Logical Predicates

These are translations from English sentences to logical statements that take an arbitrary Free Variable like P(x). The quantifiers can be evaluated with an input to return true or false They tend to paired with Logical Connectives and Logical Predicates:

Universal Quantified

For a statement $P(x)$: $\forall x,P(x)$

• True if P(x) is true for every element in the Universe of Discourse
• False otherwise

Existentially Quantified

$\exists x,P(x)$

• True if P(x) is true for some element in the Universe of Discourse
• False otherwise

Quantifier Negation Laws

$\neg \forall ,P(x)\equiv \exists x,\neg P(x)$ $\neg \exists ,P(x)\equiv \forall x,\neg P(X)$ Negate the predecates, and the quantifier

Negating Predicates with Negated Quantifiers

$\exists x\neg \forall y,C(y,x)$

1. Remove all negations in front of quantifiers
2. Flip quantifiers

Bound Quantifiers

This is when the Universe of Discourse is bounded $\forall x\in A,P(x)$ is short for $\forall x,x\in A⟹P(x)$ $\exists x\in A,P(x)$ is short for $\exists x,x\in A\wedge P(x)$

Bound Quantifiers When $A=\varnothing$

If $A$ is the empty set $\varnothing$ then, $\forall x\in \varnothing ,P(x)$ is Vacuously True $\exists x\in \varnothing ,P(x)$ is False

Quantifier Abbreviation

$\forall x(x\in A\to P(x))\equiv \forall x\in AP(x)$