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24Логические исследования 2015. Т. 21. № 2. С. 53-60 УДК 510.649
Logical Investigations 2015, vol. 21, no 2, pp. 53-60
J. ClUOIURA
A Weakly-Intuitionistic Logic 11
Janusz Ciuciura
Department of Logic, University of Lodz, 16/18 Kopcinskiego, 90-232 Lodz, Poland. E-mail: janciu@uni.lodz.pl
In 1995, Sette and Carnielli presented a calculus, I1, which is intended to be dual to the paraconsistent calculus P1. The duality between 11 and P1 is reflected in the fact that both calculi are maximal with respect to classical propositional logic and they behave in a special, non-classical way, but only at the level of variables. Although some references are given in the text, the authors do not explicitly define what they mean by 'duality' between the calculi. For instance, no definition of the translation function from the language of 11 into the language of P1 (or from P1 to 11) was provided (see [4], pp. 88-90) nor was it shown that the calculi were functionally equivalent (see [13], pp. 260-261).
The purpose of this paper is to present a new axiomatization of 11 and briefly discuss some results concerning the issue of duality between the calculi.
Keywords: weakly-intuitionistic logic, paracomplete logic, 11, paraconsistent logic, Sette's calculus, P1
1. Introduction
Suppose that L is a logic defined in a propositional language with at least the connectives: A and V. We say that (1) a logic L is weakly-intuitionistic if the law of excluded middle aV ~ a is not valid in L; (2) a logic L is weakly-paraconsistent if the law of non-contradiction ~ (aA ~ a) is not valid in L (cf. [11], p. 182).
Though this definition is intuitive enough, it may give rise to some doubts. Observe, for example, that intuitionistic logic is weakly-intuitionistic (but not vice versa) and there are some paraconsistent logics which are weakly-paraconsistent — it suffices to recall that the law of noncontradiction is not valid in da Costa's calculi Cn (see [2]). On the other hand, some paraconsistent logics such as CLuNs or Jaskowski's discursive logic are not weakly-paraconsistent at all (see [1], [6] and [7]). So, as we can see, there is a kind of asymmetry here probably caused by the lack of uniform criteria for paraconsistency (cf. [8]) — not to mention that one and only one paraconsistent logic does not exist, if any (see [12]).
Another point is that the calculi 11 and P1 are defined in a propositional language with the connectives of negation and implication
© Ciuciura J.
taken as primitives. In fact, the connectives of conjunction and disjunction are nothing but abbreviations (see [9, p. 178] and [11, p. 199], for details) which do not appear explicitly in formulas. This leads to an alternative definition of the weakly-intuitionistic (and weakly-paraconsistent) logic, viz.
Definition 1. A logic L is weakly-intuitionistic if the law of Clavius, p1 — p1) — pi, is not valid in L, for any p1 € var.
Definition 2. A logic L is weakly-paraconsistent if the law of Duns Scotus, p1 — p1 — p2), is not valid in L, for any p1, p2 € var.
where var = {p1,p2,p3, ■ ■■} and i € N.
Now let us consider the following axiom schemata: (A1) a — (0 — a)
(A2) (a — (0 — j)) — ((a — 0) — (a — j)) (PL) ((a — 0) — a) — a (DS) a — ( ~ a — 0) (CM) a — a) — a
and Detachment, (MP) a, a — 0 / 0, as the sole rule of inference.
Notice that (MP) plus (A1), (A2), (PL), (DS) and (CM) define classical implication and classical negation (cf. [5, p. 437]). This will be a starting point for our analysis, in which a new axiomatization of 11 (and P1) is proposed.
2. Weakly-Intuitionistic Calculus 11
In this section, we present a new axiomatization of the calculus 11. The axiom schemata will be chosen to show that 11 behaves in a weakly-intuitionistic way only at the level of variables, i.e. the so-called consequentia mirabilis, a — a) — a, is an 11-tautology provided that a is not a propositional variable. As will be seen, a new set of axioms for 11 is easily obtained from the set given in Section 1 by imposing an additional condition on the axiom (CM).
Let var be a non-empty denumerable set of all propositional variables. The set of all formulas, For, is inductively defined as follows:
(1) pi € For, where pi € var and i € N
(2) if a € For, then ~ a € For
(3) if a and 0 € For, then a — 0 € For.
The calculus 11 is axiomatized by means of the following axiom schemata:
(A1) a - (0 - a)
(A2) (a - (0 - 7)) - ((a - 0) - (a - y)) (A3) a - ~ 0) - a - 0) a) (A4)--(a - 0) - (a - 0).
and (MP) a, a - 0 / 0 [11, p. 182-183].
Definition 3. A formal proof (deduction) within 11 of a from the set formulas of r is a finite sequence of formulas, 01,02,...0n, where 0n = a and each of elements in that sequence is either an axiom of 11, or belongs to r, or follows from the preceding formulas in the sequence by (MP).
Definition 4. A formula a is a syntactic consequence within 11 of a set formulas of r (r \~i 1 a, in symbols) iff there is a formal proof of a from the set r within 11.
Definition 5. A formula a is a thesis of 11 iff 0 \~I 1 a.
Theorem 1. r \~I1 a - 0 iff r U {a} \~I1 0, where a, 0 € For, r C For.
Proof. By induction. Apply (A1), (A2), a - a and use (MP) as the sole rule of inference. □
Fact 1. The formulas
(TR) (a - 0) - ((0 - y) - (a - Y))
(SIM) (a - (a - 0)) - (a - 0)
(PER) (a - (0 - y)) - (0 - (a - Y))
(PL) ((a - 0) - a) - a
(DS) a - ( ~ a - 0)
(NN1) a - a
(NN2v) a - a, if a € var
(CMv) a - a) - a, if a € var
are provable in I1.
Proof. (TR), (SIM), (PER) by Theorem 1 (DT for short) and (MP). (PL). See [11, p. 188-189]. (DS). Ibid., p. 189-190. (NN2v). Ibid., p. 183-184.
(NN1).
(a) a
(b)
(c)
(d)
(e)
(f) (g)
a — (
—— a —^
a
a a
a
rsj rsj rsj
a — a)
a
— a ► ~ a) — (( a) — a
rsj rsj rsj
a — a) — — a)
(CMv). By cases. Let a £ var then Case 1: a is of the form ~ 0.
(a) — 0 — — 0
(b) — 0 — (— 0 — 0)
(c) — 0 — ( — 0 — 0)
(d) — 0 — 0
(e) (— 0 — — 0) — ((—— 0 —
(f) ( — 0 — 0) — — 0
(g) ~ 0
Case 2: a is of the form 0 — 4.
(a) - (0 — 4) — (0 — 4)
(b) — (0 — 4) — — (0 — 4)
(c) — (0 — 4) — (0 — 4)
(d) — (0 — 4)
(e) — (0 — 4) — (0 — 4)
(f) 0 — 4
by DT
{(A1)}
{(MP), (a), (b)} {(NN2v) : a / var} {(A3)}
{(MP), (d), (e)} {(MP), (c), (f )}.
by DT {(DS )}
{(TR), (MP), (a), (b)} {MP), (c), (SIM)} {(A3)}
{(MP), (a), (e)} {(MP), (d), (f)}.
by DT
{(NN2v) : a / var} {(TR), (MP), (b), (a)} {(A3), (MP), (b), (c)} {(NN2v) : a / var} {(MP), (d), (e)}.
□
Sette and Carnielli proved that 11 was complete with respect to the matrix
Mji = ({0,1, 2}, {1},-, —),
where {0,1,2} is the set of logical values, {1} contains the designated value and the connectives of implication and negation are defined by the truth-tables:
— 1 2 0 rsj
1 1 0 0 1 0
2 1 1 1 2 0
0 1 1 1 0 1
An 11-valuation is any function v : For —> {1,2,0} compatible with the above truth-tables. An 11-tautology is a formula which under any valuation v takes on the designated value {1}.
Observe that neither the formula a — a) — a nor a — a is an 11-tautology.
Let I * be a calculus axiomatized by
(A1) a — (ß — a)
(A2) (a — (ß — y)) — ((a — ß) — (a — y))
(PL) ((a — ß) — a) — a
(DS) a — ( ~ a — ß)
(CMv) a — a) — a, if a £ var
and the rule (MP), then
FACT 2. I* = 11.
Proof.
(c) (A1), (A2), (PL), (DS) and (CMv) are theorems of I1 (cf. Fact 1) and (MP) is the sole rule of inference in 11. Then, by soundness, all the formulas are 11-tautologies and (MP) preserves validity.
(d) What is desired is to demonstrate that (A3) and (A4) are provable in I *. To prove this, we first need to show that some additional formulas are provable in I*, viz. (TR), (SIM), (PER), (R) ((a — 3) — a) — ((( — a) — a), (CMn) a — ^^ a) — ^^ a, (NN1) a — ^^ a and (CON)
(a — () — (a — ()) — (a — 3) — (a — ()).
(TR), (SIM), (PER) by DT and (MP). (R) by DT, (PL) and (MP). (CMn).
(a) ~ a — ^^ a
(b) a — ~ a)
(c) a — ^^ a) ■
(d) a — ~ a)
(e) a — ^^ a) ■
(f) a
~ a
a —y ~
^^ a a
by DT
{(CMv) : ~ a/a} a) — — a) {(MP), (TR), (b)} {(MP), (a), (c)} {(MP), (R), (d)} {(MP), (a), (e)}.
(NN1).
(a) a
(b) a
(c) — a —
(d) a (f) — a
(CON).
a — —— a) -y —— a
—y —— a) —y —— a
by DT {(DS)}
{(MP), (a), (b)} {(CMn))} {(MP), (c), (d)}.
rsj rsj rsj
(a — P))
(a) — (a — P)
(b) — (a — p)
(c) —— (a — P) — (——— (a — P) — (a — P))
(d )---(a — P) — (a — P)
(e) — (a — P) — (a — P)
(f) (— (a — P) — (a — P)) — (a — P)
(g) a — P
by DT by DT {(DS)}
{(MP), (b), (c)} {(MP), (TR), (a), (d)} {(CMv) : a — P/a} {(MP), (e), (f)}.
Now we can prove that (A3) and (A4) are theses of I*. (A3).
(a)
(b)
(c)
(d)
(e)
(f) (g)
(h)
(i)
(j) (k)
—— a — — P —— a — P
(P — (— P — — a)) — (—— a — (— P — — a))
P — (— P — — a)
—— a — (— P — — a)
— P — (—— a — — a)
((— P — (—— a — — a)) — (—— a —
(—— a — — a)))
—— a — (—— a — — a) —— a — — a (—— a — — a) — — a
a
by DT by DT
{(MP), (TR), (b)} {(DS)}
{(MP), (d), (c)} {(MP), (PER), (e)}
{(MP), (TR), (a)} {(MP), (f), (g)} {(MP), (SIM), (h)} {(CMv) : — a/a} {(MP), (i), (j)}.
(A4).
(a) —— (a — P)
(b) — (a — P) — ——— (a —
(c) —— (a — P) — (a — P)
(d) a — P
by DT
P) {(NN1)}
{(MP), (CON), (b)} {(MP), (a), (c)}.
□
It is an immediate consequence of Fact 2 that the calculus 11 is axiomatizable by (A1), (A2), (PL), (DS), (CMv) and (MP).
3. The Issue of Duality
Just like 11, the calculus P1 is expressed in a language using negation and implication as primitives. In this language P1 is axiomatized by
(A1) a - (ft - a)
(A2) (a - (ft - y)) - ((a - ft) - (a - y)) (A3) a - ~ ft) - ((~ a - ft) - a) (A4) ~ (a - a) - a (A5) (a - ft) - — (a - ft).
The sole rule of inference is (MP).
It is worth mentioning that (A4) is not independent (cf. [10, p. 155]).
Fact 3. The formulas
(DSv) a - ( ~ a - ft), if a £ var (CM) a - a) - a (NN1v) a - a, if a £ var (NN2) a - a
are provable in P1.
Fact 4. (See [3]) P1 is axiomatizable by (A1), (A2), (PL), (DSv), (CM) and (MP).
The axiom (DSv) is of special interest here because it reveals that P1 behaves in a paraconsistent manner only at the level of variables, i.e. a - ( ~ a - ft) is a P1-tautology only if a is not a propositional variable. Similarly to the calculus 11, the thought behind this was to demonstrate that it is possible to obtain a new set of axioms for P1 by imposing an additional condition on one of axioms given in Section 1. At this time, however, it is the axiom (DS). In this sense, the calculi I1 and P1 may be seen as dual (at least from the axiomatic perspective).
Acknowledgements. I would like to express my gratitude to the anonymous referee, whose comments helped to improve the manuscript.
References
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