The book is about logical analysis of natural language. Since we humans communicate by means of natural language, we need a tool that helps us to understand in a precise manner how the logical and formal mechanisms of natural language work. Moreover, in the age of computers, we need to communicate both with and through computers as well. Transparent Intensional Logic is a tool that is helpful in making our communication and reasoning smooth and precise. It deals with all kinds (...) of linguistic context in a fully compositional and anti-contextual way. (shrink)

In their recent paper Bi-facial truth: a case for generalized truth values Zaitsev and Shramko [7] distinguish between an ontological and an epistemic interpretation of classical truth values. By taking the Cartesian product of the two disjoint sets of values thus obtained, they arrive at four generalized truth values and consider two “semi-classical negations” on them. The resulting semantics is used to define three novel logics which are closely related to Belnap’s well-known four valued logic. A syntactic characterization of these (...) logics is left for further work. In this paper, based on our previous work on a functionally complete extension of Belnap’s logic, we present a sound and complete tableau calculus for these logics. It crucially exploits the Cartesian nature of the four values, which is reflected in the fact that each proof consists of two tableaux. The bi-facial notion of truth of Z&S is thus augmented with a bi-facial notion of proof. We also provide translations between the logics for semi-classical negation and classical logic and show that an argument is valid in a logic for semi-classical negation just in case its translation is valid in classical logic. (shrink)

In this paper we give an analytic tableau calculus P L 1 6 for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ⊧ t, ⊧ f, ⊧ i, and ⊧ under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the first three entailment (...) relations will in general require developing four tableaux, while proving that they are in the ⊧ relation may require six. (shrink)

In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic, an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a calculus for (...) the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof. (shrink)

Systematic semantic ambiguities result from the interaction of the two operations that are involved in resolving ellipsis in the presence of scoping elements such as quantifiers and intensional operators: scope determination for the scoping elements and resolution of the elided relation. A variety of problematic examples previously noted - by Sag, Hirschbüihler, Gawron and Peters, Harper, and others - all have to do with such interactions. In previous work, we showed how ellipsis resolution can be stated and solved in equational (...) terms. Furthermore, this equational analysis of ellipsis provides a uniform framework in which interactions between ellipsis resolution and scope determination can be captured. As a consequence, an account of the problematic examples follows directly from the equational method. The goal of this paper is merely to point out this pleasant aspect of the equational analysis, through its application to these cases. No new analytical methods or associated formalism are presented, with the exception of a straightforward extension of the equational method to intensional logic. (shrink)

This paper argues for the idea that in describing language we should follow Haskell Curry in distinguishing between the structure of an expression and its appearance or manifestation . It is explained how making this distinction obviates the need for directed types in type-theoretic grammars and a simple grammatical formalism is sketched in which representations at all levels are lambda terms. The lambda term representing the abstract structure of an expression is homomorphically translated to a lambda term representing its manifestation, (...) but also to a lambda term representing its semantics. (shrink)

In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it (...) is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up. (shrink)

This paper embeds the core part of Discourse Representation Theory in the classical theory of types plus a few simple axioms that allow the theory to express key facts about variables and assignments on the object level of the logic. It is shown how the embedding can be used to combine core analyses of natural language phenomena in Discourse Representation Theory with analyses that can be obtained in Montague Semantics.

Natural languages generally distinguishes among different existence predicates for different types of entities, such as English 'exist', 'occur', and 'obtain'. The paper gives an in-depth discussion and analysis of a range of existence predicates in natural language within the general project of descriptive metaphysics, or more specifically ‘natural language ontology’.

Here, I first prove that certain families of k-valued clones have the Gupta-Belnap fixed-point property. This essentially means that all propositional languages that are interpreted with operators belonging to those clones are such that any net of self-referential sentences in the language can be consistently evaluated. I then focus on two four-valued generalisations of the Kleene propositional operators that generalise the strong and weak Kleene operators: Belnap’s clone and Fitting’s clone, respectively. I apply the theorems from the initial part of (...) the paper to analyse the fixed-point property of Belnap’s and Fitting’s clones when some special operators that reflect the semantics are added. The conclusion of the paper is that Fitting’s clone is better suited than Belnap’s to provide self-referential languages with highly expressive resources. (shrink)

The essays collected in this volume are all concerned with the connection between fiction and truth. This question is of utmost importance to metaphysics, philosophy of language, philosophical logic and epistemology, raising in each of these areas and at their intersections a large number of issues related to creation, existence, reference, identity, modality, belief, assertion, imagination, pretense, etc. All these topics and many more are addressed in this collection, which brings together original essays written from various points of view by (...) philosophers of diverse trends. These essays constitute major contributions to the current debates that the connection between truth and fiction continually enlivens, and give a sense of the directions in which research on this question is heading. Contributors: Fred Adams, Frederick Kroon, Robert Howell, Brendan Murday, Terence Parsons, Graham Priest, Erich Rast, Manuel Rebuschi, Marion Renauld, R.M. Sainsbury, Grant Tavinor, Alberto Voltolini. (shrink)

Acknowledgements Five out of the 13 contributions to this volume originate from papers which were presented at the international workshop on “Epistemology, Context, Formalism” held at the MSH-Lorraine in Nancy, France, on November the ...

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

I argue that compositionality (in the sense of homomorphic interpretation) is compatible with radical and pervasive contextual effects on interpretation. Apparent problems with this claim lose their force if we are careful in distinguishing the question of how a grammar assigns interpretations from the question of how people figure out which interpretations the grammar assigns. I demonstrate, using a simple example, that this latter task must sometimes be done not by computing a derivation defined directly by the grammar, but through (...) the use of pragmatic background knowledge and extragrammatical reasoning, even when the grammar is designed to be fully compositional. The fact that people must sometimes use global pragmatic mechanisms to identify truth conditions therefore tells us nothing about whether the grammar assigns truth conditions compositionally. Compositional interpretation (or the lack thereof) is identifiable not by the mechanisms necessary to calculating truth conditions, but by the structural relation between the interpretation of a phrase in context and the interpretations of its parts in that same context. Even if this relation varies by context, an invariant grammar is possible if grammars can “invoke” pragmatic concepts; but this does not imply that grammatical theory must explain these concepts or incorporate a theory of pragmatics. (shrink)

I present an analysis according to which the current state of the definition of substitution leads to a contradiction in the system of Transparent Intensional Logic. I entail the contradiction using only the basic definitions of TIL and standard results. I then analyse the roots of the contradiction and motivate the path I take in resolving the contradiction. I provide a new amended definition of collision-less substitution which blocks the contradiction in a non-ad hoc way. I elaborate on the consequences (...) of the amended definition, namely the invalidity of the Church-Rosser theorem. I present a counterexample to the validity of the theorem in TIL with an amended definition of substitution. (shrink)

This article points out problems in current dynamic treatments of anaphora and provides a new account that solves these by grafting Muskens' Compositional Discourse Representation Theory onto a partial theory of types. Partiality is exploited to keep track of which discourse referents have been introduced in the text (thus avoiding the overwrite problem) and to account for cases of anaphoric failure. Another key assumption is that the set of discourse referents is well-ordered, so that we can keep track of the (...) order in which they have been introduced, allowing a semantic characterization of anaphoric accessibility across stretches of discourse. Unlike other dynamic approaches, the system defines semantic values for unresolved anaphors. This leads to a clear separation of monotonic and non-monotonic content (in this case anaphoric resolution) and arguably provides a sound basis for a non-monotonic theory of anaphoric resolution. (shrink)

This paper considers some issues to do with valuational presentations of consequence relations, and the Galois connections between spaces of valuations and spaces of consequence relations. Some of what we present is known, and some even well-known; but much is new. The aim is a systematic overview of a range of results applicable to nonreflexive and nontransitive logics, as well as more familiar logics. We conclude by considering some connectives suggested by this approach.

The paper proposes a semantics for contextual (i.e., Temporal and Locative) Prepositional Phrases (CPPs) like during every meeting, in the garden, when Harry met Sally and where I’m calling from. The semantics is embodied in a multi-modal extension of Combinatory Categoral Grammar (CCG). The grammar allows the strictly monotonic compositional derivation of multiple correct interpretations for “stacked” or multiple CPPs, including interpretations whose scope relations are not what would be expected on standard assumptions about surfacesyntactic command and monotonic derivation. A (...) type-hierarchy of functional modalities plays a crucial role in the specification of the fragment. (shrink)

We present Property Theory with Curry Typing (PTCT), an intensional first-order logic for natural language semantics. PTCT permits fine-grained specifications of meaning. It also supports polymorphic types and separation types. We develop an intensional number theory within PTCT in order to represent proportional generalized quantifiers like “most.” We use the type system and our treatment of generalized quantifiers in natural language to construct a type-theoretic approach to pronominal anaphora that avoids some of the difficulties that undermine previous type-theoretic analyses of (...) this phenomenon. (shrink)

This paper distinguishes between definiteness and determinacy. Definiteness is seen as a morphological category which, in English, marks a uniqueness presupposition, while determinacy consists in denoting an individual. Definite descriptions are argued to be fundamentally predicative, presupposing uniqueness but not existence, and to acquire existential import through general type-shifting operations that apply not only to definites, but also indefinites and possessives. Through these shifts, argumental definite descriptions may become either determinate or indeterminate. The latter option is observed in examples like (...) ‘Anna didn’t give the only invited talk at the conference’, which, on its indeterminate reading, implies that there is nothing in the extension of ‘only invited talk at the conference’. The paper also offers a resolution of the issue of whether possessives are inherently indefinite or definite, suggesting that, like indefinites, they do not mark definiteness lexically, but like definites, they typically yield determinate readings due to a general preference for the shifting operation that produces them. (shrink)

It is arguably desirable to have a theory of meaning that (i) does not identify propositions with sets of worlds, (ii) enables to capture the dynamic character of semantic interpretation and (iii) provides the basis for a semantic program that incorporates and extends the achievements of Montague semantics. A theory of properties and propositions that meets these desiderata is developed and several applications to the semantic analysis of natural languages are explored.

The paper proposes a novel solution to the problem of scope posed by natural language indefinites that captures both the difference in scopal freedom between indefinites and bona fide quantifiers and the syntactic sensitivity that the scope of indefinites does nevertheless exhibit. Following the main insight of choice functional approaches, we connect the special scopal properties of indefinites to the fact that their semantics can be stated in terms of choosing a suitable witness. This is in contrast to bona fide (...) quantifiers, the semantics of which crucially involves relations between sets of entities. We provide empirical arguments that this insight should not be captured by adding choice/Skolem functions to classical first-order logic, but in a semantics that follows Independence-Friendly Logic, in which scopal relations involving existentials are part of the recursive definition of truth and satisfaction. These scopal relations are resolved automatically as part of the interpretation of existentials. Additional support for this approach is provided by dependent indefinites, a cross-linguistically common class of special indefinites that can be straightforwardly analyzed in our semantic framework. (shrink)

Combining logics has become a rapidly expanding enterprise that is inspired mainly by concerns about modularity and the wish to join together tailor made logical tools into more powerful but still manageable ones. A natural question is whether it offers anything new over and above existing standard languages. By analysing a number of applications where combined logics arise, we argue that combined logics are a potentially valuable tool in applied logic, and that endorsements of standard languages often miss the point. (...) Using the history of quantified modal logic as our main example, we also show that the use of combined structures and logics is a recurring theme in the analysis of existing logical systems. (shrink)

In this paper it is shown how a partial semantics for presuppositions can be given which is empirically more satisfactory than its predecessors, and how this semantics can be integrated with a technically sound, compositional grammar in the Montagovian fashion. Additionally, it is argued that the classical objection to partial accounts of presupposition projection, namely that they lack “flexibility,” is based on a misconception. Partial logics can give rise to flexible predictions without postulating any ad hoc ambiguities. Finally, it is (...) shown how the partial foundation can be combined with a dynamic system of common-ground maintenance to account for accommodation. (shrink)

Este artigo canta uma canção — uma canção criada ao unir o trabalho de quatro grandes nomes na história da lógica: Hans Reichenbach, Arthur Prior, Richard Montague, e Leon Henkin. Embora a obra dos primeiros três desses autores tenha sido previamente combinada, acrescentar as ideias de Leon Henkin é o acréscimo requerido para fazer com que essa combinação funcione no nível lógico. Mas o presente trabalho não se concentra nas tecnicalidades subjacentes (que podem ser encontradas em Areces, Blackburn, Huertas, e (...) Manzano [no prelo]), e sim nos instrumentos subjacentes e no modo como trabalham em conjunto. Esperamos que o leitor fique tentado a cantar junto. DOI:10.5007/1808-1711.2011v15n2p225. (shrink)

What licenses the use of a definite description? The formal and philosophical literature has approached this question in two ways. The uniqueness approach (Frege, 1892; Russell, 1905; Strawson, 1950) holds that we may use a definite determiner only if the property denoted by its complement holds of exactly one individual in some domain: Sentence (1) and (2) can only be true if there is exactly one king of France, and exactly one American governor, respectively. Since this is not the case (...) in the actual world, the sentences are either false or (on most modern accounts) fall prey to a presupposition failure. (shrink)

This paper introduces λ-grammar, a form of categorial grammar that has much in common with LFG. Like other forms of categorial grammar, λ-grammars are multi-dimensional and their components are combined in a strictly parallel fashion. Grammatical representations are combined with the help of linear combinators, closed pure λ-terms in which each abstractor binds exactly one variable. Mathematically this is equivalent to employing linear logic, in use in LFG for semantic composition, but the method seems more practicable.

Verbs such as know, believe, hope, fear, regret and desire are commonly taken to express an attitude that one may bear towards a proposition and are therefore called verbs of propositional attitude. Thus in (1) below the agent Cathy is reported to have a certain attitude.

Ordinary Language Philosophy has largely fallen out of favour, and with it the belief in the primary importance of analyses of ordinary language for philosophical purposes. Still, in their various endeavours, philosophers not only from analytic but also from other backgrounds refer to the use and meaning of terms of interest in ordinary parlance. In doing so, they most commonly appeal to their own linguistic intuitions. Often, the appeal to individual intuitions is supplemented by reference to dictionaries. In recent times, (...) Internet search engine queries for expressions of interest have become quite popular. Apparently, philosophers attempt to surpass the limits of their own linguistic intuitions by appealing to experts or to factual uses of language. I argue that this attempt is commendable but that its execution is wanting. Instead of appealing to dictionaries or Internet queries, philosophers should employ computer-based linguistic corpora in order to confirm or falsify hypotheses about the factual use of language. This approach also has some advantages over methods employed by experimental philosophers. If the importance of ordinary language is stressed, the use of linguistic corpora is hardly avoidable. (shrink)

Distributive egalitarians believe that distributive justice is to be explained by the idea of distributive equality (DE) and that DE is of intrinsic value. The socio-relational critique argues that distributive egalitarianism does not account for the “true” value of equality, which rather lies in the idea of “equality as a substantive social value” (ESV). This paper examines the socio-relational critique and argues that it fails because – contrary to what the critique presupposes –, first, ESV is not conceptually distinct from (...) DE, and second, the idea of ESV cannot serve as a “foundation” or “root” of distributive egalitarianism. (shrink)

Since its first appearance in 1966, the notion of a supervaluation has been regarded by many as a powerful tool for dealing with semantic gaps. Only recently, however, applications to semantic gluts have also been considered. In previous work I proposed a general framework exploiting the intrinsic gap/glut duality. Here I also examine an alternative account where gaps and gluts are treated on a par: although they reflect opposite situations, the semantic upshot is the same in both cases--the value of (...) some expressions is not uniquely defined. Other strategies for generalizing supervaluations are considered and some comparative facts are discussed. (shrink)

This paper shows how the dynamic interpretation of natural language introduced in work by Hans Kamp and Irene Heim can be modeled in classical type logic. This provides a synthesis between Richard Montague's theory of natural language semantics and the work by Kamp and Heim.

Italian translation of "On Logical Relativity" (2002), by Luca Morena.

In this paper it is shown how the DRT (Discourse Representation Theory) treatment of temporal anaphora can be formalized within a version of Montague Semantics that is based on classical type logic.

The ‘syntax’ and ‘combinatorics’ of my title are what Curry (1961) referred to as phenogrammatics and tectogrammatics respectively. Tectogrammatics is concerned with the abstract combinatorial structure of the grammar and directly informs semantics, while phenogrammatics deals with concrete operations on syntactic data structures such as trees or strings. In a series of previous papers (Muskens, 2001a; Muskens, 2001b; Muskens, 2003) I have argued for an architecture of the grammar in which finite sequences of lambda terms are the basic data structures, (...) pairs of terms syntax, semantics for example. These sequences then combine with the help of simple generalizations of the usual abstraction and application operations. This theory, which I call Lambda Grammars and which is closely related to the independently formulated theory of Abstract Categorial Grammars (de Groote, 2001; de Groote, 2002), in fact is an implementation of Curry’s ideas: the level of tectogrammar is encoded by the sequences of lambda-terms and their ways of combination, while the syntactic terms in those sequences constitute the phenogrammatical level. In de Groote’s formulation of the theory, tectogrammar is the level of abstract terms, while phenogrammar is the level of object terms. (shrink)

A logic is called higher order if it allows for quantification over higher order objects, such as functions of individuals, relations between individuals, functions of functions, relations between functions, etc. Higher order logic began with Frege, was formalized in Russell [46] and Whitehead and Russell [52] early in the previous century, and received its canonical formulation in Church [14].1 While classical type theory has since long been overshadowed by set theory as a foundation of mathematics, recent decades have shown remarkable (...) comebacks in the fields of mechanized reasoning (see, e.g., Benzm¨. (shrink)

Standard approaches to proper names, based on Kripke's views, hold that the semantic values of expressions are (set-theoretic) functions from possible worlds to extensions and that names are rigid designators, i.e.\ that their values are \emph{constant} functions from worlds to entities. The difficulties with these approaches are well-known and in this paper we develop an alternative. Based on earlier work on a higher order logic that is \emph{truly intensional} in the sense that it does not validate the axiom scheme of (...) Extensionality, we develop a simple theory of names in which Kripke's intuitions concerning rigidity are accounted for, but the more unpalatable consequences of standard implementations of his theory are avoided. The logic uses Frege's distinction between sense and reference and while it accepts the rigidity of names it rejects the view that names have direct reference. Names have constant denotations across possible worlds, but the semantic value of a name is not determined by its denotation. (shrink)

It is routinely observed in NLP that sentences seem to “evoke” situations (where I use this term comprehensively to cover events, episodes, eventualities, processes, etc.). Much like discourse entities evoked by explicit noun phrases, these evoked situations can be referred to anaphorically, as for instance in (1) and (3) below, and can be modified in various ways, for instance by supplying their duration and location, as in (2) and (4).

Logic has its roots in the study of valid argument, but while traditional logicians worked with natural language directly, modern approaches first translate natural arguments into an artificial language. The reason for this step is that some artificial languages now have very well developed inferential systems. There is no doubt that this is a great advantage in general, but for the study of natural reasoning it is a drawback that the original linguistic forms get lost in translation. An alternative approach (...) would be to develop a general theory of the natural logic behind human reasoning and human information processing by studying formal logics that operate directly on linguistic representations. That this is possible we will try to make plausible in this paper. It will turn out that one level of representation, that of Logical Form, can meaningfully be identified with the language of an existing and well-understood logic, a restricted form of the theory of types. It is not difficult to devise inference systems for this language, and it is thus possible to study reasoning systems that are based directly on language. (shrink)

In this paper we introduce a Gentzen calculus for (a functionally complete variant of) Belnap's logic in which establishing the provability of a sequent in general requires \emph{two} proof trees, one establishing that whenever all premises are true some conclusion is true and one that guarantees the falsity of at least one premise if all conclusions are false. The calculus can also be put to use in proving that one statement \emph{necessarily approximates} another, where necessary approximation is a natural dual (...) of entailment. The calculus, and its tableau variant, not only capture the classical connectives, but also the `information' connectives of four-valued Belnap logics. This answers a question by Avron. (shrink)

the world of phenomena is immensely large this means we can perceive only part of the world. We see, feel and hear parts of reality, not the whole of it, and it seems that a sentence containing a verb of perception like 'John sees a house burn' is most naturally treated as saying that the subject sees an incomplete world in which the embedded sentence is true (see Barwise (1981) for this analysis). But if we want to analyse perception verbs (...) thus, we must introduce some form of incompleteness into our formal system, the.. (shrink)