Notes › MATH 2130: Discrete Mathematics Lecture 5
Rules of Inference and Valid Arguments
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Table of Contents
Valid Arguments
- Definition of valid argument
- Can follow from previous statements due to rules of inference
Rules of Inference
Modus Ponens
$$ \begin{align*} P \rightarrow \ &Q, \\ &P \\ \vdash &Q \end{align*} $$Modus Tollens
$$ \begin{align*} P &\rightarrow Q, \\ &\neg Q \\ \vdash &\neg P \end{align*} $$Hypothetical Syllogism
$$ \begin{align*} P &\rightarrow Q, \\ Q &\rightarrow R \\ \vdash P &\rightarrow R \end{align*} $$Disjunctive Syllogism
$$ \begin{align*} P \ \vee \ &Q, \\ \neg &P \\ \vdash &Q \end{align*} $$Addition
$$ \begin{aligned} P \\ \vdash P \vee Q \end{aligned} $$Simplification
$$ \begin{aligned} P \wedge Q \\ \vdash P \end{aligned} $$Conjunction
$$ \begin{align*} &P, \\ &Q \\ \vdash \ P \wedge \ &Q \end{align*} $$Resolution
$$ \begin{align*} P &\vee Q, \\ \neg P &\vee R \\ \vdash Q &\vee R \end{align*} $$Replacement
- Logically equivalent statements can replace each other
- e.g. contrapositive
References
Course slides §6.1: Rules of Inference and Valid Arguments
Sources
- Course slides §6.1: Rules of Inference and Valid Arguments