Download the common rules of binary connectives are finitely based

Bulletin of the Section of Logic
Volume 18/2 (1989), pp. 87–90
reedition 2006 [original edition, pp. 87–89]
Wolfgang Rautenberg
THE COMMON RULES OF BINARY CONNECTIVES ARE
FINITELY BASED
A propositional logic (here a standard consequence relation ` in a
given propositional language) is said to be f.b. (finitely based) if all its sequential rules derive from a finite subset. A binary propositional connective
is proper if it depends on both arguments. Of the 16 binary connectives, 10
are proper ones, ∨, →, ←, ↔, ↑ (Sheffer function) and its duals. |=f denotes
the logic of the connective f in the language L with one 2-place operation
symbol whose writing will be omitted in the sequel. Left bracketing will be
used in writing formulas of L. p, q, ...,Tz denote distinct variables. If S is a
set of proper connectives then `S := {`f | f ∈ S} consists of the common
rules of the f ∈ S in L. The main result is
T
Theorem 1. `S = {|=f | f ∈ S} is f.b. for each set of proper binary
propositional connectives, S. Moreover, at most ternary rules are needed in
an axiomatization.
Example 1. p(p(pq))/pq is a unary rule common to all (including the
improper) binary connectives. p; q; pr/qr is a ternary such rule. The
unary rules p/pp3 p3 , pp3 p3 /p, p2 q 2 /q 2 p2 rule out the improper connectives. Modus ponens is a common rule for ↔, →, ∨, and the duals of →
and ↑.
Theorem 1 is interesting not only for logical or linguistical reasons
but also for systems of information processing dealing with incomplete
information (for instance, if no information is at hand which of a given
sample S of connectives was meant in a message from outside). The system
may work provisorically with the rules common to all f ∈ S. That these
are f.b. is particularly convenient for logical programming.
88
Wolfgang Rautenberg
An interesting algebraic consequence of Theorem 1 is that each variety
generated by a set of proper 2-element groupoids is finitely based in the
sense of equational logic.
Theorem 1 generalizes earlier results of the author. In [2] we showed
(as a special case) that |=f is f.b. for any f . In [3] we claimed
T that |=f
∩ |=f ∗ is f.b. where f ∗ is the dual of f . In [4] is shown that {|=f | f is
a proper semigroup connective} is f.b. None of these results can easily be
obtained because the intersection of f.b. logics in L needs not to be f.b.
even if these logics are defined by finite matrices (see [5]).
The proof of Theorem 1 is essentially based on the Theorems 2, 3
below which seem to be of considerable interest in itself. The proofs of the
Theorems are constructive, i.e., a base for `S can explicitely be exhibited
in each of the 1023 cases of S.
The following definition is similar to a definition from [1]. Logics
`1 , . . . , `n (in a fixed propositional language) are said to be independent if
there is a formula τ (p1 , . . . , pn ) with precisely the indicated variables such
that pi `i τ (p1 , . . . , pn ) `i pi for i = 1, . . . , n. An important example for the
purpose of Theorem 1 is the sequence |=→ , |=← , |=↔ , |=↑ . An independence
formula is given by
τ (p, q, r, s) = qq 2 (s2 s2 )p3 r3 (qq 2 (s2 s2 )p3 )3
as is shown by straight-forward calculation.
Theorem 2. If `1 , . . . , `n are independent and f.b. then `1 ∩ . . . ∩ `n is
f.b.
Example 2. As is well known, |=→ , |=← , |=↔ , |=↑ are f.b. Since these
logics are independent according to the above, the common rules of →,
←, ↔, ↑ are f.b., by Theorem 2. This yields some special cases of Theorem
1. The main obstacle here is that ∨ and ∧ cannot be included in this
argument.
For the rest we consider solely the language L with one 2-place operation symbol. This is only for convenience. In a more complex language
the additional rules in Theorem 3 have a somewhat more complex shape,
varϕ is the set of variables of a formula ϕ and a similar notion is used for
The Common Rules of Binary Connectives are Finitely Based
89
formula sets Φ.
Let ` be any logic in L. Define `∧ (the ∧-kernel of `), `∨ (the ∨-kernel
of `) and `s (the semilattice kernel of `) as follows:
Φ `∧ α iff ∅ =
6 Φ ` α and varα ⊆ varΦ,
Φ `∨ α iff ∅ =
6 Φ ` α and varϕ ⊆ varα for some ϕ ∈ Φ,
Φ `s α iff ∅ =
6 Φ ` α and varϕ ⊆ varα ⊆ varΦ for some ϕ ∈ Φ.
It is easily seen that `∧ , `∨ are the intersections of ` with the 2valued |=∧ , |=∨ , respectively, whereas the semilattice kernel of ` equals
` ∩ |=∧ ∩ |=∨ .
If τ (p, q) is a fixed formula put αβ := τ (α, β). If ρ is a proper standard rule (a finitary sequential rule), ρ = α1 ; . . . ; αn /α0 say, let ρ be
α1βu ; . . . ; αnβu /α0βu with β := α0 . . . αn and u 6∈ varβ. If ρ is axiomatic,
ρ = α say, let ρ be the unary rule pαu /αpu , with p, u 6∈ varα.
Theorem 3. Let ` be a logic in L based on a set of standard rules, R.
Suppose τ (p, q) is a formula with p ` τ (p, q) ` p. Then
(a) `s is based on the following set of rules, S:
R : ρ for all ρ ∈ R,
ρ1 : q; rp /rq ,
ρ2 : rpq /(rp )q , ρ20 (rp )q /rpq , ρ3 : rpq /rqp ,
pqr pqr
ρ4 : s /s , ρ5 : rp /rpp ,
ρ50 : rpp /rp , ρ6 : p/pp ,
0
p
q
pq
ρ6 : p /p,
ρ7 : p /p ,
ρ70 : ppq /pq .
(b) `∧ is based on S plus ρ8 : pq /p.
(c) `∨ is based on S plus ρ80 : p/pq .
Therefore, `s , `∧ , `∨ are f.b. provided ` is f.b.
Remark. Although `s equals (`∧ )∨ , it is impossible to derive finite
axiomalizability of `s from that of `∧ and `∨ , because there is no τ (p, q)
with p `∧ τ (p, q) and no τ (p, q) with τ (p, q) `∨ p. Notice that in the
calculus of `∧ the rules ρ50 , ρ60 , ρ70 are derivable with ρ8, and ρ5, ρ6, ρ7
are derivable in the calculus for `∨ .
Example 3. Consider the either-or logic |=+ . Then |=∨
+ = |=+ ∩ |=∨ axiomatizes the common rules of or and either-or. A rule base for |=+ can be
found in [2]. Clearly, p |=+ τ (p, q) |=+ p for τ (p, q) := q 2 p. From Theorem
3 (c) we obtain the following base for |=∨
+ (after some simplification):
90
Wolfgang Rautenberg
p; q; r/pqr,
q; p2 r/q 2 r,
pqrs/p(qr)s, p3 q/pq,
pq/qp, pqr/qpr, pqr/p(qr)
p2 /p,
p3 /p,
p/q 2 p.
The second rule is ρ1 in Theorem 3, the last two are ρ60 , ρ80 , respectively. The first rule is ternary. It is impossible to replace it by a set of at
most binary rules.
Thus, the “Moreover” part in Theorem 1 cannot be improved. As
a matter of fact, ternary rules are definitely needed in Theorem 1 only if
+ ∈ S. Otherwise, at most binary rules are sufficient. S = {∨} is the only
case such that `S is based on unary rules.
References
[1] G. Grätzer, H. Lakser, J. Plonka, Joins and direct products of
equational classes, Can. Math. Bull. 12 (1969), pp. 741–744.
[2] W. Rautenberg, 2-element matrices, Studia Logica 40 (1981),
pp. 315–353.
[3] W. Rautenberg, Consequence relations of 2-element algebras, [in:]
Foundations of Logic and Linguistics, Plenum Press, New York 1985.
[4] W. Rautenberg, Axiomatization of semigroup consequences, to
appear.
[5] A. Wroński, On finitely based consequences operations, Studia
Logica 35 (1976), pp. 453–458.