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Determine if the following set of sentence is consistent or inconsistent:

If John committed the murder, then he was in the victim's apartment and did not leave before 11. In fact, he was in the victim's apartment. If he left before 11, then the doorman saw him but it is not the case either that the doorman saw him or that he committed the murder.

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Since you are new to this site, please consider reading this: How to ask a homework question? I wrote this comment because the question sounds homework-like. –  Pedro Tamaroff Sep 18 '12 at 19:05
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1 Answer

It’s entirely possible to work the problem using only ordinary English. However, you can also approach it systematically. The statement of the problem involves four propositions. In the order that they appear they are:

$p$: John committed the murder.
$q$: John was in the victim’s apartment.
$r$: John did not leave before $11$.
$s$: The doorman saw him.

Here I’ve given them symbolic names for brevity. Now translate the assertions in the three sentences into statements in propositional logic:

$$\begin{align*} &p\to(q\land r)\\ &q\\ \lnot &r\to\big(s\land\lnot(s\lor p)\big) \end{align*}$$

If you do a little ‘algebraic’ manipulation of the third line, you may be able to see right away whether the system of statements is consistent. If not, write out a complete truth table; since you have four basic propositions, your truth table will have $2^4=16$ lines. Is there any line in which all three statements are true? If so, they’re consistent; if not, they’re inconsistent. (Notice that you really only have to construct the half of the truth table in which $q$ is true, since you don’t care about the other eight cases.)

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that was gr8 help! thx –  Khanak Sep 18 '12 at 19:51
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My natural reading of the third English sentence would have "then" binding tighter than "but", so I would represent it as $(\neg r\to s)\land\neg(s\lor p)$ –  Henning Makholm Sep 18 '12 at 22:55
    
@Henning: I thought about it, but if that were intended, I’d expect a comma before the but: one is technically required there, so its absence ought to be significant. However, I will readily agree that the intended interpretation isn’t really all that clear, and it’s entirely possible that the author was insensitive to such subtleties. (You get a free pass, even if your English is excellent. :-)) –  Brian M. Scott Sep 18 '12 at 23:01
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