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

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

Modus Tollens

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• Denying the consequent, based on the idea of Contrapositive (逆否命题)

Premise 1	$p\to q$
Premise 2	$\neg q$
Conclusion	$\neg p$

Generalisation

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

Specialisation

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

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

Elimination

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

Transitivity

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	Case 1
Premise 1	$p\to q$
Premise 2	$q\to r$
Conclusion	$p\to r$

Division into Cases

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• The table below only shoes 2 cases, we can have more than 2 cases

	Case 1
Premise 1	$p\vee q$
Premise 2	$p\to r$
Premise 3	$q\to r$
Conclusion	$r$

Contradiction Rule

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Premise 1	$\neg p\to false$
Conclusion	$p$

Important

If a assumption leads to a contradiction, then that assumption must be false. The core of Proof by Contradiction (矛盾证明法)