Quantifiers metamorphoses. Generalizations, variations, algorithmic semantics (original) (raw)

Logicism, quantifiers, and abstraction

2008

With the aid of a non-standard (but still first-order) cardinality quantifier and an extra-logical operator representing numerical abstraction, this paper presents a formalization of first-order arithmetic, in which numbers are abstracta of the equinumerosity relation, their properties derived from those of the cardinality quantifier and the abstraction operator.

Quantification in ordinary language

2011

We firstly show that the standard interpretation of natural quantification in mathematical logic does not provide a satisfying account of its original richness. In particular, it ignores the difference between generic and distributive readings. We claim that it is due to the use of a set theoretical framework. We therefore propose a proof theoretical treatment in terms of proofs and refutations. Thereafter we apply these ideas to quantifiers that are not first order definable like "the majority of".

$$\mathbb {K}$$—A Semantic Framework for Programming Languages and Formal Analysis

Engineering Trustworthy Software Systems, 2020

We give an overview on the applications and foundations of the K language framework, a semantic framework for programming languages and formal analysis tools. K represents a 20-year effort in pursuing the ideal language framework vision, where programming languages must have formal definitions, and tools for a given language, such as parsers, interpreters, compilers, semantic-based debuggers, state-space explorers, model checkers, deductive program verifiers, etc., can be derived from just one reference formal definition of the language, which is executable, and no other semantics for the same language should be needed. The correctness of the language tools is guaranteed on a case-by-case basis by proof objects, which encode rigorous mathematical proofs as certificates for every individual tasks that the tools do and can be mechanically checked by third-party proof checkers.

Stanley Peters and Dag Westerståhl: Quantifiers in language and logic

Linguistics and Philosophy, 2010

Quantifiers in Language and Logic (QLL) is a major contribution to natural language semantics, specifically to quantification. It integrates the extensive recent work on quantifiers in logic and linguistics. It also presents new observations and results. QLL should help linguists understand the mathematical generalizations we can make about natural language quantification, and it should interest logicians by presenting an extensive array of quantifiers that lie beyond the pale of classical logic. Here we focus on those aspects of QLL we judge to be of specific interest to linguists, and we contribute a few musings of our own, as one mark of a worthy publication is whether it stimulates the reader to seek out new observations, and QLL does. QLL is long and fairly dense, so we make no attempt to cover all the points it makes. But QLL has a topic index, a special symbols index and two tables of contents, a detailed one and an overview one, all of which help make it user friendly. QLL is presented in four parts: I, ''The Logical Conception of Quantifiers and Quantification'' with an introductory section ''Quantification''. II, ''Quantifiers of Natural Language'', the most extensive section in the book and of the most direct interest to linguists. III, ''Beginnings of a Theory of Expressiveness, Translation, and Formalization'' introduces notions of expressive power and definability, and IV, presents recent work and techniques concerning quantifier definability over finite domains, making accessible to linguists recent work in finite model theory.

A formal theory of intermediate quantifiers

The paper provides a logical theory of a specific class of natural language expressions called intermediate quantifiers (most, a lot of, many, a few, a great deal of, a large part of, a small part of), which can be ranked among generalized quantifiers. The formal frame is the fuzzy type theory (FTT). Our main idea lays in the observation that intermediate quantifiers speak about elements taken from a class that is made "smaller" than the original universe in a specific way. Our theory is based on the formal theory of trichotomous evaluative linguistic expressions. Thus, an intermediate quantifier is obtained as a classical quantifier "for all" or "exists" but taken over a class of elements that is determined using an appropriate evaluative expression. In the paper we will characterize the behavior of intermediate quantifiers and prove many valid syllogisms that generalize classical Aristotle's ones. . 1 In , the intermediate quantifiers are called comparative. According to the literature on generalized quantifiers, however, "comparative quantifiers" are quantifiers such as "more than about", "about three times less than", etc. We will, therefore, keep the term intermediate in this paper.

V -A Formal Mathematical Language Short paper

The V language is based on ZFC set theory and intended for use mostly in education. So every proof step is quite simple and can be checked by a human. The most important syntactic construct in V is the class term. Class terms are used for defining theories, the lambda, set builder, and restricted quantification notations. For testing the V language the following 3 modules (modules in V are similar to articles in Mizar) were written: ❼ root: elementary set theory: 5134 lines, 143 proofs, 47 equational proofs; some of the theorems were declared as axioms, they have to be proved later; ❼ group: the basics of group theory, including the isomorphism theorems: 5402 lines, 508 proofs, 326 equational proofs; ❼ altg: the basics of the permutation theory, necessary for defining the alternating group: about 1700 lines, 136 proofs, 60 equational proofs; Also, a computer program (a proof checker) has been written(26300 lines in C++), which now is working in an experimental mode. The 3 modules have been successfully checked by the proof checker. The processing times for the root, group, and altg modules are 35sec,53sec, and 14sec respectively on the laptop Lenovo ThinkPad Yoga 11e 5th Gen. The modules can be downloaded at [2].