Extending Metacompleteness to Systems with Classical Formulae

In honour of Bob Meyer, the paper extends the use of his concept of metacompleteness to include various classical systems, as much as we are able. To do this for the classical sentential calculus, we add extra axioms so as to treat the variables like constants. Further, we use a one-sorted and a two-sorted approach to add classical sentential constants to the logic DJ of my book, Universal Logic. It is appropriate to use rejection to represent classicality in the one-sorted case. We then extend these methods to the quantified logics, but we use a finite domain of individual constants to do this

A Rejection System for the First-Degree Formulae of some Relevant Logics

The standard Hilbert-style of axiomatic system yields the assertion of axioms and, via the use of rules, the assertion of theorems. However, there has been little work done on the corresponding axiomatic rejection of non-theorems. Such Hilbert-style rejection would be achieved by the inclusion of certain rejection-axioms (r-axioms) and, by use of rejection-rules (r-rules), the establishment of rejection-theorems (r-theorems). We will call such a proof a rejection-proof (r-proof). The ideal to aim for would be for the theorems and r-theorems to bemutually exclusive and exhaustive. That is, if a formula A is a theorem then it is not an r-theorem, and if A is a non-theorem then it is an r-theorem. In this paper, I present a rejecion system for the first-degree formulae of a large number of relevant logics

Some Concerns Regarding Ternary-relation Semantics and Truth-theoretic Semantics in General

This paper deals with a collection of concerns that, over a period of time, led the author away from the Routley–Meyer semantics, and towards proof- theoretic approaches to relevant logics, and indeed to the weak relevant logic MC of meaning containment

Extending Metacompleteness to Systems with Classical Formulae

In honour of Bob Meyer, the paper extends the use of his concept of metacompleteness to include various classical systems, as much as we are able. To do this for the classical sentential calculus, we add extra axioms so as to treat the variables like constants. Further, we use a one-sorted and a two-sorted approach to add classical sentential constants to the logic DJ of my book, Universal Logic. It is appropriate to use rejection to represent classicality in the one-sorted case. We then extend these methods to the quantified logics, but we use a finite domain of individual constants to do this

A Rejection System for the First-Degree Formulae of some Relevant Logics

The standard Hilbert-style of axiomatic system yields the assertion of axioms and, via the use of rules, the assertion of theorems. However, there has been little work done on the corresponding axiomatic rejection of non-theorems. Such Hilbert-style rejection would be achieved by the inclusion of certain rejection-axioms (r-axioms) and, by use of rejection-rules (r-rules), the establishment of rejection-theorems (r-theorems). We will call such a proof a rejection-proof (r-proof). The ideal to aim for would be for the theorems and r-theorems to bemutually exclusive and exhaustive. That is, if a formula A is a theorem then it is not an r-theorem, and if A is a non-theorem then it is an r-theorem. In this paper, I present a rejecion system for the first-degree formulae of a large number of relevant logics

In Support of Valerie Plumwood

This paper offers general support for what Valerie Plumwood’s paper is trying to achieve by supporting the rejection of each of her four “false laws of logic”: exportation, illegitimate replacement, commutation (aka. permutation) and disjunctive syllogism. We start by considering her general characterizations of entailment, beginning with her stated definition of entailment as the converse of deducibility. However, this applies to a wide range of relevant logics and so is not able to be used as a criterion for deciding what laws to include in a logic. In this context, we examine the two key differences between deduction from premises to conclusion and entailment from antecedent to consequent. We also consider her use of sufficiency as a general characterizing feature. We then discuss Plumwood’s syntactic criteria used to reject the first three of her false laws of logic and add the Relevance Condition in this context. We next consider semantic characterizing criteria for a logic. After making a case against using truth, we introduce Brady’s logic MC of meaning containment. We then examine the content semantics for MC and use it to reject all of Plumwood’s false laws of logic together with some others. We follow with the related Depth Relevance Condition, which is a syntactic cri- terion satisfied by MC. This clearly rejects the first three of these laws and many others, but not the fourth law. We conclude by giving our overall support for her general enterprise

Starting the Dismantling of Classical Mathematics

This paper uses the relevant logic, MCQ, of meaning containment to explore mathematics without various classical theses, in particular, without the law of excluded middle

Strong Depth Relevance

Relevant logics infamously have the property that they only validate a conditional when some propositional variable is shared between its antecedent and consequent. This property has been strengthened in a variety of ways over the last half-century. Two of the more famous of these strengthenings are the strong variable sharing property and the depth relevance property. In this paper I demonstrate that an appropriate class of relevant logics has a property that might naturally be characterized as the supremum of these two properties. I also show how to use this fact to demonstrate that these logics seem to be constructive in previously unknown ways

Recent Work in Relevant Logic

This paper surveys important work done in relevant logic in the past 10 years