The contribution of this paper lies with providing a systematically specified and intuitive interpretation pattern and delineating a class of relational structures and models providing a natural interpretation of logical operators on an underlying propositional calculus of Positive Lattice Logic and subsequently proving a generic completeness theorem for the related class of logics, sometimes collectively referred to as Generalized Galois Logics.

The aim of this paper is to define the logical system Sm4 characterised by the degree of truth-preserving consequence relation defined on the ordered set of values of Smiley’s four-element matrix MSm4. The matrix MSm4 has been of considerable importance in the development of relevant logics and it is at the origin of bilattice logics. It will be shown that Sm4 is a most interesting paraconsistent logic which encloses a sound theory of logical necessity similar to that of Anderson and (...) Belnap’s logic of entailment E. Intuitively, Sm4 can be described as a four-valued expansion of the positive fragment of Lewis’ S5 or, alternatively, as a four-valued version of S5. (shrink)

In Saving Truth from Paradox, Hartry Field presents and defends a theory of truth with a new conditional. In this paper, I present two criticisms of this theory, one concerning its assessments of validity and one concerning its treatment of truth-preservation claims. One way of adjusting the theory adequately responds to the truth-preservation criticism, at the cost of making the validity criticism worse. I show that in a restricted setting, Field has a way to respond to the validity criticism. I (...) close with some general considerations on the use of revision-theoretic methods in theories of truth. (shrink)

The rules for Core Logic are stated, and various important results about the system are summarized. We describe its relationship to other systems, such as Classical Logic, Intuitionistic Logic, Minimal Logic, and the Anderson–Belnap relevance logicR. A precise, positive explication is offered of what it is for the premises of a proof to connect relevantly with its conclusion. This characterization exploits the notion of positive and negative occurrences of atoms in sentences. It is shown that all Core proofs are relevant (...) in this precisely defined sense. We survey extant results about variable-sharing in rival systems of relevance logic, and find that the variable-sharing conditions established for them are weaker than the one established here for Core Logic (and for its classical extension). Proponents of other systems of relevance logic (such asRand its subsystems) are challenged to formulate a stronger variable-sharing condition, and to prove thatRor any of its subsystems satisfies it, but that Core Logic does not. We give reasons for pessimism about the prospects for meeting this challenge. (shrink)

This paper sets out to evaluate the claim that Aristotle’s Assertoric Syllogistic is a relevance logic or shows significant similarities with it. I prepare the grounds for a meaningful comparison by extracting the notion of relevance employed in the most influential work on modern relevance logic, Anderson and Belnap’s Entailment. This notion is characterized by two conditions imposed on the concept of validity: first, that some meaning content is shared between the premises and the conclusion, and second, that the premises (...) of a proof are actually used to derive the conclusion. Turning to Aristotle’s Prior Analytics, I argue that there is evidence that Aristotle’s Assertoric Syllogistic satisfies both conditions. Moreover, Aristotle at one point explicitly addresses the potential harmfulness of syllogisms with unused premises. Here, I argue that Aristotle’s analysis allows for a rejection of such syllogisms on formal grounds established in the foregoing parts of the Prior Analytics. In a final section I consider the view that Aristotle distinguished between validity on the one hand and syllogistic validity on the other. Following this line of reasoning, Aristotle’s logic might not be a relevance logic, since relevance is part of syllogistic validity and not, as modern relevance logic demands, of general validity. I argue that the reasons to reject this view are more compelling than the reasons to accept it and that we can, cautiously, uphold the result that Aristotle’s logic is a relevance logic. (shrink)

We examine the sense in which logic is a priori, and explain how mathematical theories can be dichotomized non-trivially into analytic and synthetic portions. We argue that Core Logic contains exactly the a-priori-because-analytically-valid deductive principles. We introduce the reader to Core Logic by explaining its relationship to other logical systems, and stating its rules of inference. Important metatheorems about Core Logic are reported, and its important features noted. Core Logic can serve as the basis for a foundational program that could (...) be called Natural Logicism, an exposition of which will build on the (meta)logical ideas explained here. (shrink)

We discuss conditions under which the following ‘truism’ does indeed express a truth: the weaker a logic is in terms of what it proves, the stronger it is as a tool for registering distinctions amongst the formulas in its language.

An elementary exercise in symbolizing an existential to universal relation reveals expressive limitations in standard first-order predicate-quantificational logic. Alternative translations of a sample some-every sentence are considered and rejected after criticism, leaving as the best choice a particular structure that demonstrably does not serve for all predicates available to the ordinary language to which the sample sentence belongs. We explain the difficulties encountered in trying to arrive at an adequate translation of the sentence in classical logic, as background to examining (...) five alternative nonclassical logics to resolve the problem. We settle finally if only provisionally in the process on a version of intensional logic that permits reference and true predication of constitutive properties to nonexistent as well as existent objects. (shrink)

In early Buddhist logic, it was standard to assume that for any state of a ff airs there were four possibilities: that it held, that it did not, both, or neither. This is the catuskoti (or tetralemma). Classical logicians have had a hard time mak­ing sense of this, but it makes perfectly good sense in the se­mantics of various paraconsistent logics, such as First Degree Entailment. Matters are more complicated for later Buddhist thinkers, such as Nagarjuna, who appear to suggest (...) that none of these options , or more than one, may hold. The point of this paper is to examine the matter, including the formal logical machinery that may be appropriate. (shrink)

Our beliefs utilize various sources: perception, memory, induction, etc. We trust these sources to provide reliable information about the world around us. My dissertation investigates how this trust could be justified. Chapter one introduces background material. I argue that justification rather than knowledge is of primary epistemological importance, discuss the internalism/externalism debate, and introduce an evidentialist thesis that provides a starting point/framework for epistemological theorizing. Chapter two introduces a puzzle concerning justification. Can a belief source provide justification absent prior justification (...) for believing it's reliable? Any answer appears to either make justifying the reliability of a source intellectually unsatisfying or all together impossible. Chapter three considers and rejects a plethora of proposed solutions to our puzzle. Investigating these solutions illustrates the need to further investigate evidence, evidence possession, and evidential support. Chapter four discusses the metaphysics of evidence. I argue that evidence always consists of a set of facts and that fact-proposition pairs stand in confirmation relations isomorphic to those holding between pairs of propositions. Chapter five argues that justification requires what I call actually connected possession of supporting evidence: a subject must be aware of supporting evidence and of the support relation itself. Chapter six argues that the relation constitutive of a set of facts being justificatory evidence is a sui generis and irreducible relation that is knowable a priori. Chapter seven begins by showing how Richard Fumerton's acquaintance theory meets the constraints on a theory of justification laid down in previous chapters. I modify the theory so as to: make room for fallible foundational justification, and allow inferential justification absent higher-order beliefs about evidential connections. Chapter eight applies the developed theory of justification to our initial puzzle. I show how my modified acquaintance theory is in a unique position to vindicate the idea that necessarily a source provides a person with justification only if she is aware of evidence for the reliability of that source. However, this awareness of evidence for a source's reliability falls short of a justified belief and thereby avoids impalement from our dilemma's skeptical horn. (shrink)

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

The implicational fragment of the logic of relevant implication, $R_\to$ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, $T_\to$ is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of $T_\to$ to the decidability problem of $R_\to$. The decidability of $T_\to$ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over (...) the base $\{\textsf{B},\textsf{B}',\textsf{I},\textsf{W}\}$. (shrink)

Taking our inspiration from modal correspondence theory, we present the idea of correspondence analysis for many-valued logics. As a benchmark case, we study truth-functional extensions of the Logic of Paradox (LP). First, we characterize each of the possible truth table entries for unary and binary operators that could be added to LP by an inference scheme. Second, we define a class of natural deduction systems on the basis of these characterizing inference schemes and a natural deduction system for LP. Third, (...) we show that each of the resulting natural deduction systems is sound and complete with respect to its particular semantics. (shrink)

This paper sets out three programmes that attempt to use relevant logic as the basis for a philosophy of mathematics. Although these three programmes do not exhaust the possible approaches to mathematics through relevant logic, they are fairly representative of the current state of the field. The three programmes are compared and their relative strengths and weaknesses set out. At the end of the paper I examine the consequences of adopting each programme for the realist debate about mathematical objects.

A collection of material on Husserl's Logical Investigations, and specifically on Husserl's formal theory of parts, wholes and dependence and its influence in ontology, logic and psychology. Includes translations of classic works by Adolf Reinach and Eugenie Ginsberg, as well as original contributions by Wolfgang Künne, Kevin Mulligan, Gilbert Null, Barry Smith, Peter M. Simons, Roger A. Simons and Dallas Willard. Documents work on Husserl's ontology arising out of early meetings of the Seminar for Austro-German Philosophy.

In the proof-theoretic semantics approach to meaning, harmony , requiring a balance between introduction-rules (I-rules) and elimination rules (E-rules) within a meaning conferring natural-deduction proof-system, is a central notion. In this paper, we consider two notions of harmony that were proposed in the literature: 1. GE-harmony , requiring a certain form of the E-rules, given the form of the I-rules. 2. Local intrinsic harmony : imposes the existence of certain transformations of derivations, known as reduction and expansion . We propose (...) a construction of the E-rules (in GE-form) from given I-rules, and prove that the constructed rules satisfy also local intrinsic harmony. The construction is based on a classification of I-rules, and constitute an implementation to Gentzen’s (and Pawitz’) remark, that E-rules can be “read off” I-rules. (shrink)

The quantified relevant logic RQ is given a new semantics in which a formula for all xA is true when there is some true proposition that implies all x-instantiations of A. Formulae are modelled as functions from variable-assignments to propositions, where a proposition is a set of worlds in a relevant model structure. A completeness proof is given for a basic quantificational system QR from which RQ is obtained by adding the axiom EC of 'extensional confinement': for all x(A V (...) B) -> (A V for all xB). with x not free in A. Validity of EC requires an additional model condition involving the boolean difference of propositions. A QR-model falsifying EC is constructed by forming the disjoint union of two natural arithmetical structures in which negation is interpreted by the minus operation. (shrink)

Tarski’s conceptual analysis of the notion of logical consequence is one of the pinnacles of the process of defining the metamathematical foundations of mathematics in the tradition of his predecessors Euclid, Frege, Russell and Hilbert, and his contemporaries Carnap, Gödel, Gentzen and Turing. However, he also notes that in defining the concept of consequence “efforts were made to adhere to the common usage of the language of every day life.” This paper addresses the issue of what relationship Tarski’s analysis, and (...) Béziau’s further generalization of it in universal logic, have to reasoning in the everyday lives of ordinary people from the cognitive processes of children through to those of specialists in the empirical and deductive sciences. It surveys a selection of relevant research in a range of disciplines providing theoretical and empirical studies of human reasoning, discusses the value of adopting a universal logic perspective, answers the questions posed in the call for this special issue, and suggests some specific research challenges. (shrink)

In "General Information in Relevant Logic" (Synthese 167, 2009), the semantics for relevant logic is interpreted in terms of objective information. Objective information is potential data that is available in an environment. This paper explores the notion of objective information further. The concept of availability in an environment is developed and used as a foundation for the semantics, in particular, as a basis for the understanding of the information that is expressed by relevant implication. It is also used to understand (...) the nature of misinformation. A form of relevant logic—called "LOI" for "logic of objective information"—is presented and the relationship between the justification of its proof theory and the semantics is discussed. This relationship is rather reciprocal. Intuitive features of the logic are used to interpret and justify aspects of the model theory and intuitive aspects of the model theory are used to interpret and justify features of the logic. Information conditions are presented for the connectives and the way that they fit into the theory of information is discussed. (shrink)

Kniha, jako je tato, nemůže být tak docela dílem jediného člověka. Dovést ji do podoby koherentního celku bych nedokázal bez pomoci svých kolegů, kteří po mně text četli a upozornili mě na spoustu chyb a nedůsledností, které se v něm vyskytovaly. Můj dík v tomto směru patří zejména Vojtěchu Kolmanovi, Liboru Běhounkovi a Martě Bílkové. Za připomínky k různým částem rukopisu jsem vděčen i Pavlu Maternovi, Milanu Matouškovi, Prokopu Sousedíkovi, Vladimíru Svobodovi, Petru Hájkovi a Grahamu Priestovi. Kniha vznikla v rámci (...) projektu podpořeného grantem AV ČR číslo A0009001/00. (shrink)

The claim that nature is self-consistent has recently been contested by a number of paraconsistent logicians. In this paper I will survey the arguments which paraconsistent logicians have presented for the thesis that nature is actually inconsistent. My conclusion is that these arguments all fail.The paraconsistency programme has to date been concerned primarily with outlining the philosophical inadequacy of classical logic, and detailed discussions of issues bearing upon the philosophical adequacy of the paraconsistency position itself are not to be found (...) as yet in the literature. This inadequacy will be illustrated here with respect to the question of the self-consistency of nature. (shrink)

It is shown that there are exactly six normal DeMorgan monoids generated by the identity element alone. The free DeMorgan monoid with no generators but the identity is characterised and shown to have exactly three thousand and eighty-eight elements. This result solves the "Ackerman constant problem" of describing the structure of sentential constants in the logic R.

A method is presented for constructing natural deduction-style systems for propositional relevant logics. The method consists in first translating formulas of relevant logics into ternary relations, and then defining deduction rules for a corresponding logic of ternary relations. Proof systems of that form are given for various relevant logics. A class of algebras of ternary relations is introduced that provides a relation-algebraic semantics for relevant logics.

What is an inference rule? This question does not have a unique answer. One usually finds two distinct standard answers in the literature; validity inference $(\sigma \vdash_\mathrm{v} \varphi$ if for every substitution $\tau$, the validity of $\tau \lbrack\sigma\rbrack$ entails the validity of $\tau\lbrack\varphi\rbrack)$, and truth inference $(\sigma \vdash_\mathrm{t} \varphi$ if for every substitution $\tau$, the truth of $\tau\lbrack\sigma\rbrack$ entails the truth of $\tau\lbrack\varphi\rbrack)$. In this paper we introduce a general semantic framework that allows us to investigate the notion of inference (...) more carefully. Validity inference and truth inference are in some sense the extremal points in our framework. We investigate the relationship between various types of inference in our general framework, and consider the complexity of deciding if an inference rule is sound, in the context of a number of logics of interest: classical propositional logic, a nonstandard propositional logic, various propositional modal logics, and first-order logic. (shrink)

In his famous work on vagueness, Russell named “fallacy of verbalism” the fallacy that consists in mistaking the properties of words for the properties of things. In this paper, I examine two (clusters of) mainstream paraconsistent logical theories – the non-adjunctive and relevant approaches –, and show that, if they are given a strongly paraconsistent or dialetheic reading, the charge of committing the Russellian Fallacy can be raised against them in a sophisticated way, by appealing to the intuitive reading of (...) their underlying semantics. The meaning of “intuitive reading” is clarified by exploiting a well-established distinction between pure and applied semantics. If the proposed arguments go through, the dialetheist or strong paraconsistentist faces the following Dilemma: either she must withdraw her claim to have exhibited true contradictions in a metaphysically robust sense – therefore, inconsistent objects and/or states of affairs that make those contradictions true; or she has to give up realism on truth, and embrace some form of anti-realistic (idealistic, or broadly constructivist) metaphysics. Sticking to the second horn of the Dilemma, though, appears to be promising: it could lead to a collapse of the very distinction, commonly held in the literature, between a weak and a strong form of paraconsistency – and this could be a welcome result for a dialetheist. (shrink)

The object of this paper is to show why recent research in the psychology of deductive and probabilistic reasoning does not have.

This is an extended critical notice of Gillian Russell's book, Barriers to Entailment.

Most of the logics of grounding that have so far been proposed contain grounding axioms, or grounding rules, for the connectives of conjunction, disjunction and negation, but little attention has been dedicated to the implication connective. The present paper aims at repairing this situation by proposing adequate grounding principles for relevant implication. Because of the interaction between negation and implication, new grounding principles concerning negation will also arise.

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

Equivalent overdetermined and underdetermined bivalent Belnap–Dunn type semantics for the logics determined by all natural implicative expansions of Kleene’s strong 3-valued matrix with only one designated value are provided.