The aim of the present book is to give a comprehensive account of the ‘state of the art’ of substructural logics, focusing both on their proof theory and on their semantics (both algebraic and relational. It is for graduate students in either philosophy, mathematics, theoretical computer science or theoretical linguistics as well as specialists and researchers.

Algebraic structures on linguistic sets associated with a linguistic variable are introduced. The linguistics with single closed binary operations are only semigroups and monoids. We describe the new notion of linguistic semirings, linguistic semifields, linguistic semivector spaces and linguistic semilinear algebras defined over linguistic semifields. We also define algebraic structures on linguistic subsets of a linguistic set associated with a linguistic variable. We define the notion of linguistic subset semigroups, (...) linguistic subset monoids and their respective substructures. We also define as in case of deals in classical semigroups, linguistic ideals in linguistic semigroups and linguistic monoids. This concept of linguistic ideals is extended to the case of linguistic subset semigroups and linguistic subset monoids. We also define linguistic substructures. (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)

This paper deals with substructural nuclear (image-based) logics and their algebraic and Kripke-style semantics. More precisely, we first introduce a class of substructural logics with connective N satisfying nucleus property, called here substructural nuclear logics, and its subclass, called here substructural nuclear image-based logics, where N further satisfies homomorphic image property. We then consider their algebraic semantics together with algebraic characterizations of those logics. Finally, we introduce operational Kripke-style semantics for those logics and provide two sorts of completeness results for (...) them, one of which is based on algebraic completeness for all the logics and the other of which is based on set-theoretic one for the nuclear image-based logics. (shrink)

This book introduces an important group of logics that have come to be known under the umbrella term 'susbstructural'. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. _An Introduction to Substrucural Logics_ is the first book to systematically survey the new results and the significant impact that this class of logics has had on a wide range of fields.The following topics are covered: * Proof Theory * Propositional Structures * Frames * Decidability * Coda Both (...) students and professors of philosophy, computing, linguistics, and mathematics will find this to be an important addition to their reading. (shrink)

This volume presents the state of the art in the algebraic investigation into substructural logics. It features papers from the workshop AsubL (Algebra & Substructural Logics - Take 6). Held at the University of Cagliari, Italy, this event is part of the framework of the Horizon 2020 Project SYSMICS: SYntax meets Semantics: Methods, Interactions, and Connections in Substructural logics. -/- Substructural logics are usually formulated as Gentzen systems that lack one or more structural rules. They have been intensively studied over (...) the past two decades by logicians of various persuasions. These researchers include mathematicians, philosophers, linguists, and computer scientists. Substructural logics are applicable to the mathematical investigation of such processes as resource-conscious reasoning, approximate reasoning, type-theoretical grammar, and other focal notions in computer science. They also apply to epistemology, economics, and linguistics. The recourse to algebraic methods -- or, better, the fecund interplay of algebra and proof theory -- has proved useful in providing a unifying framework for these investigations. The AsubL series of conferences, in particular, has played an important role in these developments. -/- This collection will appeal to students and researchers with an interest in substructural logics, abstract algebraic logic, residuated lattices, proof theory, universal algebra, and logical semantics. (shrink)

This essay is structured around the bifurcation between proofs and models: The first section discusses Proof Theory of relevant and substructural logics, and the second covers the Model Theory of these logics. This order is a natural one for a history of relevant and substructural logics, because much of the initial work — especially in the Anderson–Belnap tradition of relevant logics — started by developing proof theory. The model theory of relevant logic came some time later. As we will see, (...) Dunn’s algebraic models [76, 77] Urquhart’s operational semantics [267, 268] and Routley and Meyer’s relational semantics [239, 240, 241] arrived decades after the initial burst of activity from Alan Anderson and Nuel Belnap. The same goes for work on the Lambek calculus: although inspired by a very particular application in linguistic typing, it was developed first proof-theoretically, and only later did model theory come to the fore. Girard’s linear logic is a different story: it was discovered though considerations of the categorical models of coherence.. (shrink)

There have been recent arguments against the idea that substructural solutions are uniform. The claim is that even if the substructuralist solves the common semantic paradoxes uniformly by targeting Cut or Contraction, with additional machinery, we can construct higher-level paradoxes (e.g., a higher-level Liar, a higher-level Curry, and a meta-validity Curry). These higher-level paradoxes do not use metainferential Cut or Contraction, but rather, higher-level Cuts and higher-level Contractions. These kinds of paradoxes suggest that targeting Cut or Contraction is not enough (...) for solving semantic paradoxes; the substructuralist must target Cut of every level or Contraction of every level to solve the paradoxes. Hence, the substructuralists do not provide as uniform of a solution as they hoped they did. In response, we argue that the substructuralists need not admit these additional machineries. In fact, they are redundant in light of the validity predicate (i.e., there is no gain in terms of expressive power). The validity predicate is powerful enough to creep these paradoxes in the object level. The substructuralist does not need to ascend to metainferences to construct higher-level paradoxes. Moreover, there is a reading available to the substructuralist such that all the higher-level structural rules would collapse to instances of the object-level structural rules (e.g., meta \(_n\) Cut and meta \(_n\) Contraction would become instances of Cut and Contraction). We then address Barrio et al.’s worry that the validity predicate has its shortcomings; the substructuralist cannot internalize some of its metarules. We claim that the validity of metarules can be internalized without the need to strengthen the validity predicate. However, a problem raised by Barrio et al. is still present—the problem of internalizing unwanted instances of Cut in Cut-free approaches. We argue that this internalization problem is not unique to the validity predicate; the same problem is present with other problematic predicates, such as the truth predicate and the provability predicate. (shrink)

Formal systems seem to come in two general kinds: useful and useless. This is painting things starkly, but the point is important. Formal structures can either be used in interesting and important ways, or they can languish unused and irrelevant. Lewis' modal logics are good examples. The systems S4 and S5 are useful in many different ways. They map out structures that are relevant to a number of different applications. S1, S2 and S3 however, are not so lucky. They are (...) little studied, and used even less. It has become dear that the structures described by S4 and S5 are important in different ways, while the structures described by S1 to S3 are not so important. In this paper, we will see another formal system with a number of different uses. We will examine a substructural logic which is important in a number of different ways. The logic of Peirce monoids, inspired by the logic of relations, is useful in the independent areas of linguistic types and information flow.In what follows I will describe the logic of Peirce monoids in its various guises, sketch out its main properties, and indicate why it is important. As proofs of theorems are readily available elsewhere in the literature, I simply sketch the relevant proofs here, and point the interested reader to where complete proofs can be found. (shrink)

We introduce several restricted versions of the structural rules in the implicational fragment of Gentzen's sequent calculus LJ. For example, we permit the applications of a structural rule only if its principal formula is an implication. We investigate cut-eliminability and theorem-equivalence among various combinations of them. The results include new cut-elimination theorems for the implicational fragments of the following logics: relevant logic E, strict implication S4, and their neighbors (e.g., E-W and S4-W); BCI-logic, BCK-logic, relevant logic R, and the intuitionistic (...) logic. (shrink)

We study filters in residuated structures that are associated with congruence relations (which we call -filters), and develop a semantical theory for general substructural logics based on the notion of primeness for those filters. We first generalize Stone’s sheaf representation theorem to general substructural logics and then define the primeness of -filters as being “points” (or stalkers) of the space, the spectrum, on which the representing sheaf is defined. Prime FL-filters will turn out to coincide with truth sets under various (...) well known semantics for certain substructural logics. We also investigate which structural rules are needed to interpret each connective in terms of prime -filters in the same way as in Kripke or Routley-Meyer semantics. We may consider that the set of the structural rules that each connective needs in this sense reflects the difficulty of giving the meaning of the connective. A surprising discovery is that connectives , ⅋ of linear logic are linearly ordered in terms of the difficulty in this sense. (shrink)

Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.

We investigate the semantics of the logical systems obtained by introducing the modalities and into the family of substructural implication logics (including relevant, linear and intuitionistic implication). Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" this semantics into the classical proof system KE. This leads to the formulation of a uniform labelled refutation system for the new logics which is a natural extension of a system for substructural implication developed by the first two authors in (...) a previous paper. (shrink)

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL ew (namely, a certain variety of FL ew -algebras) are term equivalent. This answers a longstanding question of Nelson [30]. (...) Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL ew. (shrink)

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew. The main result of Part I of this series [41] shows that the equivalent variety semantics of N and the equivalent variety semantics of NFL ew are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting of deductive (...) systems to establish the definitional equivalence of the logics N and NFL ew. It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logic. (shrink)

The results of this paper extend some of the intimate relations that are known to obtain between combinatory logic and certain substructural logics to establish a general characterization theorem that applies to a very broad family of such logics. In particular, I demonstrate that, for every combinator X, if LX is the logic that results by adding the set of types assigned to X (in an appropriate type assignment system, TAS) as axioms to the basic positive relevant logic B∘T, then (...) LX is sound and complete with respect to the class of frames in the Routley-Meyer relational semantics for relevant and substructural logics that meet a first-order condition that corresponds in a very direct way to the structure of the combinator X itself. (shrink)

It is proved that MacLane's coherence results for monoidal and symmetric monoidal categories can be extended to some other categories with multiplication; namely, to relevant, affine and cartesian categories. All results are formulated in terms of natural transformations equipped with “graphs” (g-natural transformations) and corresponding morphism theorems are given as consequences. Using these results, some basic relations between the free categories of these classes are obtained.

It is proved that MacLane''s coherence results for monoidal and symmetric monoidal categories can be extended to some other categories with multiplication; namely, to relevant, affine and cartesian categories. All results are formulated in terms of natural transformations equipped with graphs (g-natural transformations) and corresponding morphism theorems are given as consequences. Using these results, some basic relations between the free categories of these classes are obtained.

Many systems of quantified modal logic cannot be characterised by Kripke's well-known possible worlds semantic analysis. This book shows how they can be characterised by a more general 'admissible semantics', using models in which there is a restriction on which sets of worlds count as propositions. This requires a new interpretation of quantifiers that takes into account the admissibility of propositions. The author sheds new light on the celebrated Barcan Formula, whose role becomes that of legitimising the Kripkean interpretation of (...) quantification. The theory is worked out for systems with quantifiers ranging over actual objects, and over all possibilia, and for logics with existence and identity predicates and definite descriptions. The final chapter develops a new admissible 'cover semantics' for propositional and quantified relevant logic, adapting ideas from the Kripke-Joyal semantics for intuitionistic logic in topos theory. This book is for mathematical or philosophical logicians, computer scientists and linguists. (shrink)

There is an intimate connection between proofs of the natural deduction systems and typed lambda calculus. It is well-known that in simply typed lambda calculus, the notion of formulae-as-types makes it possible to find fine structure of the implicational fragment of intuitionistic logic, i.e., relevant logic, BCK-logic and linear logic. In this paper, we investigate three classical substructural logics (GL, GLc, GLw) of Gentzen's sequent calculus consisting of implication and negation, which contain some of the right structural rules. In terms (...) of Parigot's -calculus with proper restrictions, we introduce a proof term assignment to these classical substructural logics. According to these notions, we can classify the -terms into four categories. It is proved that well-typed GLx--terms correspond to GLx proofs, and that a GLx--term has a principal type if stratified where x is nil, c, w or cw. Moreover, we investigate embeddings of classical substructural logics into the corresponding intuitionistic substructural logics. It is proved that the Gödel-style translations of GLx--terms are embeddings preserving substructural logics. As by-products, it is obtained that an inhabitation problem is decidable and well-typed GLx--terms are strongly normalizable. (shrink)

Substructural logics and their application to logical and semantic paradoxes have been extensively studied. In the paper, we study theories of naïve consequence and truth based on a non-reflexive logic. We start by investigating the semantics and the proof-theory of a system based on schematic rules for object-linguistic consequence. We then develop a fully compositional theory of truth and consequence in our non-reflexive framework.

Beall and Murzi :143–165, 2013) introduce an object-linguistic predicate for naïve validity, governed by intuitive principles that are inconsistent with the classical structural rules. As a consequence, they suggest that revisionary approaches to semantic paradox must be substructural. In response to Beall and Murzi, Field :1–19, 2017) has argued that naïve validity principles do not admit of a coherent reading and that, for this reason, a non-classical solution to the semantic paradoxes need not be substructural. The aim of this (...) paper is to respond to Field’s objections and to point to a coherent notion of validity which underwrites a coherent reading of Beall and Murzi’s principles: grounded validity. The notion, first introduced by Nicolai and Rossi, is a generalisation of Kripke’s notion of grounded truth, and yields an irreflexive logic. While we do not advocate the adoption of a substructural logic, we take the notion of naïve validity to be a legitimate semantic notion that points to genuine expressive limitations of fully structural revisionary approaches. (shrink)

Language in Action demonstrates the viability of mathematical research into the foundations of categorial grammar, a topic at the border between logic and linguistics. Since its initial publication it has become the classic work in the foundations of categorial grammar. A new introduction to this paperback edition updates the open research problems and records relevant results through pointers to the literature. Van Benthem presents the categorial processing of syntax and semantics as a central component in a more general dynamic logic (...) of information flow, in tune with computational developments in artificial intelligence and cognitive science. Using the paradigm of categorial grammar, he describes the substructural logics driving the dynamics of natural language syntax and semantics. This is a general type-theoretic approach that lends itself easily to proof-theoretic and semantic studies in tandem with standard logic. The emphasis is on a broad landscape of substructural categorial logics and their proof-theoretical and semantic peculiarities. This provides a systematic theory for natural language understanding, admitting of significant mathematical results. Moreover, the theory makes possible dynamic interpretations that view natural languages as programming formalisms for various cognitive activities. (shrink)

Nonclassical logics have played an increasing role in recent years in disciplines ranging from mathematics and computer science to linguistics and philosophy. _Generalized Galois Logics_ develops a uniform framework of relational semantics to mediate between logical calculi and their semantics through algebra. This volume addresses normal modal logics such as K and S5, and substructural logics, including relevance logics, linear logic, and Lambek calculi. The authors also treat less-familiar and new logical systems with equal deftness.

My model-theoretic realist account of science places linguistic systems and the corresponding non-linguistic structures at different stages of the scientific process. It is shown that science and its progress cannot be analysed in terms of only one of these strata. Philosophy of science literature offers mainly two approaches; to the structure of scientific knowledge analysed in terms of theories and their models, the "statement" and the "non-statement" approaches. In opposition to the statement approach's belief that scientific knowledge is (...) embodied in theories symbolic language) with direct interpretative links---via so-called "bridge principles"---to reality, the defenders of the non-statement approach believe in an analysis where the language in which the theory is formulated plays a much smaller role than the structures which satisfy that theory. ;The model-theoretic realism expounded here retains the notion of a scientific theory as a set of sentences, while simultaneously emphasising the interpretative role of the conceptual models of these theories. My criticism against the non-statement approach is based on the fact that merely "giving" the theory "in terms of" its mathematical structures leaves out any real interpretation of the nature and role of general terms in science. Against the statement approach's "direct" linking of general theoretical terms to reality, my approach interpolates models between theories and reality in the interpretative chain. ;The links between the general terms of scientific theories and their interpretations in the various models of the theory regulate the whole referential process. The terms of a theory are "general" in the sense that they are the result of certain abstractive conceptualisations of the object of scientific investigation and subsequent linguistic formulations of these conceptualisations. Their meanings can be "given back" only by interpreting them in the limited context, of the various conceptual models of their theory and, finally, by finding an isomorphic relation between some substructure of the conceptual model in question and some empirical conceptualisation of relevant experimental data. In this sense the notion of scientific "truth" becomes inextricably linked with that of articulated reference, as it---given its model-dependent nature---should be. (shrink)

Once the Kripke semantics for normal modal logics were introduced, a whole family of modal logics other than the Lewis systems S1 to S5 were discovered. These logics were obtained by changing the semantics in natural ways. The same can be said of the Kripke-style semantics for relevant logics: a whole range of logics other than the standard systems R, E and T were unearthed once a semantics was given (cf. Priest and Sylvan [6], Restall [7], and Routley et al. (...) [8]). In a similar way, weakening the structural rules of the Gentzen formulation of classical logic gives rise to other ‘substructural’ logics such as linear logic (as in Girard [4]). This process of ‘strategic weakening’ is becoming popular today, with the discovery of applications of these logics to areas such as linguistics and the theory of computation (cf. van Benthem [1]). Until now no-one has (to my knowledge) examined what the process of weakening does to the Kripke-style semantics of intuitionistic logic. This paper remedies the deficiency, introducing the family of subintuitionistic logics. These systems have some appealing features. Unlike other substructural logics such as linear logic (which lack distribution of extensional disjunction over conjunction) they have a very natural Kripke-style worlds semantics. Also, the difficulties with regard to modelling quantification in these systems may be able to shed some light on the difficulties in naturally modelling quantification in relevant logics, as it must be admitted that the semantics currently available for quantified relevant logics are rather baroque (cf. Fine [3]). But most importantly, delving in the undergrowth of logics such as intuitionistic logic gives us a ‘feel’ for how such systems are put together, and what job is being done by each aspect of the modelling conditions in.. (shrink)

Comics are complex multimodal documents that make for intriguing materials to analyze with computer vision and computational linguistics. This review summarizes the growing developments in computational modeling which have been progressing to analyze visual narratives across their various substructures.

We introduce a relational semantics based on poset products, and provide sufficient conditions guaranteeing its soundness and completeness for various substructural logics. We also demonstrate that our relational semantics unifies and generalizes two semantics already appearing in the literature: Aguzzoli, Bianchi, and Marra’s temporal flow semantics for Hájek’s basic logic, and Lewis-Smith, Oliva, and Robinson’s semantics for intuitionistic Łukasiewicz logic. As a consequence of our general theory, we recover the soundness and completeness results of these prior studies in a uniform (...) fashion, and extend them to infinitely-many other substructural logics. (shrink)

Algebraic work [9] shows that the deep theory of possible world semantics is available in the more general setting of substructural logics, at least in an algebraic guise. The question is whether it is also available in a relational form.This article seeks to set the stage for answering this question. Guided by the algebraic theory, but purely relationally we introduce a new type of frames. These structures generalize Kripke structures but are two-sorted, containing both worlds and co-worlds. These latter points (...) may be viewed as modelling irreducible increases in information where worlds model irreducible decreases in information. Based on these structures, a purely model theoretic and uniform account of completeness for the implication-fusion fragment of various substructural logics is given. Completeness is obtained via a generalization of the standard canonical model construction in combination with correspondence results. (shrink)

A graph-theoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as a multi-graph (m-graph) where the nodes and the m-edges include the sorts and the constructors of the signatures at hand. Fibring of two models is a multi-graph (m-graph) where the nodes and the m-edges are the values and the operations in the models, respectively. Fibring of two deductive systems is (...) an m-graph whose nodes are language expressions and the m-edges represent the inference rules of the two original systems. The sobriety of the approach is confirmed by proving that all the fibring notions are universal constructions. This graph-theoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with non-deterministic semantics, logics with an algebraic semantics, logics with partial semantics and substructural logics, among others. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems can be avoided. (shrink)

Head-driven phrase structure grammar (HPSG) is one of the most prominent theories employed in deep parsing of natural language. Many linguistic theories are arguably best formalized in extensions of modal or dynamic logic (Keller, Feature logics, infinitary descriptions and grammar, 1993; Kracht, Linguistics Philos 18:401–458, 1995; Moss and Tiede, In: Blackburn, van Benthem, and Wolther (eds.) Handbook of modal logic, 2006), and HPSG seems to be no exception. Adequate extensions of dynamic logic have not been studied in detail, however; (...) the most important aspect is the reference to sets of substructures. In this paper, an adequate extension is identified, and some important results are established: Satisfiability is highly undecidable, and model checking is shown to be in EXPTIME and PSPACE-hard. A fragment with polynomial time model checking procedures is identified; it is shown to cover considerable fragments of HPSG. (shrink)

This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these asvariant semanticsand present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we demonstrate that (...) failure via specific finite examples. Using these, we prove that the syntactic version of these translations will fail to satisfy Troelstra’s criteria in the corresponding substructural logical setting. (shrink)

In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the lattice-theoretical point of view; this set is represented by the lattice FACSs A of all algebraic closed-set systems C on A such that (A, C) ε FGModS. We relate some properties of these structures (...) with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGModS under substructures and under reduced products, and the property that for any A the lattice FACSs A is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to hold only for deductive systems in more restricted classes like the protoalgebraic or the weakly algebraizable ones. The paper ends with a section on examples and counterexamples. (shrink)

A standard formalization of a scientific theory is a system of axioms for that theory in a first-order language (possibly many-sorted; possibly with the membership primitive $$\in$$ ). Suppes (in: Carvallo M (ed) Nature, cognition and system II. Kluwer, Dordrecht, 1992) expressed skepticism about whether there is a “simple or elegant method” for presenting mathematicized scientific theories in such a standard formalization, because they “assume a great deal of mathematics as part of their substructure”. The major difficulties amount to these. (...) First, as the theories of interest are mathematicized, one must specify the underlying applied mathematics base theory, which the physical axioms live on top of. Second, such theories are typically geometric, concerning quantities or trajectories in space/time: so, one must specify the underlying physical geometry. Third, the differential equations involved generally refer to coordinate representations of these physical quantities with respect to some implicit coordinate chart, not to the original quantities. These issues may be resolved. Once this is done, constructing standard formalizations is not so difficult—at least for the theories where the mathematics has been worked out rigorously. Here we give what may be claimed to be a simple and elegant means of doing that. This is for mathematicized scientific theories comprising differential equations for $$\mathbb{R}$$ -valued quantities Q (that is, scalar fields), defined on n (“spatial” or “temporal”) dimensions, taken to be isomorphic to the usual Euclidean space $$\mathbb{R}^n$$. For illustration, I give standard (in a sense, “text-book”) formalizations: for the simple harmonic oscillator equation in one-dimension and for the Laplace equation in two dimensions. (shrink)

The basic premise of the Parallel Architecture (Jackendoff 1997, 2002) is that phonology, syntax, and semantics are independent generative components in language, each with its own primitives and principles of combination. The theory builds on insights about linguistic structure that emerged in the 1970s. First, phonology was demonstrated to have highly articulated structure that cannot be derived directly from syntax: structured units such as syllables and prosodic constituents do not correspond one-to-one with syntactic units. Moreover, phonological structure includes several (...) independent substructures or tiers, each with its own type of generative structure: segmental-syllabic structure, the metrical grid, intonation contour, and (in tone languages) the tone tier. The tiers are correlated with each other by interface rules: principles that establish optimal correspondence between structures of two independent types. Such rules are not derivational. Since these phonological structures cannot be derived from syntactic structures, the connection between syntax and phonology must also be mediated not by derivations, but by a component of interface rules. (shrink)

This paper investigates whether the fundamental linguistic insights and intuitions of Lexical Functional Grammar, which is usually presented as a constraint-based linguistic theory, can be reformulated in a resource sensitive framework using a substructural modal logic. In the approach investigated here, LFG's f-descriptions are replaced with expressions from a multi-modal propositional logic. In effect, the feature structure unification basis of LFG's f-structures is replaced with a very different resource based mechanism. It turns out that some linguistic analyses (...) that required non-monotonic devices in LFG analysis of Icelandic) can be straightforwardly expressed in the framework presented here. Moreover, a Curry– Howard correspondence between proofs in this logic and -terms provides a semantic interpretation as a by-product of the process of showing syntactic well-formedness. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Gerald Prince WORLDS WITH STYLE Whether it is taken to be a laudable characteristic of verbal artifacts (as in, "This essay is really well written"), a distinctive feature of an individual manner of speaking or writing (as in, "Jane definitely has a style of her own"), an ornamental supplement to that which is expressed (style as elocutio), or an appropriate way of using language in different contexts (there is (...) — or was — a style for tragedy, another one for comedy, and still another one for pastoral; there is a style for speaking at business meetings and other styles better suited to everyday conversations, political rallies, or Ph.D. exams), whether it is taken to be any of these four things (or many other things as well), the style of an utterance is traditionally viewed as a secondary phenomenon different from and added to some primary base. It is the mode of expressing a given matter, the way of "saying" some thing, the how of a what. The same is true when the notion style is used in connection with nonverbal artifacts — we talk about musical styles and about pictorial ones — but also in connection with any number of activities or states of affairs: walking down the street, dancing the tango, being pleasant, or playing ping-pong. Ping-pong players use rackets, a net, and a ball, in accordance with certain rules, when they engage in a casual game; but they are thought to be doing somediing more than merely playing: they are performing an activity (the what) in a particular way. They do it stylishly, idiosyncratically, with a lot of showboating, or in a manner appropriate to the informal context.1 It is somewhat surprising, therefore, that, when it comes to verbal (and nonverbal) representations, style is often thought to be irrelevant to the signified (what is represented) as opposed to the signifier (what represents). Graham Hough writes, for example: "In discussing whatever it is we mean by style we assume mat the primary choice, the choice of subject-matter in the large sense, has already been made.... We can talk about the secondary choice, the stylistic one, in various ways.... But whatever point of view we adopt, it will be the verbal ordonnance that we discuss, not the oudines of die mym, the facts of the case, the ideological or biographical substructure."2 Yet, as I have more than suggested, any what has (or can have) a corresponding how. Besides, both 59 60Philosophy and Literature in formal criticism and informal exchanges, we frequently seem to refer to the how of a signified. When we say, for instance, that a given short story is Kafkaesque, we do not, I think, have its verbal style in mind; and when we claim that a novelistic episode or character comes right out of Alice in Wonderland, we are not talking about linguistic matters. Of course, the fact mat we seem to refer to the how of signifieds (or that we do refer to the how of signifiers) does not necessarily mean mat mere are such things (or that we can describe them). But I suspect that there are (and that we can). I would like to discuss in what follows some of the ways in which we might be able to talk of the style of represented "worlds" more or less meaningfully and systematically and some of the problems attending such talk. For the sake of convenience and because of personal affinities, I will restrict myself to verbal narratives. I will not make a distinction between literary and non-literary ones, since such a distinction is formally and even pragmatically quite problematic. IfI concentrate on fictional narratives, it is, again, because of personal affinities and not because I mink that, when it comes to style, (ontological or other) differences between fiction and nonfiction make very much of a difference. As I will argue, style is in the eye ofthe beholder and, from that point of view, whether or not something is compossible with one's knowledge of the "real world" does not necessarily make it function as fictional or nonfictional; indeed, I am constandy surprised by the real world and I could easily recount... (shrink)

Lambek elegantly characterized part of natural language. As is well-known, his substructural logic L, and its non-associative version NL, handle basic function/argument composition well, but not scope taking and syntactic displacement—at least, not in their full generality. In previous work, I propose $$\text {NL}_\lambda $$, which is NL supplemented with a single structural inference rule (“abstraction”). Abstraction closely resembles the traditional linguistic rule of quantifier raising, and characterizes both semantic scope taking and syntactic displacement. Due to the unconventional form (...) of the abstraction inference, there has been some doubt that $$\text {NL}_\lambda $$ should count at a legitimate substructural logic. This paper argues that $$\text {NL}_\lambda $$ is perfectly well-behaved. In particular, it enjoys cut elimination and an interpolation result. In addition, perhaps surprisingly, it is decidable. Finally, I prove that it is sound and complete with respect to the usual class of relational frames. (shrink)

Lambek elegantly characterized part of natural language. As is well-known, his substructural logic L, and its non-associative version NL, handle basic function/argument composition well, but not scope taking and syntactic displacement—at least, not in their full generality. In previous work, I propose \, which is NL supplemented with a single structural inference rule.ion closely resembles the traditional linguistic rule of quantifier raising, and characterizes both semantic scope taking and syntactic displacement. Due to the unconventional form of the abstraction inference, (...) there has been some doubt that \ should count at a legitimate substructural logic. This paper argues that \ is perfectly well-behaved. In particular, it enjoys cut elimination and an interpolation result. In addition, perhaps surprisingly, it is decidable. Finally, I prove that it is sound and complete with respect to the usual class of relational frames. (shrink)

Given a positive universal formula in the language of residuated lattices, we construct a recursive basis of equations for a variety, such that a subdirectly irreducible residuated lattice is in the variety exactly when it satisfies the positive universal formula. We use this correspondence to prove, among other things, that the join of two finitely based varieties of commutative residuated lattices is also finitely based. This implies that the intersection of two finitely axiomatized substructural logics over FL + is also (...) finitely axiomatized. Finally, we give examples of cases where the join of two varieties is their Cartesian product. (shrink)

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follow, including various types of completeness theorems of substructural logics.

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follow, including various types of completeness theorems of substructural logics.

Combinatory logic is known to be related to substructural logics. Algebraic considerations of the latter, in particular, algebraic considerations of two distinct implications, led to the introduction of dual combinators in Dunn & Meyer 1997. Dual combinators are "mirror images" of the usual combinators and as such do not constitute an interesting subject of investigation by themselves. However, when combined with the usual combinators, the whole system exhibits new features. A dual combinatory system with weak equality typically lacks the Church-Rosser (...) property, and in general it is inconsistent. In many subsystems terms "unexpectedly" turn out to be weakly equivalent. The paper is a preliminary attempt to investigate some of these issues, as well as, briefly compare function application in symmetric -calculus and dual combinatory logic. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Myth, Truth and Literature: Towards a True Post-ModernismJames SeatonMyth, Truth and Literature: Towards a True Post-Modernism, by Colin Falck; xix & 208 pp. Cambridge: Cambridge University Press, 2nd edition, 1994, $59.95 cloth, $16.95 paper.Colin Falck has written a book that seeks to bind a critique of postmodernism to a plan for salvaging what is best about it. He wants to devise “a true post-modernism,” because until now the (...) movement’s goal has been “the Abolition of Reality.”The critique is the stronger part of the book. Falck offers several reasons why the currently fashionable versions of postmodernism are unsatisfactory. Identifying it with “post-Saussurian theory” (p. xiii), Falck traces the fundamental mistake of postmodernism back to Saussure’s rejection of one linguistic theory in favor of another, sliding from the view that language is based on “the baptismal labeling of some kind of already-differentiated and pre-baptismally identifiable item of the world’s contents” (p. 22) to seeing it “as a disembodied and contextless process” (p. 11). To Falck, though, the notion that a rejection of the first requires an acceptance of the second seems “entirely fallacious.” And yet “the whole internally interdependent substructure of post-Saussurian literary theory” depends upon the unexamined assumption that the two are inextricable.Falck finds other instances in which poststructuralists leap from true but trivial premises to otherwise unsupported conclusions. Jacques Derrida, for example, moves from an uncontroversial rejection of theories about “fully-experienced ‘presence’” (p. 21) to the very different issue of the priority of speech over writing. The goal of such non sequiturs is “in effect the Abolition of Reality” (p. 31).Such abolitions are always subject to the refutation Samuel Johnson made to Berkeley’s idealism. Kicking a stone, he vindicated the materiality of the external world. Now any textbook will point out that Johnson’s gesture was not an argument, but Falck argues that gestures do indeed possess philosophical significance. Speech is prior to writing because it is in speech and the gestures that accompany it that the location of language within “our necessary embodiedness” (p. 115) is most evident. Exploring the implications of the thesis that “all knowledge must be based in bodily awareness” (p. 119) along lines suggested by Merleau-Ponty among others, Falck argues that any satisfying account of language and, especially, literature must take account of dimensions of experience that contemporary theory peremptorily dismisses with the slogan “Nothing outside the text!”Although Falck’s refutation of postmodernism is telling and effective, it parallels the critique offered by other critics, such as John Ellis in Against Deconstruction. The true originality of his book lies in its admirably straightforward call to take up again the project of English and German Romanticism. For [End Page 264] Falck “the essential function of the literary text is one of revelation or disclosure” (p. 90). Myth, Truth and Literature argues that literature can reveal truths otherwise inexpressible through the use of myth. The romantic revival that he calls for will, however, also be a true postmodernism, because it will share the postmodernist rejection of transcendence in favor of “the essentially Romantic, but philosophically profound, recognition of the necessarily embodied and located nature of all human experiencing” (p. 84). Considering Romanticism “the third great advent of spirituality to the Western world” (p. 140) after Hellenic paganism and Christianity, Falck looks forward to a postmodernist literature that will reaffirm the truths of paganism while avoiding the “imaginative exclusiveness” of Christianity.One may share Falck’s refusal to accept postmodernism’s Abolition of Reality and yet have serious doubts about his proposed alternative. A reconsideration of the theories of Friedrich Schlegel and Samuel Taylor Coleridge offers more hope for literary criticism than an acceptance of the dominance of postmodernism and cultural studies, but it is not clear that Falck’s version of romanticism avoids the pitfalls of the contemporary approaches he rightly criticizes.He may not attempt to abolish reality, but Falck does seem to neglect the lessons of contemporary history. He opens his book by asserting that “romanticism looks forward to Marxism, psychoanalysis, and to every significant modern attempt to persuade men to take... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS 151 nuanced and cannot adequately be discussed in this short note. But we can say that Haar repreatedly comes back to phrases such as "a latent sketchof artistic configurations " (196), and a "secret outline of forms" (216) when describing the earth (both in the artwork and the world of artistic existence) as the origin and substructure of human, linguistic existence. Though Haar finds ample support (...) in Heidegger's writings, one is nevertheless led to wonder: What is the ontological status of this latent articulation immanent in earth? In what sense can earth be transepochal, ahistorical, if it contains such articulations to which the poet, artist, and thinker respond? How is the ahistorical/prelinguistic song from the earth to be compared to the historical/linguistic song about the earth? One must finally ask if the graceful and thoughtful prose Haar has given us is, as he hopes, a resonance of the former, or a spontaneous production of the latter. The success of Haar's insightful exposition of the ahistorical substructure of existence stands and falls on the answer to these questions. REGINALD LILLY Universityof New Hampshire Richard McKeon. Rhetoric: Essays in Invention and Ducovery. Edited with an Introduction by Mark Backman. Woodbridge, CT: Ox Bow Press, 1987. Pp. xxxii + 22o. $25.00. 'History of philosophy' is an ambiguous term. It has been applied to doxography, a history of doctrines apart from their systematic relations or context. It has been applied to rational reconstructions of what a thinker must have meant apart from whether the thinker ever said it and from the context that provides checks on possible interpretations, rational or not. And it has been applied to contextual interpretations, which set philosophers and philosophies in the context of the language, history, and culture that condition them. Richard McKeon practiced the history of philosophy (or of thought, a phrase he sometimes prefers) in a different way: in terms of the subject matters thinkers treat and the functions they try to perform under the rubric of philosophy (rhetoric, poetry, grammar, and so on). Thus it is a history of changing definitions and differing conceptions, and of the spread of intellectual devices and ideas from realm to realm and from age to age. McKeon's kind of history is based on the observation that the terms we use--the names of our arts and sciences and the concepts employed in them--are stable but ambiguous. Thus it might be called a semantic history. It is also based on the insight, explained in the concluding essay, "A Philosopher Meditates on Discovery," that the truth is one, though it can be expressed in many different ways. And his history is consciously intended to provide both theoretical clarifications and insights and practical guidance in the resolution of contemporary problems. McKeon's work has not been widely influential in philosophy, although he is considered a major figure in the recent "renaissance of rhetoric" to which this volume primarily directs itself. However, despite the title, the selection of articles reprinted, and the 152 JOURNAL OF THE HISTORY OF PHILOSOPHY 28: I JANUARY ~99 o editor's emphasis on the term 'rhetoric' rather than 'philosophy', these essays can well be read--in roughly the reverse of their present order--as the discovery, illustration, and application of what McKeon calls his "new philosophy" (120). It is a philosophy built on the notions of"historical semantics" and "philosophic semantics" (218-20) and aiming to reunite philosophy and rhetoric, wisdom and eloquence, theory and practice. In the concluding essay, he describes his own intellectual development and the complex matrix of subject matters, methods, principles, and aims that can be used to characterize philosophic positions unambiguously despite their ambiguous use of common terms.' The system of historical semantics is illustrated by the two preceding essays. "Rhetoric in the Middle Ages," a tour de force of medieval scholarship and famous among medievalists and historians of rhetoric, shows how rhetoric affected not only rhetoricians, but also philosophers, theologians, and logicians from Augustine into the Renaissance. "Poetry and Philosophy in the Twelfth Century" presents the varying conceptions of the two disciplines and the oscillation in their relations from identity to antithesis. In "Symbols, Myths, and... (shrink)