Quantified Rule of Inference

Abstract

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• Rule of Inference (推理规则) with Quantifier

Universal Modus Ponens

	Case 1
Premise 1	$\forall x(P(x)\to Q(x))$
Premise 2	$P(a)$
Conclusion	$Q(a)$

Universal Modus Tollens

	Case 1
Premise 1	$\forall x(P(x)\to Q(x))$
Premise 2	$\neg Q(a)$
Conclusion	$\neg P(a)$

Universal Transitivity

	Case 1
Premise 1	$\forall x(P(x)\to Q(x))$
Premise 2	$\forall x(Q(x)\to R(x))$
Conclusion	$\forall x(P(x)\to R(x))$

Existential Instantiation

$$\exists x\in D,P(x)$$ $$\therefore P(a)forsomea\in D$$

Universal Instantiation

• If some property is true of everything in the set, then it is true of any particular thing in the set
• Core tool for deductive reasoning

$$\forall x\in D,P(x)$$ $$\therefore a\in D\to P(a)$$