Im trying to understand how a theorem or statement is proved using Rules of Inference.I have this example

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I really don't understand how they say.Now p->q may be true with p being false.Then conclusion of p is false.Hence the argument is not valid

What i know is the proposition p->q is only false when p is true and q is false.From that how can they say that if p>q is true when p is false then conclusion is false.Can someone explain the logic in layman's terms please.

  • Im referring Discrete Mathematical Structures by Kolman,Busby and Ross

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But he assume that $p \to q$ is true !

This is possible also with $p$ false and $q$ true.

In this case, we have both : $p \to q$ true and $q$ true ("assume that $p \to q$ and $q$ are both true"), and the conclusion $p$ will be false.

Thus :

the argument is invalid.

  • $\begingroup$ thanks.I get that now.But if a question like this comes up,what is the correct method to approach such problems and identify whether the argument is valid/invalid. $\endgroup$ – techno Sep 15 '15 at 9:45
  • $\begingroup$ @techno - exactly that used in the text : to found a counterexample. In the case of prop logic, to find an assignment of truth values such that all the premises are true with it and the conclusion is false; this will be enough to show that the argument is invalid. In order to find it, work backward : start with a "partial" assignment of t-values falsifying the conclusion (in this case, start with $p$ false) and try to "extend" it to the premises. If it works, you have find the counterexample: if you find a contradiction, there is none. $\endgroup$ – Mauro ALLEGRANZA Sep 15 '15 at 9:53
  • $\begingroup$ Please see the edit.I have added an example.It says that the argument is valid but the conclusion is incorrect. What does the book mean by it. $\endgroup$ – techno Sep 15 '15 at 9:59
  • $\begingroup$ @You have to review the def of valid argument... An argument is valid when : it is not possible that all the premises are true and the conclusion is false (see invalidity of the first example). But the validity of an argument is not falsified if applied to a set of premises that are not all true (see second example). $\endgroup$ – Mauro ALLEGRANZA Sep 15 '15 at 10:15
  • $\begingroup$ okay.But in the second example where do we get the information that smoking is unhealthy? $\endgroup$ – techno Sep 15 '15 at 10:25

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