Concept
Universal quantification
Overview
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.
It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("∀x", "∀(x)", or sometimes by "(x)" alone). Universal quantification is distinct from existential quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain.
Quantification in general is covered in the article on quantification (logic). The universal quantifier is encoded as U+2200 ∀ FOR ALL in Unicode, and as \forall in LaTeX and related formula editors.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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This improvement also is due to Cantor, and it is one which has revolutionised mathematics. Only order is now relevant to limits. Thus, for instance, the smallest of the infinite integers is the limit of the finite integers, though all finite integers are at an infinite distance from it. The study of different types of series is a general subject of which the study of ordinal numbers (mentioned above) is a special…
Bertrand Russell, Mysticism and Logic and Other Essays
States what Universal quantification is; the author is tied to Universal quantification in the knowledge graph.
We thus obtain, on the straight line in question, a continuous series of points, which, since it is continuous, we regard as constituting our straight line. It is thus solely from the hypostatizing of relations, which metrical Geometry requires, that the view of straight lines and planes as _composed_ of points arises, and it is from this hypostatizing that the difficulties of metrical Geometry spring. =132.= The…
Bertrand Russell, An essay on the foundations of geometry
States what Universal quantification is; the author is tied to Universal quantification in the knowledge graph.
The copula is the sign denoting that there is an affirmation or denial; and thereby enabling the hearer or reader to distinguish a proposition from any other kind of discourse. Thus, in the proposition, The earth is round, the Predicate is the word _round_, which denotes the quality affirmed, or (as the phrase is) predicated: _the earth_, words denoting the object which that quality is affirmed of, compose the…
John Stuart Mill, A System of Logic
States what Universal quantification is.
Philosophers who discussed Universal quantification
Related ideas
- Exclusive or
- First-order logic
- Giuseppe Peano
- Counterexample
- Negation
- Logical conjunction
- Logical connective
- Mathematical logic
- Logical NOR
- Natural number
- Logical biconditional
- Domain of discourse
- False (logic)
- Power set
- Symbol (formal)
- Rule of inference
- Set (mathematics)
- Predicate (logic)
- A
- Sheffer stroke
- Saunders Mac Lane
- Type theory
- Logical constant
- Bertrand Russell
Opposing perspectives
Place in the tradition
Part of First-order logic.