We collect together some misgivings about the logic R of relevant inplication, and then give support to a weak entailment logic $DJ^{d}$ . 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 $DJ^{d}$ with respect (...) to a content semantics, which embraces the concept of meaning containment. (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)

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)

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)

This is a general account of metavaluations and their applications, which can be seen as an alternative to standard model-theoretic methodology. They work best for what are called metacomplete logics, which include the contraction-less relevant logics, with possible additions of Conjunctive Syllogism, & →.A→C, and the irrelevant, A→.B→A, these including the logic MC of meaning containment which is arguably a good entailment logic. Indeed, metavaluations focus on the formula-inductive properties of theorems of entailment form A→B, splintering into two types, M1- (...) and M2-, according to key properties of negated entailment theorems. Metavaluations have an inductive presentation and thus have some of the advantages that model theory does, but they represent proof rather than truth and thus represent proof-theoretic properties, such as the priming property, if ├ A $\vee$ B then ├ A or ├ B, and the negated-entailment properties, not-├ ∼ and ├ ∼ iff ├ A and ├ ∼ B. Topics to be covered are their impact on naive set theory and paradox solution, and also Peano arithmetic and Godel’s First and Second Theorems. Interesting to note here is that the familiar M1- and M2-metacomplete logics can be used to solve the set-theoretic paradoxes and, by inference, the Liar Paradox and key semantic paradoxes. For M1-logics, in particular, the final metavaluation that is used to prove the simple consistency is far simpler than its correspondent in the model-theoretic proof in that it consists of a limit point of a single transfinite sequence rather than that of a transfinite sequence of such limit points, as occurs in the model-theoretic approach. Additionally, it can be shown that Peano Arithmetic is simply consistent, using metavaluations that constitute finitary methods. Both of these results use specific metavaluational properties that have no correspondents in standard model theory and thus it would be highly unlikely that such model theory could prove these results in their final forms. (shrink)

We start by noting that the set-theoretic and semantic paradoxes are framed in terms of a definition or series of definitions. In the process of deriving paradoxes, these definitions are logically represented by a logical equivalence. We will firstly examine the role and usage of definitions in the derivation of paradoxes, both set-theoretic and semantic. We will see that this examination is important in determining how the paradoxes were created in the first place and indeed how they are to be (...) solved in a uniform way. There are three features that are special about these definitions used in the derivation of most of the above paradoxes. The first is the use of self-reference between the definiens and the definiendum, the second is the generality of the definiendum, and the third is the under-determination and over-determination of concepts that usually occur as a result of these definitions. We will examine the impact of these three features on the logical representation of definitions and show how this representation then leads to a uniform paradox solution using an appropriate logic that is both paraconsistent and paracomplete. However, it is the paracompleteness, exhibited through the rejection of the Law of Excluded Middle, together with the rejection of contraction principles, that enables the solution of the paradoxes to go through. We characterize definitions as involving syntactic identity and/or meaning identity. We point out that some paradoxes do not have explicit self-reference or circularity and some may not utilize the generality of the definiendum, but the general characterizations of definitions that we give will still apply. We also look beyond all this to paradoxes that rely on illicit definitions between objects that are essentially different to start with. (shrink)

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)

We provide five semantic preservation properties which apply to the various rules -- primitive, derived and admissible -- of Hilbert-style axiomatizations of relevant logics. These preservation properties are with respect to the Routley-Meyer semantics, and consist of various truth- preservations and validity-preservations from the premises to the conclusions of these rules. We establish some deduction theorems, some persistence theorems and some soundness and completeness theorems, for these preservation properties. We then apply the above ideas, as best we can, to the (...) classical sentential and predicate calculi, to normal and non- normal modal logics, and to many- valued logics. (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)

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).

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. (shrink)

Science requires postformal capabilities to compare competing explanations and conceptualize how to coordinate or integrate them. With conflicts thus reconciled, science advances. The Model of Hierarchical Complexity facilitates the coordination of current arguments about intelligence. A cross-species measurement theory of comparative cognition is proposed. It has potential to overcome the lack of a general measurement theory for the science of comparative cognition, and the lack of domain-general mechanisms for evolutionary psychologists. The hierarchical complexity of concepts and debates as well as (...) the new theory are scored, and demonstrate the postformal hierarchical complexity of the proposed theory. (shrink)

This paper proves decidability of a range of weak relevant logics using decision procedures based on the Routley-Meyer semantics. Logics are categorized as F-logics, for those proved decidable using a filtration method, and U-logics, for those proved decidable using a direct (unfiltered) method. Both of these methods are set out as reductio methods, in the style of Hughes and Cresswell. We also examine some extensions of the U-logics where the method fails and infinite sequences of worlds can be generated.

This is the culmination of a discussion on Berry's Paradox with Graham Priest, over an extended period from 1983 to 2019, the central point being whether the Paradox can be avoided or not by removal of the Law of Excluded Middle (LEM). Priest is of the view that a form of the Paradox can be derived without the LEM, whilst Brady disputes this. We start by conceptualizing negation in the logic MC of meaning containment and introduce the LEM as part (...) of the classical recapture. We then examine the usage of the LEM in some other paradoxes and see that it is applied to cases of self-reference. In relation to Priest's [2019] paper, we go on to find a similar use of the LEM in Priest's derivation of Berry's Paradox. However, it is found to be deeper and trickier than other paradoxes, requiring a special effort to untangle the relationships between the LEM, self-reference and meta-theoretic influence. We then examine Brady's previous formalization of Berry's Paradox, considering Brady's most recent view of restricted quantification and his recursive account of the least number satisfying a property. We show that neither of these methods can be used to formalize the paradox. We also examine the inductive shapes for the Substitution of Identity Rule for intensional and extensional identities, while conceding Priest's point, made in [2019], that his shape of Substitution of Identity is not relevant to his proof of Berry's Paradox. (shrink)

Four Basic Logical Issues: The paper addresses what we see as the four major issues in logic. The overriding issue is that of the choice of logic. We start with some discussion of the preliminary issue of whether there is such a 'one true logic,' but we reserve the main discussion for the first issue of 'classical logic versus nonclassical logic.' Here, we discuss the role of meaning and truth, the relation between classical logic and classical negation, and whether and, (...) if so, how classical logic should reside at the base world. Given the argument in favor of an overall use of nonclassical logic, the second issue is that of the choice of nonclassical logic. Brady's logic MC of meaning containment is argued for, with some comparison made with other relevant logics. For the remaining two issues, we make a case for relevant deduction, in comparison with classical deduction, and we explore possibilities for the appropriate metalogic, comparing classical and nonclassical approaches. (shrink)

We argue that formal logical systems are four-valued, these four values being determined by the four deductive outcomes: A without ~A, ~A without A, neither A nor ~A, and both A and ~A. We further argue that such systems ought to be three-valued, as any contradiction, A and ~A, should be removed by reconceptualisation of the concepts captured by the system. We follow by considering suitable conditions for the removal of the third value, neither A nor ~A, yielding a classically (...) valued system. We then consider what values are appropriate for the meta-theory, arguing that it should be three-valued, but reducible to the two classical values upon the decidability of the object system. (shrink)

Institutional investors, policy makers, and researchers have advocated for greater director independence in hopes of improving corporate governance and discouraging unethical behaviors such as corporate frauds, accounting irregularities, and other organizational failures. However, increasing demands upon directors and sitting CEOs, as well as constraints on the number of boards on which they can serve, has resulted in a dramatic increase in the use of retired independent directors. Compared to other directors with full-time job demands, we argue that RIDs have lesser (...) time constraints and greater attentional capacities with which to discharge their responsibilities, thereby improving overall board “bandwidth.” Using S&P 1500 firms for the period of 2000–2012, we find that enhanced board bandwidth associated with an increased proportion of RIDs on the board relates to greater resource provisioning through reducing costs of capital, improved monitoring through reducing disclosure-related weaknesses, and better accounting and market performance. We thereby advance traditional board bandwidth research by contemplating the available time that independent directors would have by virtue of being retired and relating this to board effectiveness. (shrink)

ABSTRACTThis special issue of Social Epistemology has its origin in two symposia organised by the Contemporary Philosophy of Technology Research Group at the University of Birmingham. These we...

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. (shrink)

A collection of modern Zen teaching stories, or koans, compiled and commented on by an Australian Zen teacher.

George MacDonald Ross; Philosophy in Schools, Journal of Philosophy of Education, Volume 22, Issue 2, 30 May 2006, Pages 207–219, https://doi.org/10.1111/j.1467.

In this incisive political analysis, Kristin Ross thinks through everyday existence across a range of practices-from philosophy to history, from the visual arts to popular fiction-and across the forms taken by collective political action in contemporary struggles. Ross returns to Henri Lefebvre's powerful intuition that ordinary life is both residue and resource, the site of profound alienation and, by the same token, the origin of all emancipatory initiatives and desires.

Many studies have attempted to identify measures that predict reading abilities. The results of these studies may be inclined to over-identification of children considered at risk in kindergarten but who achieve parity in reading by the end of first grade. Therefore, the current study sought to analyze the specific cognitive and linguistic predictors of reading accuracy and reading speed separately. Additionally, the study examined if it is possible to use empirically validated measures to distinguish between children who are not ready (...) to learn how to read in kindergarten but manage to acquire reading skills by the end of first grade, and those who continue to exhibit difficulties. The study followed 98 kindergarteners aged 4 years 10 months to six years from three different schools, who were taught how to read in kindergarten. Multiple measures of general cognitive skills, linguistic abilities, and reading abilities were measured at three different points in time: the beginning of kindergarten, the end of kindergarten, and the end of first grade. The study found that most of the children with good literacy and cognitive abilities learned how to read by the end of kindergarten. The analysis revealed a significant difference in cognitive abilities, such as executive functions and memory, which distinguished between the ability to acquire fluent reading and accurate reading. The study was able to successfully distinguish between “children with difficulties” and “un-ready” children. These results have various implications, especially in regard to the identification of and intervention with kindergarten children at risk for reading disabilities. (shrink)