Download 1. Introduction 2. Curry algebras

Bulletin of the Section of Logic
Volume 16/4 (1987), pp. 151–156
reedition 2005 [original edition, pp. 151–158]
Jair Minoro Abe
A NOTE ON CURRY ALGEBRAS
1. Introduction
In one of its possible formulations, the principle of the excluded middle
says that, from two propositions A and ¬A (the negation of A), one is
true. A paracomplete logic is a logic which can be the basis of theories
in which there are propositions A such that A and ¬A are both false. So,
we may assert that in a paracomplete logic the law of the excluded middle
fails. For a discussion of such kind of logic, as well as for the study of some
paracomplete systems, see da Costa and Marconi 1987a and 1987b. In the
first of these papers, the authors investigate a hierarchy Pn , l ≤ n ≤ ω,
of paracomplete propositional calculi; these calculi are the “duals” of the
paraconsistent propositional logics Cn , l ≤ n ≤ ω, introduced be da Costa
(cf. da Costa 1963). In this note we present an algebraization of P1 ,
developing some ideas of da Costa and Marconi 1987a, and study some of
the main properties of the resulting algebraic system. It is not difficult
to verify that the negation operator, in such an algebraic system, is not
compatible with the basis equivalence relation. Thus, our algebraic systems
constitute Curry algebras in the sense of da Costa 1966.
2. Curry algebras
The algebraic structures here considered are of the following type:
Definition 1. An algebra is a structure
R = < S, ≡, (fk )k∈K , (Cn )n∈N >, where
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(i) S is a nonempty set;
(ii) ≡ is an equivalence relation defined on S;
(iii) (Fk )k∈K is a finite nonempty family of operations on S; and
(iv) (Cn )n∈N is a finite family of elements of S.
Definition 2. An algebra R is called a Curry algebra if at least one of
its operations is non-monotone in the sense Curry 1977. (The expressions
“monotone operation relative to ≡” and “operation compatible with ≡”
have the same meaning.)
It is easy to define the concepts of lattice, Boolean algebra, etc. when
the basic relation of the system is an equivalence relation, instead of the
relation of equality. When the operations are all monotone, we can pass to
the quotient, and obtain a usual algebraic structure; otherwise, we have to
deal with a more general species of structures.
Definition 3. A CP1 -algebra (or CP1 -lattice) is a classical implicative
lattice < S, ≡, ⊃, ∧, ∨,0 > with a greatest element and with an operator 0
satisfying the following conditions, where p∗ =def p ∨ p0 :
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
p ≤ p00
(p ∧ p0 ) ≡ 1
p∗ ∧ q ∗ ≤ (p ⊃ q)∗
p∗ ∧ q ∗ ≤ (p ∧ q)∗
p∗ ∧ q ∗ ≤ (p ∨ q)∗
p∗ ∧ q ∗ ≤ (p0 )∗
p∗ ≤ (p ⊃ q) ⊃ ((p ⊃ q 0 ) ⊃ p0 )
p ≤ (p0 ⊃ q).
3. The calculus P1
Da Costa and Marconi constructed a hierarchy of paracomplete propositional calculi P1 , . . . , Pω (da Costa and Marconi 1987a). The primitive
connectives of these calculi are: ⊃, ∧, ∨, and ¬. The symbol ∼ for equivalence is introduced as usual. Here we shall only treat the calculus P1 , but
our methods can also be applied to Pn , 2 ≤ n ≤ ω.
Definition 4. A∗ ≡def A ∨ ¬A, where A is a formula.
A Note on Curry Algebras
153
The postulates (axiom schemes and primitive rules of inference) of P1
are those of classical positiove logic plus the following:
(¬1 )
(¬2 )
(¬3 )
(¬4 )
(¬5 )
A∗ ⊃ ((A ⊃ B) ⊃ ((A ⊃ ¬B) ⊃ ¬A))
A∗ ∧ B ∗ ⊃ (A ⊃ N )∗ ∧ (A ∧ B)∗ ∧ (A ∨ B)∗ ∧ (¬A)∗
¬(A ∧ ¬A)
A ⊃ (¬A ⊃ B)
A ⊃ ¬¬A.
Theorem 1. In P1 , ¬ is not compatible with the equivalence relation ∼,
and we have ` A ∧ ¬A ⊃ B.
Definition 5.
B ≤ A.
In P1 : A ≤ B =def ` A ⊃ B, A ≡ B =def A ≤ B and
Theorem 2. ≤ is a quasi-order, and ≡ is an equivalence relation.
4. The CP1 -algebras
From the algebraic point of view, P1 is a classical implicative lattice. By
Theorem 1, it follows that this lattice has a first element. Due to ¬1 − ¬5 ,
we conclude that in this lattice there is an operator, denoted by 0 , possissing
some properties of the Boolean complement. Summarizing, P1 is a CP1 algebra.
Theorem 3. A CP1 -algebra has a greatest element and is distributive.
Definition 6. ¬∗ A =def A ⊃ (B ∧ B 0 ), where B is a fixed formula.
Theorem 4. In a CP1 -lattice, ¬∗ A is a Boolean complement of A; so,
A ∨ ¬∗ A ≡ 1 and A ∧ ¬∗ A ≡ 0, where 1 and 0 are greatest element and
least element of the lattice, in the sense of Curry 1977.
Theorem 5. In a CP1 -lattice, the structure composed by the underlying
set and by operations ∧, ∨, and ¬∗ is a Boolean algebra.
Definition 7. Let R = < S, ≡, ⊃, ∧, ∨,0 > be a CP1 -algebra and <
S, ≡, ∧, ∨, ¬∗ > the Boolean algebra obtained as in the above theorem.
Any Boolean algebra that is isomorphic to the quotient algebra of < S, ≡
, ∧, ∨, ¬∗ > by ≡, is called Boolean algebra associated with the CP1 -algebra
R.
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We have the following representation theorem:
Theorem 6. Any CP1 -algebra is associated with a field of sets. Moreover,
any CP1 -algebra is associated with the field of sets simultaneously open and
closed of a totally disconnected compact Hausdorff space.
5. Filters in a CP1 -algebra
We easily define the notions of filter and ultrafilter in a CP1 -algebra. The
filters are partially ordered by set inclusion. Filters which are maximal
with respect to this ordering are called ultrafilters, and it is easy to prove
that every filter in a CP1 -algebra can be extended to an ultrafilter.
Definition 8. If R1 = < S1 , ≡1 ,⊃1 , ∧1 ,∨1 ,01> and R2 = < S2 , ≡2 ,⊃2 ,
∧2 , ∨2 ,02> are two CP1 -algebras, a homomorphism of R1 into R2 is a map
f of S1 into S2 which preserves the algebraic operations, i.e. such that for
x, y ∈ S1 :
(1) f (x ∧1 y) ≡2 f (x) ∧2 f (y)
(2) f (x ∨1 y) ≡2 f (x) ∨2 f (y)
(3) f (x ⊃1 y) ≡2 f (x) ⊃2 f (y)
(4) f (x01 ) ≡2 f (x)02
(5) if x ≡1 y, then f (x) ≡2 f (y).
Theorem 7. Let R1 and R2 be CP1 -algebras and f a homomorphism
from R1 into R2 . The set {x ∈ S1 | f (x) ≡ 1} (the shell of f ) is a filter.
6. Soundness and completeness theorems
Theorem 8. If the shell of a homomorphism v from F (the class of all
formulas of O1 ) into a CP1 -algebra is an ultrafilter, then:
(1) v(A) ≡ 1 entails v(¬A) ≡ 0,
(2) v(A) ≡
6 1v(¬A) entails v(¬A) 6≡ v(¬¬A),
(3) if v(A) 6≡ v(¬A) and v(B) 6≡ v(¬B), then v(A ⊃ B) 6≡
v(¬(A ⊃ B)), v(A ∧ B) 6≡ v(¬(A ∧ B)), and v(A ∨ B) 6≡ v(¬(A ∨ B)).
(4) v(¬(A ∧ ¬A)) ≡ 1.
A Note on Curry Algebras
155
Definition 9. Let F be the class of all formulas of P1 and v a homomorphism from F into a CP1 -algebra. We write v |= Γ where Γ is a subset of
F , if for each A ∈ Γ, v(A) ≡ 1, Γ |= A means that for all homomorphisms
v from F into an arbitrary CP1 -algebra, if v |= Γ then v(A) ≡ 1.
Theorem 9. (Soundness) If A is a provable formula of P1 , then v(A) ≡ 1
for any homomorphism v from P1 into R, where R is an arbitrary CP1 algebra (that is, |= A).
Proof. By induction on the length of proofs.
Theorem 10. Let U be an ultrafilter in F . Then, there is a homomorphism v from P1 into 2 such that the shell of v is U (where 2 = {0, 1} is
the two-element Boolean algebra).
Theorem 11. (Completeness) Let F be the set of all formulas of P1 , and
A ∈ F . Let us suppose that v(A) ≡ 1 for any homomorphism v from P1
into an arbitrary CP1 -algebra. Then, A is a provable formula of P1 .
Proof. If A is not provable, then A 6≡ 1 and so A0 6≡ 0. Therefore
there is an ultrafilter U in F that contains A0 . By Theorem 10, there is
a homomorphism v from P1 into 2 and thus v(A0 ) ≡ 1. If follows that
v(A) ≡ 0, which is a contradiction.
We think that the results presented here, as well as many others from
the references, justify the study of certain “pre-algebraical” structures,
without passing to the quotient by an appropriate equivalence relation. It
is worthwhile to observe that this is true not only for non-classical logics,
but also for classical logic, as shown by Eytan 1975.
References
[1] H. Curry, Foundations of Mathematical Logic, Dover, New
York 1977.
[2] N. C. A. da Costa, Calculus propositional pour les systemes formels
inconsistents, CR Acad. Sc. Paris 257 (1963), pp. 3790–3792.
[3] N. C. A. da Costa, Filtres et Idéaux d’une algébre Cn , CR Acad.
Sc. Paris 264 (1967), pp. 549–552.
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[4] N. C. A. da Costa, Opérations non-monotones dans les treillis,
CR Acad. Sc. Paris 263 (1966), pp. 429–432.
[5] N. C. A. da Costa and D. Marconi, A note on paracomplete logic, to
appear in Rendiconti dell’ Accademia Nazionale dei Lincei (1987).
[6] N. C. A. da Costa and D. Marconi, Paraconsistent and Paracomplete logics, to appear in Rendiconti dell’ Accademia Nazionale dei
Lincei (1987).
[7] M. Eytan, Tableaux de Smullyan, ensembles de Hintikka et tout ça:
un point de vue algébrique, Math. Sci. Humaines 48 (1975), pp. 21–27.
[8] S. C. Kleene, Introduction to Metamathematics, Van Nostrand, New York 1952.
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Department of Sciences
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