his paper provides a sound and complete axiomatisation for constant domain modal logics without Boolean negation. This is a simpler case of the difficult problem of providing a sound and complete axiomatisation for constant-domain quantified relevant logics, which can be seen as a kind of modal logic with a two-place modal operator, the relevant conditional. The completeness proof is adapted from a proof for classical modal predicate logic (I follow James Garson’s 1984 (...) presentation of the completeness proof quite closely), but with an important twist, to do with the absence of Boolean negation. (shrink)

The main object of this article is to give two novel proofs of the admissibility of Ackermann’s rule (γ) for the propositional relevant logic R. The results are established as corollaries of cut elimination for systems of tableaux for R. Cut elimination, in turn, is established both nonconstructively (as a corollary of completeness) and constructively (using Gentzen-like methods). The extensibility of the techniques is demonstrated by showing that (γ) is admissible for RQ* (R with constant domain (...) quantifiers). The status of the admissibility of (γ) for RQ* was, to the best of the author’s knowledge, an open problem. Further extensions of these results will be explored in the sequel(s). (shrink)

Here I show that the one-variable fragment of several first-order relevant logics corresponds to certain S5ish extensions of the underlying propositional relevant logic. In particular, given a fairly standard translation between modal and one-variable languages and a permuting propositional relevant logic L, a formula $$\mathcal {A}$$ A of the one-variable fragment is a theorem of LQ (QL) iff its translation is a theorem of L5 (L.5). The proof is model-theoretic. In one direction, semantics based on (...) the Mares-Goldblatt [15] semantics for quantified L are transformed into ternary (plus two binary) relational semantics for S5-like extensions of L (for a general presentation, see Seki [26, 27]). In the other direction, a valuation is given for the full first-order relevant logic based on L into a model for a suitable S5 extension of L. I also discuss this work’s relation to finding a complete axiomatization of the constant domain, non-general frame ternary relational semantics for which RQ is incomplete [11]. (shrink)

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. (shrink)

Mares and Goldblatt, 163–187, 2006) provided an alternative frame semantics for two quantified extensions of the relevant logic R. In this paper, I show how to extend the Mares-Goldblatt frames to accommodate identity. Simpler frames are provided for two zero-order logics en route to the full logic in order to clarify what is needed for identity and substitution, as opposed to quantification. I close with a comparison of this work with the Fine-Mares models for relevant logics (...) with identity and a discussion of constant and variable domains. (shrink)

In spite of a powerful tradition, more than two thousand years old, that in a valid argument the premises must be relevant to the conclusion, twentieth-century logicians neglected the concept of relevance until the publication of Volume I of this monumental work. Since that time relevance logic has achieved an important place in the field of philosophy: Volume II of Entailment brings to a conclusion a powerful and authoritative presentation of the subject by most of the top people (...) working in the area. Originally the aim of Volume II was simply to cover certain topics not treated in the first volume--quantification, for example--or to extend the coverage of certain topics, such as semantics. However, because of the technical progress that has occurred since the publication of the first volume, Volume II now includes other material. The book contains the work of Alasdair Urquhart, who has shown that the principal sentential systems of relevance logic are undecidable, and of Kit Fine, who has demonstrated that, although the first-order systems are incomplete with respect to the conjectured constant domain semantics, they are still complete with respect to a semantics based on "arbitrary objects." Also presented is important work by the other contributing authors, who are Daniel Cohen, Steven Giambrone, Dorothy L. Grover, Anil Gupta, Glen Helman, Errol P. Martin, Michael A. McRobbie, and Stuart Shapiro. Robert G. Wolf's bibliography of 3000 items is a valuable addition to the volume. (shrink)

In spite of a powerful tradition, more than two thousand years old, that in a valid argument the premises must be relevant to the conclusion, twentieth-century logicians neglected the concept of relevance until the publication of Volume I of this monumental work. Since that time relevance logic has achieved an important place in the field of philosophy: Volume II of Entailment brings to a conclusion a powerful and authoritative presentation of the subject by most of the top people (...) working in the area. Originally the aim of Volume II was simply to cover certain topics not treated in the first volume--quantification, for example--or to extend the coverage of certain topics, such as semantics. However, because of the technical progress that has occurred since the publication of the first volume, Volume II now includes other material. The book contains the work of Alasdair Urquhart, who has shown that the principal sentential systems of relevance logic are undecidable, and of Kit Fine, who has demonstrated that, although the first-order systems are incomplete with respect to the conjectured constant domain semantics, they are still complete with respect to a semantics based on "arbitrary objects." Also presented is important work by the other contributing authors, who are Daniel Cohen, Steven Giambrone, Dorothy L. Grover, Anil Gupta, Glen Helman, Errol P. Martin, Michael A. McRobbie, and Stuart Shapiro. Robert G. Wolf's bibliography of 3000 items is a valuable addition to the volume. Originally published in 1992. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

Sedlár and Vigiani [18] have developed an approach to propositional epistemic logics wherein (i) an agent’s beliefs are closed under relevant implication and (ii) the agent is located in a classical possible world (i.e., the non-modal fragment is classical). Here I construct first-order extensions of these logics using the non-Tarskian interpretation of the quantifiers introduced by Mares and Goldblatt [12], and later extended to quantified modal relevant logics by Ferenz [6]. Modular soundness and completeness are proved for (...) class='Hi'>constant domain semantics, using non-general frames with Mares–Goldblatt truth conditions. I further detail the relation between the demand that classical possible worlds have Tarskian truth conditions and incompleteness results in quantified relevant logics. (shrink)

The aim of this paper is to provide a proof-theoretic characterization of relevant logics including fusion and fission connectives, as well as Ackermann’s truth constant. We achieve this by employing the well-established methodology of labelled sequent calculi. After having introduced several systems, we will conduct a detailed proof-theoretic analysis, show a cut-admissibility theorem, and establish soundness and completeness. The paper ends with a discussion that contextualizes our current work within the broader landscape of the proof theory of (...) class='Hi'>relevant logics. (shrink)

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and (...) the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work. (shrink)

This paper gives an account of Anderson and Belnap’s selection criteria for an adequate theory of entailment. The criteria are grouped into three categories: criteria pertaining to modality, those pertaining to relevance, and those related to expressive strength. The leitmotif of both this paper and its prequel is the relevant legitimacy of disjunctive syllogism. Relevant logics are commonly held to be paraconsistent logics. It is shown in this paper, however, that both E and R can be extended to (...) explosive logics which satisfy all of Anderson and Belnap’s selection criteria, provided the truth-constant known as the Ackermann constant is available. One of the selection criteria related to expressive strength is having an “enthymematic” conditional for which a deduction theorem holds. I argue that this allows for a new interpretation of Anderson and Belnap’s take on logical consequence, namely as committing them to pluralism about logical consequence. (shrink)

I build a canonical model for constant domain basic first-order logic (BQLCD), the constant domain first-order extension of Visser’s basic propositional logic, and use the canonical model to verify that BQLCD satisfies the disjunction and existence properties.

The logic CD is an intermediate logic which exactly corresponds to the Kripke models with constant domains. It is known that the logic CD has a Gentzen-type formulation called LD and rules are replaced by the corresponding intuitionistic rules) and that the cut-elimination theorem does not hold for LD. In this paper we present a modification of LD and prove the cut-elimination theorem for it. Moreover we prove a “weak” version of cut-elimination theorem for LD, saying (...) that all “cuts” except some special forms can be eliminated from a proof in LD. From these cut-elimination theorems we obtain some corollaries on syntactical properties of CD: fragments collapsing into intuitionistic logic. Harrop disjunction and existence properties, and a fact on the number of logical symbols in the axiom of CD. (shrink)

For each intermediate propositional logicJ, J * denotes the least predicate extension ofJ. By the method of canonical models, the strongly Kripke completeness ofJ *+D(=x(p(x)q)xp(x)q) is shown in some cases including:1. J is tabular, 2. J is a subframe logic. A variant of Zakharyashchev's canonical formulas for intermediate logics is introduced to prove the second case.

The completeness w.r.t. Kripke frames with equality (or, equivalently, w.r.t. Kripke sheaves, [ 8 ] or [4, Sect. 3.6]) is established for three superintuitionistic predicate logics: ( Q - H + D *), ( Q - H + D *&K), ( Q - H + D *& K & J ). Here Q - H is intuitionistic predicate logic, J is the principle of the weak excluded middle, K is Kuroda’s axiom, and D * (cf. [ 12 ]) is (...) a weakened version of the well-known constant domains principle D . Namely, the formula D states that any individual has ancestors in earlier worlds, and D * states that any individual has $${\neg\neg}$$ -ancestors (i.e., ancestors up to $${\neg\neg}$$ -equality) in earlier worlds. In particular, the logic ( Q - H + D *& K & J ) is the Kripke sheaf completion of ( Q - H + E & K & J ), where E is a version of Markov’s principle (cf. [ 12 ]). On the other hand, we show that the logic ( Q - H + D *& J ) is incomplete w.r.t. Kripke sheaves. (shrink)

We introduce a restricted second-order logic $\textrm{SO}^{\textit{plog}}$ for finite structures where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. We demonstrate the relevance of this logic and complexity class by several problems in database theory. We then prove a Fagin’s style theorem showing that the Boolean queries which can be expressed in the existential fragment of $\textrm{SO}^{\textit{plog}}$ correspond exactly to the class of decision problems that can be computed by a (...) non-deterministic Turing machine with random access to the input in time $O^k)$ for some $k \ge 0$, i.e. to the class of problems computable in non-deterministic poly-logarithmic time. It should be noted that unlike Fagin’s theorem which proves that the existential fragment of second-order logic captures NP over arbitrary finite structures, our result only holds over ordered finite structures, since $\textrm{SO}^{\textit{plog}}$ is too weak as to define a total order of the domain. Nevertheless, $\textrm{SO}^{\textit{plog}}$ provides natural levels of expressibility within poly-logarithmic space in a way which is closely related to how second-order logic provides natural levels of expressibility within polynomial space. Indeed, we show an exact correspondence between the quantifier prefix classes of $\textrm{SO}^{\textit{plog}}$ and the levels of the non-deterministic poly-logarithmic time hierarchy, analogous to the correspondence between the quantifier prefix classes of second-order logic and the polynomial-time hierarchy. Our work closely relates to the constant depth quasipolynomial size AND/OR circuits and corresponding restricted second-order logic defined by David A. Mix Barrington in 1992. We explore this relationship in detail. (shrink)

We consider a labelled tableau presentation of constant domain first-order S5 and prove a strong cut-elimination theorem.

The aim of this paper is to construct a tableau decision algorithm for the modal description logic K ALC with constant domains. More precisely, we present a tableau procedure that is capable of deciding, given an ALC-formula with extra modal operators (which are applied only to concepts and TBox axioms, but not to roles), whether is satisfiable in a model with constant domains and arbitrary accessibility relations. Tableau-based algorithms have been shown to be practical even for logics (...) of rather high complexity. This gives us grounds to believe that, although the satisfiability problem for K ALC is known to be NEXPTIME-complete, by providing a tableau decision algorithm we demonstrate that highly expressive description logics with modal operators have a chance to be implementable. The paper gives a solution to an open problem of Baader and Laux [5]. (shrink)

Sentential constants have been part of the R environment since Church [1]. They have had diverse uses in explicating relevant ideas and in sim- plifying them technically. Of most interest have been the Ackermann pair of constants t; f, functioning conceptually as a least truth, and as a greatest , under the ordering of propositions under true impli- cation. Also interesting have been the Church constants F; T, functioning similarly as least greatest propositions.

We exhibit infinitely many semisimple varieties of semilinear De Morgan monoids (and likewise relevant algebras) that are not tabular, but which have only tabular proper subvarieties. Thus, the extension of relevance logic by the axiom $$(p\rightarrow q)\vee (q\rightarrow p)$$ ( p → q ) ∨ ( q → p ) has infinitely many pretabular axiomatic extensions, regardless of the presence or absence of Ackermann constants.

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. (shrink)

The main aim of this paper is to introduce first-order versions of logics of evidence and truth, together with corresponding sound and complete Kripke semantics with variable and constant domains. According to the intuitive interpretation proposed here, these logics intend to represent possibly inconsistent and incomplete information bases over time. The paper also discusses the connections between Belnap-Dunn’s and da Costa’s approaches to paraconsistency, and argues that the logics of evidence and truth combine them in a very natural way.

It is shown that the classes of Routley-Meyer models which are axiomatizable by a theory in a propositional relevant language with fusion and the Ackermann constant can be characterized by their closure under certain model-theoretic operations involving prime filter extensions, relevant directed bisimulations and disjoint unions.

This paper presents completeness and conservative extension results for the boolean extensions of the relevant logic T of Ticket Entailment, and for the contractionless relevant logics TW and RW. Some surprising results are shown for adding the sentential constant t to these boolean relevant logics; specifically, the boolean extensions with t are conservative of the boolean extensions without t, but not of the original logics with t. The special treatment required for the semantic normality of (...) T is also shown along the way. (shrink)

ABSTRACT My aim in this paper is to present a pluralist thesis about the inferential role of logical constants, which embraces classical, relevant, linear and ordered logic. That is, I defend that a logical constant c has more than one correct inferential role. The thesis depends on a particular interpretation of substructural logics' vocabulary, according to which classical logic captures the literal meaning of logical constants and substructural logics encode a pragmatically enriched sense of those connectives. (...) The paper is divided into four parts: first, I introduce the motivation for the pluralist thesis of the paper; second I introduce the calculus for the different logics endorsed in the pluralist thesis; third, I motivate how the different behaviors of the logical vocabulary of these logics can be pragmatically interpreted, and fourth, I motivate how the different inferential roles that each substructural logic attribute to logical constants can coexist, and how we should reason with a logical constant c given this plurality. (shrink)

We provide Routley-Meyer type semantics for relevant logics including Contractionless Ticket Entailment TW (without the truth constant t and o) plus reductio R and Ackermann’s rule γ (i.e., disjunctive syllogism). These logics have the following properties. (i) All have the variable sharing property; some of them have, in addition, the Ackermann Property. (ii) They are stable. (iii) Inconsistent theories built upon these logics are not necessarily trivial.

Under a proper translation, the languages of propositional (and quantified relevant logic) with an absurdity constant are characterized as the fragments of first order logic preserved under (world-object) relevant directed bisimulations. Furthermore, the properties of pointed models axiomatizable by sets of propositional relevant formulas have a purely algebraic characterization. Finally, a form of the interpolation property holds for the relevant fragment of first order logic.

The problem of logical constants consists in finding a principled way to draw the line between those expressions of a language that are logical and those that are not. The criterion of invariance under permutation, attributed to Tarski, is probably the most common answer to this problem, at least within the semantic tradition. However, as the received view on the matter, it has recently come under heavy attack. Does this mean that the criterion should be amended, or maybe even that (...) it should be abandoned? I shall review the different types of objections that have been made against invariance as a logicality criterion and distinguish between three kinds of objections, skeptical worries against the very relevance of such a demarcation, intensional warnings against the level at which the criterion operates, and extensional quarrels against the results that are obtained. I shall argue that the first two kinds of objections are at least partly misguided and that the third kind of objection calls for amendment rather than abandonment. (shrink)

Relevant propositional dynamic logics have been sporadically discussed in the broader context of modal relevant logics, but have not come up for sustained investigation until recently. In this paper, we develop a philosophical motivation for these systems, and present some new results suggested by the proposed motivation. Among these, we’ll show how to adapt some recent work to show that the extensions of relevant logics by the extensional truth constants \ are complete with respect to a natural (...) class of ternary relation models to show a similar result for the constant-free versions of the logic. In addition, we prove that the logics in question satisfy the variable sharing property, vindicating the claim that they really are relevant logics. (shrink)

What is logical relevance? Anderson and Belnap say that the “modern classical tradition [,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and (...) Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein’s Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains. (shrink)

In the domain of the Ukrainian history of philosophy one can clearly observe symptoms of eclecticism, contamination, and methodological ambiguity. Such situation is rooted in continuous insufficiency of the methodological framework of Ukrainian philosophy, split by competing ideological structures. Critical condition of humanitarian epistemology is aggravated by the urgent need to overtake foreign studies in the corresponding field, which constantly increase in number and evolve in character. The article aims at critical assessment of methodological approaches to historical philosophical analysis (...) within the light of current philosophical studies. Analysis of the relevant literature on the problem of how to identify the character of interference between the target philosophical discourse and the discourse of the historical reconstruction allowed revealing the critical lines of relativist archeology and traditional historiography. Relativist approach is averted from the dimension of concrete texts to logical anachronistic speculations over seemingly evident contextual links. Such strategy is legitimate to sustain logical descriptions of the categories exempted from the certain paradigm but it is useless to reveal factual genealogical links between the discourses of the two philosophers. Descriptive studies seem to fall prey to hermeneutic ambiguity in referring to a set of ideas forming the canon of perennial philosophy. It is proved that historiographical approach enables to objectively examine genetic relations between various corpora of philosophical texts. Nowadays the Ukrainian historico-philosophical science is going through a harsh crisis of methodology, caused by distortion of universal coordinates of scientific thinking allegedly providing the condition of the so-called methodological pluralism. Numerous discussions focus on the question about the very scientific nature of history of philosophy as a branch of knowledge. The tendency which appears to prevail in such disputes defines history of philosophy as philosophy in contrast to the opinion that history of philosophy should be rather seen as approximate to social history. (shrink)

Timothy Williamson’s Modal Logic as Metaphysics is a book-length defense of necessitism about objects—roughly put, the view that, necessarily, any object that exists, exists necessarily. In more formal terms, Williamson argues for the validity of necessitism for objects (NO: ◻︎∀x◻︎∃y(x=y)). NO entails both the (first-order) Barcan formula (BF: ◇∃xΦ → ∃x◇Φ, for any formula Φ) and the (first-order) converse Barcan formula (CBF: ∃x◇Φ → ◇∃xΦ, for any formula Φ). The purpose of this essay is not to assess Williamson’s arguments (...) either for necessitism (although discussion of these arguments will play a central role in the dialectic) or for necessitism’s two famous corollaries. Instead, the focus shall be a general principle governing abstract objects—the abstract of principle (or AOP) —instances of which seems to be at work in some of Williamson’s central arguments for necessitism. The AOP can be straightforwardly formulated and applied within the neo-logicist framework—in fact, arguably the principle is most naturally formulated in neo-logicist terms. -/- After closely examining, and carefully formalizing, the AOP, the remainder of the paper focuses on arguments for necessitism-like claims (the exact meaning of “necessitism-like” will become clearer as the essay progresses) based on the AOP. In particular, we shall focus on the instance of the AOP that applies to the abstract objects governed by the most well-known and most fully studied abstraction principle: Hume’s Principle (HP). It turns out that, although we cannot reconstruct a valid argument for necessitism based on this numerical instance of the AOP, we can obtain valid arguments for weaker, but equally interesting conclusions. In particular, we shall show that, although HP combined with the AOP (and some additional, related assumptions) allows the contents of the domains of possible worlds to vary, the size of those domains must remain constant. The paper concludes by developing and critiquing some related arguments for necessitism based on applying relevant instances of the AOP to abstraction principles governing sets (or extensions), and to a simple objectual abstraction principle. (shrink)

This paper uses Avron’s algebraic semantics for the logic RMI to model some ideas in the philosophy of science. Avron’s relevant disjunctive structures are each partitioned into relevance domains. Each relevance domain is a boolean algebra. I employ this semantics to act as a formal framework to represent what Nancy Cartwright calls the “dappled world”. On the dappled world hypothesis, local scientific theories each represent restricted aspects and regions of the universe. I use relevance domains to represent (...) the domains of each of these local theories and I provide a formalisation of the salient relationships between so-called fundamental theories and local theories. I also examine ways in which the paraconsistent nature of RMI can be used to deal with inconsistencies within and between theories adopted by scientists. The paper ends with some suggestions about updating RDS given changes in the theories that science adopts. (shrink)

It is known that many relevant logics can be conservatively extended by the truth constant known as the Ackermann constant. It is also known that many relevant logics can be conservatively extended by Boolean negation. This essay, however, shows that a range of relevant logics with the Ackermann constant cannot be conservatively extended by a Boolean negation.

The logical constants are technical terms, invented and precisely defined by logicians for the purpose of producing rigorous formal proofs. Mathematics virtually exhausts the domain of deductive reasoning of any complexity, and it is there that the benefits of this refined form of language are felt. Pragmatic issues may arise — issues concerning the point of making a certain statement — for there will be more or less perspicuous and illuminating ways of presenting proofs in this language, and we (...) may be puzzled or misled when we wonder why the mathematician is taking some particular step. But this is hardly a compulsory topic in the philosophy of language. (shrink)

What is the relationship between logic and reasoning? How do logical norms guide inferential performance? This paper agrees with Gilbert Harman and most of the psychologists that logic is not directly relevant to reasoning. It argues, however, that the mental model theory of logical reasoning allows us to harmonise the basic principles of deductive reasoning and inferential perfomances, and that there is a strong connexion between our inferential norms and actual reasoning, along the lines of Peacocke’s conception (...) of inferential role. (shrink)

We show that as long as the propositional constants t and f are not included in the language, any language-preserving extension of any important fragment of the relevance logics R and RMI can have only classical tautologies as theorems . This property is not preserved, though, if either t or f is added to the language, or if the contraction axiom is deleted.

Domain constraint, the requirement that analogues be selected from "the same category," inheres in the popular saying "you can't compare apples and oranges" and the textbook principle "the greater the number of shared properties, the stronger the argument from analogy." I identify roles of domains in biological, linguistic, and legal analogy, supporting the account of law with a computer word search of judicial decisions. I argue that the category treatments within these disciplines cannot be exported to general informal (...) class='Hi'>logic, where the relevance of properties, not their number, must be the logically prior criterion for evaluating analogical arguments. (shrink)

The new area of logic and computation is now undergoing rapid development. This has affected the social pattern of research in the area. A new topic may rise very quickly with a significant body of research around it. The community, however, cannot wait the traditional two years for a book to appear. This has given greater importance to thematic collections of papers, centred around a topic and addressing it from several points of view, usually as a result of a (...) workshop, summer school, or just a scientific initiative. Such a collection may not be as coherent as a book by one or two authors yet it is more focused than a collection of key papers on a certain topic. It is best thought of as a thematic collection, a study in the area of logic and computation. The new series Studies in Logic and Computation is intended to provide a home for such thematic collections. Substructural logics are nonclassical logics, which arose in response to problems in foundations of mathematics and logic, theoretical computer science, mathematical linguistics, and category theory. They include intuitionistic logic, relevant logic, BCK logic, linear logic, and Lambek's calculus of syntactic categories. Substructural logics differ from classical logics, and from each other, in their presuppositions about Gentzen's structural rules, although their presuppositions about the deductive role of logical constants are invariant. Substructural logics have been a subject of study for logicians during the last sixty years. Specialists have often worked in isolation, however, largely unaware of the contributions of others. This book brings together new papers by some of the most eminent authorities in these varioustraditions to produce a unified view of substructural logics. (shrink)

An Ackermann constant is a formula of sentential logic built up from the sentential constant t by closing under connectives. It is known that there are only finitely many non-equivalent Ackermann constants in the relevant logic R. In this paper it is shown that the most natural systems close to R but weaker than it-in particular the non-distributive system LR and the modalised system NR-allow infinitely many Ackermann constants to be distinguished. The argument in each (...) case proceeds by construction of an algebraic model, infinite in the case of LR and of arbitrary finite size in the case of NR. The search for these models was aided by the computer program MaGIC (Matrix Generator for Implication Connectives) developed by the author at the Australian National University. (shrink)

Preface The status of sovereignty as a highly ambiguous concept is well established. Pointing out or deploring, the ambiguity of the idea has itself become a recurring motif in the literature on sovereignty. As the legal theorist and international lawyer Alf Ross put it, “there is hardly any domain in which the obscurity and confusion is as great as here.” 1 The concept of sovereignty is often seen as a downright obstacle to fruitful conceptual analysis, carried over from its (...) proper setting in history to “plague and befog contemporary thought.” 2 . . . So contested is the concept that, rather than pursuing the contestation, many political theorists think we should give up so protean a notion. Granting that the debate on the relevance of sovereignty frustratingly oscillates between claims that it will either continue to exist or that it is about to disappear, forgetting it altogether, and thereby escaping this seemingly endless argument, can easily appear as the most urgent task for political theory . The following argument makes a case that the “urgent task” is not the abandonment of the concept of sovereignty, but an understanding of its essential philosophical nature as an integrated and evolving expression of practical reason. Sovereignty is neither ambiguous nor obscure once its fundamental presuppositions are laid bare and its many philosophical and historical manifestations shown to be the product, in actuality, of a single, dialectally dynamic but integrated set of metaphysical elements. This is the first of three arguments describing the evolution of international law as a manifestation of practical reason through an application of philosophical method to the source , locus , and scope of the concept of sovereignty. It moves from a dialectic balance favoring utility to a balance dominated by legal right to a dialectic of duty to humanity and nature. All three arguments are meant to be a contribution to the new field of International Legal Philosophy as defined by Phillip Allott. 4 This field combines a sensitivity to legal practice with an effort to understand the underlying philosophical determinants of empirical choice and behavior. One purpose of international legal philosophy is to “remove” from the minds of those who study the law what Diderot defined as “the sophism of the ephemeral,” and what Allott calls “the disempowering idea that what xii Preface happens to exist now is inevitable and permanent.” 5 A core imperative is to “reunderstand what it is to be a thinking being” 6 and to rediscover the dialectic between the private and the public as it determines, and is redetermined by, legal practice. This requires a “revolution in the human mind” 7 so that we may transcend the current dependence on positivist methods and empirical fact as an end-in-itself, and try to understand the underlying and more constant and essential ideas and inherent dialectics that constitute the substructure or “metaphysics” of international law. I will approach this “revolution” with the use of R. G. Collingwood’s philosophical method 8 and the philosophy of David Hume, applied to international law as an expression of practical reason. The goal of philosophical method is the construction of a comprehensive policy argument (CPA) for a public policy or legal issue. In addition to the conventional use of empirical models and their logic of investigation in the study of policy and law, CPA requires that an underlying philosophical logic of concepts be deciphered to identify the ideas within the issue, and their definition, overlap, and systematic interdependence. Philosophical method is a means with which to interpret and understand competing systematic and complete conceptual logics, existing at the core of an issue and pertinent to policy change. Philosophical method is therefore not meant to be a replacement for the empirical investigation of a policy or legal issue, or the use of scientific method in social studies. Rather, it is a complimentary and prerequisite method that seeks to transcend the limitations of positivism and present a more complete understanding of the philosophical presuppositions of positivist ideas like power, interest, or strategic rationality. Philosophical method is meant to be used with the facts of the policy or legal issue to match an illuminating logic of concepts with a pertinent logic of investigation . Within the CPA, the use of philosophical method and the metaphysics of a policy or legal issue is assumed to be critical to the full understanding of the overlapping concepts, dialectics, and scale of forms that determine, and are determined by, the empirical context of the policy or legal topic. Specifically, instead of utilizing bits and pieces of various theoretical arguments to address narrowly focused empirical questions, as positivism prescribes, I will address the evolution of international law as practical reason in three phases. Each will be approached through a single integrated logic of philosophical concepts from a particular philosopher (i.e., David Hume, G. W. F. Hegel, Immanuel Kant). This philosophically holistic approach to the law is based on the assumption that only through the use of a single integrated argument in legal analysis can sovereignty, or any concept, be understood as a truly systematic and logical whole. A complete philosophical paradigm has a dialectic integrity and systematic logic that can more adequately describe the evolving essence of a concept like sovereignty. This approach also has the advantage of generating a number of distinct holistic descriptions of the law through the application of different philosophical systems, one at a time, to its factual structure. 9 Positivism does not seek Preface xiii holism, and rejects the idea that “theory” has such a characteristic. The essential or comprehensive substructure of any idea is therefore ignored in a method that recommends the observation of empirical problems through the use of whatever hodgepodge of theoretical elements is seen fit to frame its superstructure. This failure to deal with metaphysics has retarded both an essential understanding of international law as a species of legal system, and any holistic and dialectical conceptualization of its inherent concepts, like sovereignty. A second positivist convention expects modern theorists to create new theory rather than to refine and apply that of existing philosophy. This predisposition is driven by the positivist goal of discovery that ignores refinement as a possible purpose of philosophical analysis. Collingwood argues that philosophy must take that set of ideas already known and utilize existing systematic philosophical arguments to refine them so that they evolve closer to their essence as concepts. Considering this imperative, the idea of sovereignty can be assumed to have had valid usage for hundreds of years, over which time, the concept has evolved to mean different things, each a refinement of the definition that preceded it. Transcending positivism means that the scholar’s search is not for “new” material, but to decipher the metaphysical essence of a concept as it has been made manifest over time and context. These manifestations are rooted, and refined from, the known terms of that concept’s inherent idea(s). Rather than depending exclusively on positivism and its conventions, my work utilizes, in addition to Collingwood, the intact philosophical systems of Hume, Hegel, and Kant to trace the refinement of international law as a product of human practical reason. These paradigms, or integrated systems of logical concepts, will be applied to legal practice individually, so that each CPA can be deciphered separately. This provides a set of integrated and logically intact paradigms for the evolutionary stages of practical reason in international law. Because each argument is applied systematically, a deeper understanding of the source, locus, and scope in the development of law in general, and international law in particular, is possible where it is not with the application of various disconnected components of many theories. Each CPA based on Hume, Hegel, or Kant can then be used to describe a distinct context that its logic of concepts best illuminates; specifically, the (1) genesis, (2) contemporary dilemmas, and (3) future of the international legal system. By widening the perspective of international lawyers and policymakers, they can more easily perceive the dialectic of ideas that has created, and is refined by, the legal practice in which they participate. We also move toward Allott’s goal of “human self-perfecting.” 10 And, in addition, by providing a more complete knowledge of the origins of legal practice and its evolution, we illuminate the practical possibilities for what we might “choose to be” 11 in the future. To achieve this, the essential metaphysical elements of state sovereignty and its inherent evolutionary scale of forms will be deciphered and described. This will transform what appears to be a multitude of definitions and xiv Preface practical realizations of the concept of sovereignty into a set of interdependent manifestations of a single substructure, made of a single set of dialectic elements. The interpretation of international law through practical reason sorts and integrates a diverse and discordant literature and defines state sovereignty as a single concept evolving on a scale of forms that allows it to exhibit diverse character traits, all arising from different combinations of common and essential metaphysical elements. This approach, compared to positivist methods and legal realism, allows one to transcend current agreement that sovereignty is, at best, a narrowly focused set of empirical characteristics or, at worst, “organized hypocrisy.” 12 This method also encourages the scholar and practitioner to understand the predispositions and pitfalls of the concept of sovereignty, as well as its potential future paths, more effectively. The use of philosophical method to create policy paradigms out of preexisting philosophical systems and apply these to international law will be called Philosophical-Policy & Legal Design . This approach allows the use of preexisting and complete philosophical arguments that provide an adequate logic of concepts to chart the evolution of the idea of sovereignty along its scale of forms. An examination of the source of practical reason in human social convention with the employment of a philosophical-policy drawn from Hume’s logic of concepts about human nature will demonstrate this new approach. Why Hume? Because, up to now, without an adequate substructure we have arguments, like Brunne é’ s and Troope’s, 13 that may correctly identify international law as an “interactional” system, but cannot present any argument as to why it is, where this empirical reality comes from, or what its implications are for the future. Comparatively, Hume provides a logic of philosophical concepts that answers these concerns. First, he fulfills the requirements for a fuller understanding of the origin and evolution of law from social convention and the dependence of social convention on the human imperative for society. Second, he offers a more adequate delineation of the overlapping concepts of the law in terms of the ideas and institutions that deal with norms and justice (e.g., principle, process, practice, rule, power, interest). Third, he provides a fundamental understanding of the essential dialectics at the core of a conceptualization of the law with both unconscious and conscious human participation (i.e., passion. (shrink)

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan (...) Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics. (shrink)

I discuss paradoxes of implication in the setting of a proof-conditional theory of meaning for logical constants. I argue that a proper logic of implication should be not only relevant, but also constructive and nonmonotonic. This leads me to select as a plausible candidate LL, a fragment of linear logic that differs from R in that it rejects both contraction and distribution.

The system B+ is the minimal positive relevant logic. B+ is trivially extended to B+T on adding a greatest truth (Church constant) T. If we leave ∨ out of the formation apparatus, we get the fragment B∧T. It is known that the set of ALL B∧T theories provides a good model for the combinators CL at Level-I, which is the theory level. Restoring ∨ to get back B+T was not previously fruitful at Level-I, because the set of (...) all B+T theories is NOT a model of CL. It was to be expected from semantic completeness arguments for relevant logics that basic combinator laws would hold when restricted to PRIME B+T theories. Overcoming some previous difficulties, we show that this is the case, at Level I. But this does not form a model for CL. This paper also looks for corresponding results at Level-II, where we deal with sets of theories that we call propositions. We adapt work by Ghilezan to note that at Level-II also there is a model of CL in B∧T propositions. However, the corresponding result for B+T propositions extends smoothly to Level-II only in part. Specifically, only some of the basic combinator laws are proved here. We accordingly leave some work for the reader. (shrink)

In the article we study the existence predicate \(\varepsilon\) in the context of semantics for first-order modal logic. For a formula \(\varphi\) we define \(\varphi^{\varepsilon}\) - the so called existence relativization. We point to a gap in the work of Fitting and Mendelsohn concerning the relationship between the truth of \(\varphi\) and \(\varphi^{\varepsilon}\) in classes of varying- and constant-domain models. We introduce operations on models which allow us to fill the gap and provide a more general perspective (...) on the issue. As a result we obtain a series of theorems describing the logical connection between the notion of truth of a formula with the existence predicate in constant-domain models and the notion of truth of a formula without the existence predicate in varying-domain models. (shrink)