Online research in philosophy

This paper gives an outline of three different approaches to the four-valued semantics for relevant logics (and other non-classical logics in their vicinity). The first approach borrows from the Australian Plan semantics, which uses a unary operator for the evaluation of negation. This approach can model anything that the two-valued account can, but at the cost of relying on insights from the Australian Plan. The second approach is natural, well motivated, independent of the Australian Plan, and it provides a semantics for the contraction-free relevant logicC (orRW). Unfortunately, its approach seems to model little else. The third approach seems to capture a wide range of formal systems, but at the time of writing, lacks a completeness proof.

Once the Kripke semantics for normal modal logics were introduced, a whole family of modal logics other than the Lewis systems S1 to S5 were discovered. These logics were obtained by changing the semantics in natural ways. The same can be said of the Kripke-style semantics for relevant logics: a whole range of logics other than the standard systems R, E and T were unearthed once a semantics was given (cf. Priest and Sylvan [6], Restall [7], and Routley et al. [8]). In a similar way, weakening the structural rules of the Gentzen formulation of classical logic gives rise to other ‘substructural’ logics such as linear logic (as in Girard [4]). This process of ‘strategic weakening’ is becoming popular today, with the discovery of applications of these logics to areas such as linguistics and the theory of computation (cf. van Benthem [1]). Until now no-one has (to my knowledge) examined what the process of weakening does to the Kripke-style semantics of intuitionistic logic. This paper remedies the deficiency, introducing the family of subintuitionistic logics. These systems have some appealing features. Unlike other substructural logics such as linear logic (which lack distribution of extensional disjunction over conjunction) they have a very natural Kripke-style worlds semantics. Also, the difficulties with regard to modelling quantification in these systems may be able to shed some light on the difficulties in naturally modelling quantification in relevant logics, as it must be admitted that the semantics currently available for quantified relevant logics are rather baroque (cf. Fine [3]). But most importantly, delving in the undergrowth of logics such as intuitionistic logic gives us a ‘feel’ for how such systems are put together, and what job is being done by each aspect of the modelling conditions in..

We introduce Kripke semantics for modal substructural logics, and provethe completeness theorems with respect to the semantics. Thecompleteness theorems are proved using an extended Ishihara's method ofcanonical model construction (Ishihara, 2000). The framework presentedcan deal with a broad range of modal substructural logics, including afragment of modal intuitionistic linear logic, and modal versions ofCorsi's logics, Visser's logic, Méndez's logics and relevant logics.

This paper sets out two semantics for the relevant logic R based on Dunn's four-valued semantics for first-degree entailments. Unlike Routley's semantics for weak relevant logics, they do not use two ternary accessibility relations. Unlike Restall's semantics, they capture all of R. But there is a catch. Both of the present semantics are neighbourhood semantics, that is, they include sets of propositions in the specification of their frames.

The main goal of this paper is to explain the link between the algebraic models and the Kripke-style models for certain classes of propositional non-classical logics. We consider logics that are sound and complete with respect to varieties of distributive lattices with certain classes of well-behaved operators for which a Priestley-style duality holds, and present a way of constructing topological and non-topological Kripke-style models for these types of logics. Moreover, we show that, under certain additional assumptions on the variety of the algerabic models of the given logics, soundness and completeness with respect to these classes of Kripke-style models follows by using entirely algebraical arguments from the soundness and completeness of the logic with respect to its algebraic models.

This paper continues the work of Priest and Sylvan inSimplified Semantics for Basic Relevant Logics, a paper on the simplified semantics of relevant logics, such asB + andB. We show that the simplified semantics can also be used for a large number of extensions of the positive base logicB +, and then add the dualising* operator to model negation. This semantics is then used to give conservative extension results for Boolean negation.

The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀ pA true when there is some true admissible proposition that entails all p -instantiations of A . It is also shown that without the admissibility qualification many of the systems considered are semantically incomplete, including all those that are sub-logics of the quantified version of Anderson and Belnap’s system E of entailment, extended by the mingle axiom and the Ackermann constant t . The incompleteness proof involves an algebraic semantics based on atomless complete Boolean algebras.

A system FDQ of first degree entailment with quantification, extending classical quantification logic Q by an entailment connective, is axiomatised, and the choice of axioms defended and also, from another viewpoint, criticised. The system proves to be the equivalent to the first degree part of the quantified entailmental system EQ studied by Anderson and Belnap; accordingly the semantics furnished are alternative to those provided for the first degree of EQ by Belnap. A worlds semantics for FDQ is presented, and the soundness and completeness of FDQ proved, the main work of the paper going into the proof of completeness. The adequacy result is applied to yield, as well as the usual corollaries, weak relevance of FDQ and the fact that FDQ is the common first degree of a wide variety of (constant domain) quantified relevant logics. Finally much unfinished business at the first degree is discussed.

We provide a semantics for relevant logics with addition of Aristotle's Thesis, ∼(A→∼A) and also Boethius,(A→B)→∼(A→∼B). We adopt the Routley-Meyer affixing style of semantics but include in the model structures a regulatory structure for all interpretations of formulae, with a view to obtaining a lessad hoc semantics than those previously given for such logics. Soundness and completeness are proved, and in the completeness proof, a new corollary to the Priming Lemma is introduced (c.f.Relevant Logics and their Rivals I, Ridgeview, 1982).

We present an algebraic-style of semantics, which we call a content semantics, for quantified relevant logics based on the weak system BBQ. We show soundness and completeness for all quantificational logics extending BBQ and also treat reduced modelling for all systems containing BB d Q. The key idea of content semantics is that true entailments AB are represented under interpretation I as content containments, i.e. I(A)I(B) (or, the content of A contains that of B). This is opposed to the truth-functional way which represents true entailments as truth-preservations over all set-ups (or worlds), i.e. (VaK) (if I(A, a) = T then I(B, a)= T).