I have the following:
Premise: {$p \lor q, q\rightarrow r,p \land s \rightarrow t, \lnot r, \lnot q \rightarrow u \land s$}, conclusion: $t$
I'm having a real hard understanding how to prove the above using rules of inference. I can't seem to see the broader picture of how to use these rules to prove anything. Here's my attempt.
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$q \rightarrow r$ Premise
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$\lnot r$ Premise
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$\lnot q$ Modus Tollens using (2) and (1)
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$p \lor q$ Premise
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$q \lor r$ Resolution using (1) and (4)
and this is the point I get stuck. I have two more premises but can't see where or how they would resolve anywhere. Any hints?
Best Answer
$(1) + (4)$ do not imply $q\lor r$, so undo line $(5)$ We do have that $(3) + (4)$ imply $p$. If $p \lor q$, and $\lnot q$, then $p$. Perhaps this can be line $(5)$.
Suggestion for the next step $(6)$: from $(3),$ along with the premise $\lnot q \rightarrow (u \land s)$, it follows by modus ponens that $u \land s$.
(7) Now, extract $s$ from $u \land s$.
(8) Then introduce the conjunction: $p \land s$. $p$ is from your new (5th) step, and $s$ from step (7).
Now we can use the premise $p\land s \implies t$ and $p \land s$ from step (8) to conclude by modus ponens that $t$, as desired
Key point: If you haven't used a premise, think of how it might help get you from what you have to what you need to establish. Always keep the goal or target proposition in your mind.