The paper deals with the problem of logical adequacy of language knowledge with cognition of reality. A logical explication of the concept of language knowledge conceived of as a kind of codified knowledge is taken into account in the paper. Formal considerations regarding the notions of meta-knowledge (logical knowledge about language knowledge) and truth are developed in the spirit of some ideas presented in the author’s earlier papers (1991, 1998, 2001a,b, 2007a,b,c) treating about the notions of meaning, denotation and truthfulness (...) of well-formed expressions (wfes) of any given categorial language. Three aspects connected with knowledge codified in language are considered, including: 1) syntax and two kinds of semantics: intensional and extensional, 2) three kinds of non-standard language models and 3) three notions of truthfulness of wfes. Adequacy of language knowledge to cognitive objects is understood as an agreement of truthfulness of sentences in these three models. (shrink)

This paper uses classical logic for a simultaneous description of the syntax and semantics of a fragment of English and it is argued that such an approach to natural language allows procedural aspects of linguistic theory to get a purely declarative formulation. In particular, it will be shown how certain construction rules in Discourse Representation Theory, such as the rule that indefinites create new discourse referents and definites pick up an existing referent, can be formulated declaratively if logic is used (...) as a metalanguage for English. In this case the declarative aspects of a rule are highlighted when we focus on the model theory of the description language while a procedural perspective is obtained when its proof theory is concentrated on. Themes of interest are Discourse Representation Theory, resolution of anaphora, resolution of presuppositions, and underspecification. (shrink)

We study mathematical and algorithmic properties of Lambek's pregroups and illustrate them by the French noun phrase. An algorithm of complexity n3 to solve the reduction problem in an arbitrary free pregroup as well as recognition by a pregroup grammar is presented. This algorithm is then specified to run in linear time. A sufficient condition for a language fragment that makes the linear algorithm complete is given.

The relationship between Lexical-Functional Grammar (LFG) functional structures (f-structures) for sentences and their semanticinterpretations can be formalized in linear logic in a way thatcorrectly explains the observed interactions between quantifier scopeambiguity, bound anaphora and intensionality.Our linear-logic formalization of the compositional properties ofquantifying expressions in natural language obviates the need forspecial mechanisms, such as Cooper storage, in representing thescoping possibilities of quantifying expressions. Instead, thesemantic contribution of a quantifier is recorded as a linear-logicformula whose use in a proof will establish the (...) scope of thequantifier. Different proofs can lead to different scopes. In eachcomplete proof, the properties of linear logic ensure thatquantifiers are properly scoped. (shrink)

The Distributional Compositional Categorical model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that (...) could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, something which the other methods lack, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambekʼs pregroup grammar. Pregroup types form a posetal compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambekʼs previous type-logic, the Lambek calculus. The latter and its extensions have been extensively used to formalise and reason about various linguistic phenomena. Hence, the apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings, and also reflects the structure of the parse trees of the Lambek calculus. (shrink)

We discuss the logic of pregroups, introduced by Lambek [34], and its connections with other type logics and formal grammars. The paper contains some new ideas and results: the cut-elimination theorem and a normalization theorem for an extended system of this logic, its P-TIME decidability, its interpretation in L1, and a general construction of (preordered) bilinear algebras and pregroups whose universe is an arbitrary monoid.

We prove the finite model property (fmp) for BCI and BCI with additive conjunction, which answers some open questions in Meyer and Ono [11]. We also obtain similar results for some restricted versions of these systems in the style of the Lambek calculus [10, 3]. The key tool is the method of barriers which was earlier introduced by the author to prove fmp for the product-free Lambek calculus [2] and the commutative product-free Lambek calculus [4].

Involutive Nonassociative Lambek Calculus is a nonassociative version of Noncommutative Multiplicative Linear Logic, but the multiplicative constants are not admitted. InNL adds two linear negations to Nonassociative Lambek Calculus ; it is a strongly conservative extension of NL Logical aspects of computational linguistics. LNCS, vol 10054. Springer, Berlin, pp 68–84, 2016). Here we also add unary modalities satisfying the residuation law and De Morgan laws. For the resulting logic InNLm, we define and study phase spaces. We use them to prove (...) the cut elimination theorem for a one-sided sequent system for InNLm, introduced here. Phase spaces are also employed in studying auxiliary systems InNLm, assuming the k-cyclic law for negation. The latter behave similarly as Classical Nonassociative Lambek Calculus, studied in de Groote and Lamarche :355–388, 2002) and Buszkowski. We reduce the provability in InNLm to that in InNLm. This yields the equivalence of type grammars based on InNLm with -free) context-free grammars and the PTIME complexity of InNLm. (shrink)

Action logic of Pratt [21] can be presented as Full Lambek Calculus FL [14, 17] enriched with Kleene star *; it is equivalent to the equational theory of residuated Kleene algebras (lattices). Some results on axiom systems, complexity and models of this logic were obtained in [4, 3, 18]. Here we prove a stronger form of *-elimination for the logic of *-continuous action lattices and the –completeness of the equational theories of action lattices of subsets of a finite monoid and (...) action lattices of binary relations on a finite universe. We also discuss possible applications in linguistics. (shrink)

Pentus (1992) proves the equivalence of LCG's and CFG's, and CFG's are equivalent to BCG's by the Gaifman theorem (Bar-Hillel et al., 1960). This paper provides a procedure to extend any LCG to an equivalent BCG by affixing new types to the lexicon; a procedure of that kind was proposed as early, as Cohen (1967), but it was deficient (Buszkowski, 1985). We use a modification of Pentus' proof and a new proof of the Gaifman theorem on the basis of the (...) Lambek calculus. (shrink)

The Lambek calculus introduced in Lambek [6] is a strengthening of the type reduction calculus of Ajdukiewicz [1]. We study Associative Lambek Calculus L in Gentzen style axiomatization enriched with a finite set Γ of nonlogical axioms, denoted by L(Γ).It is known that finite axiomatic extensions of Associative Lambek Calculus generate all recursively enumerable languages (see Buszkowski [2]). Then we confine nonlogical axioms to sequents of the form p → q, where p and q are atomic types. For calculus L(Γ) (...) we prove interpolation lemma (modifying the Roorda proof for L [10]) and the binary reduction lemma (using the Pentus method [9] with modification from [3]). In consequence we obtain the weak equivalence of the Context-Free Grammars and grammars based on L(Γ). (shrink)

Nonassociative Lambek Calculus (NL) is a syntactic calculus of types introduced by Lambek [8]. The polynomial time decidability of NL was established by de Groote and Lamarche [4]. Buszkowski [3] showed that systems of NL with finitely many assumptions are decidable in polynomial time and generate context-free languages; actually the P-TIME complexity is established for the consequence relation of NL. Adapting the method of Buszkowski [3] we prove an analogous result for Nonassociative Lambek Calculus with unit (NL1). Moreover, we show (...) that any Lambek grammar based on NL1 (with assumptions) can be transformed into an equivalent context-free grammar in polynomial time. (shrink)

This paper uses a partially ordered set of syntactic categories to accommodate optionality and licensing in natural language syntax. A complex but well-studied data set pertaining to the syntax of quantifier scope and negative polarity licensing in Hungarian is used to illustrate the proposal. The presentation is geared towards both linguists and logicians. The paper highlights that the main ideas can be implemented in different grammar formalisms, and discusses in detail an implementation where the partial ordering on categories is given (...) by the derivability relation of a calculus with residuated and Galois-connected unary operators. (shrink)

A number of general points behind the story of this paper may be worth setting out separately, now that we have come to the end.There is perhaps one obvious omission to be addressed right away. Although the word “information” has occurred throughout this paper, it must have struck the reader that we have had nothing to say on what information is. In this respect, our theories may be like those in physics: which do not explain what “energy” is (a notion (...) which seems quite similar to “information” in several ways), but only give some basic laws about its behaviour and transmission.The eventual recommendation made here has been to use a broad type-theoretic framework for studying various more classical and more dynamic notions of proposition in their interaction. This is not quite the viewpoint advocated by many current authors in the area, who argue for a whole-sale switch from a ‘static’ to a ‘dynamic’ perspective on propositions. This is not the place, however, to survey the conceptual arguments for and against such a more radical move.This still leaves many questions about possible reductions from one perspective to another. For instance, it would seem that classical systems ought to serve as a ‘limiting case’, which should still be valid after procedural details of some cognitive process have been forgotten. There are various ways of implementing the desired correspondence: e.g. by considering extreme cases with ⫅ equal to identity, or, in the pure relational algebra framework by considering only pairs (x, x). What we shall want then are reductions of dynamic logics, in those special cases, to classical logic. But perhaps also, more sophisticated views are possible. How do we take a piece of ‘dynamic’ prose, remove control instructions and the like, and obtain a piece of ‘classical’ text, suitable for inference ‘in the light of eternity’?There is also a more technical side to the matter of ‘reduction’. By now, Logic has reached such a state of ‘inter-translatability’ that almost all known variant logics can be embedded into each other, via suitable translations. In particular, once an adequate semantic has been given for a new system, this usually induces an embedding into standard logic: as we know, e.g., for the case of Modal Logic. Like-wise, all systems of dynamic interpretation or inference proposed so far admit of direct embedding into an ordinary ‘static’ predicate logic having explicit transition predicates (cf. van Benthem 1988b). Thus, our moral is this. The issue is not whether the new systems of information structure or processing are essentially beyond the expressive resources of traditional logical systems: for, they are not. The issue is rather which interesting phenomena and questions will be put into the right focus by them.The next broad issue concerns the specific use of the perspective proposed here, vis-à-vis concrete proposals for information-oriented or dynamic semantics. The general strategy advocated here is to locate some suitable base calculus and then consider which ‘extras’ are peculiar to the proposal. For instance, this is the spirit in which modal S4 would be a base logic of information models, and intuitionistic logicthe special theory devoting itself to upward persistent propositions. Or, with the examples in Section 4.1, the underlying base logic is our relational algebra, whereas, say, ordinary updates then impose special properties, such as ‘idempotence’: $$xRy \Rightarrow yRy$$ Does this kind of application presuppose the existence of one distinguished base logic, of which all others are extensions? This would be attractive-and some form of relational algebra or linear logic might be a reasonable candidate. Nevertheless, the enterprise does not rest on this outcome. What matters is an increased sensitivity to the ‘landscape’ of dynamic logics, just as with the ‘Categorial Hierarchy’ in Categorial Grammar (cf. van Benthem 1989a, 1991) where the family of logics with their interconnections seems more important than any specific patriarch.Finally, perhaps the most important issue in the new framework is the possibility of new kinds of questions arising precisely because of its differences from standard logic. Notably, given the option of regarding propositions as programs, it will be of interest to consider systematically which major questions about programming languages now make sense inside logic too.EXAMPLE. Correctness. When do we have $$\left[\kern-0.15em\left[ \pi \right]\kern-0.15em\right](\left[\kern-0.15em\left[ A \right]\kern-0.15em\right]) \subseteq \left[\kern-0.15em\left[ B \right]\kern-0.15em\right]$$ for (s, t) propositions A, B and a dynamic (s, (s, t)) proposition π?Program Synthesis. Which dynamic proposition will take us from an information state satisfying A to one satisfying B? (This question needs refinement, lest there be trivial answers.)Determinism. Which propositions as programs are deterministic, in the sense of defining single-valued functions from states to states?Querying. What does it mean to ask for information in the present setting? (Again, individual types referring to e will be crucial here.)This is not merely an agenda for wishful thinking. Within Logic, there are various ways of introducing such concerns into semantics, especially, using tools from Automata Theory. (See van Benthem 1989c for further discussion of such computational perspectives in ‘cognitive programming’.)At least if one believes that ‘dynamics’ is of the essence in cognition (rather than a mere interfacing problem between the halls of eternal truth and the noisy streets of reality), the true test for the present enterprise is the development of a significant new research program not merely copying the questions of old. (shrink)

The paper presents a way to transform pregroup grammars into contextfree grammars using functional composition. The same technique can also be used for the proof-nets of multiplicative cyclic linear logic and for Lambek calculus allowing empty premises.

Using the programming-language concept of continuations, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of in-situ quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as continuations are the tool of choice for reasoning about evaluation order and side effects in programming languages, our system provides a principled, type-logical way to model evaluation order and side effects in natural language. We illustrate with an (...) improved account of quantificational binding, weak crossover, wh-questions, superiority, and polarity licensing. (shrink)

I propose the first strictly compositional semantic account of same. New data, including especially NP-internal uses such as two men with the same name, suggests that same in its basic use is a quantificational element taking scope over nominals. Given type-lifting as a generally available mechanism, I show that this follows naturally from the fact that same is an adjective. Independently-motivated assumptions extend the analysis to standard examples such as Anna and Bill read the same book via a mechanism I (...) call PARASITIC SCOPE, in which the scope of same depends on the scope of some other scopetaking element in the sentence. Although I will initially discuss the analysis in terms of a familiar Quantifier Raising framework, I go on to implement the analysis within an innovative continuation-based Type-Logical Grammar. The empirical payoff for dealing in continuations is that a simple generalization of the basic analysis gives the first ever formal account of cases in which same distributes over objects other than NP denotations, as in the relevant interpretation of John hit and killed the same man. (shrink)

This work presents a computational interpretation of the construction process for cyclic linear logic and non-commutative logic sequential proofs. We assume a proof construction paradigm, based on a normalisation procedure known as focussing, which efficiently manages the non-determinism of the construction. Similarly to the commutative case, a new formulation of focussing for NL is used to introduce a general constraint-based technique in order to dealwith partial information during proof construction. In particular, the procedure develops through construction steps propagating constraints in (...) intermediate objects called abstract proofs. (shrink)

The problem of whether Lambek Calculus is complete with respect to (w.r.t.) relational semantics, has been raised several times, cf. van Benthem (1989a) and van Benthem (1991). In this paper, we show that the answer is in the affirmative. More precisely, we will prove that that version of the Lambek Calculus which does not use the empty sequence is strongly complete w.r.t. those relational Kripke-models where the set of possible worlds,W, is a transitive binary relation, while that version of the (...) Lambek Calculus where we admit the empty sequence as the antecedent of a sequent is strongly complete w.r.t. those relational models whereW=U×U for some setU. We will also look into extendability of this completeness result to various fragments of Girard's Linear Logic as suggested in van Benthem (1991), p. 235, and investigate the connection between the Lambek Calculus and language models. (shrink)

We define proof nets for cyclic multiplicative linear logic as edge bi-coloured graphs. Our characterization is purely graph theoretical and works without further complication for proof nets with cuts, which are usually harder to handle in the non-commutative case. This also provides a new characterization of the proof nets for the Lambek calculus (with the empty sequence) which simply are a restriction on the formulas to be considered (which are asked to be intuitionistic).

We present a geometrical analysis of the principles that lay at the basis of Categorial Grammar and of the Lambek Calculus. In Abrusci it is shown that the basic properties known as Residuation laws can be characterized in the framework of Cyclic Multiplicative Linear Logic, a purely non-commutative fragment of Linear Logic. We present a summary of this result and, pursuing this line of investigation, we analyze a well-known set of categorial grammar laws: Monotonicity, Application, Expansion, Type-raising, Composition, Geach laws (...) and Switching laws. (shrink)

This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. -/- The chapters of which this book is composed discuss the (...) subject from a rich variety of angles, including the history of logic, the proper interpretation of logical validity, natural deduction rules, the notions of harmony and of synonymy, the structure of proofs, the logical status of equality, intentional phenomena, and the proof theory of second-order arithmetic. All chapters relate directly to questions that have driven Schroeder-Heister's own research agenda and to which he has made seminal contributions. The extensive autobiographical chapter not only provides a fascinating overview of Schroeder-Heister's career and the evolution of his academic interests but also constitutes a contribution to the recent history of logic in its own right, painting an intriguing picture of the philosophical, logical, and mathematical institutional landscape in Germany and elsewhere since the early 1970s. The papers collected in this book are illuminatingly put into a unified perspective by Schroeder-Heister's comments at the end of the book. Both graduate students and established researchers in the field will find this book an excellent resource for future work in proof-theoretic semantics and related areas. (shrink)

Prioritized bases, i.e., weakly ordered sets of sentences, have been used for specifying an agent’s ‘basic’ or ‘explicit’ beliefs, or alternatively for compactly encoding an agent’s belief state without the claim that the elements of a base are in any sense basic. This paper focuses on the second interpretation and shows how a shifting of priorities in prioritized bases can be used for a simple, constructive and intuitive way of representing a large variety of methods for the change of belief (...) states—methods that have usually been characterized semantically by a system-of-spheres modeling. Among the methods represented are ‘radical’, ‘conservative’ and ‘moderate’ revision, ‘revision by comparison’ in its raising and lowering variants, as well as various constructions for belief expansion and contraction. Importantly, none of these methods makes any use of numbers. (shrink)

This book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones using the latter. Covering modal logics, many-valued logics, superintuitionistic and substructural logics, together with their algebraic semantics, the book also provides an introduction to nonclassical logic for undergraduate (...) or graduate level courses.The book is divided into two parts: Proof Theory in Part I and Algebra in Logic in Part II. Part I presents sequent systems and discusses cut elimination and its applications in detail. It also provides simplified proof of cut elimination, making the topic more accessible. The last chapter of Part I is devoted to clarification of the classes of logics that are discussed in the second part. Part II focuses on algebraic semantics for these logics. At the same time, it is a gentle introduction to the basics of algebraic logic and universal algebra with many examples of their applications in logic. Part II can be read independently of Part I, with only minimum knowledge required, and as such is suitable as a textbook for short introductory courses on algebra in logic. (shrink)

In this note, we review paradoxes like Russell’s, the Liar, and Curry’s in the context of intuitionistic logic. One may observe that one cannot blame the underlying logic for the paradoxes, but has to take into account the particular concept formations. For proof-theoretic semantics, however, this comes with the challenge to block some forms of direct axiomatizations of the Liar. A proper answer to this challenge might be given by Schroeder-Heister’s definitional freedom.

The paper introduces a new type of rules into Natural Deduction, elimination rules by composition. Elimination rules by composition replace usual elimination rules in the style of disjunction elimination and give a more direct treatment of additive disjunction, multiplicative conjunction, existence quantifier and possibility modality. Elimination rules by composition have an enormous impact on proof-structures of deductions: they do not produce segments, deduction trees remain binary branching, there is no vacuous discharge, there is only few need of permutations. This new (...) type of rules fits especially to substructural issues, so it is shown for Lambek Calculus, i.e. intuitionistic non-commutative linear logic and to its extensions by structural rules like permutation, weakening and contraction. Natural deduction formulated with elimination rules by composition from a complexity perspective is superior to other calculi. (shrink)

A formulation of Full Lambek Calculus in the framework of natural deduction is given.

The calculus DNL results from the non-associative Lambek calculus NL by splitting the product functor into the right (⊲) and left (⊳) product interacting respectively with the right (/) and left () residuation. Unlike NL, sequent antecedents in the Gentzen-style axiomatics of DNL are not phrase structures (i.e., bracketed strings) but functor-argument structures. DNL − is a weaker variant of DNL restricted to fa-structures of order ≤ 1. When axiomatized by means of introduction/elimination rules for / and , it shows (...) a perfect analogy to NL which DNL lacks. (shrink)

The paper continues a series of results on cut-rule axiomatizability of the Lambek calculus. It provides a complete solution of a problem which was solved partially in one of the author''s earlier papers. It is proved that the product-free Lambek Calculus with the empty string (L 0) is not finitely axiomatizable if the only rule of inference admitted is Lambek''s cut rule. The proof makes use of the (infinitely) cut-rule axiomatized calculus C designed by the author exactly for this purpose.

An axiomatics of the product-free syntactic calculus L ofLambek has been presented whose only rule is the cut rule. It was alsoproved that there is no finite axiomatics of that kind. The proofs weresubsequently simplified. Analogous results for the nonassociativevariant NL of L were obtained by Kandulski. InLambek's original version of the calculus, sequent antecedents arerequired to be nonempty. By removing this restriction, we obtain theextensions L 0 and NL 0 ofL and NL, respectively. Later, the finiteaxiomatization problem for L (...) 0 andNL 0 was partially solved, viz., for formulas withoutleft (or, equivalently, right) division and an (infinite) cut-ruleaxiomatics for the whole of L 0 has been given. Thepresent paper yields an analogous axiomatics forNL 0. Like in the author's previous work, the notionof rank of an axiom is introduced which, although inessentialfor the results given below, may be useful for the expectednonfinite-axiomatizability proof. (shrink)

In Zielonka (1981a, 1989), I found an axiomatics for the product-free calculus L of Lambek whose only rule is the cut rule. Following Buszkowski (1987), we shall call such an axiomatics linear. It was proved that there is no finite axiomatics of that kind. In Lambek's original version of the calculus (cf. Lambek, 1958), sequent antecedents are non empty. By dropping this restriction, we obtain the variant L0 of L. This modification, introduced in the early 1980s (see, e.g., Buszkowski, 1985; (...) Zielonka, 1981b), did not gain much popularity initially; a more common use of L0 has only occurred within the last few years (cf. Roorda, 1991: 29). In Zielonka (1988), I established analogous results for the restriction of L0 to sequents without left (or, equivalently, right) division. Here, I present a similar (cut-rule) axiomatics for the whole of L0.This paper is an extended, corrected, and completed version of Zielonka (1997). Unlike in Zielonka (1997), the notion of rank of an axiom is introduced which, although inessential for the results given below, may be useful for the expected non-finite-axiomatizability proof.The paper follows the same way of subject exposition as Zielonka (2000) but it is technically much less complicated. I restrict myself to giving bare results; all the ideological background is exactly the same as in case of the non-associative calculusNL0 and those who are interested in it are requested to consult the introductory section of Zielonka (2000). (shrink)

This paper develops an inference system for natural language within the ‘Natural Logic’ paradigm as advocated by van Benthem, Sánchez and others. The system that we propose is based on the Lambek calculus and works directly on the Curry-Howard counterparts for syntactic representations of natural language, with no intermediate translation to logical formulae. The Lambek -based system we propose extends the system by Fyodorov et~al., which is based on the Ajdukiewicz/Bar-Hillel calculus Bar Hillel,. This enables the system to deal with (...) new kinds of inferences, involving relative clauses, non-constituent coordination, and meaning postulates that involve complex expressions. Basing the system on the Lambek calculus leads to problems with non-normalized proof terms, which are treated by using normalization axioms. (shrink)

This paper reminds, puts in order, sketches and also initiates some researches from the field of logic and philosophy of language. It lays emphasis on the logical-linguistic and ontological developmental lines originated with Polish researchers. The author discusses two opposite orientations of the former line in the process of formalization of language, called here nominalistic and Platonistic. The paper mentions the author's result (1989; 1991) concerning theoretical equivalence of two axiomatic approaches to language syntax which take into consideration these two (...) different philosophical orientations, respectively. It also introduces certain formal-logical foundations of language semantics, discusses and develops the problem of the syntactic and semantic categorial agreement, i.e. the agreement of the syntactic category of each language expression with the ontological category assigned to the reference of this expression. The author, in particular, gives a solution to this problem for quantifiers. (shrink)

In the paper, original formal-logical conception of syntactic and semantic: intensional and extensional senses of expressions of any language L is outlined. Syntax and bi-level intensional and extensional semantics of language L are characterized categorically: in the spirit of some Husserl’s ideas of pure grammar, Leśniewski-Ajukiewicz’s theory syntactic/semantic categories and in accordance with Frege’s ontological canons, Bocheński’s famous motto—syntax mirrors ontology and some ideas of Suszko: language should be a linguistic scheme of ontological reality and simultaneously a tool of its (...) cognition. In the logical conception of language L, its expressions should satisfy some general conditions of language adequacy. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic, and semantic compatibility, correspondence guaranteed by the acceptance of a postulate of categorial compatibility syntactic and semantic categories of expressions of L. From this postulate, three principles of compositionality follow: one syntactic and two semantic already known to Frege. They are treated as conditions of homomorphism partial algebra of L into algebraic models of L: syntactic, intensional, and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, of course, an idealization, but only expressions with high degrees of precision of their senses, after due justification, may become theorems of science. (shrink)

The main objective of the paper is to provide a conceptual apparatus of a general logical theory of language communication. The aim of the paper is to outline a formal-logical theory of language in which the concepts of the phenomenon of language communication and language communication in general are defined and some conditions for their adequacy are formulated. The theory explicates the key notions of contemporary syntax, semantics, and pragmatics. The theory is formalized on two levels: token-level and type-level. As (...) such, it takes into account the dual – token and type – ontological character of linguistic entities. The basic notions of the theory: language communication, meaning and interpretation are introduced on the second, type-level of formalization, and their required prior formalization of some of the notions introduced on the first, token-level; among others, the notion of an act of communication. Owing to the theory, it is possible to address the problems of adequacy of both empirical acts of communication and of language communication in general. All the conditions of adequacy of communication discussed in the presented paper, are valid for one-way communication (sender-recipient); nevertheless, they can also apply to the reverse direction of language communication (recipient-sender). Therefore, they concern the problem of two-way understanding in language communication. (shrink)

The paper is an attempt at a logical explication of some crucial notions of current general semantics and pragmatics. A general, axiomatic, formal-logical theory of meaning and interpretation is outlined in this paper.In the theory, accordingto the token-type distinction of Peirce, language is formalised on two levels: first as a language of token-objects (understood as material, empirical, enduring through time-and space objects) and then – as a language of type-objects (understood as abstract objects, as classes of tokens). The basic concepts (...) of the theory, i.e. the notions: meaning, denotation and interpretation of well-formed expressions (wfes) of the language are formalised on the type-level, by utilising some semantic-pragmatic primitive notions introduced on the token-level. The paper is divided into two parts.In Part Ia theoryof meaningand denotation is proposed, and in Part II - its expansion to the theory of meaning and interpretation is presented.The meaninga wfe is defined as an equivalence class of the relation possessing the same manner of using types (cf. Ajdukiewicz [1934], Wittgenstein [1953]). The concept of denotation is defined by means of the relation of referring which holds between wfe-types and objects of reality described by the given language. (shrink)

The paper enriches the conceptual apparatus of the theory of meaning and denotation that was presented in Part I (Section 3). This part concentrates on the notion of interpretation, which is defined as an equivalence class of the relation possessing the same manner of interpreting types. In this part, some relations between meaning and interpretation, as well as one between denotation an interpretational denotation are established. In the theory of meaning and interpretation, the notion of language communication has been formally (...) introduced and some conditions of correctness of communication have been formulated. (shrink)

The paper concentrates on the problem of adequate reflection of fragments of reality via expressions of language and inter-subjective knowledge about these fragments, called here, in brief, language adequacy. This problem is formulated in several aspects, the most being: the compatibility of language syntax with its bi-level semantics: intensional and extensional. In this paper, various aspects of language adequacy find their logical explication on the ground of the formal-logical theory T of any categorial language L generated by the so-called classical (...) categorial grammar, and also on the ground of its extension to the bi-level, intensional and extensional semantic-pragmatic theory ST for L. In T, according to the token-type distinction of Ch.S. Peirce, L is characterized first as a language of well-formed expression-tokens (wfe-tokens) - material, concrete objects - and then as a language of wfe-types - abstract objects, classes of wfe-tokens. In ST the semantic-pragmatic notions of meaning and interpretation for wfe-types of L of intensional semantics and the notion of denotation of extensional semanics for wfe-types and constituents of knowledge are formalized. These notions allow formulating a postulate (an axiom of categorial adequacy) from which follow all the most important conditions of the language adequacy, including the above, and a structural one connected with three principles of compositionality. (shrink)

We prove completeness for some language-theoretic models of the full Lambek calculus and its various fragments. First we consider syntactic concepts and syntactic concepts over regular languages, which provide a complete semantics for the full Lambek calculus \. We present a new semantics we call automata-theoretic, which combines languages and relations via closure operators which are based on automaton transitions. We establish the completeness of this semantics for the full Lambek calculus via an isomorphism theorem for the syntactic concepts lattice (...) of a language and a construction for the universal automaton recognizing the same language. Finally, we use automata-theoretic semantics to prove completeness of relation models of binary relations and finite relation models for the Lambek calculus without and with empty antecedents and \), thus solving a problem left open by Pentus. (shrink)

This paper deals with various substructural propositional logics, in particular with substructural subsystems of Nelson's constructive propositional logics N– and N. Doen's groupoid semantics is extended to these constructive systems and is provided with an informational interpretation in terms of information pieces and operations on information pieces.

This paper studies the relation between some extensions of the non-associative Lambek Calculus NL and their interpretation in tree models (free groupoids). We give various examples of sequents that are valid in tree models, but not derivable in NL. We argue why tree models may not be axiomatizable if we add finitely many derivation rules to NL, and proceed to consider labeled calculi instead.We define two labeled categorial calculi, and prove soundness and completeness for interpretations that are almost the intended (...) one, namely for tree models where some branches of some trees may be resp. all branches of all trees must be infinitely extending. Extrapolating from the experiences in our quite simple systems, we briefly discuss some problems involved with the introduction of labels in categorial grammar, and argue that many of the basic questions are not yet understood. (shrink)

This paper is concerned with De Morgan’s explanation of the validity of arguments that involve relational notions. It discusses De Morgan’s expansion of traditional logic aimed at accommodating those inferences, and makes the point that his endeavour is not successful in that the rules that made up his new logic are not sound. Nevertheless, the most important scholarly work on De Morgan’s logic, and contrary to that De Morgan’s mistake is not beyond repair. The rules that determine his new logic (...) are in fact monotonie replacement rules. And provided they are restricted in the correct way, these rules are demonstrably sound. (shrink)

The idea that natural language grammar and planned action are relatedsystems has been implicit in psychological theory for more than acentury. However, formal theories in the two domains have tendedto look very different. This article argues that both faculties sharethe formal character of applicative systems based on operationscorresponding to the same two combinatory operations, namely functional composition and type-raising. Viewing them in thisway suggests simpler and more cognitively plausible accounts of bothsystems, and suggests that the language faculty evolved in the (...) speciesand develops in children by a rather direct adaptation of a moreprimitive apparatus for planning purposive action in the world bycomposing affordances of objects or tools. Theknowledge representation that underlies such planning is alsoreflected in the natural language semantics of tense, mood, andaspect, which the paper begins by arguing provides the key tounderstanding both systems. (shrink)

In this paper, we propose a game semantics for the (associative) Lambek calculus . Compared to the implicational fragment of intuitionistic propositional calculus, the semantics deals with two features of the logic: absence of structural rules, as well as directionality of implication. We investigate the impact of these variations of the logic on its game semantics.