Quantified Rule of Inference

Abstract

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• Rule of Inference (推理规则) with Quantifiers

Universal Modus Ponens

• Premise: For all x, if x makes P(x) true, then x makes Q(x) true
• Premise: a is an element of x, a makes P(x) true
• Conclusion: a makes Q(x) true

Universal Modus Tollens

• Premise: If x makes P(x) true, then x makes Q(x) true
• Premise: a doesn’t make Q(x) true

Universal Transitivity

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

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