We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multi-quantifier sentences. First, we show that the standard constructions that turn simple determiners into complex quantifiers, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provide an insight into branching operation yielding intractable natural language multi-quantifier expressions. Next, we focus on a linguistic case study. We use computational complexity results (...) to investigate semantic distinctions between quantified reciprocal sentences. We show a computational dichotomy<br>between different readings of reciprocity. Finally, we go more into philosophical speculation on meaning, ambiguity and computational complexity. In particular, we investigate a possibility to<br>revise the Strong Meaning Hypothesis with complexity aspects to better account for meaning shifts in the domain of multi-quantifier sentences. The paper not only contributes to the field of the formal<br>semantics but also illustrates how the tools of computational complexity theory might be successfully used in linguistics and philosophy with an eye towards cognitive science. (shrink)

We study generalized quantifiers on finite structures.With every function : we associate a quantifier Q by letting Q x say there are at least (n) elementsx satisfying , where n is the sizeof the universe. This is the general form ofwhat is known as a monotone quantifier of type .We study so called polyadic liftsof such quantifiers. The particular lifts we considerare Ramseyfication, branching and resumption.In each case we get exact criteria fordefinability of the lift in terms (...) of simpler quantifiers. (shrink)

We consider countable so‐called rich subsemigroups of ; each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, if and only if is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in polyadic (...) equality set algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEAT is canonical and atom‐canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEAT are completely representable and that CPEAT has the super amalgamation property. If T is rich and finitely represented, it is shown that CPEAT is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic with equality if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented. (shrink)

In this paper, I have proposed a compositional semantic analysis of exception NPs from which three core properties of exception constructions could be derived. I have shown that this analysis overcomes various empirical and conceptual shortcomings of prior proposals of the semantics of exception sentences. The analysis was first formulated for simple exception NPs, where the EP-complement was considered a set-denoting term and the EP-associate was a monadic quantifier. It was then generalized in two steps: first, in order to account (...) for quantified EP-complements, and second, in order to account for polyadic quantifiers as the EP-associates. An additional assumption that was made in several places was that EPs may operate at the level of implications. The consequences of this assumption, though, still have to be investigated. (shrink)

In the dissertation we study the complexity of generalized quantifiers in natural language. Our perspective is interdisciplinary: we combine philosophical insights with theoretical computer science, experimental cognitive science and linguistic theories. -/- In Chapter 1 we argue for identifying a part of meaning, the so-called referential meaning (model-checking), with algorithms. Moreover, we discuss the influence of computational complexity theory on cognitive tasks. We give some arguments to treat as cognitively tractable only those problems which can be computed in polynomial (...) time. Additionally, we suggest that plausible semantic theories of the everyday fragment of natural language can be formulated in the existential fragment of second-order logic. -/- In Chapter 2 we give an overview of the basic notions of generalized quantifier theory, computability theory, and descriptive complexity theory. -/- In Chapter 3 we prove that PTIME quantifiers are closed under iteration, cumulation and resumption. Next, we discuss the NP-completeness of branching quantifiers. Finally, we show that some Ramsey quantifiers define NP-complete classes of finite models while others stay in PTIME. We also give a sufficient condition for a Ramsey quantifier to be computable in polynomial time. -/- In Chapter 4 we investigate the computational complexity of polyadic lifts expressing various readings of reciprocal sentences with quantified antecedents. We show a dichotomy between these readings: the strong reciprocal reading can create NP-complete constructions, while the weak and the intermediate reciprocal readings do not. Additionally, we argue that this difference should be acknowledged in the Strong Meaning hypothesis. -/- In Chapter 5 we study the definability and complexity of the type-shifting approach to collective quantification in natural language. We show that under reasonable complexity assumptions it is not general enough to cover the semantics of all collective quantifiers in natural language. The type-shifting approach cannot lead outside second-order logic and arguably some collective quantifiers are not expressible in second-order logic. As a result, we argue that algebraic (many-sorted) formalisms dealing with collectivity are more plausible than the type-shifting approach. Moreover, we suggest that some collective quantifiers might not be realized in everyday language due to their high computational complexity. Additionally, we introduce the so-called second-order generalized quantifiers to the study of collective semantics. -/- In Chapter 6 we study the statement known as Hintikka's thesis: that the semantics of sentences like ``Most boys and most girls hate each other'' is not expressible by linear formulae and one needs to use branching quantification. We discuss possible readings of such sentences and come to the conclusion that they are expressible by linear formulae, as opposed to what Hintikka states. Next, we propose empirical evidence confirming our theoretical predictions that these sentences are sometimes interpreted by people as having the conjunctional reading. -/- In Chapter 7 we discuss a computational semantics for monadic quantifiers in natural language. We recall that it can be expressed in terms of finite-state and push-down automata. Then we present and criticize the neurological research building on this model. The discussion leads to a new experimental set-up which provides empirical evidence confirming the complexity predictions of the computational model. We show that the differences in reaction time needed for comprehension of sentences with monadic quantifiers are consistent with the complexity differences predicted by the model. -/- In Chapter 8 we discuss some general open questions and possible directions for future research, e.g., using different measures of complexity, involving game-theory and so on. -/- In general, our research explores, from different perspectives, the advantages of identifying meaning with algorithms and applying computational complexity analysis to semantic issues. It shows the fruitfulness of such an abstract computational approach for linguistics and cognitive science. (shrink)

This chapter offers a logical, linguistic, and philosophical account of modern quantification theory. Contrasting the standard approach to quantifiers (according to which logical quantifiers are defined by enumeration) with the generalized approach (according to which quantifiers are defined systematically), the chapter begins with a brief history of standard quantifier theory and identifies some of its logical, linguistic, and philosophical strengths and weaknesses. It then proceeds to a brief history of generalized quantifier theory and explains how it overcomes (...) the weaknesses of the standard theory. One of the main philosophical advantages of the generalized theory is its philosophically informative criterion of logicality. The chapter describes the work done so far in this theory, highlights some of its central logical results, offers an overview of its main linguistic contributions, and discusses its philosophical significance. (shrink)

We prove that PTIME generalized quantifiers are closed under Boolean operations, iteration, cumulation and resumption. -/- .

This paper is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics. The paper has a survey character, too. The most frequently used algebras like cylindric-, relation-, polyadic-, and quasi-polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively comprehensive. (...) In some directions we tried to give an overview of the most recent results and research trends, too. (shrink)

Following research initiated by Tarski, Craig and Nemeti, and futher pursued by Sain and others, we show that for certain subsets G of $^\omega \omega $ , atomic countable G poiyadic algebras are completely representable. G polyadic algebras are obtained by restricting the similarity type and axiomatization of ω-dimensional polyadic algebras to finite quantifiers and substitutions in G. This contrasts the cases of cylindric and relation algebras.

We prove several interpolation theorems for many-valued infinitary logic with quantifiers by studying expansions of MV-algebras in the spirit of polyadic and cylindric algebras. We prove for various reducts of polyadic MV-algebras of infinite dimensions that if is the free algebra in the given signature,, is in the subalgebra of generated by, is in the subalgebra of generated by and, then there exists an interpolant in the subalgebra generated by and such that. We call this a varying (...) interpolation property because the integer depends on the inequality. We also address cases where this interpolation property fails, but other weaker ones hold. One such interpolation theorem says that though an interpolant may not be found as above, an interpolant can always be found if finitely many universal quantifiers are applied to making it smaller and the same number of existential quantifiers are applied to making it bigger. This number of quantifiers also varies; it depends on the inequality. Se.. (shrink)

We argue that a comprehensive theory of reciprocals must rely on a general taxonomy of restrictions on the interpretation of relational expressions. Developing such a taxonomy, we propose a new principle for interpreting reciprocals that relies on the interpretation of the relation in their scope. This principle, the Maximal Interpretation Hypothesis (MIH), analyzes reciprocals as partial polyadic quantifiers. According to the MIH, the partial quantifier denoted by a reciprocal requires the relational expression REL in its scope to denote (...) a maximal relation in REL’s interpretation domain. In this way the MIH avoids a priori assumptions on the available readings of reciprocal expressions, which are necessary in previous accounts. Relying extensively on the work of Dalrymple et al. (Ling Philos 21:159–210, 1998), we show that the MIH also exhibits some observational improvements over Dalrymple et al.’s Strongest Meaning Hypothesis (SMH). In addition to deriving some attested reciprocal interpretations that are not expected by the SMH, the MIH offers a more restrictive account of the way context affects the interpretation of reciprocals through its influence on relational domains. Further, the MIH generates a reciprocal interpretation at the predicate level, which is argued to be advantageous to Dalrymple et al.’s propositional selection of reciprocal meanings. More generally, we argue that by focusing on restrictions on relational domains, the MIH opens the way for a more systematic study of the ways in which lexical meaning, world knowledge and contextual information interact with the interpretation of quantificational expressions. (shrink)

Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they can take their knowledge further and (...) the philosophical questions around logic. -/- Logic: the Basics is essential reading for first-year undergraduate philosophy students on standard introductory logic courses. (shrink)

The Korean particle -nun combined with an accent indicates contrast :269–320, 1972; Heycock, in: Merce Proceedings of NELS, vol 24, pp 159–187, 1993; in: Miyagawa, Saito Handbook of Japanese linguistics, Oxford University Press, Cambridge, 2007; Hara, in: Dekker, Franke Fifteenth Amsterdam colloquium, Universiteit van Amsterdam, pp 101–106, 2006; Lee, in: Lee, Gordon, Büring Topic and focus: meaning and intonation from a crosslinguistic perspective. Springer, Berlin, 2003; Tomioka, in: Zimmermann, Fery Information structure, Oxford University Press, Cambridge, pp 115–138, 2009, among many (...) others). While this is not controversial, what it means to be contrastive remains unclear. In this paper, instead of analyzing contrastive -nun as a discourse device in the sense of information structure, as has been done in previous studies, I explore how the contrastive meaning is derived compositionally. I treat contrastive -nun as a focus sensitive particle that associates with prosodic accents in two places, generating a non-at-issue meaning. The non-at-issue meaning contains a polyadic quantifier that establishes a contrastive relationship between two elements in a given set. This analysis explains how focus following contrastive -nun is associated with -nun. It also provides an explanation for the uncertainty implicature that contrastive -nun gives rise to. Finally, it clarifies the logical relationship among different focus particles in Korean. (shrink)

Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. GMV-algebras are a non-commutative generalization of MV-algebras and are an algebraic counterpart of the non-commutative Łukasiewicz infinite valued logic. We introduce monadic GMV-algebras and describe their connections to certain couples of GMV-algebras and to left adjoint mappings of canonical embeddings of GMV-algebras. Furthermore, functional MGMV-algebras are studied and polyadic GMV-algebras are introduced and discussed.

The semantic automata framework, developed originally in the 1980s, provides computational interpretations of generalized quantifiers. While recent experimental results have associated structural features of these automata with neuroanatomical demands in processing sentences with quantifiers, the theoretical framework has remained largely unexplored. In this paper, after presenting some classic results on semantic automata in a modern style, we present the first application of semantic automata to polyadic quantification, exhibiting automata for iterated quantifiers. We also discuss the role (...) of semantic automata in linguistic theory and offer new empirical predictions for sentence processing with embedded quantifiers. (shrink)

Sentences with multiple occurrences of plural definites give rise to certain effects suggesting that distributivity should be modeled by polyadic operations. Yet in this paper it is argued that the simpler treatment of distributivity using unary universal quantification should be retained. Seemingly polyadic effects are claimed to be restricted to definite NPs. This fact is accounted for by the special anaphoric (dependent) use of definites. Further evidence concerning various plurals, island constraints, and cumulative quantification is shown to support (...) this claim. In addition, it is shown that the evidence against a simple atomic version of unary distributivity is not decisive either. In the (uncommon) cases where distributivity with definites is not strictly atomic, they can be analyzed as dependent on implicit quantifiers. (shrink)

The semantic automata framework, developed originally in the 1980s, provides computational interpretations of generalized quantifiers. While recent experimental results have associated structural features of these automata with neuroanatomical demands in processing sentences with quantifiers, the theoretical framework has remained largely unexplored. In this paper, after presenting some classic results on semantic automata in a modern style, we present the first application of semantic automata to polyadic quantification, exhibiting automata for iterated quantifiers. We also discuss the role (...) of semantic automata in linguistic theory and offer new empirical predictions for sentence processing with embedded quantifiers. (shrink)

In this paper, we study natural language constructions which were first examined by Barwise: The richer the country, the more powerful some of its officials. Guided by Barwise’s observations, we suggest that conceivable interpretations of such constructions express the existence of various similarities between partial orders such as homomorphism or embedding. Semantically, we interpret the constructions as polyadic generalized quantifiers restricted to finite models. We extend the results obtained by Barwise by showing that similarity quantifiers are not (...) expressible in elementary logic over finite models. We also investigate whether the proposed readings are sound from the cognitive perspective. We prove that almost all similarity quantifiers are intractable. This leads us to first-order variants, which only approximate the strong readings, but are cognitively more plausible. Driven by the question of ambiguity, we recall Barwise’s argumentation in favour of strong readings, enriching it with some arguments of our own. Given that Barwise-like sentences are indeed ambiguous, we use a generalized Strong Meaning Hypothesis to derive predictions for their verification. Finally, we propose a hypothesis according to which conflicting pressures of communication and cognition might give rise to an ambiguous construction, provided that different semantic variants of the construction withstand different pressures involved in its usage. (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)

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)

There were already confusions in the Middle Ages with the reading of Aristotle on negative terms, and removing these confusions shows that the four traditional Syllogistic forms of statement can be readily generalised not only to handle polyadic relations (for long a source of difficulty), but even other, more measured quantifiers than just ‘all’, ‘some’, and ‘no’. But these historic confusions merely supplement the main confusions, which arose in more modern times, regarding the logic of singular statements. These (...) main confusions originate in the inability of the mainline modern tradition to supply the ‘logically proper names’ which alone have the right to replace individual variables; an inability which has resulted in the widespread, but erroneous replacement of individual variables with ordinary proper names, i.e. names for contingent beings, in many if not most contemporary logic texts. The paper includes the exhibition and grammatical characterisation of the logically proper names that are required instead, specifying just how they differ syntactically from ordinary proper names. It also shows how ontologically significant is the distinction, since not only do logically proper names refer to necessarily existent objects (showing there are no ‘empty domains’ for Classical Logic to fail to apply to), but also thereby central features of Realism become considerably clarified. (shrink)

There are enormous differences between quantifying name-variables only, quantifying verb-variables only, and quantifying both. These differences are found only in the logic of polyadic predication; and this presumably is why Richard Gaskin thinks that they distinguish names from transitive verbs only, and not from verbs generally. But that thought is mistaken: these differences also distinguish names from intransitive verbs. They thus vindicate the common idea that on the difference between names and verbs we may base grandiose metaphysical distinctions, and (...) undermine Gaskin's idea that both names and verbs may be said to designate objects. (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)

Many arguments for flexible type assignment to syntactic categories have to do with the need to account for the various scopings resulting from the interaction of quantified DPs with other quantified DPs or with intensional or negated verb contexts. We will define a type for arbitrary arity relations in polymorphic type theory. In terms of this, we develop the Boolean algebra of relations as far as needed for natural language semantics. The type for relations is flexible: it can do duty (...) for the whole family of types t, e → t, e → e → t, and so on. If we use this flexible type for the interpretations of verbs, we can perform a ‘flexible lift’ on DP interpretations, so that DPs get interpreted as operations on verb meanings. This leads to an elegant implementation of Keenan’s proposal for polyadic quantification in natural language (‘Beyond the Frege Boundary’). It turns out that scope reorderings are particularly easy to implement in this framework. (shrink)

Deductive Logic is an introductory textbook in formal logic. The book is divided into four parts covering (i) truth-functional logic, (ii) monadic quantifi- cation, (iii) polyadic quantification and (iv) names and identity, and there are exercises for all these topics at the end of the book. In the truth-functional logic part, the reader learns to produce paraphrases of English statements and arguments in logical notation (this subsection is called “analysis”), then about the semantic properties of such paraphrased statements and (...) arguments, such as satisfiability, implication and equivalence (“logical assessment”) and finally (“reflection”) there is an axiomatic proof method and some important extras such as disjunctive normal form and expressive adequacy. Parts two and three mirror this analysis/assessment/reflection structure for monadic and polyadic quantification, though this time the proof system is a natural deduction one, and part three contains a completeness proof for that system. The fourth part of the book introduces names, the identity predicate and descriptions and examines the additional expressive power which these provide. (shrink)

Section 1 provides a brief summary of the pair-list literature singling out some points that are particularly relevant for the coming discussion. -/- Section 2 shows that the dilemma of quantifi cation versus domain restriction arises only in extensional complement interrogatives. In matrix questions and in intensional complements only universals support pairlist readings, whence the simplest domain restriction treatment suffices. Related data including conjunction, disjunction, and cumulative readings are discussed -/- Section 3 argues that in the case of extensional complements (...) the domain restriction treatment is inadequate for at least two independent reasons. One has to do with the fact that not only upward monotonic quantifi ers support pairlist readings, and the other with the derivation of apparent scope out readings. The reasoning is supplemented with some discussion of the semantic properties of layered quantifi ers.  The above will establish the need for quantifi cation, so the question arises how the objections explicitly enlisted in the literature against quantifi cation can be answered. Section 4 considers the de dicto reading of the quantifi er s restriction, quanti cational variability, and the absence of pairlist readings with whether questions, and argues that they need not militate against the quanti ficational analysis. -/- Section 5 summarizes the emergent proposal -/- Finally, section 6 discusses the signifi cance of the above findings for the behavior of weak islands. (shrink)

The question of the origin of polyadic expressivity is explored and the results are brought to bear on Bertrand Russell's 1903 theory of denoting concepts, which is the main object of criticism in his 1905 "On Denoting". It is shown that, appearances to the contrary notwithstanding, the background ontology of the earlier theory of denoting enables the full-blown expressive power of first-order polyadic quantification theory without any syntactic accommodation of scopal differences among denoting phrases such as 'all φ', (...) 'every φ', and 'any φ' on the one hand, and 'some φ' and 'a φ' on the other. The case provides an especially vivid illustration of the general point that structural (or ideological) austerity can be paid for in the coin of ontological extravagance. (shrink)

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)

Rough polyadic modal logics, introduced in the paper, contain modal operators of many arguments with a relational semantics, based on the Pawlak's rough set theory. Rough set approach is developed as an alternative to the fuzzy set philosophy, and has many applications in different branches in Artificial Intelligence and theoretical computer science.

It is known that every α-dimensional quasi polyadic equality algebra (QPEA α ) can be considered as an α-dimensional cylindric algebra satisfying the merrygo- round properties . The converse of this proposition fails to be true. It is investigated in the paper how to get algebras in QPEA from algebras in CA. Instead of QPEA the class of the finitary polyadic equality algebras (FPEA) is investigated, this class is definitionally equivalent to QPEA. It is shown, among others, that (...) from every algebra in a β-dimensional algebra can be obtained in QPEA β where , moreover the algebra obtained is representable in a sense. (shrink)

I think it would be fair to say that, until about 1900, philosophers were generally reluctant to admit the existence of what are nowadays called polyadic properties.1 It is important to recognize, however, that this reluctance on the part of pre-twentieth-century philosophers did not prevent them from theorizing about relations. On the contrary, philosophers from the ancient through the modern period have had much to say about both the nature and the ontological status of relations. In this paper I (...) examine the views of one such philosopher, namely, Albert the Great. (shrink)

The growth of self-tracking and personal surveillance has given rise to the Quantified Self movement. Members of this movement seek to enhance their personal well-being, productivity, and self-actualization through the tracking and gamification of personal data. The technologies that make this possible can also track and gamify aspects of our interpersonal, romantic relationships. Several authors have begun to challenge the ethical and normative implications of this development. In this article, we build upon this work to provide a detailed ethical analysis (...) of the Quantified Relationship. We identify eight core objections to the QR and subject them to critical scrutiny. We argue that although critics raise legitimate concerns, there are ways in which tracking technologies can be used to support and facilitate good relationships. We thus adopt a stance of cautious openness toward this technology and advocate the development of a research agenda for the positive use of QR technologies. (shrink)

Quantification is a topic which brings together linguistics, logic, and philosophy. Quantifiers are the essential tools with which, in language or logic, we refer to quantity of things or amount of stuff. In English they include such expressions as no, some, all, both, many. Peters and Westerstahl present the definitive interdisciplinary exploration of how they work - their syntax, semantics, and inferential role.

We define an interpretation of modal languages with polyadic operators in modal languages that use monadic operators (diamonds) only. We also define a simulation operator which associates a logic $\Lambda^{sim}$ in the diamond language with each logic Λ in the language with polyadic modal connectives. We prove that this simulation operator transfers several useful properties of modal logics, such as finite/recursive axiomatizability, frame completeness and the finite model property, canonicity and first-order definability.

We investigate quantifier-free induction for Lisp-like lists constructed inductively from the empty list $$ nil $$ nil and the operation $${\textit{cons}}$$ cons, that adds an element to the front of a list. First we show that, for $$m \ge 1$$ m ≥ 1, quantifier-free $$m$$ m -step induction does not simulate quantifier-free $$(m + 1)$$ ( m + 1 ) -step induction. Secondly, we show that for all $$m \ge 1$$ m ≥ 1, quantifier-free $$m$$ m -step induction does not (...) prove the right cancellation property of the concatenation operation on lists defined by left-recursion. (shrink)

In this paper, we provide a comprehensive survey of the space of possible analyses of the phenomenon of quantifier domain restriction, together with a set of considerations which militate against all but our own proposal. Among the many accounts we consider and reject are the ‘explicit’ approach to quantifier domain restric‐tion discussed, for example, by Stephen Neale, and the pragmatic approach to quantifier domain restriction proposed by Kent Bach. Our hope is that the exhaustive discussion of this special case of (...) context‐dependence will provide guidelines for how to decide, for an arbitrary case of context‐dependent discourse, whether it should be treated syntactically, semantically, or pragmatically. (shrink)

This paper extends the investigations into logical properties of the quantified argument calculus (Quarc) by suggesting a series of proper subsystems which, although retaining the entire vocabulary of Quarc, restrict quantification in such a way as to make the result decidable. The proof of decidability is via a procedure that prunes the infinite branches of a derivation tree in what is a syntactic counterpart of semantic filtration. We demonstrate an application of one of these systems by showing that Aristotle’s assertoric (...) syllogistic is embeddable within, thus also providing another method of showing its decidability. (shrink)

In 2015, A.V. Figallo and G. Pelaitay introduced tense n×m-valued Łukasiewicz–Moisil algebras, as a common generalization of tense Boolean algebras and tense n-valued Łukasiewicz–Moisil algebras. Here we initiate an investigation into the class tpLMn×m of tense polyadic n × m-valued Łukasiewicz–Moisil algebras. These algebras constitute a generalization of tense polyadic Boolean algebras introduced by Georgescu in 1979, as well as the tense polyadic n-valued Łukasiewicz–Moisil algebras studied by Chiriţă in 2012. Our main result is a representation theorem (...) for tense polyadic n × m-valued Łukasiewicz–Moisil algebras. (shrink)

Quantifier variance is a well-known view in contemporary metaontology, but it remains very widely misunderstood by critics. Here we briefly and clearly explain the metasemantics of quantifier variance and distinguish between modest and strong forms of variance (Section I), explain some key applications (Section II), clear up some misunderstandings and address objections (Section III), and point the way toward future directions of quantifier-variance-related research (Section IV).

Quantified expressions in natural language generally are taken to act like quantifiers in logic, which either range over entities that need to satisfy or not satisfy the predicate in order for the sentence to be true or otherwise are substitutional quantifiers. I will argue that there is a philosophically rather important class of quantified expressions in English that act quite differently, a class that includes something, nothing, and several things. In addition to expressing quantification, such expressions act like (...) nominalizations, introducing a new domain of objects that would not have been present in the semantic structure of the sentence otherwise. The entities those expressions introduce are of just the same sort as those that certain ordinary nominalizations refer to (such as John's wisdom or John's belief that S), namely they are tropes or entities related to tropes. Analysing certain quantifiers as nominalizing quantifiers will shed a new light on philosophical issues such as the status of properties and the nature of propositional attitudes. (shrink)