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Given any logical formulas $\alpha, \beta, \gamma$. Then:

$$ (\alpha \lor \beta) \land (\neg \beta \lor \gamma) \models (\alpha \lor \gamma) $$

Unlike for Modus Ponens and the chain rule, we were not given a name for this law.

mafu
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  • I wouldn't consider this a "basic" law. – David H Mar 03 '14 at 02:50
  • @DavidH I'm not sure. Why not? – mafu Mar 03 '14 at 02:53
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    The name for this, when viewed as a rule of inference, is "resolution". I would recommend http://en.wikipedia.org/wiki/Resolution_(logic) but it seems to be particularly unclear now that I look at it. – Carl Mummert Mar 03 '14 at 03:01
  • @CarlMummert I agree that this can be seen as a resolution, but I was under the impression that this is an independent and more basic concept. Good chance I was wrong, though - Resolution would capture the idea pretty closely imo. – mafu Mar 03 '14 at 03:10

1 Answers1

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If we work with classical logic, this a a propositional logic Hypothetical syllogism "in disguise".

In classical logic we have that $\lnot p \lor q$ is equivalent to : $p \rightarrow q$ is equivalent to and so that $p \lor q$ is equivalent to : $\lnot p \rightarrow q$.

So, your example :

$ (\alpha \lor \beta) \land (\lnot \beta \lor \gamma) \vDash (\alpha \lor \gamma)$

can be rewritten as :

$(\lnot \alpha \rightarrow \beta) \land (\beta \rightarrow \gamma) \vDash (\lnot \alpha \rightarrow \gamma)$.

Mauro ALLEGRANZA
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