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Given the implication $[(p\vee (q\wedge r) \wedge (p\to s)] \to (r\vee s)$, establish the validity of the argument using resolution.

This is the answer my textbook gave:

$$\begin{array}{rll}1&p\vee(q\wedge r)&\textrm{Premise}\\ 2&(p\vee q)\wedge(p\vee r)&\textrm{Step 1 and Distributive Law}\\ 3&p\vee r&\textrm{Step 2 Rule of Conjunctive Simplification}\\ 4&p\to s&\textrm{Premise}\\ 5&!p\vee s&\textrm{Step 4, }p\to s\Leftrightarrow\,!p\vee s\\ 6&r\vee s&\textrm{Steps 3 and 5, Rule of the Conjunction and Resolution}\end{array}$$

Why is $p\vee q$ not used in step 6, with the resolvent as $r\vee s\vee q$?

Mauro ALLEGRANZA
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Simon
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    I don't understand your question. Why would you want to get $R\vee s\vee q$, when you are asked to get $r\vee s$? – frabala Feb 27 '14 at 05:19

2 Answers2

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What is called in Step 6 : Rule of the Conjunction and Resolution, is called also Resolution (see Wiki : List of rules of inference:

from $p \lor r$ and $\lnot p \lor s$, $\quad r \lor s$ follows.

So in your proof, in Step 5 you use the

equivalence between $p \rightarrow s$ and $\lnot p \lor s$,

and in Step 6 you must use Resolution to get , from $p \lor r$ (Step 3) and $\lnot p \lor s$ (Step 5) to :

$r \lor s$.

Mauro ALLEGRANZA
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The clauses $p \vee r$ (from step 3) and $\neg p \vee s$ (from step 5) together imply $r \vee s$, which gives you what you need. $q$ was not really needed.

angryavian
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