Propositional quantifiers are quantifiers binding proposition letters, understood as variables. This Element introduces propositional quantifiers and explains why they are especially interesting in the context of propositional modal logics. It surveys the main results on propositionally quantified modal logics which have been obtained in the literature, presents a number of open questions, and provides examples of applications of such logics to philosophical problems.

In discussing propositional quantifiers we have considered two kinds of variables: variables occupying the argument places of connectives, and variables occupying the argument places of predicates.We began with languages which contained the first kind of variable, i.e., variables taking sentences as substituends. Our first point was that there appear to be no sentences in English that serve as adequate readings of formulas containing propositional quantifiers. Then we showed how a certain natural and illuminating extension of English by prosentences did provide (...) perspicuous readings. The point of introducing prosentences was to provide a way of making clear the grammar of propositional variables: propositional variables have a prosentential character — not a pronominal character. Given this information we were able to show, on the assumption that the grammar of propositional variables in philosopher's English should be determined by their grammar in formal languages (unless a separate account of their grammar is provided), that propositional variables can be used in a grammatically and philosophically acceptable way in philosophers' English. According to our criteria of well-formedness Carnap's semantic definition of truth does not lack an ‘essential’ predicate - despite arguments to the contrary. It also followed from our account of the prosentential character of bound propositional variables that in explaining propositional quantification, sentences should not be construed as names.One matter we have not discussed is whether such quantification should be called ‘propositional’, ‘sentential’, or something else. As our variables do not range over (they are not terms) either propositions, or sentences, each name is inappropriate, given the usual picture of quantification. But we think the relevant question in this context is, are we obtaining generality with respect to propositions, sentences, or something else?Because people have argued that all bound variables must have a pronominal character, we presented and discussed in the third section languages in which the variables take propositional terms as substituends. In our case we included names of propositions, that-clauses, and names of sentences in the set of propositional terms. We made a few comparisons with the languages discussed in the second section. We showed among other things how a truth predicate could be used to obtain generality. In contrast, the languages of the second section, using propositional variables, obtain generality without the use of a truth predicate. (shrink)

In this paper ontological implications of the Barcan formula and its converse will be discussed at the conceptual and technical level. The thesis that will be defended is that sentential moods are not ontologically neutral since the rejection of ontological implications of Barcan formula and its converse is a condition of a possibility of the imperative mood. The paper is divided into four sections. In the first section a systematization of semantical systems of quantified modal logic is introduced for (...) the purpose of making explicit their ontological presuppositions. In this context Jadacki's ontological difference between being and existence is discussed and analyzed within the framework of hereby proposed system of quantified modal logic. The second section discusses ontological implications of the Barcan formula and its converse within the system accommodating the difference between being and existence. The third section presents a proof of incompatibility of the Barcan formula and its converse with the use of imperatives. In the concluding section, a thesis on logical pragmatics foreclosing the dilemma between necessitism and contingentism is put forward and defended against some objections. (shrink)

Classical logic, of first or higher order, is extended with sentential operators and quantifiers, interpreted substitutionally over unrestricted substitution class. Operators mark a single layered, consistent metalanguage. Self-reference, arising from substitutional quantification over sentences, allows to express paradoxes which, unlike contradictions, do not lead to explosion. Semantics of the resulting language, using semi-kernels of digraphs, is non-explosive yet two-valued and has classical semantics as a special case for clasically consistent theories. A complete reasoning is obtained by extending LK with (...) two rules for sentential quantifiers. Adding (cut) yields a complete system for the explosive semantics. (shrink)

One well known problem regarding quantifiers, in particular the 1storder quantifiers, is connected with their syntactic categories and denotations. The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial languages generated (...) by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functorfunction defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility. (shrink)

Exhumation and study of the 1945 paradox of confirmation brings out the defect of its formulation. In the context of quantifier conditional-probability logic it is shown that a repair can be accomplished if the truth-functional conditional used in the statement of the paradox is replaced with a connective that is appropriate to the probabilistic context. Description of the quantifier probability logic involved in the resolution of the paradox is presented in stages. Careful distinction is maintained between a formal (...) logic language and its semantics, as the same language may be outfitted with different semantics. An acquaintance with sections 1? 5 of Hailperin (2006) covering the sentential aspects of probability logic is assumed as background information for quantifier probability logic. (shrink)

One well known problem regarding quantifiers, in particular the 1st order quantifiers, is connected with their syntactic categories and denotations.The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial languages generated (...) by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functor-function defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility. (shrink)

In this paper we present a logical system able to compute the semantics of both declarative and interrogative sentences. Our proposed analysis takes place at both the sentential and at the discourse level. We use syntactic inference on the sentential level for declarative sentences, while the discourse level comes into play for our treatment of questions. Our formalization uses a type logic sensitive to both the syntactic and semantic properties of natural language. We will show how an account (...) of the linguistic data follows naturally from the logical relations inherent in the type logic. (shrink)

The paper proposes an account of the contrast (noticed in Karttunen 1976) between the interpretations of the following two discourses: Harvey courts a girl at every convention. {She is very pretty. vs. She always comes to the banquet with him.}. The initial sentence is ambiguous between two quantifier scopings, but the first discourse as a whole allows only for the wide-scope indefinite reading, while the second allows for both. This cross-sentential interaction between quantifier scope and anaphora is (...) captured by means of a new dynamic system couched in classical type logic, which extends Compositional DRT (Muskens 1996) with plural information states (modeled, following van den Berg 1996, as sets of variable assignments). Given the underlying type logic, compositionality at sub-clausal level follows automatically and standard techniques from Montague semantics become available. The paper also shows that modal subordination (A wolf might come in. It would eat Harvey first) can be analyzed in a parallel way, i.e. the system captures the anaphoric and quantificational parallels between the individual and modal domains argued for in Stone (1999) (among others). In the process, we see that modal / individual-level quantifiers enter anaphoric connections as a matter of course, usually functioning simultaneously as both indefinites and pronouns. (shrink)

A new approach to truth is offered which dispenses with the truth predicate, and replaces it with a special kind of quantifier which simultaneously binds variables in sentential and nominal positions. The resulting theory of truth for a (first-order) language is shown to be able to handle blind truth ascriptions, and is shown to be compatible with a characterization of the semantic and syntactic principles governing that language. Comparisons with other approaches to truth are drawn. An axiomatization of (...) AU-quantifiers and a model theory for them is given, and an appendix contains a completeness proof. (shrink)

Combinatory logic (Curry and Feys 1958) is a “variable-free” alternative to the lambda calculus. The two have the same expressive power but build their expressions differently. “Variable-free” semantics is, more precisely, “free of variable binding”: it has no operation like abstraction that turns a free variable into a bound one; it uses combinators—operations on functions—instead. For the general linguistic motivation of this approach, see the works of Steedman, Szabolcsi, and Jacobson, among others. The standard view in linguistics is that reflexive (...) and personal pronouns are free variables that get bound by an antecedent through some coindexing mechanism. In variable free semantics the same task is performed by some combinator that identifies two arguments of the function it operates on (a duplicator). This combinator may be built into the lexical semantics of the pronoun, into that of the antecedent, or it may be a free-floating operation applicable to predicates or larger chunks of texts, i.e. a typeshifter. This note is concerned with the case of cross-sentential anaphora. It adopts Hepple’s and Jacobson’s interpretation of pronouns as identity maps and asks how this can be extended to the cross-sentential case, assuming the dynamic semantic view of anaphora. It first outlines the possibility of interpreting indefinites that antecede non-ccommanded pronouns as existential quantifiers enriched with a duplicator. Then it argues that it is preferable to use the duplicator as a type-shifter that applies “on the fly”. The proposal has consequences for two central ingredients of the classical dynamic semantic treatment: it does away with abstraction over assignments and with treating indefinites as inherently existentially quantified. However, cross-sentential anaphora remains a matter of binding, and the idea of propositions as context change potentials is retained. (shrink)

The paper proposes a novel analysis of quantificational subordination, e.g. Harvey courts a woman at every convention. {She is very pretty. vs. She always comes to the banquet with him.} (Karttunen 1976), in particular of the fact that the indefinite in the initial sentence can have wide or narrow scope, but the first discourse as a whole allows only for the wide scope reading, while the second discourse allows for both readings. The cross-sentential interaction between scope and anaphora is (...) captured in terms of structured anaphora to quantifier domains, formalized in a new dynamic system couched in classical type logic; given the underlying type logic, Montague-style compositionality at sub-clausal level follows automatically. Modal subordination (Roberts 1987) is analyzed in a parallel way, thereby capturing the parallels between the individual and modal domains argued for in Stone (1999). Several other phenomena are analyzed in terms of structured anaphora: exceptional wide scope, weak / strong donkey readings, anaphoric / uniqueness-implying definite descriptions and interactions between same / different and quantifier scope. (shrink)

ABSTRACT In several publications, François Recanati argues that time, world, location, and similar constituents are not arguments of the verb, although they do affect truth conditions. However, he points out that this fact does not decide the debate regarding whether these notions are represented as sentential operators variables bound by quantifiers, as both approaches can be made compatible with such non-arguments. He makes these points using philosophical arguments; in this paper I use linguistic evidence from a variety of languages. (...) Specifically, I demonstrate that time, world, location, and person behave syntactically and semantically the same across languages. Hence, either all are arguments, or neither are; and the evidence points to the latter position. The cross-linguistic data give rise to a generalization, which connects the availability of quantifying over times, worlds, locations, or persons with the availability of indexical expressions of the corresponding type. For example, a language that does not have temporal indexicals cannot have tense. I demonstrate that this generalization follows naturally from the quantifier approach, but would remain a mystery under the operator view. Therefore, the cross-linguistic evidence argues in the favor of the quantificational approach. (shrink)

Inspired by the grammar of natural language, the paper presents a variant of first-order logic, in which quantifiers are not sentential operators, but are used as subnectors . A quantified term formed by a subnector is an argument of a predicate. The logic is defined by means of a meaning-conferring natural-deduction proof-system, according to the proof-theoretic semantics program. The harmony of the I/E-rules is shown. The paper then presents a translation, called the Frege translation, from the defined logic to (...) standard first-order logic, and shows that the proof-theoretic meanings of both logics coincide. The paper criticizes Frege’s original regimentation of quantified sentences of natural language, and argues for advantages of the proposed variant. (shrink)

x1. In this paper we are concerned with some special theories of non-Fregean logic which determine correspondingly spe- cial elementary, i.e. axiomatic, strengthenings of that logic. The construc- tions presented here extend what has been done in [1] and [2] within the SCI-language. They also apply to the non-Fregean logic in comprehensive languages of kind W involving quantiers binding sentential variables and nominal variables, as well, . However, for the sake of sim- plicity, the underlying language considered here is (...) the SCI-language with quantiers 8 and 9. The reader may inspect the case when the numerical quantiers also occur in the language. (shrink)

We study the non-Fregean propositional logic with propositional quantifiers, denoted by $\mathsf{SCI}_{\mathsf{Q}}$. We prove that $\mathsf{SCI}_{\mathsf{Q}}$ does not have the finite model property and that it is undecidable. We also present examples of how to interpret in $\mathsf{SCI}_{\mathsf{Q}}$ various mathematical theories, such as the theory of groups, rings, and fields, and we characterize the spectra of $\mathsf{SCI}_{\mathsf{Q}}$-sentences. Finally, we present a translation of $\mathsf{SCI}_{\mathsf{Q}}$ into a classical two-sorted first-order logic, and we use the translation to prove some model-theoretic properties of (...) $\mathsf{SCI}_{\mathsf{Q}}$. (shrink)

Alethic pluralists often claim that accommodating certain alethic platitudes motivates rejecting deflationism in favour of a pluralist inflationism about truth. Deflationists claim that the logical role of the truth predicate, viz providing something equivalent to variables for sentence-in-use positions and quantifiers governing them, is sufficient to account for the appeal to truth in the alethic platitudes. Surprisingly, however, most deflationists face an insufficiently acknowledged problem with respect to explaining how this mode of generalizing works. The standard substitutional or higher-order interpretations (...) of sentential quantifiers and variables do not meet two desiderata that we claim any adequate account of them must satisfy. To address this issue, we review and extend A. N. Prior's adverbial understanding of sentential quantification, explain how it satisfies the desiderata, and respond to some objections. This shows that deflationists can accommodate and account for the alethic platitudes by applying this non-nominal understanding of generalizing on sentence positions. (shrink)

This chapter contains sections titled: Sentential Connectives Quantifiers and Quantified Noun Phrases Proper Names and Individual Constants Adjectives Adverbs and Events Utterance Modifiers Logical Form as Grammatical Form Summary.

Classical modalism about essence is the view that essence can be analysed in modal terms. Despite Kit Fine's influential critique, no general refutation of classical modalism has yet been given. In the first part of the paper, I provide such a refutation by showing that the notion of essence cannot be analysed in terms of any sentential operator definable in the language of standard quantified modal logic. As a reaction to Fine's critique, some have defended sophisticated modalism, which attempts (...) to analyse essence in an enriched modal language quantifying over both possible and impossible worlds. In the second part of the paper, I argue that sophisticated modalism falls prey to variations on Fine's counterexamples to classical modalism. I conclude that the most promising approaches to understanding the notion of essence consist in taking essence either as primitive or as analysable via a combination of modal and non-modal notions. (shrink)

The main goal of this paper is to present and compare two approaches to formalizing cross-world comparisons like John might have been taller than he is in quantified modal logics. One is the standard method employing degrees and graded positives, according to which the example just given is to be paraphrased as something like The height that John has is such that he might have had a height greater than it, which is amenable to familiar formalization strategies with respect to (...) quantified modal logic. The other approach, based on subjunctive modal logic, mimics the mixed indicative-subjunctive patterns typical of cross-world comparisons in many natural languages by means of explicit mood markers. This latter approach is new and should, for various reasons, appeal to linguists and philosophers. Along the way, I argue that attempts to capture cross-world comparison by means of sentential operators are either inadequate or subject to substantive logical and philosophical objections. (shrink)

In the opening sections of this paper, we defined ambiguity in terms of distinct sentences (for a single sentence-string) with, in particular, distinct sets of truth conditions for the corresponding negative sentence-string. Lexical vagueness was defined as equivalent to disjunction, for under conditions of the negation of a sentence-string containing such an expression, all the relevant more specific interpretations of the string had also to be negated. Yet in the case of mixed quantification sentences, the strengthened, more specific, interpretations of (...) some such positive string are not all of them necessarily implied to be false if the corresponding negative sentence-string is asserted. On the contrary, as we saw in section 6, a negative sentence-string can be used to deny one of the more specific interpretations of the corresponding positive string without also denying other weaker interpretations of that same string. One might therefore argue that the only empirical evidence availble for assessing quantified sentences suggests clearly that these sentence-strings are ambiguous. Indeed logicians, many of whom restrict their attention to propositions, MUST recognise logical ambiguity at this point. For the contextualisation of the negative sentences in section 6 showed that it was possible to assert the falsity of some proposition P expressed by the sentence S while asserting a further proposition which was compatible with the truth of S. However the corresponding conclusion that such sentence-strings are sententially ambiguous is not a necessary conclusion for the linguist: for the alternative account of postulating a single semantic representation plus a set of semantic procedures is also compatible with the negation evidence. Moreover we have seen independent reasons for thinking that if sentential ambiguity is assumed to be in one-to-one correspondence with what we should now call logical ambiguity, a considerable body of generalisations is lost. For the maximal ambiguity account, it should be recalled, is committed to assigning at least thirteen distinct propositions and hence thirteen distinct sentence outputs for every sentence-string containing no more than two quantifiers, for three out of the four interpretations originally outlined in this paper can be understood with each numeral taken either in an ’exactly’ sense or in an ’at least’ sense. Moreover there is no explanation of why just these interpretations are available-they are merely an arbitrary list, no more connected than are the two interpretations of John saw her duck, with no reason to predict that the ambiguity would carry over from language to language. If then it is granted that an ambiguity account fails to capture appropriate generalisations, only two alternative accounts of mixed quantification sentence-strings remain viable-an analysis proposing an initial co-ordinate logical form like the logical form III, which is the strongest form compatible with each of the propositional interpretations of sentence-string Two examiners marked six scripts, and the radically weak form in which only existential quantification (both over sets and over members of those sets) is invoked. Since there are strong arguments to suggest that the procedures which both analyses require are semantic, there seems no reason not to adopt the radical vagueness account, with its considerably greater simplicity. (shrink)

This book is an examination of the foundations and applications of the program of truth-theoretic semantics for natural languages introduced in 1967 by Donald Davidson in his classic paper “Truth and Meaning.” This is the second of two books on Donald Davidson’s central philosophical project. The first, Donald Davidson: Meaning, Truth, Language and Reality (Oxford: Oxford University Press, 2005), dealt with the basic framework of Davidson’s truth-theoretic approach to providing a meaning theory for a natural language, and then with his (...) development of his general project in the philosophy of language and theory of meaning by way of the analysis of the empirical confirmation of a meaning theory for a speaker, from the stance of the radial interpreter, that is, the interpreter who starts out without any detailed knowledge of the speakers’ meanings or attitudes. This book turns to the development of details of Davidson’s semantic program, specifically, the pursuit of the project of providing a compositional semantic theory for natural languages by making strategic use of an appropriately adapted Tarski-style axiomatic truth theory. Such a theory aims to provide a specification of the truth conditions of any felicitous utterance of a natural language sentence which serves to interpret it, and exhibits in the proof of the theorem that gives the truth conditions the compositional structure of the sentence and how its truth depends on its context of utterance. The book develops the basic framework, applies it to quantifiers, proper names, indexicals, simple and complex demonstratives, quotation, adjectives and adverbs, simple tenses of state and event verbs, temporal adverbs and quantifiers, tense in sentential complements and perfect tenses, opaque contexts, especially indirect discourse and attitude sentences, non-declarative sentences, semantic structure and logical form, and discusses the relation of the framework the metaphysics of truth. (shrink)

‘Quantified pure existentials’ are sentences (e.g., ‘Some things do not exist’) which meet these conditions: (i) the verb EXIST is contained in, and is, apart from quantificational BE, the only full (as against auxiliary) verb in the sentence; (ii) no (other) logical predicate features in the sentence; (iii) no name or other sub-sentential referring expression features in the sentence; (iv) the sentence contains a quantifier that is not an occurrence of EXIST. Colin McGinn and Rod Girle have alleged (...) that standard first-order logic cannot adequately deal with some such existentials. The article defends the view that it can. (shrink)

Bare plural generic sentences pervade ordinary talk. And yet it is extremely controversial what semantics to assign to such sentences. In this paper, I achieve two tasks. First, I develop a novel classification of the various standard uses to which bare plurals may be put. This “variety data” is important—it gives rise to much of the difficulty in systematically theorizing about bare plurals. Second, I develop a novel account of bare plurals, the radical account. On this account, all bare plurals (...) fail to express propositions. The content of a bare plural has to be pragmatically “completed” by a speaker in order for her to make an assertion. At least the content of a quantifier expression has to be supplied. But sometimes, the content of a sentential operator or modal verb is also supplied. The radical account straightforwardly explains the variety data: Speakers’ communicative intentions vary wildly across different contexts. (shrink)

Consider two standard quantified modal languages A and P whose vocabularies comprise the identity predicate and the existence predicate, each endowed with a standard S5 Kripke semantics where the models have a distinguished actual world, which differ only in that the quantifiers of A are actualist while those of P are possibilist. Is it possible to enrich these languages in the same manner, in a non-trivial way, so that the two resulting languages are equally expressive-i.e., so that for each sentence (...) of one language there is a sentence of the other language such that given any model, the former sentence is true at the actual world of the model iff the latter is? Forbes (1989) shows that this can be done by adding to both languages a pair of sentential operators called Vlach-operators, and imposing a syntactic restriction on their occurrences in formulas. As Forbes himself recognizes, this restriction is somewhat artificial. The first result I establish in this paper is that one gets sameness of expressivity by introducing infinitely many distinct pairs of indexed Vlach-operators. I then study the effect of adding to our enriched modal languages a rigid actuality operator. Finally, I discuss another means of enriching both languages which makes them expressively equivalent, one that exploits devices introduced in Peacocke (1978). Forbes himself mentions that option but does not prove that the resulting languages are equally expressive. I do, and I also compare the Peacockian and the Vlachian methods. In due course, I introduce an alternative notion of expressivity and I compare the Peacockian and the Vlachian languages in terms of that other notion. (shrink)

Some philosophers argue that the principles of human reasoning are impeccable, and that mistakes are no more than momentary lapses in “information processing”. This article makes a case to the contrary. It shows that human reasoners commit systematic fallacies. The theory of mental models predicts these errors. It postulates that individuals construct mental models of the possibilities to which the premises of an inference refer. But, their models usually represent what is true in a possibility, not what is false. This (...) procedure reduces the load on working memory, and for the most part it yields valid inferences. However, as a computer program implementing the theory revealed, it leads to fallacious conclusions for certain inferences—those for which it is crucial to represent what is false in a possibility. Experiments demonstrate the variety of these fallacies and contrast them with control problems, which reasoners tend to get right. The fallacies can be compelling illusions, and they occur in reasoning based on sentential connectives such as “if” and “or”, quantifiers such as “all the artists” and “some of the artists”, on deontic relations such as “permitted” and “obligated”, and causal relations such as “causes” and “allows”. After we have reviewed the principal results, we consider the potential for alternative accounts to explain these illusory inferences. And we show how the illusions illuminate the nature of human rationality. (shrink)

An introduction to sentential logic and first-order predicate logic with identity, logical systems that significantly influenced twentieth-century analytic philosophy. After working through the material in this book, a student should be able to understand most quantified expressions that arise in their philosophical reading. -/- This books treats symbolization, formal semantics, and proof theory for each language. The discussion of formal semantics is more direct than in many introductory texts. Although forall x does not contain proofs of soundness and completeness, (...) it lays the groundwork for understanding why these are things that need to be proven. -/- The book highlights the choices involved in developing sentential and predicate logic. Students should realize that these two are not the only possible formal languages. In translating to a formal language, we simplify and profit in clarity. The simplification comes at a cost, and different formal languages are suited to translating different parts of natural language. (shrink)

The meaning of 'most' can be described in many ways. We offer a framework for distinguishing semantic descriptions, interpreted as psychological hypotheses that go beyond claims about sentential truth conditions, and an experiment that tells against an attractive idea: 'most' is understood in terms of one-to-one correspondence. Adults evaluated 'Most of the dots are yellow', as true or false, on many trials in which yellow dots and blue dots were displayed for 200 ms. Displays manipulated the ease of using (...) a 'one-to-one with remainder' strategy, and a strategy of using the Approximate Number System to compare of (approximations of) cardinalities. Interpreting such data requires care in thinking about how meaning is related to verification. But the results suggest that 'most' is understood in terms of cardinality comparison, even when counting is impossible. (shrink)

Russell argued, famously, that definite descriptions are not logical constituents of the sentences in which they appear. In neither of the following should we suppose that the definite description picks anything out: The King of France is bald The Prince of Wales is bald Since France is a republic, nothing could be picked out by the first; and if the semantic structures of each are the same, it cannot be the function of the second to pick anything out either. On (...) the alternative semantics developed in his 1905 article 'On Denoting', definite descriptions do not have meaning in isolation; they have meaning only in the context of a whole sentence. Andrew Botterell and Robert Stainton have pointed out that this conclusion appears to be at odds with the phenomenon of unembedded definite descriptions, in which definite descriptions are uttered, meaningfully, without accompanying predicates. For example, it is possible to utter 'The last temptation' on its own and in doing so express a proposition. Since definite descriptions can be used in this way, how can it be right to claim, with Russell, that they lack meaning in isolation? The present paper seeks to show how a Russellian semantics for definite descriptions is entirely compatible with the phenomenon of unembedded definite descriptions. In particular, Botterell and Stainton are wrong to think that generalized quantifier semantics is better able to cope with the phenomenon than a more authentically Russellian syncategorematic semantics. (shrink)

Two of the main debates in philosophy of language concerning time and tense are the debate about the semantics of the tenses in the English language and the debate over whether propositions can be transiently true or false as opposed to always being eternally true or false. The latter quarrel is also known as the "temporalism–eternalism debate." Given standard semantics, the two debates are not logically independent, as we will see. Those who believe propositions are eternally true or false needn’t (...) treat the tenses as operators. Their opponents, on the other hand, appear to be committed to an operator theory of the tenses, given a standard semantic framework. In this chapter I will focus primarily on these two debates. (shrink)

It is argued that propositions cannot be the compositional semantic values of sentences (in context) simply due to issues stemming from the compositional semantics of modal operators (or modal quantifiers). In particular, the fact that the arguments for double indexing generalize to multiple indexing exposes a fundamental tension in the default philosophical conception of semantic theory. This provides further motivation for making a distinction between two sentential semantic contents—what (Dummett 1973) called “ingredient sense” and “assertoric content”.

Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and to (...) computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives. Predicates are constructed from two atomic binary predicates (which denote the relations of identity and set-theoretic membership) by repeated applications of four operators that are analogues of the well-known operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics. The book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; first-order logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and decision problems in equational logic. (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)

This set brings together the most important contributions to semantic theory ranging from Gottlob Frege's 1892 essay "On Sense and Reference" to recent cutting-edge scholarship from leading journals in the field. The collection is structured around three major themes: * Fundamental notions, the relations between semantics and grammar and the relations between meaning and cognition * The semantics of basic grammatical constructions and structures, such as the semantics of determiners, nouns, adjectives and related topics including quantifier scope and the (...) definiteness/indefiniteness distinction. * Topics related to sentential and discourse semantics: events and thematic roles, tense, aspect and modality, negation, ellipsis, questions, pronouns and the semantics/pragmatics interface. (shrink)

Bare plurals (dogs) behave in ways that quantified plurals (some dogs) do not. For instance, while the sentence John owns dogs implies that John owns more than one dog, its negation John does not own dogs does not mean “John does not own more than one dog”, but rather “John does not own a dog”. A second puzzling behavior is known as the dependent plural reading; when in the scope of another plural, the ‘more than one’ meaning of the plural (...) is not distributed over, but the existential force of the plural is. For example, My friends attend good schools requires that each of my friends attend one good school, not more, while at the same time being inappropriate if all my friends attend the same school. This paper shows that both these phenomena, and others, arise from the same cause. Namely, the plural noun itself does not assert ‘more than one’, but rather the plural denotes a predicate that is number neutral (unspecified for cardinality). The ‘more than one’ meaning arises as an scalar implicature, relying on the scalar relationship between the bare plural and its singular alternative, and calculated in a sub-sentential domain; namely, before existential closure of the event variable. Finally, implications of this analysis will be discussed for the analysis of the quantified noun phrases that interact with bare plurals, such as indefinite numeral DPs (three boys), and singular universals (every boy). (shrink)

In this paper, I will discuss boulesic and deontic logic and the relationship between these branches of logic. By ‘boulesic logic,’ or ‘the logic of the will,’ I mean a new kind of logic that deals with ‘boulesic’ concepts, expressions, sentences, arguments and systems. I will concentrate on two types of boulesic expression: ‘individual x wants it to be the case that’ and ‘individual x accepts that it is the case that.’ These expressions will be symbolised by two sentential (...) operators that take individuals and sentences as arguments and give sentences as values. Deontic logic is a relatively well-established branch of logic. It deals with normative concepts, sentences, arguments and systems. In this paper, I will show how deontic logic can be grounded in boulesic logic. I will develop a set of semantic tableau systems that include boulesic and alethic operators, possibilist quantifiers and the identity predicate; I will then show how these systems can be augmented by a set of deontic operators. I use a kind of possible world semantics to explain the intended meaning of our formal systems. Intuitively, we can think of our semantics as a description of the structure of a perfectly rational will. I mention some interesting theorems that can be proved in our systems, including some versions of the so-called hypothetical imperative. Finally, I show that all systems that are described in this paper are sound and complete with respect to their semantics. (shrink)

Some say that presentism precludes time travel into the past since it implies that the past does not exist, but this is a bad argument. Presentism says that only currently existing entities exist, and that the only properties and relations those entities instantiate are those that they currently instantiate. This does in a sense imply that the past does not exist. But if that precluded time travel into the past, it would also preclude the one-second-per-second “time travel” into the future (...) that is ordinary persistence, for presentism accords the future the same ontological status as the past. Instead of quantifying over past and future objects and events, presentists speak a tensed language, regimented with primitive sentential tense operators. For a presentist, a persisting person is one who did exist, and who will exist. Regimented, these claims become: it was the case that she exists, and it will be the case that she exists. The presentist may then apply the same strategy to time travel proper. Suppose Katy travels back to the time of the dinosaurs. The presentist can say that it was the case two hundred million years ago that Katy exists. This claim, which consists of a present-tense statement “Katy exists” embedded within the past tense operator it was the case two hundred million years ago that, is exactly the sort of statement about time that a presentist is free to accept. This has all been made clear by Simon Keller and Michael Nelson ( ). In addition to rebutting the bad argument against the consistency of presentism and time travel, Keller and Nelson argue positively in favor of consistency by showing how to translate David Lewis’s ( ) account of time travel into the presentist’s tensed language. The appearance of con ict between presentism and time travel, they argue, is due only to the fact that most defenders of time travel (for example Lewis) have tended to phrase their defenses in nonpresentist terms. As much as I applaud their rebuttal of the bad argument, I wish to sound a note of caution.. (shrink)

This paper introduces the logic _Q__L__E__T_ _F_, a quantified extension of the logic of evidence and truth _L__E__T_ _F_, together with a corresponding sound and complete first-order non-deterministic valuation semantics. _L__E__T_ _F_ is a paraconsistent and paracomplete sentential logic that extends the logic of first-degree entailment (_FDE_) with a classicality operator ∘ and a non-classicality operator ∙, dual to each other: while ∘_A_ entails that _A_ behaves classically, ∙_A_ follows from _A_’s violating some classically valid inferences. The semantics of (...) _Q__L__E__T_ _F_ combines structures that interpret negated predicates in terms of anti-extensions with first-order non-deterministic valuations, and completeness is obtained through a generalization of Henkin’s method. By providing sound and complete semantics for first-order extensions of _FDE_, _K3_, and _LP_, we show how these tools, which we call here the method of _anti-extensions + valuations_, can be naturally applied to a number of non-classical logics. (shrink)

The paper proposes a semantic value for the logical constants (connectives and quantifiers) within the framework of proof-theoretic semantics, basic meaning on the introduction rules of a meaning conferring natural deduction proof system. The semantic value is defined based on Fregecontributions” to sentential meanings as determined by the function-argument structure as induced by a type-logical grammar. In doing so, the paper proposes a novel proof-theoretic interpretation of the semantic types, traditionally interpreted in Henkin models. The compositionality of the resulting (...) attribution of semantic values is discussed. Elsewhere, the same method was used for defining proof-theoretic meaning of subsentential phrases in a fragment of natural language. Doing the same for (the simpler and clearer case of) logic sheds more light on the proposal. (shrink)

In philosophical logic necessity is usually conceived as a sentential operator rather than as a predicate. An intensional sentential operator does not allow one to express quantified statements such as 'There are necessary a posteriori propositions' or 'All laws of physics are necessary' in first-order logic in a straightforward way, while they are readily formalized if necessity is formalized by a predicate. Replacing the operator conception of necessity by the predicate conception, however, causes various problems and forces one (...) to reject many philosophical accounts involving necessity that are based on the use of operator modal logic. We argue that the expressive power of the predicate account can be restored if a truth predicate is added to the language of first-order modal logic, because the predicate 'is necessary' can then be replaced by 'is necessarily true'. We prove a result showing that this substitution is technically feasible. To this end we provide partial possible-worlds semantics for the language with a predicate of necessity and perform the reduction of necessities to necessary truths. The technique applies also to many other intensional notions that have been analysed by means of modal operators. (shrink)

Deontic logic is an area of logic which investigates normative concepts, systems of norms, and normative reasoning. The word ‘deontic’ is derived from the Greek expression ‘déon’, which means ‘what is binding’ or ‘proper’. Thus, Jeremy Bentham used the word ‘deontology’ for “the science of morality,” and the Austrian philosopher Ernst Mally, who developed in the 1920s a system of the “fundamental principles the logic of ought,” called his theory ‘Deontik’. Normative concepts include the concepts of obligation (ought), permission (may), (...) prohibition (may not), and related notions, such as the concept of right. Systems of deontic logic contain, in addition to the usual sentential connectives and quantifiers, logical constants which represent some of these normative concepts. (shrink)

This paper addresses the two interpretations that a combination ofnegative indefinites can get in concord languages like French:a concord reading, which amounts to a single negation, and a doublenegation reading. We develop an analysis within a polyadic framework,where a sequence of negative indefinites can be interpreted as aniteration of quantifiers or via resumption. The first option leadsto a scopal relation, interpreted as double negation. The secondoption leads to the construction of a polyadic negative quantifiercorresponding to the concord reading. Given that (...) sentential negationparticipates in negative concord, we develop an extension of thepolyadic approach which can deal with non-variable binding operators,treating the contribution of negation in a concord context assemantically empty. Our semantic analysis, incorporated into agrammatical analysis formulated in HPSG, crucially relies on theassumption that quantifiers can be combined in more than one wayupon retrieval from the quantifier store. We also considercross-linguistic variation regarding the participation ofsentential negation in negative concord. (shrink)

In this paper, I provide a compositional semantics for sentences with enough and too followed by a to-infinitive clause and for resultative constructions with so... that within the framework of possible world semantics. It is proposed that the sentential complement of these constructions denotes an incomplete conditional and is explicitly or implicitly modalized, as if it were the consequent of a complete conditional. Enough, too, and so are quantifiers that relate an extent predicate and the incomplete conditional (expressed by (...) the sentential complement) and are interpreted as comparisons between two extents. The first extent is the maximal extent that satisfies the extent predicate. The second extent is the minimal or maximal extent of a set of extents that satisfy the (hidden) conditional, where the sentential complement supplies the consequent and the main clause the antecedent. This approach permits to predict (a) the context-dependent interpretation of these constructions and (b) the duality relations between the degree words. (shrink)