Universal Statement

Universal Statement

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$$\forall x\in D,Q(x)$$ $$\forall x(x\in D\to Q(x))$$

• Made of Predicate Symbol, Predicate Variable & Quantifier
• true iff Q(x) is true for every x in D
• false iff Q(x) is false for at least one x in D
• For simple ones, usually universal comes with Conditional Statement

Important

$\forall x\in D,Q(x)$ is same as $Q(x_1)\wedge Q(x_2)\wedge \dots \wedge Q(x_n)$.

Negation of Universal

$$\neg (\forall x\in D,Q(x))\equiv \exists x\in D,\neg Q(x)$$

• Negation of Universal is Logical Equivalence to Existential Statement
• The above Predicate means There is AT LEAST ONE that IS NOT

Vacuous Truth of Universal

• Given the statement: All balls in the bowl are blue, however no balls in the bowl. The statement is vacuously true, because the Negation of Universal is One of the balls in the bowl isn't blue which is obviously false

Another perspective

If you consider a hypothesis as a set, when the set is an empty set (aka the hypothesis is false), the negation of the statement is that there exists at least one element in the set that contradicts the given universal statement. Since it is an empty set, there isn’t such an element. Thus, an empty set or a false hypothesis will always result in a universal statement that is true.

Universal Conditional

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$$\forall x(P(x)\to Q(x))$$

• Made of Universal Statement & Conditional Statement

Simplified to universal statement

The above universal conditional statement can be reduced to $\forall x\in P,Q(x)$ by narrowing down the Domain of Predicate Variable $x$ with respect to $P(x)$. In essence, the new $x$ is the Truth Set of $P(x)$.

Negation of Universal Conditional

• Make use of Negation of Universal & Implication Law

Important

Useful for Proof by Contradiction (矛盾证明法)!

Vacuous Truth of Universal Conditional

• Given $\forall x\in D,P(x)\to Q(x)$
• It is Vacuously True if and only if P(x) is false for every x in D
• Vacuous Truth of Universal also applies here for the Quantifier part