Wolfgang Rautenberg

2-Element matrices

Sections 1, 2 and 3 contain the main result, the strong finite axiomatizability of all 2-valued matrices. Since non-strongly finitely axiomatizable 3-element matrices are easily constructed the result reveals once again the gap between 2-valued and multiple-valued logic. Sec. 2 deals with the basic cases which include the important Fi∞ from Posts classification. The procedure in Sec. 3 reduces the general problem to these cases. Sec. 4 is a study of basic algebraic properties of 2-element algebras. In particular, we show that equational completeness is equivalent to the Stone-property and that each 2-element algebra generates a minimal quasivariety. The results of Sec. 4 will be applied in Sec. 5 to maximality questions and to a matrix free characterization of 2-valued consequences in the lattice of structural consequences in any language. Sec. 6 takes a look at related axiomatization. problems for finite algebras and matrices. We study the notion of a propositional consequence with equality and, among other things, present explicit axiomatizations of 2-valued consequences with equality.

On reduced matrices

It is shown that the class of reduced matrices of a logic ⊢ is a 1st order ∀∃-class provided the variety associated with ⊢ has the finite replacement property in the sense of [7]. This applies in particular to all 2-valued logics. For 3-valued logics the class of reduced matrices need not be 1st order.

Axiomatizing logics closely related to varieties

Let V be a s.f.b. (strongly finitely based, see below) variety of algebras. The central result is Theorem 2 saying that the logic defined by all matrices (A, d) with d ε A ε V is finitely based iff the A ε V have 1st order definable cosets for their congruences. Theorem 3 states a similar axiomatization criterion for the logic determined by all matrices (A, εA), A ∈ V, ε a term which is constant in V. Applications are given in a series of examples.

Applications of weak Kripke semantics to intermediate consequences

abstractSection 1 contains a Kripke-style completeness theorem for arbitrary intermediate consequences. In Section 2 we apply weak Kripke semantics to splittings in order to obtain generalized axiomatization criteria of the Jankov-type. Section 3 presents new and short proofs of recent results on implicationless intermediate consequences. In Section 4 we prove that these consequences admit no deduction theorem. In Section 5 all maximal logics in the 3rd counterslice are determined. On these results we reported at the 1980 meeting on Mathematical Logic at Oberwolfach. This paper concerns propositional logic only.

Finite replacement and finite Hilbert-style axiomatizability

We define a property for varieties V, the f.r.p. (finite replacement property). If it applies to a finitely based V then V is strongly finitely based in the sense of [14], see Theorem 2. Moreover, we obtain finite axiomatizability results for certain propositional logics associated with V, in its generality comparable to well-known finite base results from equational logic. Theorem 3 states that each variety generated by a 2-element algebra has the f.r.p. Essentially this implies finite axiomatizability of a 2-valued logic in any finite language.

A calculus for the common rules of ∧ and ∨

We provide a finite axiomatization of the consequence ⊢∧∪⊢∨, i.e. of the set of common sequential rules for ∧ and ∨. Moreover, we show that ⊢∧∪⊢∨ has no proper non-trivial strengthenings other than ⊢∧ and ⊢∨. A similar result is true for ⊢↔∪⊢→, but not, e.g., for ⊢↔∪⊢+.

Axiomatization of the De Morgan type rules

In Section 1 we show that the De Morgan type rules (= sequential rules in L(∨, ⌝) which remain correct if ∧ and ∨ are interchanged) are finitely based. Section 2 contains a similar result for L(→). These results are essentially based on special properties of some equational theories.

Axiomatization of semigroup consequences

We show (1) the consequence determined by a variety V of algebraic semigroup matrices is finitely based iffV is finitely based, (2) the consequence determined by all 2-valued semigroup connectives, Λ, ∨, ↔, +, in other words the collection of common rules for all these connectives, is finitely based. For possible applications see Sect. 0.

CONSEQUENCE RELATIONS OF 2-ELEMENT ALGEBRAS

We show that the consequence determined by a 2-element algebra has at most 7 proper non-trivial extensions and that it is finitely axiomatizable with sequential rules. This implies among other things the finite axiomatizability of each 2-valued consequence in a finite language.

Strongly finitely based equational theories

Let ~ denote the result of discarding the replacement rules from the Birkhoff calculus [- for equational logic. ~ is not necessarily incomplete, i.e., for suitable sets F of equations F ]- e may imply F ~ e, for all equations e. Call an equational theory T strongly based on F __ Tiff each equation e in T is derivable from F in the rudimentary Birkhoff calculus ~, that is, F ~ e. T is said to be strongly finitely based (s.f.b.) provided T is strongly based on some finite F. Clearly, a s.f.b, theory isfb. (finitely based). Conversely, many familiar lb. equational theories turn out to be sf.b., others not. The calculus ~ looks artificial only at first glance. It has interesting applications in general propositional logic. ~ links equational logic with Hilbert style axiomatization in a clear way: A variety V (more precisely, its equation theory) is sf.b. iff the propositional consequence defined by all logical matrices whose underlaying algebras belong to V is finitely axiomatizable, i.e., based on finitely many inference rules (cf. [10], also for the mentioned applications). Roughly speaking, replacement, if translated into propositional logic, is not expressible by Hilbert style inference rules. It has a particular status. A referee of the present paper also suggested the application to computerized proof systems since in a s.f.b, theory the time spent for searching matches is greatly reduced. Section 1 contains some examples and counterexamples of sf.b. theories and a generalization of a well-known result from [7], saying that the regularization T ~ of a strongly irregular lb. equational theory T is again f.b. Theorem 1 states the corresponding for the s.f.b, case. This is one of the steps in getting our main result, Theorem 2, saying that each variety of groupoids generated by its proper 2-element members is s.f.b. (A groupoid (A, �9 ) is called proper here if. depends effectively on both arguments. There are five proper 2-element groupoids, up to isomorphism.) As