We present a semantics for interpreting probabilistic statements expressed in a first-order quantifier-free language. We show how this semantics places constraints on the probabilities which can be associated with such statements. We then consider its use in the area of story understanding. We show that for at least simple models of stories (equivalent to the script/plan models) there arc ways ...

The main result of this paper is (a slightly stronger form of) the following theorem: let T be a countable complete first-order theory which is stable. If, for some a > 1, the Malitz quantifier Ql is eliminable in T then all Malitz quantifiers Qg (β > 0, m > 1) are eliminable in T. This complements results of Baldwin-Kueker [1J and Rothmaler-Tuschik [3]. 1. Introduction. In this paper we consid...

Generalizing Cooper’s method of quantifier elimination for Presburger arithmetic, we give a new proof that all parametric Presburger families {St : t ∈ N} (as defined by Woods in [8]) are definable by formulas with polynomially bounded quantifiers in an expanded language with predicates for divisibility by f(t) for every polynomial f ∈ Z[t]. In fact, this quantifier bounding method works more g...

Although grammarians traditionally consider names and quantifier expressions (“something”, “everything”, etc.) as belonging to the noun phrases category, logicians usually treat quantifiers as operators, i.e., as expressions of a different category from the name category. In this brief and valuable paper, logical differences between names and quantifiers are studied on a deductive level while a...

Gaifman’s normal form theorem showed that every first order sentence of quantifier rank n is equivalent to a Boolean combination of “scattered local sentences”, where the local neighborhoods have radius at most 7n−1. This bound was improved by Lifsches and Shelah to 3 · 4n−1. We use Ehrenfeucht-Fräıssé type games with a “shrinking horizon” to get a spectrum of normal form theorems of the Gaifma...

We consider categorical logic on the category of Hilbert spaces. More generally, in fact, any pre-Hilbert category suffices. We characterise closed subobjects, and prove that they form orthomodular lattices. This shows that quantum logic is just an incarnation of categorical logic, enabling us to establish an existential quantifier for quantum logic, and conclude that there cannot be a universa...

We study tree languages that can be defined in ∆2. These are tree languages definable by a first-order formula whose quantifier prefix is ∃ ∗ ∀ ∗, and simultaneously by a first-order formula whose quantifier prefix is ∀∗∃∗, both formulas over the signature with the descendant relation. We provide an effective characterization of tree languages definable in ∆2. This characterization is in terms ...

Quantifiers, unlike proper names or definite descriptions, cannot be given the semantics of referring expressions. This fact has triggered a long-standing debate in formal semantics and syntax as to the combinatorial means by which quantifiers are integrated into a sentence. The present paper contributes to this debate through an investigation of quantifier comprehension during real-time senten...

We consider inclusion relations among a multitude of classical complexity classes and classes with probabilistic components. A key tool is a method for characterizing such classes in terms of the ordinary quantifiers 3 and V together with a quantifier 3+, which means roughly “for most,” applied to polynomial-time predicates. This approach yields a uniform treatment which leads to easier proofs ...

We prove a local normal form theorem of the Gaifman type for the infinitary logic L∞ω(Qu) whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local EhrenfeuchtFräıssé type game similar to the one in [KL04]. A consequence is that every sentence of L∞ω(Qu) of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form (∃≥iy)ψ(y) ...