# universal quantification

In predicate logic, universal quantification is an attempt to formalize the notion that something (a logical predicate) is true for everything, or every relevant thing. The resulting statement is a universally quantified statement, and we have universally quantified over the predicate. In symbolic logic, the universal quantifier (typically ) is the symbol used to denote universal quantification, and is often informally read as "given any" or "for all".

Quantification in general is covered in the article on quantification, while this article discusses universal quantification specifically.

Compare this with existential quantification, which says that something is true for at least one thing.

## Basics

Suppose you wish to say
2·0 = 0 + 0, and 2·1 = 1 + 1, and 2·2 = 2 + 2, etc.
This would seem to be a logical conjunction because of the repeated use of "and." But the "etc" can't be interpreted as a conjunction in formal logic. Instead, rephrase the statement as
For all natural numbers n, 2·n = n + n.
This is a single statement using universal quantification.

Notice that this statement is really more precise than the original one. It may seem obvious that the phrase "etc" is meant to include all natural numbers, and nothing more, but this wasn't explicitly stated, which is essentially the reason that the phrase couldn't be interpreted formally. In the universal quantification, on the other hand, the natural numbers are mentioned explicitly.

This particular example is true, because you could put any natural number in for n and the statement "2·n = n + n" would be true. In contrast, "For all natural numbers n, 2·n > 2 + n" is false, because if you replace n with, say, 1 you get the false statement "2·1 > 2 + 1". It doesn't matter that "2·n > 2 + n" is true for most natural numbers n: even the existence of a single counterexample is enough to prove the universal quantification false.

On the other hand, "For all composite numbers n, 2·n > 2 + n" is true, because none of the counterexamples are composite numbers. This indicates the importance of the domain of discourse, which specifies which values n is allowed to take. Further information on using domains of discourse with quantified statements can be found in the Quantification article. But in particular, note that if you wish to restrict the domain of discourse to consist only of those objects that satisfy a certain predicate, then for universal quantification, you do this with a logical conditional. For example, "For all composite numbers n, 2·n > 2 + n" is logically equivalent to "For all natural numbers n, if n is composite, then 2·n > 2 + n". Here the "if ... then" construction indicates the logical conditional.

In symbolic logic, we use the universal quantifier symbol (an upside-down letter "A" in a sans-serif font) to indicate universal quantification. Thus if P(n) is the predicate "2·n > 2 + n" and N is the set of natural numbers, then

is the (false) statement
For all natural numbers n, 2·n > 2 + n.

Similarly, if Q(n) is the predicate "n is composite", then

is the (true) statement
For all composite numbers n, 2·n > 2 + n.

Several variations in the notation for quantification (which apply to all forms) can be found in the quantification article. But there is a special notation used only for universal quantification, which we also give here:

The parentheses indicate universal quantification by default.

## Properties

### Negation

Note that a quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The notation mathematicians and logicians utilize to denote negation is: .

For example, let P(x) be the propositional function "x is married"; then, for a Universe of Discourse X of all living human beings, consider the universal quantification "Given any living person x, that person is married":

A few second's thought demonstrates this as irrevocably false; then, truthfully, we may say, "It is not the case that, given any living person x, that person is married", or, symbolically:
.

Take a moment and consider what, exactly, negating the universal quantifier means: if the statement is not true for every element of the Universe of Discourse, then there must be at least one element for which the statement is false. That is, the negation of is logically equivalent to "There exists a living person x such that he is not married", or:

Generally, then, the negation of a propositional function's universal quantification is an existential quantification of that propositional function's negation; symbolically,

A common error is writing "all persons are not married" (i.e. "there exists no person who is married") when one means "not all persons are married" (i.e. "there exists a person who is not married"):

### Rules of Inference

A rule of inference is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the universal quantifier.

Universal instantiation concludes that, if the propositional function is known to be universally true, then it must be true for any arbitrary element of the Universe of Discourse. Symbolically, this is represented as

where c is a completely arbitrary element of the Universe of Discourse.

Universal generalization concludes the propositional function must be universally true if it is true for any arbitrary element of the Universe of Discourse. Symbolically, for an arbitrary c,

It is especially important to note c must be completely arbitrary; else, the logic does not follow: if c is not arbitrary, and is instead a specific element of the Universe of Discourse, then P(c) only implies an existential quantification of the propositional function.

## References

• Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-568-81262-0.
predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulas contain variables which can be quantified.
An open sentence is a sentence which contains variables. Unlike an ordinary sentence, which contains constants, open sentences do not express propositions; they are neither true nor false. Hence, the open sentence:

(1) x is a number

Has no truth-value.
Symbolic logic is the area of mathematics which studies the purely formal properties of strings of symbols. The interest in this area springs from two sources. First, the symbols used in symbolic logic can be seen as representing the words used in philosophical logic.
The term quantification has several meanings, general and specific. Primarily it covers all those acts which quantify observations and experiences by converting them into numbers through counting and measuring. It is thus the basis for mathematics and for science.
In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. The logical operator symbol ∃ called the existential quantifier is used to denote existential quantification.
In logic and/or mathematics, logical conjunction or and is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false!

## Definition

Logical conjunction
Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration.
In mathematics, a natural number can mean either an element of the set (i.e the positive integers or the counting numbers) or an element of the set (i.e. the non-negative integers).
Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration.
Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration.
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i.e., a specific instance of the falsity of a universal quantification (a "for all" statement).
A composite number is a positive integer which has a positive divisor other than one or itself. By definition, every integer greater than one is either a prime number or a composite number. The number one is considered to be neither prime nor composite.
The domain of discourse, sometimes called the universe of discourse, is an analytic tool used in deductive logic, especially predicate logic. It indicates the relevant set of entities that are being dealt with by quantifiers.
The term quantification has several meanings, general and specific. Primarily it covers all those acts which quantify observations and experiences by converting them into numbers through counting and measuring. It is thus the basis for mathematics and for science.
The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain conditionals in logic. In propositional logic, it expresses a binary truth function ⊃ from truth-values to truth-values.
In logic, statements p and q are logically equivalent if they have the same logical content.

Syntactically, p and q are equivalent if each can be proved from the other.
Symbolic logic is the area of mathematics which studies the purely formal properties of strings of symbols. The interest in this area springs from two sources. First, the symbols used in symbolic logic can be seen as representing the words used in philosophical logic.
A is the first letter in the Latin alphabet. Its name in English is a[1] (IPA: /eɪ/), plural aes, as, or a's.
In typography, a sans-serif or sans serif (sometimes just sans) typeface is one that does not have the small features called "serifs" at the end of strokes. The term comes from the French word sans, meaning "without".
SET may stand for:
• Sanlih Entertainment Television, a television channel in Taiwan
• Secure electronic transaction, a protocol used for credit card processing,

The term quantification has several meanings, general and specific. Primarily it covers all those acts which quantify observations and experiences by converting them into numbers through counting and measuring. It is thus the basis for mathematics and for science.
In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. The logical operator symbol ∃ called the existential quantifier is used to denote existential quantification.
In logic, a rule of inference is a function from sets of formulae to formulae. The argument is called the premise set (or simply premises) and the value the conclusion.
In logic universal instantiation (UI, also called "Dictum de omni") is an inference from a truth about each member of a class of individuals to the truth about a particular individual of that class.