David Makinson

David Makinson works in logic. Over the years he has made contributions to modal logic, deontic logic, the logic of belief change (as one of the AGM trio), uncertain inference both qualitative and probabilistic, input/output logics, intelim rules for classical and non-classical connectives, and other areas of the subject. Before his association with LSE he worked at King’s College London, following a long stint outside academia in Unesco and early years at the American University of Beirut, Lebanon. Australian by nationality, he obtained his Bachelor’s in Philosophy at Sydney University and his doctorate at Oxford. His books include Topics in Modern Logic, Bridges from Classical to Nonmonotonic Logic, and Sets Logic and Maths for Computing. Currently he is working on a knotty problem in relevance logic. More details on his personal webpage.

Project Title: Relevance via Decomposition

Project Description

Background

In its standard Hilbertian axiomatization, the well-known relevance logic R has many axiom schemes and two derivation rules. The derivation rules are straightforward: conjunction and detachment with respect to the non-classical propositional connective intended to represent relevant implication. But the dozen axiom schemes form a rather motley and untidy crew. Along with several of its neighbours, R has been characterized by the Routley-Meyer possible-worlds semantics with a three-place relation satisfying various constraints. Notwithstanding its technical usefulness and versatility, however, the semantics can hardly be said to provide a satisfying motivation for relevance logic since there is no clear understanding of the meaning of the relation employed, even less so of the constraints placed upon it. A more convincing rationale is provided by the standard natural deduction system for R, which is centred on a meaningful restriction imposed on the rule of arrow introduction. But the system is still not entirely transparent, notably in its use of a ‘same suppositions’ proviso on the rules of conjunction introduction and disjunction elimination. That proviso appears to be overkill, for it has the effect of blocking derivation of the classical principle of distribution of conjunction over disjunction, forcing an ad hoc and inelegant addition of distribution as a separate inference rule. 

Project goal 

Our goal is to obtain a clearer picture by suitably refining the classical procedure of semantic decomposition trees.

Results so far obtained 

Using suitably articulated decomposition trees, we have defined a set of ‘directly acceptable’ formulae, which is closed under substitution and conjunction and contains all the standard axioms of R together with certain additional items, notably some attractive formula noticed by Urquhart and Humberstone. The set also satisfies the well-known letter-sharing condition, usually regarded as a sine qua non for relevance, and successfully excludes formulae such as mingle which, whilst satisfying that condition, are notoriously repugnant to relevantists.However, the set of directly acceptable formulae is not closed under detachment. We close it under that rule to define a set of ‘acceptable’ formulae. In effect, this is to treat the directly acceptable formulae as constituting the axioms of a Hilbertian system with detachment as sole derivation rule – which is legitimate since direct acceptability is decidable, indeed for short formulae can quickly be determined by manual computation. The set of acceptable formulae continues to be closed under substitution and conjunction and thus properly includes R. 

Problems under investigation

The central problem is to determine whether the set of acceptable formulae continues to satisfy the letter-sharing condition and to exclude other undesirable items such as mingle. In the affirmative case, we have an intuitively transparent construction of a relevance logic in the neighbourhood of R and the project may then be regarded as complete. In the negative case it becomes of interest to examine a certain strengthened notion of direct acceptability that is not closed under detachment but is nevertheless very well-behaved in other respects.   

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