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.
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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, or many. This book presents the definitive interdisciplinary exploration of how they work — their syntax, semantics, and inferential role. Don't have an account? Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use for details see www.
University Press Scholarship Online. If D is a domain of x and P x is a predicate dependent on x , then the universal proposition can be expressed as. This notation is known as restricted or relativized or bounded quantification. Together with negation, only one of either the universal or existential quantifier is needed to perform both tasks:. Every quantification involves one specific variable and a domain of discourse or range of quantification of that variable. The range of quantification specifies the set of values that the variable takes.
In the examples above, the range of quantification is the set of natural numbers. Specification of the range of quantification allows us to express the difference between, asserting that a predicate holds for some natural number or for some real number. Expository conventions often reserve some variable names such as " n " for natural numbers and " x " for real numbers, although relying exclusively on naming conventions cannot work in general since ranges of variables can change in the course of a mathematical argument.
A more natural way to restrict the domain of discourse uses guarded quantification. For example, the guarded quantification.
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In some mathematical theories a single domain of discourse fixed in advance is assumed. For example, in Zermelo—Fraenkel set theory , variables range over all sets.
- Quantifier (logic) - Wikipedia;
- Stanley Peters and Dag Westerståhl.
- Quantifier (logic)!
In this case, guarded quantifiers can be used to mimic a smaller range of quantification. Thus in the example above to express. Mathematical Semantics is the application of mathematics to study the meaning of expressions in a formal language. It has three elements: A mathematical specification of a class of objects via syntax , a mathematical specification of various semantic domains and the relation between the two, which is usually expressed as a function from syntactic objects to semantic ones.
This article only addresses the issue of how quantifier elements are interpreted. Given a model theoretical logical framework, the syntax of a formula can be given by a syntax tree. Quantifiers have scope and a variable x is free if it is not within the scope of a quantification for that variable.
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An interpretation for first-order predicate calculus assumes as given a domain of individuals X. A formula A whose free variables are x 1 , Boolean-valued means that the function assumes one of the values T interpreted as truth or F interpreted as falsehood. The interpretation of the formula. Similarly the interpretation of the formula. The semantics for uniqueness quantification requires first-order predicate calculus with equality. One possible interpretation mechanism can be obtained as follows: If A is a formula with free variables x 1 , A few other quantifiers have been proposed over time.
Term logic , also called Aristotelian logic, treats quantification in a manner that is closer to natural language, and also less suited to formal analysis. Term logic treated All , Some and No in the 4th century BC, in an account also touching on the alethic modalities. In , George Bentham published his Outline of a new system of logic, with a critical examination of Dr Whately's Elements of Logic , describing the principle of the quantifier, but the book was not widely circulated.
William Hamilton claimed to have coined the terms "quantify" and "quantification", most likely in his Edinburgh lectures c. Augustus De Morgan confirmed this in , but modern usage began with De Morgan in where he makes statements such as "We are to take in both all and some-not-all as quantifiers". The book contains many of the grand old themes, as well as several new ones.
Part I explains how quantifiers like "a", "some", "three", or "most" are among the most pervasive structures by means of which we formulate information, and at the same time, they underlie the basic cognitive processes of counting and computation.
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Thus, the quantificational repertoire of natural and formal languages has always been a shared concern between linguistics, logic, and the foundations of mathematics -- and insights from one area travel to another: And, despite initial appearances in first-order logic, there is much more to quantification than just simple enumeration of properties of individual objects. Semantic discussion continues right until today about the intricacies of quantification in non-countable measure settings "waiting all of my life" , quantified collective properties and actions, as described by plurals, or just the meaning of interactive quantification like 'doing things together'.
Indeed, a sense of such subtleties is also entering computer science, with newer query languages in data base theory, and my sense is that complex forms of quantification will even enter theories of interaction in game theory and social choice theory. Part II of the book is a masterful survey of major quantificational structures in natural language. We get an in-depth study of the main types of determiner expression, as well as a lucid account of the major formal semantic tools developed over the years to describe them, including set-theoretic structures and numerical representations.
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Special care is given to detailed semantic accounts of constructions whose precise general analysis has defied researchers for many years, such as possessives and reciprocals. Chapter 7 on possessives presents the most sophisticated general definition to date for the ubiquitous genitive ending " 's". The authors' solution uses a higher-order operator performing predicate restrictions of two quantifier arguments to some binary possessive relation, which is logically interesting, and which seems to fit the known facts rather well.
Finally, Chapter 8 on exceptive constructions "All numbers are man-made, except for the natural ones" examines the major semantic accounts of today, and then proposes a new view on what an 'exception' is which in fact dates back to Antiquity, as the authors note which solves some major current descriptive problems. In summary, Chapters 6, 7, and 8 definitely improve on the state of the art in practising semantics.
In addition to searching for specific truth conditions for important expressions, Part II of the book also studies general types of semantic behaviour in depth.
Quantifiers in Language and Logic
In particular, it tells the story of 'logicality' of quantifiers in terms of invariance of their truth value across isomorphisms between individual domains sometimes called 'permutation invariance'. Also, it gives a comprehensive treatment of what might be called inferential stability under increasing or decreasing denotations for set arguments, making "all penguins walked" imply that all penguins moved, and also, that all female penguins walked.
These famous 'monotonicity inferences', upward or downward, eventually go back right to medieval logic. Intertwined with this sophisticated descriptive apparatus are more theoretical themes which give generalized quantifier theory its distinctive flavour, viz. Many of these have to do with the expressive power of natural languages.