Quine’s most important charge against second-, and more generally, higher-order logic is that it carries massive existential commitments. The force of this charge does not depend upon Quine’s questionable assimilation of second-order logic to set theory. Even if we take second-order variables to range over properties, rather than sets, the charge remains in force, as long as properties are individuated purely extensionally. I argue that if we interpret them as ranging over properties more reasonably construed, in accordance with an abundant (...) or deflationary conception, Quine’s charge can be resisted. This interpretation need not be seen as precluding the use of model-theoretic semantics for second-order languages; but it will preclude the use of the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach I recommend has revisionary implications which some may find unpalatable; it is, however, compatible with the quite different special case in which the second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, some important metalogical results obtainable under the standard semantics may still be obtained. In my final section, I discuss the relations between second-order logic, interpreted as I recommend, and a strong version of schematic ancestral logic promoted in recent work by Richard Heck. I argue that while there is an interpretation on which Heck’s logic can be contrasted with second-order logic as standardly interpreted, when it is so interpreted, its differences from the more modest form of second-order logic I advocate are much less substantial, and may be largely presentational. (shrink)

The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (that is, logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous aproaches to quantified LFIs presented in the literature. The case of QmbC, (...) the simpler quantified LFI expanding classical logic, will be analyzed in detail. An axiomatic extension of QmbC called QLFI1o is also studied, which is equivalent to the quantified version of da Costa and D'Ottaviano 3-valued logic J3. The semantical structures for this logic turn out to be Tarkian structures based on twist structures. The expansion of QmbC and QLFI1o with a standard equality predicate is also considered. (shrink)

The existence of non-standard numbers in first-order arithmetics is a semantic obstacle for modelling our arithmetical skills. This article argues that so far there is no adequate approach to overcome such a semantic obstacle, because we can also find out, and deal with, non-standard elements in Turing machines.

The present paper deals with the predicate version MTL of the logic MTL by Esteva and Godo. We introduce a Kripke semantics for it, along the lines of Ono''s Kripke semantics for the predicate version of FLew (cf. [O85]), and we prove a completeness theorem. Then we prove that every predicate logic between MTL and classical predicate logic is undecidable. Finally, we prove that MTL is complete with respect to the standard semantics, i.e., with respect to (...) Kripke frames on the real interval [0,1], or equivalently, with respect to MTL-algebras whose lattice reduct is [0,1] with the usual order. (shrink)

This article addresses and resolves some issues of relational, Kripke-style, semantics for the logics of bounded lattice expansions with operators of well-defined distribution types, focusing on the case where the underlying lattice is not assumed to be distributive. It therefore falls within the scope of the theory of Generalized Galois Logics, introduced by Dunn, and it contributes to its extension. We introduce order-dual relational semantics and present a semantic analysis and completeness theorems for non-distributive lattice logic with n (...) -ary additive or multiplicative operators, with negation operators modally interpreted as impossibility and unnecessity, as well as with implication connectives. Order-dual relational semantics shares with the generalized Kripke frames, or the bi-approximation semantics approach, the use of both a satisfaction and a co-satisfaction relation, but it also responds to the recently voiced concerns of Craig, Haviar and Conradie about the relative non-intuitiveness of the 2-sorted semantics of the aforementioned approaches. In this article, we provide a standard interpretation of modalities and natural interpretations of both negation and implication, despite the absence of distribution. Thereby, our results contribute in creating the necessary background for research in non-distributive logics with modalities variously interpreted as dynamic, temporal etc, by analogy to the classical case. (shrink)

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms . We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering (...) canonical real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind , where φ has no additional truth-constants. (shrink)

The paper presents Property Theory with Curry Typing (PTCT) where the language of terms and well-formed formulæ are joined by a language of types. In addition to supporting fine-grained intensionality, the basic theory is essentially first-order, so that implementations using the theory can apply standard first-order theorem proving techniques. The paper sketches a system of tableau rules that implement the theory. Some extensions to the type theory are discussed, including type polymorphism, which provides a useful analysis of conjunctive terms. (...) Such terms can be given a single polymorphic type that expresses the fact that they can conjoin phrases of any one type, yielding an expression of the same type. (shrink)

The main aim of this work is to evaluate whether Boolos’ semantics for second-order languages is model-theoretically equivalent to standard model-theoretic semantics. Such an equivalence result is, actually, directly proved in the “Appendix”. I argue that Boolos’ intent in developing such a semantics is not to avoid set-theoretic notions in favor of pluralities. It is, rather, to prevent that predicates, in the sense of functions, refer to classes of classes. Boolos’ formal semantics differs from a (...) semantics of pluralities for Boolos’ plural reading of second-order quantifiers, for the notion of plurality is much more general, not only of that set, but also of class. In fact, by showing that a plurality is equivalent to sub-sets of a power set, the notion of plurality comes to suffer a loss of generality. Despite of this equivalence result, I maintain that Boolos’ formal semantics does not committ second-order languages to second-order entities, contrary to standard semantics. Further, such an equivalence result provides a rationale for many criticisms to Boolos’ formal semantics, in particular those by Resnik and Parsons against its alleged ontological innocence and on its Platonistic presupposition. The key set-theoretic notion involved in the equivalence proof is that of many-valued function. But, first, I will provide a clarification of the philosophical context and theoretical grounds of the genesis of Boolos’ formal semantics. (shrink)

The paper presents Property Theory with Curry Typing where the language of terms and well-formed formulæ are joined by a language of types. In addition to supporting fine-grained intensionality, the basic theory is essentially first-order, so that implementations using the theory can apply standard first-order theorem proving techniques. The paper sketches a system of tableau rules that implement the theory. Some extensions to the type theory are discussed, including type polymorphism, which provides a useful analysis of conjunctive terms. Such (...) terms can be given a single polymorphic type that expresses the fact that they can conjoin phrases of any one type, yielding an expression of the same type. (shrink)

First we have individual variables, as usual in first-order logics. (We do not have individual constants, but this is a minor point.) The propositional logic LP has justification constants, but in FOLP these are generalized to allow individual variables as arguments. Thus we have as justification constants c, c(x), c(x, y), . . . . Similarly LP has justification variables, but in FOLP these can be parametrized with individual variables p, p(x), p(x, y), . . . . To keep terminology (...) in line with past papers, we will still refer to things as justification constants and justification variables, even though they have structure to them. As in LP, justification terms are built up from justification constants and justification variables using ·, +, ! as usual. In addition there is a new constructor, genx, introduced by Artemov, and there is one further new constructor, exsx, introduced in this paper. If t is a justification term and x is an individual variable, genxt and exsxt are justification terms. An individual variable x is free in a justification term unless it is bound by genx or exsx. More specifically, the free variables of p(x, y, . . .) and of c(x, y, . . .) are {x, y, . . .}, the free variables of s · t and of s + t are the free variables of s together with the free variables of t, the free variables of !s are the free variables of s, and the free variables of genxt and of exsxt are the free variables of t except for x. Formulas are built up from atomic formulas, including ⊥, in the way standard in first-order logic, together with the additional formation rule: t:X is a formula provided t is a justification term, X is a formula, and all free variables of X occur in t. We assume ⊃, ⊥, and ∀ are basic, with other connectives and quantifier defined. The axiomatization used here is a combination of an LP axiomatization and a standard axiomatization of first-order logic, together with a version of the Barcan formula, and one additional axiom that corresponds to the converse Barcan formula.. (shrink)

In the context of life sciences, we are constantly confronted with information that possesses precise semantic values and appears essentially immersed in a specific evolutionary trend. In such a framework, Nature appears, in Monod’s words, as a tinkerer characterized by the presence of precise principles of self-organization. However, while Monod was obliged to incorporate his brilliant intuitions into the framework of first-order cybernetics and a theory of information with an exclusively syntactic character such as that defined by Shannon, research advances (...) in recent decades have led not only to the definition of a second-order cybernetics but also to an exploration of the boundaries of semantic information. As H. Atlan states, on a biological level "the function self-organizes together with its meaning". Hence the need to refer to a conceptual theory of complexity and to a theory of self-organization characterized in an intentional sense. There is also a need to introduce, at the genetic level, a distinction between coder and ruler as well as the opportunity to define a real software space for natural evolution. The recourse to non-standard model theory, the opening to a new general semantics, and the innovative definition of the relationship between coder and ruler can be considered, today, among the most powerful theoretical tools at our disposal in order to correctly define the contours of that new conceptual revolution increasingly referred to as metabiology. This book focuses on identifying and investigating the role played by these particular theoretical tools in the development of this new scientific paradigm. Nature "speaks" by means of mathematical forms: we can observe these forms, but they are, at the same time, inside us as they populate our organs of cognition. In this context, the volume highlights how metabiology appears primarily to refer to the growth itself of our instruments of participatory knowledge of the world. (shrink)

(1) This is Part 2 of the semantic theory I call TM. In Part 1, I developed TM as a theory in the analytic philosophy of language, in lexical semantics, and in the sociology of relating occasions of statement production and comprehension to formal and informal lexicographic conclusions about statements and lexical items – roughly, as showing how synchronic semantics is a sociological derivative of diachronic, person-relative acts of linguistic behavior. I included descriptions of new cognitive psychology experimental (...) paradigms which would allow us to precisely measure the two constituents of semantics – meaning and reference – both at the level of individual speech acts and at the level of societal convergences, i.e. at both the token and type levels. -/- (2) In the Introduction, I recapitulate the arguments of Part 1. The Introduction also develops some analytic philosophical and lexical semantics themes not discussed in Part 1. -/- (3) After the Introduction, I present neural TM (nTM) as a theory of the neural mechanisms and processes which give rise to these person/occasion-relative acts of linguistic behavior. I develop nTM at three levels, the first two of which describe linguistic/semantic functions independently of their cortical locations. At the first level, I describe individual word-to-word and word-to-object connections. At the second level, I describe the corresponding structuralist networks of which they are the individual components. At this level, I introduce some key linguistic concepts of TM – its graded meaning, reference, and generalization sets, and the types of statements which express various levels of word-to-word and word-to-object relationships among lexical items which, because of the constraints they impose on the use of those lexical items in statements we produce and comprehend, are concepts. This constitutes the second structural level of nTM. -/- (4) At the third level, I associate the non-localized structures of the previous levels with cortically located neural structures and with the fasciculi that connect them. I distinguish neural areas in which primary (phonetic) and secondary (orthographic) lexicons are stored in long-term memory. I also describe the embodied concepts which co-exist in the anterior temporal lobes with the images they lexicalize. These concepts are often said to name physical objects and their features, although what they in fact name are kinds of physical objects and features. I describe how conceptual constraints and referential constraints interact to channel our intentions to say how things are into statements which are semantically well-formed, and which consequently successfully communicate information. -/- (5) Following this presentation of nTM, I examine five prominent neural semantic theories. I point out what is wrong with each of them as far as their explanations of semantics are concerned, and I also indicate how nTM can replace the “semantic cores” of those theories. -/- (6) The two basic mistakes made by neuroscience semantic theories, as I will explain, are (i) that all but one of them regard semantics as a matter of the association of words with perceptual images, and of generalizations from those associations; and (ii) that they all rely on an unspecified set of neural structures which purportedly encode the meaning of concepts in abstraction from their phonological and orthographic forms. nTM maintains, in contrast, that there are no abstract neural representations of semantic content. Neural constraints on our linguistic behavior, especially on our ascriptive and co-ascriptive use of words, express the semantic constraints on those words which make them concepts. That is the semantic content of words. -/- (7) I next consider several results from neuroscience experimental data which have been given one interpretation by one or another of the standard neurosemantic theories, but to which nTM gives a different interpretation. I include several predictions which I have found neither confirmed nor disconfirmed in the experimental neuroscience literature. -/- (8) After a concluding section in which I summarize the major changes to neurosemantic theory introduced by TM, and the analytic philosophy of language and lexical semantics contexts within which TM is situated, there follows an appendix in which I discuss neural net AI, and make some recommendations for implementing nTM in silicon. (shrink)

This document presents a Gentzen-style deductive calculus and proves that it is complete with respect to a 3-valued semantics for a language with quantifiers. The semantics resembles the strong Kleene semantics with respect to conjunction, disjunction and negation. The completeness proof for the sentential fragment fills in the details of a proof sketched in Arnon Avron (2003). The extension to quantifiers is original but uses standard techniques.

This book offers a state-of-the-art introduction to the basic techniques and results of neighborhood semantics for modal logic. In addition to presenting the relevant technical background, it highlights both the pitfalls and potential uses of neighborhood models – an interesting class of mathematical structures that were originally introduced to provide a semantics for weak systems of modal logic. In addition, the book discusses a broad range of topics, including standard modal logic results ; bisimulations for neighborhood models (...) and other model-theoretic constructions; comparisons with other semantics for modal logic ; neighborhood semantics for first-order modal logic, applications in game theory ; applications in epistemic logic ; and non-normal modal logics with dynamic modalities. The book can be used as the primary text for seminars on philosophical logic focused on non-normal modal logics; as a supplemental text for courses on modal logic, logic in AI, or philosophical logic ; or as the primary source for researchers interested in learning about the uses of neighborhood semantics in philosophical logic and game theory. (shrink)

The purpose of the present paper is to provide a way of understanding systems of logic of essence by introducing a new semantic framework for them. Three central results are achieved: first, the now standard Fitting semantics for the propositional logic of evidence is adapted in order to provide a new, simplified semantics for the propositional logic of essence; secondly, we show how it is possible to construe the concept of necessary truth explicitly by using the concept (...) of essential truth; finally, Fitting semantics is adapted in order to present a simplified semantics for the quantified logic of essence. (shrink)

Linear logic is a new logic which was recently developed by Girard in order to provide a logical basis for the study of parallelism. It is described and investigated in Gi]. Girard's presentation of his logic is not so standard. In this paper we shall provide more standard proof systems and semantics. We shall also extend part of Girard's results by investigating the consequence relations associated with Linear Logic and by proving corresponding str ong completeness theorems. Finally, (...) we shall investigate the relation between Linear Logic and previously known systems, especially Relevance logics. (shrink)

As part of an approach to the liar paradox and the other paradoxes affecting truth, I have proposed replacing our concept of truth with two concepts: ascending truth and descending truth.1 I am not going to discuss why I think this is the best approach or how it solves the paradoxes; instead, I concentrate on the theory of ascending and descending truth. I formulate an axiomatic theory of ascending truth and descending truth (ADT) and provide a possible-worlds semantics for (...) it (which I dub xeno semantics). Xeno semantics is a generalization of the familiar neighborhood semantics, which itself is a generalization of the standard relational semantics. Once the details of ADT have been presented, it is easy to show that neither relational semantics nor neighborhood semantics will work for it; thus, the move to a more general framework is required. The main result is a fixed point theorem that guarantees the existence of an acceptable first-order constant-domain xeno model. From this result it follows that ADT is sound with respect to the class of such models. The upshot is that ADT is consistent relative to the background set theory. (shrink)

Standard first-order logic plus quantifiers of all finite orders ("SFOLω") faces four well-known difficulties when used to characterize the behavior of certain English quantifier phrases. All four difficulties seem to stem from the typed structure of SFOLω models. The typed structure of SFOLω models is in turn a product of an asymmetry between the meaning of names and the meaning of predicates, the element-set asymmetry. In this paper we examine a class of models in which this asymmetry of meaning (...) is removed. The models of this class permit definitions of the quantifiers which allow a desirable flexibility in fixing the domain of quantification. Certain SFOLω type restrictions are thereby avoided. The resulting models of English validate all of the standard first-order logical truths and are free of the four deficiencies of SFOLω models. (shrink)

Most descriptions of higher-order vagueness in terms of traditional modal logic generate so-called higher-order vagueness paradoxes. The one that doesn't is problematic otherwise. Consequently, the present trend is toward more complex, non-standard theories. However, there is no need for this.In this paper I introduce a theory of higher-order vagueness that is paradox-free and can be expressed in the first-order extension of a normal modal system that is complete with respect to single-domain Kripke-frame semantics. This is the system QS4M+BF+FIN. (...) It corresponds to the class of transitive, reflexive and final frames. With borderlineness defined logically as usual, it then follows that something is borderline precisely when it is higher-order borderline, and that a predicate is vague precisely when it is higher-order vague.Like Williamson's, the theory proposed here has no clear borderline cases in Sorites sequences. I argue that objections that there must be clear borderline cases ensue from the confusion of two notions of borderlineness—one associated with genuine higher-order vagueness, the other employed to sort objects into categories—and that the higher-order vagueness paradoxes result from superimposing the second notion onto the first. Lastly, I address some further potential objections. (shrink)

Chomsky’s (1995, 2000a) Minimalist Program (MP) invites a perspective on semantics that is distinctive and attractive. In section one, I discuss a general idea that many theorists should find congenial: the spoken or signed languages that human children naturally acquire and use— henceforth, human languages—are biologically implemented procedures that generate expressions, whose meanings are recursively combinable instructions to build concepts that reflect a minimal interface between the Human Faculty of Language (HFL) and other cognitive systems. In sections two and (...) three, I develop this picture in the spirit of MP, in part by asking how much of the standard Frege-Tarski apparatus is needed in order to provide adequate and illuminating descriptions of the “concept assembly instructions” that human languages can generate. I’ll suggest that we can make do with relatively little, by treating all phrasal meanings as instructions to assemble number-neutral concepts that are monadic and conjunctive. But the goal is not to legislate what counts as minimal in semantics. Rather, by pursuing one line of Minimalist thought, I hope to show how such thinking can be fruitful. (shrink)

Semantic noise, the effect ensuing from the denotative and thus functional variability exhibited by different terms in different contexts, is a common concern in natural language processing (NLP). While unarguably problematic in specific applications (e.g., certain translation tasks), the main argument of this paper is that failing to observe this linguistic matter of fact as a generative effect rather than as an obstacle, leads to actual obstacles in instances where language model outputs are presented as neutral. Given that a common (...) and long-standing challenge in NLP is the interpretation of ambiguous – i.e., semantically noisy – cases, this article focuses on an exemplar ambiguity-resolution task in NLP: the problem of anaphora in Winograd schemas. The main question considered is: to what extent is the standard approach to disambiguation in NLP subject to a stagnant “image of language”? And, can a transdisciplinary, dynamic approach combining linguistics and philosophy elucidate new perspectives on these possible conceptual shortcomings? In order to answer these questions we explore the term and concept of noise, particularly in its presentation as semantic noise. Owing to its definitional plurality, and sometimes even desirable unspecificity, the term noise is thus used as proof of concept for semantic generativity being an inherent characteristic in linguistic representation, and its concept is used to interrogate assumptions admitted in the resolution of Winograd schemas. The argument is speculative and theoretical in method, and the result is an analysis which provides an account of the fundamentally dialogical and necessarily open-ended effects of semantic noise in natural language. (shrink)

In order to avoid the paradoxes of standard deontic logic, we have to give up the semantic construction that identifies obligatory status with presence in all elements of a subset of the set of possible worlds. It is proposed that deontic logic should instead be based on a preference relation, according to the principle that whatever is better than something permitted is itself permitted. Close connections hold between the logical properties of a preference relation and those of the deontic (...) logics that are derived from it in this way. The paradoxes of SDL can be avoided with this construction, but it is still an open question what type of preference relation is best suited to be used as a basis for deontic logic. (shrink)

This multiplex semantics incorporates multiple relations of deontic accessibility or multiple preference rankings on alternative worlds to represent distinct normative standards. This provides a convenient framework for deontic logic that allows conflicts of obligation, due either to conflicts between normative standards or to incoherence within a single standard. With the multiplex structures, two general senses of "ought" may be distinguished, an indefinite sense under which something is obligatory when it is enjoined by some normative standard and a (...) core sense for when something is enjoined by all normative standards. Multiple normative standards may themselves be given a preferential order; this leads to a concept of ranked obligation. This paper presents the foundations of this multiplex semantics and the propositional deontic logics they define. (shrink)

In this paper, we outline an approach to giving extensional truth-theoretic semantics for what have traditionally been seen as opaque sentential contexts. We outline an approach to providing a compositional truth-theoretic semantics for opaque contexts which does not require quantifying over intensional entities of any kind, and meets standard objections to such accounts. The account we present aims to meet the following desiderata on a semantic theory T for opaque contexts: (D1) T can be formulated in a (...) first-order extensional language; (D2) T does not require quantification over intensional entities­i.e., meanings, propositions, properties, relations, or the like­in its treatment of opaque contexts; (D3) T captures the entailment relations that hold in virtue of form between sentences in the language for which it is a theory; (D4) T has a finite number of axioms. If the approach outlined here is correct, it resolves a longstanding complex of problems in metaphysics, the philosophy of mind and the philosophy of language. (shrink)

Davidson approaches the notions of meaning and interpretation with the aim of characterizing semantic competence in the syntactically characterized natural language. The objective is to provide a truth-theory for a language, generating T-sentences expressed in the semantic metalanguage, so that each sentence of the object language receives an appropriate interpretation. Proceeding within the constraints of referential semantics, I will argue for the viability of reconstructing the notion of linguistic meaning within the Tarskian theory of reference. However, the view proposed (...) here involves a revision of Davidson’s con-ception of the object of semantic investigation. Taking (idealized) language-theories as the proper object of semantic characterization, provides solutions to outstanding problems in Davidson’s views, better approximates the practice in standard model-theoretic semantics, and incorporates the elements of semantic competence sought for in tradi¬tional theories of lexical analysis. Sources of evidence beyond those emphasized by Davidson will be invoked in order to allow for the selection of interpre¬tive T-sentences. In the final section, possible Quinean objections will be considered. (shrink)

In this paper we will discuss some epistemological aspects of lexical and terminological usage in the international arena, with special reference to the different rhetorics of the social and natural sciences. Sociolinguistic research confined to monolingual communities suggests that close-knit network structure is an important mechanism of language maintenance, in that speakers are able to form a cohesive group capable of resisting pressure, linguistic and social, from outside the group (MILROY, 1987). The concept of a linguistic norm in sociolinguistic theory (...) can be viewed here as the product of "cultural focusing". In the same way, intercultural communication is characterized by the spreading of standardized terminologies (ISO) and conceptual modes of thinking associated with a common scientific and technological practice in cross-border communities. A basic feature is frequent language interaction, based upon the need for scientific exchange and upon the extralinguistic models conveyed by powerful inter/transnational links and shared scientific paradigms. In this context, an international standard language could be seen as another case of cultural focusing, where intercultural contact would evolve along monocultural lines. However, a formally standardized language may conceal conceptual fuzziness, as has been shown for example in analyses of the kind of language used by social scientists (INTERCOCTA) or found in political rhetoric (ECCRDSS). The semantic and pragmatic differentiations of these latter forms of intercultural communication, and more generally, the interactions between the ordering of knowledge, the transfer of this knowledge through standardized language and the variety of cultural perceptions, call for a polylogic framework that accounts for their coexistence. (shrink)

This paper introduces and explores a conservative extension of inquisitive logic. In particular, weak negation is added to the standard propositional language of inquisitive semantics, and it is shown that, although we lose some general semantic properties of the original framework, such an enrichment enables us to model some previously inexpressible speech acts such as weak denial and ‘might’-assertions. As a result, a new modal logic emerges. For this logic, a Fitch-style system of natural deduction is formulated. The (...) main result of this paper is a theorem establishing the completeness of the system with respect to inquisitive semantics with weak negation. At the conclusion of the paper, the possibility of extending the framework to the level of first order logic is briefly discussed. (shrink)

A hyperintensional semantics for natural language is proposed which is agnostic about the question of whether propositions are sets of worlds or worlds are sets of propositions. Montague’s theory of intensional senses is replaced by a weaker theory, written in standard classical higher-order logic, of fine-grained senses which are in a many-to-one correspondence with intensions; Montague’s theory can then be recovered from the proposed theory by identifying the type of propositions with the type of sets of worlds and (...) adding an axiom to the effect that each world is the set of propositions which are true there. Senses are compositionally assigned to linguistic expressions by a categorial grammar with only two rule schemas, based on the implicative fragment of intuitionistic linear propositional logic, and a fully explicit grammar fragment is provided that illustrates the compositional assignment of sense to a variety of constructions, including dummy-subject constructions, infinitive complements, predicative adjectives and nominals, raising to subject, ‘tough-movement’, and quantifier scope ambiguities. Notably, the grammar and the derivations that it licenses never make reference to either worlds or to the extensions of senses. (shrink)

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due (...) in large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

-/- A variable binding term operator (vbto) is a non-logical constant, say v, which combines with a variable y and a formula F containing y free to form a term (vy:F) whose free variables are exact ly those of F, excluding y. -/- Kalish-Montague proposed using vbtos to formalize definite descriptions, set abstracts {x: F}, minimalization in recursive function theory, etc. However, they gave no sematics for vbtos. Hatcher gave a semantics but one that has flaws. We give a (...) correct semantic analysis of vbtos. We also give axioms for using them in deductions. And we conjecture strong completeness for the deductions with respect to the semantics. The conjecture was later proved independently by the authors and by Newton da Costa. -/- The expression (vy:F) is called a variable bound term (vbt). In case F has only y free, (vy:F) has the syntactic propreties of an individual constant; and under a suitable interpretation of the language vy:F) denotes an individual. By a semantic analysis of vbtos we mean a proposal for amending the standard notions of (1) "an interpretation o f a first -order language" and (2) " the denotation of a term under an interpretation and an assignment", such that (1') an interpretation o f a first -order language associates a set-theoretic structure with each vbto and (2') under any interpretation and assignment each vb t denotes an individual. (shrink)

The purpose of this article is show that second-order logic, as understood through standard semantics, is intimately bound up with set theory, or some other general theory of interpretations, structures, or whatever. Contra Quine, this does not disqualify second-order logic from its role in foundational studies. To wax Quinean, why should there be a sharp border separating mathematics from logic, especially the logic of mathematics?

In 1988, J. Ivlev proposed some (non-normal) modal systems which are semantically characterized by four-valued non-deterministic matrices in the sense of A. Avron and I. Lev. Swap structures are multialgebras (a.k.a. hyperalgebras) of a special kind, which were introduced in 2016 by W. Carnielli and M. Coniglio in order to give a non-deterministic semantical account for several paraconsistent logics known as logics of formal inconsistency, which are not algebraizable by means of the standard techniques. Each swap structure induces naturally (...) a non-deterministic matrix. The aim of this paper is to obtain a swap structures semantics for some Ivlev-like modal systems proposed in 2015 by M. Coniglio, L. Fariñas del Cerro and N. Peron. Completeness results will be stated by means of the notion of Lindenbaum–Tarski swap structures, which constitute a natural generalization to multialgebras of the concept of Lindenbaum–Tarski algebras. (shrink)

It is now a quarter of a century ago that Wolfgang Stegmfiller wrote his monograph 'Das Wahrheitsproblem und die Idee der Semantik' (1957) which dealt with Tarski's and Carnap's foundational work in the field of semantics. While this book is about the definition of the basic semantical concepts in artificial formal languages there is an article written a year earlier (1956) in which Stegmfiller addresses himself specifically to the relation between logic and natural language. Here he gives a logical (...) analysis of the standard structural expressions in language that are still of primary concern for current semantics: quantifiers, pronouns, articles, etc. The motives for such an analysis at that time were mainly philosophical: the aim was to expose the misconceptions and pitfalls of traditional philosophy arising from the disregard of various systematic semantic ambiguities in everyday language. Or as Stegmfiller puts it: Ober sie [i.e. einige nicht triviale F~ille von Vagheit in der A11tagssprache] Klarheit zu gewinnen, ist schon deshalb yon auBerordentlicher Bedeutung, weil Unkenntnis fiber sie zu schwersten philosophischen Verirrungen ffihren kann, namlich entweder der Unterlassung von berechtigten Fragestellungen, oder, was weit h~ufiger vorgekommen ist, der Formulierung yon falsch gestellten Fragen, denen gegenfiber man dann nur die Wahl hat, entweder fiberhaupt keine oder nur sinnlose Antworten zu geben. So logic was to regiment language. Sentences involving the copula and the above-mentioned structure words are assigned one or more unambiguous formal representations in an already interpreted formal language, usually the first order predicate calculus. The natural language expressions thereby receive a precise meaning, since the semantics of the formal language has been specified in advance, as is always assumed. This procedure has, of course, always been common practice, witness the typical syntactical argot of the mathematicians. It carries, however, an obvious methodological presupposition: it is the idea that the logic of our choice to which the formal representations belong is in some sense an adequate framework to express our thoughts. (shrink)

The history of building automated theorem provers for higher-order logic is almost as old as the field of deduction systems itself. The first successful attempts to mechanize and implement higher-order logic were those of Huet [13] and Jensen and Pietrzykowski [17]. They combine the resolution principle for higher-order logic (first studied in [1]) with higher-order unification. The unification problem in typed λ-calculi is much more complex than that for first-order terms, since it has to take the theory of αβη-equality into (...) account. As a consequence, the higher-order unification problem is undecidable and sets of solutions need not even always have most general elements that represent them. Thus the mentioned calculi for higher-order logic have take special measures to circumvent the problems posed by the theoretical complexity of higher-order unification. In this paper, we will exemplify the methods and proof- and model-theoretic tools needed for extending first-order automated theorem proving to higherorder logic. For the sake of simplicity take the tableau method as a basis (for a general introduction to first-order tableaux see part I.1) and discuss the higherorder tableau calculi HT and HTE first presented in [19]. The methods in this paper also apply to higher-order resolution calculi [1, 13, 6] or the higher-order matings method of Peter [3], which extend their first-order counterparts in much the same way. Since higher-order calculi cannot be complete for the standard semantics by Gödel’s incompleteness theorem [11], only the weaker notion of Henkin models [12] leads to a meaningful notion of completeness in higher-order logic. It turns out that the calculi in [1, 13, 3, 19] are not Henkin-complete, since they fail to capture the extensionality principles of higher-order logic. We will characterize the deductive power of our calculus HT (which is roughly equivalent to these calculi) by the semantics of functional Σ-models. To arrive at a calculus that is complete with respect to Henkin models, we build on ideas from [6] and augment HT with tableau construction rules that use the extensionality principles in a goal-oriented way.. (shrink)

This paper presents the first possible world semantics for concessive conditionals (i.e., even if A, C conditionals) constructed in a compositional way. First, the meaning of if is formalized through a semantics that builds on the proposal given by Stalnaker (1968). A major difference from Stalnaker’s approach is that irrelevant conditionals (i.e., conditionals where the antecedent and the consequent have no connection) are false in this new setting. Second, the meaning of even is analyzed through a formal (...) class='Hi'>semantics based on the notion of scale. This analysis overcomes the problems arising in standard approaches, in which even is analyzed with the help of pragmatic presuppositions. Finally, the two particles are combined in order to provide a formal analysis of even if. This theory predicts the major phenomena concerning the behavior of concessive conditionals and without any call to pragmatic explanations. More generally, this approach creates the possibility of a compositional analysis of other conditionals such as if then or only if forms. (shrink)

In order to improve the translation quality of complex English sentences, this paper investigated unknown words. First, two baseline models, the recurrent neural machine translation (RNMT) model and the transformer model, were briefly introduced. Then, the unknown words were identified and replaced based on WordNet and the semantic environment and input to the neural machine translation (NMT) model for translation. Finally, experiments were conducted on several National Institute of Standards and Technology (NIST) datasets. It was found that the transformer model (...) significantly outperformed the RNMT model, its average bilingual evaluation understudy (BLEU) value was 42.14, which was 6.96 higher than the RNMT model, and its translation error rate (TER) was also smaller. After combining the intelligent algorithm, the BLEU values of both models improved, and the TER became smaller; the average BLEU value of the transformer model combined with the intelligent algorithm was 43.7, and the average TER was 57.68. The experiment verifies that the transformer model combined with the intelligent algorithm is reliable in translating complex sentences and can obtain higher-quality translation results. (shrink)

Some initial motivations for the Guarded Fragment still seem of interest in carrying its program further. First, we stress the equivalence between two perspectives: (a) satisfiability on standard models for guarded first-order formulas, and (b) satisfiability on general assignment models for arbitrary first-order formulas. In particular, we give a new straightforward reduction from the former notion to the latter. We also show how a perspective shift to general assignment models provides a new look at the fixed-point extension LFP(FO) of (...) first-order logic, making it decidable. Next, we relate guarded syntax to earlier quantifier restriction strategies for achieving effective axiomatizability in second-order logic – pointing at analogies with ‘persistent’ formulas, which are essentially in the Bounded Fragment of many-sorted first-order logic. Finally, we look at some further unexplored directions, including the systematic use of ‘quasi-models’ as a semantics by itself. (shrink)

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard (...) and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula in terms of neighborhood frames with constant domains. (...) The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics. We conclude by introducing general first order neighborhood frames and we offer a general completeness result in terms of them which circumvents some well-known problems of propositional and first order neighborhood semantics. We argue that the semantical program that thus arises surpasses both in expressivity and adequacy the standard Kripkean approach, even when it comes to the study of first order normal systems. (shrink)

Cylindric set algebras are algebraizations of certain logical semantics. The topic surveyed here, i.e. probabilities defined on cylindric set algebras, is closely related, on the one hand, to probability logic (to probabilities defined on logical formulas), on the other hand, to measure theory. The set algebras occuring here are associated, in particular, with the semantics of first order logic and with non-standard analysis. The probabilities introduced are partially continous, they are continous with respect to so-called cylindric sums.

William Whewell’s account of induction differs dramatically from the one familiar from twentieth-century debates. I argue that Whewell’s induction can be usefully understood by comparing the difference between his views and more standard accounts to contemporary debates between semantic and syntactic views of theories: rather than understanding inductive inference as capturing a relationship between sentences or propositions, Whewell understands it as a method for constructing a model of the world. The difference between this view and the more familiar picture (...) of induction is reflected in other aspects of Whewell’s philosophy of science, particularly his treatment of consilience and the order of discovery. (shrink)

In the past couple of decades, there were a few major attempts to establish the thesis of pragmatic infringement – that a significant pragmatic ingredient figures significantly in the truth-conditions for knowledge-ascriptions. As candidates, epistemic contextualism and Relativism flaunted conversational standards, and Stanley's SSI promoted stakes. These conceptions were propelled first and foremost by obviously pragmatic examples of knowledge ascriptions that seem to require a pragmatic component in the truth-conditions of knowledge ascriptions in order to be accounted for. However, if (...) such examples can be adequately explained purely pragmatically, the need that such examples seem to invoke in such a pragmatic component is undermined. Here I lay out a new pragmatic account – of action-directed pragmatics, offering a different account of such examples and their pragmatic flavor. If adequate, it obviates the above need for pragmatic enrichment. Specifically, I develop and I argue for a well-entrenched pragmatic feature – that of a steering role. The assertions of knowledge ascriptions and their denials as well as of simple assertions (that don't invoke knowledge) play a pragmatic role of steering audiences in joint deliberational setups toward, or away from, the speaker's preferred action (as well as in assertions of 'I am sure', of epistemic modals, taste assertions, causal assertions, and more.) Various features and consequences of this account are drawn. Specifically, I explain why in the bank example (and related ones) the husband, in denying the knowledge ascription, neither lies nor misleads. (shrink)

Classical first-order logic can be extended in two different ways to serve as a foundation for mathematics: introduce higher orders, type theory, or introduce sets. As it happens, both approaches have natural analogs for quantified modal logics, both approaches date from the 1960’s, one is not very well-known, and the other is well-known as something else. I will present the basic semantic ideas of both higher order intensional logic, and intensional set theory. Before doing so, I’ll quickly sketch some necessary (...) background material from quantified modal logic. Except for standard material concerning propositional modal logics, the paper is essentially self-contained. (shrink)

In a recent paper Lee Walters criticizes a number of philosophers – including Gundersen – for committing a ‘failure in the argumentative strategy’ when they attempt to amend the standard Lewis semantics for counterfactuals in order to avoid the so‐called principle of Conjunction Conditionalization. In this article we defend a Gundersen‐style probability‐based semantics against Walter's major misgivings: that it is not logically conservative, that it is committed to the Connection Hypothesis, and that it cannot deal satisfactory with (...) irrelevant semi‐factuals. (shrink)

This paper argues that the innovation of an ordering source parameter in the standard Lewis-Kratzer semantics for modals was a mistake, at least for English auxiliaries like ‘ought’, and that a simpler dyadic semantics (as proposed in my earlier work) provides a superior account of normative uses of modals. I programmatically investigate problems arising from (i) instrumental conditionals, (ii) gradability and “weak necessity”, (iii) information-sensitivity, and (iv) conflicts, and show how the simpler semantics provides intuitive (...) solutions given three basic moves: (1) an end-relational analysis of normative modality, (2) a “most” analysis of “weak necessity”, and (3) well-motivated appeals to pragmatics. I conclude with metasemantic observations about the desiderata for a semantics for ‘ought’. (shrink)

The aim of this paper is to show that it’s not a good idea to have a theory of truth that is consistent but ω-inconsistent. In order to bring out this point, it is useful to consider a particular case: Yablo’s Paradox. In theories of truth without standard models, the introduction of the truth-predicate to a first order theory does not maintain the standard ontology. Firstly, I exhibit some conceptual problems that follow from so introducing it. Secondly, I (...) show that in second order theories with standard semantics the same procedure yields a theory that doesn’t have models. So, while having an ω- inconsistent theory is a bad thing, having an unsatisfiable theory of truth is actually worse. This casts doubts on whether the predicate in question is, after all, a truthpredicate for that language. Finally, I present some alternatives to prove an inconsistency adding plausible principles to certain theories of truth. (shrink)

The monoidal t-norm based logic MTL is obtained from Hájek''s Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard (...) and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)