Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension $L_{*}^{1}$ (H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close $L_{*}^{1}$ (H) with respect to Boolean operations, and obtain the language L¹(H). At the next level, we consider an extension $L_{*}^{2}$ (H) of L¹(H) in which every sentence is (...) an L¹(H)-sentence prefixed with a Henkin quantifier. We repeat this construction to infinity. Using the (un)definability of truth - in - N for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996). (shrink)

We investigate some logics with Henkin quantifiers. For a given logic L, we consider questions of the form: what is the degree of the set of L–tautologies in a poor vocabulary (monadic or empty)? We prove that the set of tautologies of the logic with all Henkin quantifiers in empty vocabulary L*∅ is of degree 0’. We show that the same holds also for some weaker logics like L ∅(Hω) and L ∅(Eω). We show that each (...) logic of the form L ∅ (k)(Q), with the number of variables restricted to k, is decidable. Nevertheless – following the argument of M. Mostowski from [Mos89] – for each reasonable set theory no concrete algorithm can provably decide L (k) (Q), for some (Q). We improve also some results related to undecidability and expressibility for logics L(H4) and L(F2) of Krynicki and M. Mostowski from [KM92]. (shrink)

It is known that various complexity-theoretical problems can be translated into some special spectra problems. Thus, questions about complexity classes are translated into questions about the expressive power of some languages. In this paper we investigate the spectra of some logics with Henkin quantifiers in the empty vocabulary.

It is known that various complexity-theoretical problems can be translated into some special spectra problems (see e.g. Fagin [Fa74] or Blass and Gurevich, [Bl-Gu86]). So questions about complexity classes are translated into questions about the expressive power of some languages. In this paper we investigate the spectra of some logics with Henkin quanti fiers in the empty vocabulary. This problem has been investigated fi rstly by Krynicki and Mostowski in [Kr-Mo 92] and [Kr- Mo 95]. All presented results can (...) be also treated as results about the expressive power of certain languages in fi nite models. (shrink)

Krynicki, M. and M. Mostowski, Decidability problems in languages with Henkin quantifiers, Annals of Pure and Applied Logic 58 149–172.We consider the language L with all Henkin quantifiers Hn defined as follows: Hnx1…xny1…yn φ iff f1…fnx1. ..xn φ, ...,fn). We show that the theory of equality in L is undecidable. The proof of this result goes by interpretation of the word problem for semigroups.Henkin quantifiers are strictly related to the function quantifiers Fn defined (...) as follows: Fnx1…xny1…yn φ iff fx1…xn φ,...,f). In contrast with the first result we show that the theory of equality with all quantifiers Fn is decidable.We also consider decidability problems for other theories in languages L and L. (shrink)

We here examine the expressive power of first order logic with generalized quantifiers over finite ordered structures. In particular, we address the following problem: Given a family Q of generalized quantifiers expressing a complexity class C, what is the expressive power of first order logic FO(Q) extended by the quantifiers in Q? From previously studied examples, one would expect that FO(Q) captures L C , i.e., logarithmic space relativized to an oracle in C. We show that this (...) is not always true. However, after studying the problem from a general point of view, we derive sufficient conditions on C such that FO(Q) captures L C . These conditions are fulfilled by a large number of relevant complexity classes, in particular, for example, by NP. As an application of this result, it follows that first order logic extended by Henkin quantifiers captures L NP . This answers a question raised by Blass and Gurevich [Ann. Pure Appl. Logic, vol. 32, 1986]. Furthermore we show that for many families Q of generalized quantifiers (including the family of Henkin quantifiers), each FO(Q)-formula can be replaced by an equivalent FO(Q)-formula with only two occurrences of generalized quantifiers. This generalizes and extends an earlier normal-form result by I. A. Stewart [Fundamenta Inform. vol. 18, 1993]. (shrink)

Connections between partially ordered connectives and Henkin quantifiers are considered. It is proved that the logic with all partially ordered connectives and the logic with all Henkin quantifiers coincide. This implies that the hierarchy of partially ordered connectives is strongly hierarchical and gives several nondefinability results between some of them. It is also deduced that each Henkin quantifier can be defined by a quantifier of the form equation imagewhat is a strengthening of the Walkoe result. (...) MSC: 03C80. (shrink)

That the result of flipping quantifiers and negating what comes after, applied to branching-quantifier sentences, is not equivalent to the negation of the original has been known for as long as such sentences have been studied. It is here pointed out that this syntactic operation fails in the strongest possible sense to correspond to any operation on classes of models.

We present an embedding of quantified multimodal logics into simple type theory and prove its soundness and completeness. A correspondence between QKπ models for quantified multimodal logics and Henkin models is established and exploited. Our embedding supports the application of off-the-shelf higher-order theorem provers for reasoning within and about quantified multimodal logics. Moreover, it provides a starting point for further logic embeddings and their combinations in simple type theory.

A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take amodelto be one of these structures, α, together with some function or relation which associates with every formulaAa subset ofα. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and Meyer [1973], (...) a formula can only be associated with subsets which are closed upwards. It is natural to take apropositionof α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers theprimaryinterpretation associated with the semantics. (shrink)

Following Henkin's discovery of partially-ordered (branching) quantification (POQ) with standard quantifiers in 1959, philosophers of language have attempted to extend his definition to POQ with generalized quantifiers. In this paper I propose a general definition of POQ with 1-place generalized quantifiers of the simplest kind: namely, predicative, or "cardinality" quantifiers, e.g., "most", "few", "finitely many", "exactly α", where α is any cardinal, etc. The definition is obtained in a series of generalizations, extending the original, (...) class='Hi'>Henkin definition first to a general definition of monotone-increasing (M↑) POQ and then to a general definition of generalized POQ, regardless of monotonicity. The extension is based on (i) Barwise's 1979 analysis of the basic case of M↑ POQ and (ii) my 1990 analysis of the basic case of generalized POQ. POQ is a non-compositional Ist-order structure, hence the problem of extending the definition of the basic case to a general definition is not trivial. The paper concludes with a sample of applications to natural and mathematical languages. (shrink)

Following Henkin’s discovery of partially-ordered (branching) quantification (POQ) with standard quantifiers in 1959, philosophers of language have attempted to extend his definition to POQ with generalized quantifiers. In this paper I propose a general definition of POQ with 1-place generalized quantifiers of the simplest kind: namely, predicative, or “cardinality” quantifiers, e.g., “most”, “few”, “finitely many”, “exactly α ”, where α is any cardinal, etc. The definition is obtained in a series of generalizations, extending the original, (...) Henkin definition first to a general definition of monotone-increasing (M↑) POQ and then to a general definition of generalized POQ, regardless of monotonicity. The extension is based on (i) Barwise’s 1979 analysis of the basic case of M↑ POQ and (ii) my 1990 analysis of the basic case of generalized POQ. POQ is a non-compositional 1st-order structure, hence the problem of extending the definition of the basic case to a general definition is not trivial. The paper concludes with a sample of applications to natural and mathematical languages. (shrink)

The Quantified Argument Calculus (Quarc) is a formal logic system, first developed by Hanoch Ben-Yami in (Ben-Yami 2014), and since then extended and applied by several authors. The aim of this paper is to further these contributions by, first, providing a philosophical motivation for the truth-valuational, substitutional approach of (Ben-Yami 2014) and defending it against a common objection, a topic also of interest beyond its specific application to Quarc. Second, we fill the formal lacunae left in the original presentation, which (...) did not incorporate identity systematically into Quarc, and although it proved the soundness of the system did not prove its completeness. (shrink)

Hebrew and Arabic in Asymmetric Contact in Israel Israeli Hebrew and Palestinian Arabic 1 have existed side by side for well over a century in extremely close contact, accompanied by social and ideological tension, often conflict, between two communities: PA speakers, who turned from a majority to a minority following the establishment of the State of Israel in 1948, and IH speakers, the contemporary majority, representing the dominant culture. The Hebrew-speaking Jewish group is heterogeneous in terms of lands of origin (...) and includes a large community of Judeo-Arabic speakers. In this paper I discuss the ‘when,’ ‘where,’ how,' and especially ‘why’ of mutual borrowing2 between Hebrew and Arabic in Israel since the 1880s.I stress that the contact is asymmetric. The majority of Hebrew speakers today do not speak Arabic. They do, however, have access to borrowed lexicon in the domain of slang and vernacular speech that both enriches these registers and provides the speakers with a feeling of being ‘indigenous’ and ‘belonging’ to the social and cultural milieu of the larger environment. In contrast, most young adult PA-speakers are bilingual. For this minority, borrowing fulfills fundamental sociolinguistic and pragmatic needs. IH lexical items have permeated all areas of professional, educational, administrative, and social activities in everyday life discourse in all styles and registers of the spoken language and, to a lesser extent, written language. They fill lexical gaps and lacunae representing modern and western concepts that did not exist in PA. They also symbolize modernism, education, and social mobility. In other words, borrowing serves as a means of ‘stepping in’ for the Jewish majority and conversely as a means for ‘stepping out and up’ for the Arab minority. (shrink)

Though some Christian theologians have argued that Western human rights theory is grounded in religious faith, human rights morality is, in fact, autonomous. The ideologies of religion and of human rights differ in their sources, the bases of their authority, their forms of expression, and even their substantive norms. Moreover, historically, religious communities have often themselves violated human rights norms-and such violations persist today in some geographical regions and with respect to some norms. On the other hand, religious communities have (...) an obvious interest in the development and defense of some human rights, particularly religious human rights, and religious communities that once were hostile to the human rights ideology have more recently, particularly in the West, coopted the human rights idea and claimed it as their own. An "approchement" between religion and human rights is possible and highly desirable. (shrink)

Volume I provides a detailed analysis of cylindric algebras, starting with a formulation of their axioms and a development of their elementary properties, and proceeding to a deeper study of their interrelationships by means of general algebraic notions such as subalgebras, homomorphisms, direct products, free algebras, reducts and relativized algebras.