CSCE 222 Chapter 1.5

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Rules of Inference

argument

a sequence of statements (premises) followed by a conclusion

valid

the conclusion (final statement of the argument) must follow from the truth of the premises

premises

any of the preceding statements of an argument

fallacies

logic errors and incorrect reasoningwithin an argument

Rule	Tautology	Name
$\begin{array}{l}p\\ p\to q\\ \therefore q\end{array}$	$[p\wedge (p\to q)]\to q$	Modus ponens
$\begin{array}{l}\mathrm{\neg }q\\ p\to q\\ \therefore \mathrm{\neg }p\end{array}$	$[\mathrm{\neg }q\wedge (p\to q)]\to \mathrm{\neg }p$	Modus tollens
$\begin{array}{l}p\to q\\ q\to r\\ \therefore p\to r\end{array}$	$[(p\to q)\wedge (q\to r)]\to (p\to r)$	Hypothetical syllogism
$\begin{array}{l}p\vee q\\ \mathrm{\neg }p\\ \therefore q\end{array}$	$[(p\vee q)\wedge \mathrm{\neg }p]\to q$	Disjunctive syllogism
$\begin{array}{l}p\\ \therefore p\vee q\end{array}$	$p\to (p\vee q)$	Addition
$\begin{array}{l}p\wedge q\\ \therefore p\end{array}$	$(p\wedge q)\to p$	Simplification
$\begin{array}{l}p\\ q\\ \therefore p\wedge q\end{array}$	$[(p)\wedge (q)]\to (p\wedge q)$	Conjunction
$\begin{array}{l}p\vee q\\ \mathrm{\neg }p\vee r\\ \therefore q\vee r\end{array}$	$[(p\vee q)\wedge (p\vee r)]\to (q\vee r)$	Resolution
Rules of Inference for Quantified Statements
$\begin{array}{l}\mathrm{\forall }xP(x)\\ \therefore P(c)\end{array}$		Universal instantiation
$\begin{array}{l}P(c)\text{for an arbitrary}c\\ \therefore \mathrm{\forall }xP(x)\end{array}$		Universal generalization
$\begin{array}{l}\mathrm{\exists }xP(x)\\ \therefore P(c)\text{for some element}c\end{array}$		Existential instantiation
$\begin{array}{l}P(c)\text{for some element}c\\ \therefore \mathrm{\exists }xP(x)\end{array}$		Existential generalization

Common fallacies

1. Fallacy of affirming the conclusion

   $\begin{array}{l}p\to q\\ q\\ \therefore p\end{array}$

2. Fallacy of denying the hypothesis

   $\begin{array}{l}p\to q\\ \mathrm{\neg }p\\ \therefore \mathrm{\neg }q\end{array}$