The symmetry rule in propositional logic

AbstractThe addition of the symmetry rule to the resolution system sometimes allows considerable shortening in the length of refutations. We prove exponential lower bounds on the size of resolution refutations using two forms of a global symmetry rule. The paper also discusses the relationship of symmetry rules to the extension rule that allows the use of abbreviative definitions in proofs

Relevant Implication and Ordered Geometry

This paper shows that model structures for R+, the system of positive relevant implication, can be constructed from ordered geometries. This extends earlier results building such model structures from projective spaces. A final section shows how such models can be extended to models for the full system R

First degree formulas in quantified S5

This note provides a proof that the formula L(Ex)( Fx & ~LFx) is not equivalent to any first degree formula in the context of the quantified version of the modal logic S5. This solves a problem posed by Max Cresswell

The Geometry of Relevant Implication II

This note extends earlier results on geometrical interpretations of the logic KR to prove some additional results, including a simple undecidability proof for the four-variable fragment of KR

Relevance Logic: Problems Open and Closed

I discuss a collection of problems in relevance logic. The main problems discussed are: the decidability of the positive semilattice system, decidability of the fragments of R in a restricted number of variables, and the complexity of the decision problem for the implicational fragment of R. Some related problems are discussed along the way

Ehrenfeucht-Fraïssé games without identity

This note defines Ehrenfeucht-Fraïssé games where identity is not present in the basic language.  The formulation is applied to show that there is no elementary theory in the language of one binary relation that exactly characterizes models in which the relation is the identity relation

Ehrenfeucht-Fraïssé games without identity

This note defines Ehrenfeucht-Fraïssé games where identity is not present in the basic language.  The formulation is applied to show that there is no elementary theory in the language of one binary relation that exactly characterizes models in which the relation is the identity relation

Relevant Implication and Ordered Geometry

This paper shows that model structures for R+, the system of positive relevant implication, can be constructed from ordered geometries. This extends earlier results building such model structures from projective spaces. A final section shows how such models can be extended to models for the full system R

Four Variables Suffice

What I wish to propose in the present paper is a new form of “career induction” for ambitious young logicians. The basic problem is this: if we look at the n-variable fragments of relevant propositional logics, at what point does undecidability begin? Focus, to be definite, on the logic R. John Slaney showed that the 0-variable fragment of R (where we allow the sentential con- stants t and f) contains exactly 3088 non-equivalent propositions, and so is clearly decidable. In the opposite direction, I claimed in my paper of 1984 that the five variable fragment of R is undecidable. The proof given there was sketchy (to put the matter charitably), and a close examination reveals that although the result claimed is true, the proof given is incorrect. In the present paper, I give a detailed and (I hope) correct proof that the four variable fragments of the principal relevant logics are undecidable. This leaves open the question of the decidability of the n-variable fragments for n = 1, 2, 3. At what point does undecidability set in

First degree formulas in quantified S5

This note provides a proof that the formula L(Ex)( Fx & ~LFx) is not equivalent to any first degree formula in the context of the quantified version of the modal logic S5. This solves a problem posed by Max Cresswell