Negation of Quantified Statement

Negation of Universal Quantification Statement

Negation statement "For all x applicable p(x)" is "It is not true that for all x apply p(x)" or in other words "there is at least one x such that p(x) is not applicable". By using the symbol, we write as follows:

~(∀x) p(x) ≡ (∃x) ~p(x)

Example:
p: All cats have a tail
~p: Not really all cats have a tail.
~p: There is a cat that has no tail.
~p: Some cats do not have tails.

Negation of Existential Quantification Statement

Negation statement "There is x applicable p(x)" is "not true that there x applies p(x)" or in other words "For all x such that p(x) not applicable". By using the symbol we write as follows:

~(∃x) p(x) ≡ (∀x) ~p(x)

Example:
p: There is a child who likes to play ball.
~p: Not true There is a child who likes to play ball.
~p: All children do not like to play ball.

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