This document discusses quantification in logic. Quantification transforms a propositional function into a proposition by expressing the extent to which a predicate is true. There are two main types of quantification: universal quantification and existential quantification. Universal quantification expresses that a predicate is true for every element, while existential quantification expresses that a predicate is true for at least one element. The document provides examples and pros and cons of each type of quantification and notes that quantification operators like ∀ and ∃ take precedence over logical operators.

1. Manash Kumar Mondal

2. What is Quantification??
=>> Quantification is a method to transform a
propositional function into a proposition.
Express the extent
to which
a predicate is true.

3. Types
1. Universal quantification
2. Existential quantification

4. Universal quantification??
A predicate is true for every element under
consideration
denoted as ∀ p(x)
Universal Quantification allows us to capture
statements of the form “for all” or “for every”.

5. Pros and cons of
Universal quantification
• “p(x) for all values of x in the domain”
• Read it as “for all x p(x)” or “for every x p(x)”
• A statement is false if and only if p(x) is not always
true.
• An element for which p(x) is false is called a
counterexample of ∀ p(x)
• A single counterexample is all we need to establish
that is not true .

6. Existential quantification??
There exists an element x in the universe of discourse.
denoted as ∃x P (x )

7. Pros & Cons of
Existential quantification
• “There exists an element x in the domain such that p(x) (is
true)”
• Denote that as ∃x P (x ) where ∃ is the existential quantifier.
• In English, “for some”, “for at least one”, or “there is”.
• Read as “There is an x such that p(x)”, “There is at least one x
such that p(x)”, or “For some x, p(x)”.

8. Example
Universal ::
∀x (X2 ≥ 0) --> "the square of any number is not negative.''
Existential ::
∃x (X ≥ X2 ) --> is true since x=0 is a solution and there are
many others.

9. Precedence of quantifiers
∀ and ∃ have higher precedence than all logical
operators from propositional calculus
For example
∀x P (x ) ∨ Q (x ) means (∀x P (x )) ∨ Q (x ), not
∀x (P (x ) ∨ Q (x )).

10. Thank You 