congruence of H is also a congruence of (H, f ) and yields a quotient (H2,fl),where HZ is the two elements boolean algebra and f satisfies E ( f ). By functional completeness of this algebra, f coincides with a boolean term u in H2. Since u satisfies the equations of E ( f ) in Hz , it must satisfy them in all boolean algebras. On the other hand, 17f satisfies E ( f ) in the boolean algebra Reg(H) of regular elements of any Heyting algebra H where it is defined, because the onto homomorphism of Heyting algebras 17:H + Reg(H) is also a homomorphism from (H, f H ) onto (Reg (H) ,17f H ) by Corollary 2.6. By uniqueness,

f H ~ ~ ~ g ( ~ ) . 7 7 u H r Reg(H) ,because of the inter- 71 1 Reg(H) = But ~ ~ ~= g ( ~ )

pretation of V in Reg(H)as 7 7 ( avb ) in H . By regularity of - 1 7 xand Corollary 2.6 again, 77f H ( X ) = 17f H ( 7 1 ~ )= 1 7 u H ( 7 l x )= 7 7 u H( x )for any x E HI7.Take t = 17U. -I

In spite of Theorem 2.4, it is not possible to improve the previous theorem to have t < f < 7 - d . There is no Heyting term t ( x ) such that t ( x ) < y (x)< 1 7 t ( x ) in all algebras (H , yH) , for the operation y introduced in Example 3.1. Indeed, in the three-element chain H3,we have y (0 )= a , while t (0 ) = lit (0) E {0 ,1 ) for any Heyting term t .


§4. Axiomatic extensions of intuitionistic calculus by implicit connectives. The language L of formulas of the intuitionistic propositional calculus is built in the usual way from the connective symbols +, A, V, 7,corresponding to implication, conjunction, disjunction and negation, respectively, and the propositional variables n,, i = 0,1, . . . AS in the language of Heyting algebras, cp o y will stand for an abbreviation of (cp -+ I//) A ( y + cp).

Given a set @ of new connective symbols (of arbitrary arities), L(@) will denote the propositional language obtained by allowing the symbols of @ in the formation rules of formulas. We write L(V) for L({V)).

To each set of formulas d ( @ ) c_ L(@), associate the axiomatic system having d ( @ )U Int for axiom schemas, where Int is a complete system of schemas for intuitionistic propositional calculus (as given for example in [16, 17]), and substitution in axiom schemas and Modus Ponens as only rules. Only this kind of systems will be considered. If @ is empty and d is consistent, the system is an intermediate logic.

Given r U {cp) C L(@), the notation

l- t,(,) cp

will indicate that cp is deducible from r in this calculus. We write t,(,) cp if l- = 8, and r t cp for deducibility in pure intuitionistic calculus. It is immediate that the deduction theorem is satisfied:

r U { a )t,(,) cp implies r kg(,) a -+ cp,

Each formula cp E L(@) may be seen as a term in the type zU@ of Heyting algebras enlarged with the set @ of operation symbols, in the variables 71,. Therefore, to each extension d ( @ ) of intuitionistic calculus we may associate the system of equations E ( @ )= {cp = 1: cp E d ( @ )U Znt), and the corresponding variety of Heyting algebras

V(d (@))= V(E(@))>

Since L ( 0 ) with the syntactical operations is the absolute free algebra of type z u @ on the set of propositional variables I'I = {nl, n2, . . . ), any function v : II +

Domain(A) with A E V(d(@)) (called an A-valuation) may be extended to a unique homomorphism E: L(@) -+ A. Then we may define for any set r U {cp) C_ L(@) an algebraic consequence relation as follows.

DEFINITION cp if and only if for any A E V(d(@)) and A-valuation 4.1. l- It,(,) v:~ ( y )= 1for ally E r implies ~ ( c p )= 1.

It is easy to check, by induction on the length of proofs, that I,(,)is sound with respect to this semantics. That is,

(4.1) r Ed(,) cp implies r It,(,) cp.

In particular, Ed(,) cp implies that cp = 1 is an equation of the variety V(d(@)) . The reciprocal of (4. I ) , strong ulgebraic completeness oft,(,), is not generally true. The following result characterizes those extensions for which it holds.

THEOREM4.1. The following conditions are equivalent for any d ( @ )G L(@): (1) t,(,) is strongly complete for the algebraic consequence relation. That is,

l- IF,(,) cp implies l-Id(,) C L(@).cp, for any r U {cp)

mailto:V(d(@))

1629AN ALGEBRAIC APPROACH TO INTUITIONISTIC CONNECTIVES

(2 ) kg(,) A:='=, pi) -t ( V ( ~ I , V(Pl,. . . ,P n ) ) , f o r each V E @.(ai ++ . . . , a,,)H

PROOF,( 1 + 2) If G ( A ~ , ~ P i ) ) =(aiH = 1 for some A-valuation v. then G(ai) -v(Pi)for 1' = 1 : . . . , n , and so v (V(a l , .. . , an )++ V(P1,.. . , p f l ) )= VA(G(al),. . . ,

-~ ( a , ) ) . . . ,v(Pn)) 1 , for each V E @. Therefore,C ) VA(;ii(P1), =

From the strong completeness hypothesis and the deduction theorem we con- clude ( 2 ) .

(2 =+ 1) Since Id(,)includes the rules of the intuitionistic calculus, given r C L(@)and O = { (a ,p ) :r I-&(,) a H p), then L(@)/Ois a Heyting algebra. Denoting by [a]the equivalence class of a E L(@j,( 2 )implies that the operation

is well defined for each V E @, and thus (L(@)/O,F ) becomes an algebra of V ( d ( @ ) ) ,where F = {f : V E @). The (L(@)/O,F)-valuation v(n,)= [n,] extends to ~ ( a )[a]for any a E L(@). = 1 = [xl-+ nl]if and = Therefore, ~ ( a ) only if Id(,)a H (xl -+ n l ) ,if and only if, F kd(,) a. It follows that G(y)= 1 for all y E r.Hence, r Itd(,)p implies G ( p )= 1 ; that is, r td(,)(P -I

DEFINITION4.2. A set of formulas sd ( V )will be said to define axiomatically a connective V provided that

where V 1is a new n ary connective symbol and s?(V1)= { p(V/V1): p E sd(V)).

THEOREM4.2. I f d ( V )defines axiomatically a connective, then it satisfies condition ( 2 ) in Theorem 4.1.

PROOF. For notational simplicity we consider just the unary case. Given two fixed propositional variables x and y define by simultaneous induction two transforma- tions * and + from L(V, V ' ) into L(V)as follows:

p* = y if p = x;and p* = p for other propositional variables, [ p CE ty]*= (P* CE y* for CE = A , V, -+, [ 1 p ] *= l p * , [Vpl*= Vp+, [V1p]*= Vp*.

(P+ = p , for propositional variables, [ p69 y]+= p+ CB y+ for CB = A , V, -+, [ y 1 + = 7 p + > [Vpl+= Vp', [V'p]' = Vp*.

Informally, to obtain p* change each occurrence of x in p to y , except those occurring under the immediate scope of V (that is, occurring in a subformula VO of p but not in a subformula V1pof VO).Then, change all V' to V . In particular, the occurrences of V x are not changed. But the occurrences of V'x are changed to Vy.


Claim]: x H y tp+ hip*. By an easy induction on the complexity of p E L(V,V').Indeed, for p atomic

we have two trivial cases: x hi y t x hi y and x H y t p hi cp. For a Heyting connective the induction step follows from intuitionistic rules, and for V and V1it is trivial since by definition [V1p]*= [Vfcp]+ =and [Vpl* [Vp]+.

Claim 2: A proof of tsl(V)U,(v,)cp may be transformed into a proof of x H y b(,)P*.

By induction on the length of the proof. Let d(V),,,,be the set of substitutions of formulas of L(V,V')in schemas of d(V),and d(V),the set of substitutions of formulas of L(V)in schemas of d(V).Define similarly d(Vf),,,,.Now assume t,(,),,(,,) p. If p is a intuitionistic axiom, then p* is a intuitionistic axiom because * respects the Heyting formula structure. If p is an axiom in d(V1)g,~/, B(V1, . . , /ty,] with B E d(Vf) L(V, V'). then cp = nl,. ~,,)[n, and ty, E

So p* = 711,. . . , nk)[nl/ E d(V),because * does not change the formula O(V, ty:] B(Vf,711,.. . , nk),except for changing V'to V.Therefore, t,(,) p*.By a similar argument, if cp is an axiom in B?(V)~,~,then pt is an axiom in d(V).. Therefore,, t,(,) cpt, and so, x H y t,(,, p* by Claim 1. If t,(,),,(,,) p follows by Modus Ponens from t,(V)U,(v,) ty +ty and td(o)u,(ol) cp,we have by induction hypothesis x hi y t,(,) ty* and x hi y t,(,) (ty + p)* = ty* + p*,and so X H Y tsl(0)P*.

Finally, let p be the formula Vx hi V1x.Then p* is: (Vx)*H (V1x)*=

V(x)+hi V(X)*= VXhi Vy,and the theorem follows from Claim 2 and the hypothesis. The same idea works for the n-ary case if one defines: [V(cpl,., p,)l* . . . cp,+),. . = V(cpT, , [Vf(pl,...,pn)l+= V(pT,...,p,*). From Lemma 2.1, and Theorems 4.1 and 4.2, we obtain:

COROLLARY4.3. Ifaset of formulas d(V)dejines axiomatically a connective then the corresponding set of equations dejines an implicit compatible operation of Heyting algebras. Moreover, the system t,(,)is strongly complete.

By Theorem 3.3 we have, using completeness:

COROLLARY dejines axiomatically a n-ary connective V,then there is 4.4. Ifd(V) a formula p(n1,. . . , n,)E L such that

Therefore, in the context of classical propositional calculus, V collapses to a classical propositional formula. More precisely,

Recall that d(V)is a conservative extension of intuitionistic calculus or, more generally, of an intermediate logic I, if t,(,) I,and t,(,) p implies tIcp for any cp E L. This is not a restrictive condition because any consistent set of axiom schemas d(V)is a conservative extension of a unique intermediate logic, namely the logic I(d(V))= ( 4 E L : t,(,) 4 ) .

DEFINITION4.3. If d(V)defines axiomatically a connective V and it is a conservative extension of an intermediate logic I, then we say that V is an implicit

-I

AN ALGEBRAIC APPROACH TO INTUITIONISTIC CONNECTIVES

connective of I. If, in addition,

y,(v) V(711, . . . , 7 % )~ c p , foranycp E L ,

then we say that V is a new implicit connective of I.

Notice that Corollary 4.4 rules out the existence of new implicit connectives for classical propositional calculus.

Due to strong completeness, the conservativity condition in Definition 4.3 means that R e d ( V ( d ( V ) ) )and V ( 1 )satisfy the same equations and thus the class of reducts generates the variety. That is,

because R e d ( V ( d ( V ) ) )is closed under products. Due to compatibility of V we may improve this to:

(4.3) V (1 )= S (Red (V(d ( V ) ) ) ) . Indeed, if H = H ' / F for a filter F in H' 5 H'l with (H" , f ,) E V ( d ( V ) ) ,then F may be extended to F" in HI' so that F = F" nH'. Thus, H ' / F = H'/ (Ft ' n H ' ) is isomorphic to a subalgebra of HI1/F1'.But ( H " ,f o) IF'' E V ( d ( V ) ) by compatibility off , .

THEOREM4.5. If d ( V ) defines an implicit connective of an intermediate logic I, then V is new fand only f i t is not defined in all algebras of V(Z) .

PROOF. By Corollary 3.2 and strong completeness, V is new if and only if R e d ( V ( d ( V ) ) )is not a variety of Heyting algebras. After (4.3), this means R e d ( V ( d ( V ) ) )5V(1) . i

In the case of pure intuitionistic propositional calculus, Definition 4.3 of a new implicit connective includes three of the conditions in Gabbay's definition of in-tuitionistic connective [7];namely: uniqueness, conservativity, and being new. In addition, Gabbay requires condition (4.2) of Corollary 4.4, which is obviously redundant, and the disjunction property:

This last property can not be required in general if we wish to consider connectives over arbitrary intermediate logics. We do not know if it is automatically inherited by the implicit connectives of pure intuitionistic calculus.

It should be clear that the disjunction property holds if and only if 1 is join- irreducible in the free algebras of the variety V ( d ( V ) ) .We use this fact in the next examples.

$5. Some examples. Corollary 4.3 shows that the dual pseudo-complement p considered in Example 3.2 can not be defined axiomatically, in spite of the fact that it is determined univocally by equations. No sound axiomatic system for this connective consisting of axiom schemas and Modus Ponens only may be complete. or prove uniqueness of p .

Other conservative axiomatic extensions of intuitionistic calculus by connectives found in the literature contain schema (1.2),and so they are strongly complete for algebraic semantics, but do not satisfy the uniqueness property of Definition 4.2.


For instance, the connective C introduced by Kaminski in [13]admits at least two different sound interpretations: the identity and double negation.

Let us consider more positive examples.

EXAMPLE5.1. The following axiom system d ( y) defines a new implicit connective of intuitionistic calculus with the disjunction property. Hence, an intuitionistic connective in the sense of Gabbay.

C1. l l y a C2. a -+ ya, C3. ya ( a V P v l P ) ,-+

C4. ( a P ) (ya yP).-+ -+ -+

We let the reader check that the corresponding equational system E ( y ) is equivalent to the one considered in Example 3.1. Then, by C4 and Theorem 4.1, we know that y is axiomatically defined by d ( y ) and interpretable exactly in those algebras where the minimum dense exists. To show that this is a conservative extension of intuitionistic calculus, assume that t,(,)p,where p does not contain y. Then p holds in all finite Heyting algebras and thus t p,by the finite model property of intuitionistic propositional calculus. Since y does not exist in all Heyting algebras, d ( y ) defines a new implicit connective of intuitionistic calculus by Theorem 4.5.

The following algebraic argument shows that k,(?) has the disjunction property. Given a Heyting algebra H , denote by H' the Heyting algebra obtained by adding a new greatest element 1' to H (see, for instance, [ I ] ) .It is clear that y (0)is defined in H' whenever y (0)is defined in H . Moreover, the prescription

x for x E H, =f ~ b ) 1 f o r x = l ' ,

defines a homomorphism from y ) onto (H,y ) . Let F be a free algebra in ( H I ,

V ( d ( y ) ) . It is easy to check that the identity i d F :F -+ F can be lifted to a homomorphism h : F -+ F' in such a way that f h = idF. If a , b are elements of F such that a V b = 1, then h ( a )V h ( b ) = 1'. Since 1' is join-irreducible in F1we have h ( a ) = 1' or h ( b ) = 1'. Hence a = f ( h ( a ) )= 1 or b = f F (h ( b ) ) = 1 . This shows that 1 is join-irreducible in F.

After completing the first draft of this paper, we learned that the constant y(0) has been already proposed by Smetanich as an example of a new intuitionistic connective in the sense of Novikov (see [21]).

EXAMPLE5.2. The following set d ( S ) of schemas also defines a new implicit connective of intuitionistic calculus satisfying the disjunction property:

Indeed, S a ts,S 'a V (S 'a -+ a ) ts;S 1 a V a ts;S 'a v S ' a t S 'a , and so t,(s)u,is,iS a + S 'a , proving uniqueness. The other properties may be verified by algebraic means. Taking into account that in a Heyting algebra x < y iff x -+ y = 1, the corresponding system of equations E ( S ) can be expressed as follows:


There is a (necessarily unique) operation satisfying E l , E2, E3 in each complete well founded Heyting algebra H . For each x E H , define s (x) := {p E H : y < p implies y < x) and set S (x ) := V s(x). Since x E s (x) , we have x 5 S(x) and condition E l holds for S . To prove E2, suppose S(x) $ y V (y + x) . Then p $ y V (y + x) for some p E s(x). Therefore, p $ y and p $ y -+ x. The first inequality implies p A y < p and so p A y < x (since p E s(x)) . The second inequality implies p A y $ x , a contradiction. To prove E3, suppose that for some x E H , S(x) -+ x $ x. Since H is well founded, we may take p E H minimal such that p 5 S(x) ix and p $ x; then p $ S(x) . If q < p, then trivially q < S(x) + x, and by minimality we must have q < x. Therefore p E s(x) and so p 5 S (x), a contradiction.

It follows that in a finite chain H,, endowed with its natural Heyting algebra structure,

where x+ denotes the successor of x. As a matter of fact, S exists in a chain if and only if each element distinct from

1 has an immediate successor, in which case S is defined by (5.1). Indeed, if 0 < x < 1, then S(x) -+ x = x < 1. Therefore x < S(x). Now, if x < y , then S(x) Iy v (y ix) = y v x = y. This shows that S(x) is the immediate successor of x.

Since S exists in all finite Heyting algebras, conservativity over intuitionistic calculus will follow as in Example 5.1. Since S does not exists in [0, 11, we get from Theorem 4.5 that S is a new implicit connective of intuitionistic calculus.

To prove the disjunction property, suppose that S is defined on a Heyting algebra H and let HI be the Heyting algebra obtained by adding a new top element 1' to H . Then it is easy to check that the prescription S(1') = 1' extends S to HI in such a way that conditions El - E j are preserved. Then, argue as in Example 5.1.

EXAMPLE5.3. Gabbay shows in [6, 71 that the following schemas satisfy his definition of an intuitionistic connective and have a complete semantics in finite Kripke models (it is proven in [21] that G2 is a consequence of the other axioms).

GI. Ga + (P v (P -+ a ) ) , G2. (a P) 4 (Ga -+ GP),-+

G3. a + Ga, G4. Ga-+--a, G5. (Ga a ) -+ ( l l a -+ a ) .

This connective, as well as the connective y of Example 5.1. are definable from S . The reader may verify easily that if we set

then we may deduce the axioms of G in s?(S). Only Gs, that takes the form

((Sa A l l a ) + a) + ( l l a -+ a ) ,


needs a little checking. Indeed, (Sa A ??a) + a t i l a -+ (Sa -+ a ) ts, l?a 4 a;then apply the deduction theorem. Similarly, the definition

allows us to prove easily from s2 (S) the axioms of y . S is not definable from G or y because the Heyting subalgebra {O,l) of the chain

H3is closed under G but not under S,and the Heyting subalgebra {O,O+, 1) of Hq

is closed under y but not under S . Similarly, one may show that G and y are not mutually definable. However, S is definable from G and y together, since setting:

allows us to deduce the axioms of S in the system in d ( y ) U d ( G ) . For example, axiom S3becomes:

((ya V Ga) 4 a) + a .

By pure intuitionistic calculus: (ya V Ga) 4 a t (ya -+ a) A (Ga + a ) . On the other hand, y a -+ a t l l y a 4 i i a kc, i i a , and so ( y a V Ga) -+ a Id(,)-a a .A (Ga -+ a) td(G)

From the algebraic point of view, this means that the varieties V ( d ( S ) ) and V ( d ( y )u d ( G ) ) are mutually interpretable.

§6. The implicit connectives of intuitionistic n-valued logic. Let 9,be an axiomatization of the intermediate logic with values in H,, the Heyting chain of length n, n 2 3. For instance, we can add to Int the following axiom schemas ([lo], see [I91 for a different axiomatization):

Heyting three-valued logic, 9 3 , may be axiomatized alternatively by adding the single axiom: ((x + y) + z ) + (((z + x) -+ z) -+ z) (cf. [14]).

We show next that all implicit connectives of 9, are generated by the single connective S of Example 5.2. In fact, the logic 9,+S, given by the union of 9, and the axiom system s2 (S)for the connective S,does not admit extensions by new implicit connectives, even if we allow S to appear in the new axioms.

S is new over 9,for n 2 3 because the Heyting subalgebra {O,l) of Hnis not closed under the successor operation S, defined on H, by d ( S ) .

THEOREM6.1. The system 9,+ S is a conservative extension of 9,,strongly complete for valuations in the algebra (H,, S,). Moreover; any implicit connective of 9,+ S is equivalent in this system to a combination of A, V, +, and S . 1,

PROOE Since the variety V(_E",) is generated by H,, then, by Jonsson's lemma, the subdirectly irreducible algebras of this variety are exactly the chains HI,i < n. By compatibility, their respective expansions (HI,S,)are the subdirectly irreducible algebras of the variety V* = V ( 9 , + S ) . Therefore, by the subdirect decomposition theorem, the algebras of V(9,) are embedded in reducts of algebras in V*, and thus the extension is conservative. To prove completeness with respect to valuations into


(H,, S,), it is enough to notice that this algebra generates V * . Indeed, by compatibility of S, and uniqueness of S ,, the natural Heyting algebra homomorphism from H, onto H,, for i < n , is also a homomorphisms from (H,, S,) onto ( H , , S , ) .

Now, if an axiom system d ( S , V ) defines an implicit k-ary connective V over Yn+S , then the subdirectly irreducible algebras of V ( d (S , V ) ) have reducts among the subdirectly irreducible algebras of V * ;that is, they are of the form ( H I , S , , V , ) for some i < n. Let m be the maximum such i . By affine completeness of finite Heyting algebras, V , ( x l , . . . , XI,) is a Heyting polynomial p(x1, . . . , XI,, al , . . . , a,) with a, E H,. Since S , ( x ) is the successor function for 0 x < 1, by (5. l ) , it follows that each a, is definable in (H,, S,, V,) by one of the closed terms 0, S(O), S (S (O) ) , . . . , s"-' (0) , 1. Therefore, V , (xl , . . . , xk) = t ( X I , . . . , xic),a term of type {A, V, +, 7,S ) . Again by compatibility of V , and uniqueness of V , ,the other irreducible algebras (H ,, S ,,V , )are homomorphic images of (H,,,, S,, V,) , and the last equation holds in all of them. Therefore, it holds in all algebras of V ( d ( S , V ) ) and by completeness V(n1,. . . ,nk) * t (711,. . . ,711,). -I

The proof of the theorem shows that the class of implicit connectives of 9,coin-cides with the set of Heyting polynomials of H,; identical, by affine completeness, to the set of compatible functions of H,. It may be shown (cf. [3]) that these are exactly the functions f : ~ , k-+ H, satisfying, for some a E H,:

The unary implicit connectives of 9 3 , which constitute also the free algebra in one generator of the variety V ( Y 3+ S ) ,are depicted in Figure 2, in terms of their generator S .

It would be interesting to have answers to the following general questions. Does any implicit connective of pure intuitionistic calculus satisfies the disjunction

property? Does every intermediate logic have a unique completion by implicit connectives,

as 3,does?


Acknowledgment. The authors thank Pedro Massey for his opportune comments on a previous version of this paper.

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DEPARTAMENTO DE MATEMATICAS UNIVERSIDAD DE LOS ANDES

APARTADO AEREO 4976 BOGOTA. D.C. -COLOMBIA

E-mail: [email protected]

DEPARTAMENTO DE MATEMATICA FACULTAD DE CIENCIAS EXACTAS Y NATURALES

UNIVERSIDAD DE BUENOS AIRES -CONICET CIUDAD UNIVERSITARIA

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