Download An Algebraic Approach to Intuitionistic Connectives Xavier Caicedo

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
OF S ~ ~ B OLOGIC
LIC
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~lof 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~lin 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 calculi 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
0022-48lZ/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~pletenessof 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
+ S is equivalent to a formula
connective S , so that any implicit connective of 9,
of this calculus.
52. Compatible functions in Heyting algebras. We assume that the reader is familiar 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 pseudocomplement, 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 nto~[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 adjunction between A and -+:
LEMMA
2.1. The following ave equivalent for any map f : H n
algebra H .
-+
H in a Heyting
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(xi ~ y i L)f ( x l , . . . , x n ) *f(~l>...>yn),fovallxi,yi
E H
c) f is a compatible function of H . -
PROOF.It is enough to consider unary functions. Assuming (a), then by (2.2):
( x )A ( x
y) = f ( xA ( x H y))A ( x H Y ) = f (yA (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
-1
f ( x A a ) A a = f ( x ) A a , by (2.1).
Condition (a) of the lemma is equivalent to the apparently stronger equation:
f
<
<
A compatible function which is not a polynomial is given in the next example
(cf. [12, 5 4.21):
EXAMPLE
2.1. 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
f
( x )=
a
/3
if x is even or x = a,
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 Iis 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.
THEOREM
2.2. Let f : H n + H be a compatible function. Then for any subset
S s H a n d x l ,. . . , x , E S :
V
f ( x l , . . . , x n )=
f ( a 1 , . . . , a,!n(xl - U ~ ) A . . . A ( o~n, , , ) .
( ul . . . . , a , ) ~ S n
PROOF.Fix ( x I , . . , x,)
E
S n . Then by (2.3)and (2.2),for all a l , . . . , a, E S :
. f ( a l ,. . . ,a , ) ~ ( a -l x ~ ) A + . . A (o~x ,, )
= f(x1 ,..., x,)A(al o x l ) ~ . . . A ( a , o x , ) L
f ( x l , . . . , x,).
-
. . . , 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 , )
Therefore f
(XI,
But,
E
Sn).
-1
Hence, f ( x l , .. . , x , ) = maxT.
COROLLARY
2.3. 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 threevalued 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
a
l O
a
l
f
0
a
1
A compatible function is always dense in a polynomial, as shown in the next
theorem.
2.4. For any n-ary compatible function f in a Heyting algebra H , there
THEOREM
is a Heytingpolynomial p ( x l , . . . , x,) of H such that
1624
XAVIER CAICEDO AND ROBERTO CIGNOLI
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
x0 = ( x H 0 ) . By (2.2) and (2.3),
V
P ( X I , .. . > x n )=
( S I....
=
(x
H
I),
f ( ~ ,1. . . ,x , ) A x ; ' A . . . A x ~ ' ~ "
,s,,)E{O,I)"
V
= f ( x l , . . . ,X , ) A
x SI 1A . . . A x ; : "
.... . S n ) ~ { 0 . 1 ) "
(51
Thus,
(
x.
x )f
( x . .x )
V
7
x ; A...Ax'"
( s,..... S , ) € { O , l ) "
L f (xl,...,xn),
considering that
-
V
3
x;' A . . . A x : ) = 1
sn)€{O.l)"
( S I ,...,
in any Heyting algebra.
COROLLARY
2.5. [9] Any boolean algebra is afine complete. The next two observations will be useful. 2.6. For any compatiblejilnction f ,
COROLLARY
PROOF.From Theorem 2.4, 77f = 7 7 p , and the statement holds for any Heyt-1
ing polynomial, as may be shown by induction on complexity.
COROLLARY
2.7. 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)
1
1
0
0
f(0)
1
0
1
0
x v 1 x 5 f ( x )5 1-(x v 1 x ) = 1
x 5 f ( x )< 1 1 x
1 X < f ( x )5
=1 x
0 < f ( x ) <_ 7-0 = 0.
7
- --
- --
-
7
1
~
-I
EXAMPLE
2.2. 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
1 = 0 and 0 = 1 . Thus, in case is compatible, Corollary 2.7 implies
that
x = l x . Hence,
A Heyting algebra admitting a compatible De Morgan negation -- must be
a boolean algebra, where x = - x .
- -
-
AN ALGEBRAIC APPROACH TO INTUITIONISTIC CONNECTIVES
1625
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.
EXAMPLE
3.1. The system E(y) consisting of the following three equations defines 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.
EXAMPLE
3.2. The following set of equations E (p) defines an implicit non compatible 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 P 1 ,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.
1626
XAVIER CAICEDO AND ROBERTO CIGNOLI
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.
3.3. Let H be a Heyting algebra where d H = A{TXV T T X : x E H )
EXAMPLE
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
( e ) = AH ( 1 ) = i . Hence, the Heyting
dH = i . Thus, AH ( x ) = 1 for x < i and
subalgebra S = ( 0 , i, 1) is closed under A ~but
, in this subalgebra
= 1 > i.
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.
3.1. I f a system ofequations E ( f ) is satisfied by a unique n-ary comnpatTHEOREM
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 ( x 1 , .. . , 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
(a7).
F)
onto
1627
AN ALGEBRAIC APPROACH TO INTUITIONISTIC CONNECTIVES
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 H does
not depend on a l , . . . , a, or h. Moreover, f ( a l ,. . . , a,) = f ( h ( % l )., . . , h (x,))=
h ( f ~ ( %, .l. . , x,)) = h ( q F ( x l ., . . , x I , , . . . ,x k ) ) = q H ( h ( % l, ). . . , A(x,,)
,...,
h ( ~ , + ~=
) ) q H ( a l, . . . , a,,..., a,). Hence, f~ = t H , where t ( x l , . . . , x,) =
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 Lowenheimi
Skolem theorem, and the identity f = t follows by uniqueness.
f
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:
COROLLARY
3.2. 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.
THEOREM
3.3. 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
f = t in any algebra
where f is dejined. Moreover, f is defined in all boolean algebras.
11
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 ( H 2 , where HZ
is the two elements boolean algebra and f satisfies E ( f ). By functional comcoincides with a boolean term u in H2. Since u
pleteness of this algebra,
satisfies the equations of E ( f ) in H z , it must satisfy them in all boolean algef satisfies E ( f ) in the boolean algebra Reg(H)
bras. On the other hand,
of regular elements of any Heyting algebra H where it is defined, because the
H + R e g ( H ) is also a homoonto homomorphism of Heyting algebras
morphism from ( H ,f H ) onto ( R e g ( H ) , f H ) by Corollary 2.6. By uniqueness,
f H 1 R e g ( H ) = ~ ~ ~ But
g ( ~ ~ ~ ) .~ = g7 7 u( H~ r R) e g ( H ) ,because of the interpretation of V in Reg(H) as 7 7 ( av b ) in H . By regularity of - 1 7 xand Corollary 2.6
again, 77f H ( X ) = f H ( 7 1 ~ =
) 1 7 u H ( 7 l x )= 7 7 u H( x )for any x E HI7. Take
t = 17U.
-I
fl),
f
17
17:
17
71
17
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 ,y H ) ,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 .
1628
XAVIER CAICEDO AND ROBERTO CIGNOLI
§4. Axiomatic extensions of intuitionistic calculus by implicit connectives. The
language L of formulas of the intuitionistic propositional calculus is built in the
corresponding to implication,
usual way from the connective symbols +, A, V, 7 ,
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
4.1. l- It,(,)
cp if and only if for any A E V ( d ( @ ) )and A-valuation
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.
THEOREM
4.1. The following conditions are equivalent for any d ( @ ) G L(@):
(1) t,(,) is strongly complete for the algebraic consequence relation. That is,
l- )F,I(,
cp implies l- Id
cp, (
for ,
any
)
r U {cp) C L(@).
AN ALGEBRAIC APPROACH TO INTUITIONISTIC CONNECTIVES
1629
( 2 ) kg(,) A:='=,
(ai ++ p i ) -t ( V ( ~. .I. ,, a,,) H V(Pl,. . . , P n ) ) , f o r each V E @.
PROOF,( 1 + 2 ) If G ( A ~(ai
, ~H Pi)) = 1 for some A-valuation v. then G(ai)=
v(Pi)for 1' = 1 : . . . , n , and so v ( V ( a l ,... , a n )++ V(P1,.. . , pfl))= VA(G(al),
.. . ,
~ ( a , )C)) VA(;ii(P1),
. . . , v(Pn))= 1 , for each V E @. Therefore,
From the strong completeness hypothesis and the deduction theorem we conclude ( 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
F ) becomes an algebra of
is well defined for each V E @, and thus (L(@)/O,
V ( d ( @ ) where
),
F = { f : V E @). The (L(@)/O,
F)-valuation v(n,) = [n,]
extends to ~ ( a=)[a]for any a E L(@).Therefore, ~ ( a=) 1 = [xl -+ n l ]if and
a H (xl -+ n l ) ,if and only if, F kd(,) a. It follows that G(y)= 1
only if Id(,)
for all y E r. Hence, r Itd(,)p implies G ( p ) = 1 ; that is, r td(,)(P
-I
DEFINITION
4.2. 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?( V 1 =
) { p ( V / V 1:)p
E sd(V)).
THEOREM
4.2. 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 transformations * 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*= V p + ,
[V1p]*
= Vp*.
= p , for propositional variables,
[ p 69 y]+= p+ CB y+ for CB = A , V, -+,
[ y 1 + = 7p+>
[Vpl+= Vp',
[V'p]' = V p * .
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 V 1 pof 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.
(P+
1630
XAVIER CAICEDO AND ROBERTO CIGNOLI
Claim]: x H y t p+ 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 V 1it
= [Vfcp]+
and [Vpl*= [Vp]+.
is trivial since by definition [V1p]*
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
Now
of formulas of L(V)in schemas of d(V).Define similarly d(Vf),,,,.
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,~/,
then cp = B(V1,
nl,
. . . , ~,,)[n,
/ty,] with B E d(Vf)
and ty, E L(V,
V').
So p* = O(V,
711, . . . , nk)[nl/
ty:] E d(V),because * does not change the formula
B(Vf,
711,. . . , nk),except for changing V' to V.Therefore, t,(,) p*. By a similar
, then
, ~pt
, is an axiom in d(V).. Therefore,
argument, if cp is an axiom in B ? ( V ) ~
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 and td(o)u,(ol)
ty + 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)*= V X hi 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*= V(cpT,
. . . , cp,+),
[Vf(pl,...,pn)l+
= V(pT,...,p,*).
-I
From Lemma 2.1, and Theorems 4.1 and 4.2, we obtain:
COROLLARY
4.3. 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: 4.4. Ifd(V)
COROLLARY
dejines
axiomatically a n-ary connective V ,then there is 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
= ( 4 E L : t,(,) 4 ) .
the logic I(d(V))
DEFINITION
4.3. 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
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 conservativitycondition in Definition 4.3 means
that R e d ( V ( d ( V ) ) a) nd 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 ( R e d ( V ( d( V ) ) ).)
Indeed, if H = H ' / F for a filter F in H' 5 H'l with ( H " , f ,) E V ( d ( V ) ) t,hen
F may be extended to F" in HI' so that F = F" n H'. Thus, H ' / F = H ' / ( F t ' n
H ' ) is isomorphic to a subalgebra of HI1/F1'. But ( H " , f o ) IF'' E V ( d( V ) )by
compatibility off,.
THEOREM
4.5. 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 ) )5
) V(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 intuitionistic 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. e use this fact in the next
irreducible in the free algebras of the variety V ( d ( V ) ) W
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.
1632
XAVIER CAICEDO AND ROBERTO CIGNOLI
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.
5.1. The following axiom system d ( y ) defines a new implicit connective
EXAMPLE
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
(aV P v lP),
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
f ~ b =)
x
1
for x E H,
forx=l',
defines a homomorphism from
y ) onto ( H ,y ) . Let F be a free algebra in
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 F 1 we
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]).
( H I ,
EXAMPLE
5.2. 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,i
S a + S ' a , proving uniqueness. The other properties may be verified
y iff
by algebraic means. Taking into account that in a Heyting algebra x
x -+ y = 1, the corresponding system of equations E ( S ) can be expressed as
follows:
<
AN ALGEBRAIC APPROACH TO INTUITIONISTIC CONNECTIVES
1633
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 ) i x 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 E l - E j are preserved. Then, argue as in Example 5.1.
EXAMPLE
5.3. 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 ) ,
1634
XAVIER CAICEDO AND ROBERTO CIGNOLI
needs a little checking. Indeed, (Sa A ??a) + a t i l a -+ (Sa
l ? a 4 a ; then apply the deduction theorem. Similarly, the definition
-+
a ) ts,
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
a
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)
Id(,)
-a A (Ga
a) td(G
a .)
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 Hn is not
closed under the successor operation S, defined on H, by d ( S ) .
THEOREM
6.1. The system 9, + S is a conservative extension of 9,,strongly
completefor 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,
PROOESince 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
<
AN ALGEBRAIC APPROACH TO INTUITIONISTIC CONNECTIVES
1635
(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, )
x < 1, by (5.l ) ,
with a, E H,. Since S , ( x ) is the successor function for 0
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
S ) . Again by compatibility of V , and uniqueness of V , , the
of type {A, V, +, 7 ,
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
-I
V(d(SV
, ) )and by completeness
V ( n 1 ,. . . , nk) * t (711,. . . ,711,).
<
The proof of the theorem shows that the class of implicit connectives of 9,coincides 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
k H, satisfying, for some a E H,:
exactly the functions f : ~ , -+
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?
1636
XAVIER CAICEDO AND ROBERTO CIGNOLI
Acknowledgment. The authors thank Pedro Massey for his opportune comments
on a previous version of this paper.
REFERENCES
[l] R. BALBES
and P. DWINGER,
Distvib~ttivelattices, University of Missouri Press, Columbia, Missouri. 1974.
Znvestigaciones acerca de 10s conectivos intuicionistas, Revista de la Academia Colom[2] X. CAICEDO,
biana de Ciencias Exactas, Fisicas y Natrtrales, vol. 19 (1995), pp. 705-716.
[3]
, Conectivos sobre espacios topoldgicos. Revista de la Academia Colombiana de Ciencias
Exactas, Fisicas y Naturales. vol. 21 (1997). pp. 521-534.
[4] M. C. FITTING.
Intuitionistic logic, model theory, and forcing. North-Holland, Amsterdam. 1969.
Aspectsof topoi, Bulletin ofthe Australian MathematicalSociety. vol. 7 (1972), pp. 1-76.
[5] P. FREUD.
On some nebv intuitionistic propositional connectives, I, Studia Logica, vol. 36
[6] D. M. GABBAY.
(1977). pp. 127-139.
[7]
, Semantical investigations in Heyting intuitionistic logic. Reidel Publishing Company.
Dordrecht. 1981.
[8] R. GOLDBLATT.
Topoi, the categovical analysis of logic, North Holland, Amsterdam, 1984.
[9] G. GRATZER,O n boolean jilnctions, (Notes on lattice theory 11). Revue Roumaine de
Mathematiques Pures et Appliquees, vol. 7, 1962, pp. 693-697.
Eqtlational classes of relative Stone algebras. Notve Dame Journal of
[lo] T. HECHTand T. KATRINAK.
Formal Logic. vol. 13 (1972), pp. 248-254.
[ l l ] K. KAARLY
and A. F. PIXLEY.ABne complete varieties. Algebra Universalis, vol. 24 (1987),
pp. 7 4 9 0 .
[I21 -. Polynomial completeness in algebraic systems, Chapman and Hall, Boca Raton, 2000.
[13] M. KAMINSKI.
Nonstandard connectives of intuitionistic propositional logic, Notve Dame Journal
ofFormal Logic, vol. 29 (1988). pp. 309-331.
[14] L. MONTEIRO.
Algibre du calct~lpvopositionneltrivalent de Heyting. F~tnhmentaMathematicae.
vol. 74 (1972), pp. 99-109.
[15] A. PIXLEY,
Completeness in avithmeticalalgebras, Algebra Universalis. vol. 2 (1972), pp. 179-196.
[16] H. RASIOWA,
An algebraic appvoach to non-classical logics. North Holland. Amsterdam. 1974.
[17] H. RASIOWA
and R. SIKORSKI,
The
mathematics ofmetamathematics, Polish Scientific Publishers.
Warsaw. 1963. Third edition. 1970. [18] G. E. REYESand H. ZOLFAGHARI,
Bi-Heyting Algebras, topos and modalities, Journal of Philosophical Logic. vol. 25 (1996). pp. 25-43.
[19] I. THOMAS.
Finite limitations on Dtlnzmett's LC, Notre Dame Journal of Fovmal Logic, vol. 3
(1962). pp. 170-174.
[20] A. TOURAILLE.
The word pvoblenz for Heyting* algebras. Algebra Universalis. vol. 24 (1987).
pp. 120-127.
[21] A. D. YASHIN,
New solutions to Novilcov'spvoblem,fov intuitionistic connectives. Journal of Logic
and Computation. vol. 8 (1998). pp. 637-664.
p
p
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 1428 BUENOS AIRES -ARGENTINA E-mail: [email protected]