The main aim is to extend the range of logics which solve the set-theoretic paradoxes, over and above what was achieved by earlier work in the area. In doing this, the paper also provides a link between metacomplete logics and those that solve the paradoxes, by finally establishing that all M1-metacomplete logics can be used as a basis for naive set theory. In doing so, we manage to reach logics that are very close in their axiomatization to that of the (...) logic R of relevant implication. A further aim is the use of metavaluations in a new context, expanding the range of application of this novel technique, already used in the context of negation and arithmetic, thus providing an alternative to traditional model theoretic approaches. (shrink)
Free Semantics is based on normalized natural deduction for the weak relevant logic DW and its near neighbours. This is motivated by the fact that in the determination of validity in truth-functional semantics, natural deduction is normally used. Due to normalization, the logic is decidable and hence the semantics can also be used to construct counter-models for invalid formulae. The logic DW is motivated as an entailment logic just weaker than the logic MC of meaning containment. DW is the logic (...) focussed upon, but the results extend to MC. The semantics is called ‘free semantics’ since it is disjunctively and existentially free in that no disjunctive or existential witnesses are produced, unlike in truth-functional semantics. Such ‘witnesses’ are only assumed in generality and are not necessarily actual. The paper sets up the free semantics in a truth-functional style and gives a natural deduction interpetation of the meta-logical connectives. We then set out a familiar tableau-style system, but based on natural deduction proof rather than truth-functional semantics. A proof of soundness and completeness is given for a reductio system, which is a transform of the tableau system. The reductio system has positive and negative rules in place of the elimination and introduction rules of Brady’s normalized natural deduction system for DW. The elimination-introduction turning points become closures of threads of proof, which are at the points of contradiction for the reductio system. (shrink)
One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley–Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing (...) a general conception of conditionality that may unify the three given conceptions. (shrink)
We first consider the entailment logic MC, based on meaning containment, which contains neither the Law of Excluded Middle (LEM) nor the Disjunctive Syllogism (DS). We then argue that the DS may be assumed at least on a similar basis as the assumption of the LEM, which is then justified over a finite domain or for a recursive property over an infinite domain. In the latter case, use is made of Mathematical Induction. We then show that an instance of the (...) LEM is intrumental in the proof of Cantor's Theorem, and we then argue that this is based on a more general form than can be reasonably justified. We briefly consider the impact of our approach on arithmetic and naive set theory, and compare it with intuitionist mathematics and briefly with recursive mathematics. Our "Four Basic Logical Issues" paper would provide useful background, the current paper being an application of the some of the ideas in it. (shrink)
The paper reviews a number of approaches for handling restricted quantification in relevant logic, and proposes a novel one. This proceeds by introducing a novel kind of enthymematic conditional.
We collect together some misgivings about the logic R of relevant inplication, and then give support to a weak entailment logic DJd. The misgivings centre on some recent negative results concerning R, the conceptual vacuousness of relevant implication, and the treatment of classical logic. We then rectify this situation by introducing an entailment logic based on meaning containment, rather than meaning connection, which has a better relationship with classical logic. Soundness and completeness results are proved for DJd with respect to (...) a content semantics, which embraces the concept of meaning containment. (shrink)
This paper surveys the various forms of Deduction Theorem for a broad range of relevant logics. The logics range from the basic system B of Routley-Meyer through to the system R of relevant implication, and the forms of Deduction Theorem are characterized by the various formula representations of rules that are either unrestricted or restricted in certain ways. The formula representations cover the iterated form,A 1 .A 2 . ... .A n B, the conjunctive form,A 1&A 2 & ...A n (...) B, the combined conjunctive and iterated form, enthymematic version of these three forms, and the classical implicational form,A 1&A 2& ...A n B. The concept of general enthymeme is introduced and the Deduction Theorem is shown to apply for rules essentially derived using Modus Ponens and Adjunction only, with logics containing either (A B)&(B C) .A C orA B .B C .A C. (shrink)
In part I, we presented an algebraic-style of semantics, which we called “content semantics,” for quantified relevant logics based on the weak systemBBQ. We showed soundness and completeness with respect to theunreduced semantics ofBBQ. In part II, we proceed to show soundness and completeness for extensions ofBBQ with respect to this type of semantics. We introducereduced semantics which requires additional postulates for primeness and saturation. We then conclude by showing soundness and completeness forBB d Q and its extentions with respect (...) to this reduced semantics. (shrink)
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). (shrink)
The paper essentially shows that the paraconsistent logicDR satisfies the depth relevance condition. The systemDR is an extension of the systemDK of [7] and the non-triviality of a dialectical set theory based onDR has been shown in [3]. The depth relevance condition is a strengthened relevance condition, taking the form: If DR- AB thenA andB share a variable at the same depth, where the depth of an occurrence of a subformulaB in a formulaA is roughly the number of nested ''s (...) required to reach the occurrence ofB inA. The method of proof is to show that a model structureM consisting of {M 0 , M1, ..., M}, where theM i s are all characterized by Meyer''s 6-valued matrices (c. f, [2]), satisfies the depth relevance condition. Then, it is shown thatM is a model structure for the systemDR. (shrink)
A major question for the relevant logics has been, “Under what conditions is Ackermann's ruleγ from -A ∨B andA to inferB, admissible for one of these logics?” For a large number of logics and theories, the question has led to an affirmative answer to theγ problem itself, so that such an answer has almost come to be expected for relevant logics worth taking seriously. We exhibit here, however, another large and interesting class of logics-roughly, the Boolean extensions of theW — (...) free relevant logics (and, precisely, the well-behaved subsystems of the 4-valued logicBN4) — for which γ fails. (shrink)
