Exponential lower bounds are proved for the length-of-resolution refutations of sets of disjunctions constructed from expander graphs, using the method of Tseitin. Since these sets of clauses encode biconditionals, they have short (polynomial-length) refutations in a standard axiomatic formulation of propositional calculus.
In this paper we show that the propositional logics E of entailment, R of relevant implication and T of ticket entailment are undecidable. The decision problem is also shown to be unsolvable in an extensive class of related logics. The main tool used in establishing these results is an adaptation of the von Neumann coordinatization theorem for modular lattices.Interest in the decision problem for these systems dates from the late 1950s. The earliest result was obtained by Anderson and Belnap who proved that the first degree fragment of all these logics is decidable. Kripke [11] proved that the pure implicational fragments R→ and E→ of R and E are decidable. His methods were extended by Belnap and Wallace to the implication-negation fragments of these systems [3]; Kripke's methods also extend easily to include the implication-conjunction fragments of R and E. Meyer in his thesis [14] extended the result for R to include a primitive necessity operator. He also proved decidable the system R-mingle, an extension of R, and ortho-R (OR), the logic obtained from R by omitting the distribution axiom. Various weak relevant logics are also known to be decidable by model-theoretic proofs of the finite model property (see Fine [5]). Finally, S. Giambrone [7] has solved the decision problem for various logics obtained by the omission of the contraction axiom (A → ⦁ A → B) → ⦁ A → B, including R+ − W (R+ minus contraction). It is worth noting that even where positive results were obtained, the decision methods were usually of a complexity considerably greater than in the case of other nonclassical logics such as intuitionistic logic or modal logic, a fact which already indicates the difficulty of the decision problem.
No abstract
In what follows there is presented a unified semantic treatment of certain “paradox-free” systems of entailment, including Church's weak theory of implication (Church [7]) and logics akin to the systems E and R of Anderson and Belnap (Anderson [3], Belnap [6]). We shall refer to these systems generally as relevant logics.The leading idea of the semantics is that just as in modal logic validity may be defined in terms of certain valuations on a binary relational structure so in relevant logics validity may be defined in terms of certain valuations on a semilattice—interpreted informally as the semilattice of possible pieces of information. Completeness theorems can be given relative to these semantics for the implicational fragments of relevant logics. The semantical viewpoint affords some insights into the structure of the systems—in particular light is thrown upon admissible modes of negation and on the assumptions underlying rejection of the “paradoxes of material implication”.The systems discussed are formulated in fragments of a first-order language with → (entailment), &, ⋁, ¬,(x) and (∃x) primitive, omitting identity but including a denumerable list of propositional variables (p, q, r, p1,…etc.), and (for each n > 0), a denumerable list of n-ary predicate letters. The schematic letters A, B, C, D, A1,… are used on the meta-level as variables ranging over formulas. The conventions of Church [9] are followed in abbreviating formulas. The semantics of the systems are given in informal terms; it is an easy matter to turn the informal descriptions into formal set-theoretical definitions.
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