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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.