Rule of Inference (推理规则)

Abstract

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• A form of Mathematical Argument that is guaranteed to be Valid
• Tools that enable Deductive Reasoning (演繹推理)

Modus Ponens

Modus Tollens

• Denying the consequent

Generalisation

	Case 1	Case 2
Premise	$p$	$q$
Conclusion	$p\vee q$	$p\vee q$

Specialisation

• Allow us to discard some information to focus on things that we are interested

	Case 1	Case 2
Premise	$p\cap q$	$p\cap q$
Conclusion	$p$	$q$

Elimination

	Case 1	Case 2
Premise	$p\cup q$, $\neg q$	$p\cup q$, $\neg p$
Conclusion	$p$	$q$

Transitivity

• Premise: p -> q, q -> r
• Conclusion: p -> r

Division into Cases

• Premise: p v q, p -> r, q -> r
• Conclusion: r
• The above only 2 cases, we can have more than 2 cases

Contradiction Rule

• Premise: ~p -> false
• Conclusion: p
• The core of Proof by Contradiction (反证法)
• If a assumption leads to a contradiction, then that assumption must be false