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I'm really having trouble with this question. I already wasted many hours trying to think of what to do. I have to get $E$ on the right side given the set of premises $1$, $2$, $3$. I can add in $F$ later when I have $E$, so can someone please help me get $E$? I know the question is extremely easy (or should be), but I honestly just started learning this, so I'm not good at it (at all) yet.

By the way the proofs look something like this enter image description here

But obviously the question is different.


2 Answers 2


(A) The first premiss gives you

$A \lor B\quad$ (a)

$\neg C$

The latter and the second premiss gives you

$D \land \neg A$

and hence of course


$\neg A\quad$ (b).

Put together (a) and (b) and you get


which together with the third premiss gives us

$A \lor E\quad$ (c).

(b) and (c) now give us


Fill in the steps using modus ponens and disjunctive syllogism, massage it into your preferred way of laying out proofs, and we are done!

(B) In the immortal words of the Hitchhikers Guide to the Galaxy, Don't Panic! There really was nothing magic going on here in finding the proof -- a bit of calm strategic thinking will see you through with this kind of example.

You did the right thing in working from the end and asking "How can I get $E \lor F$?", and noting that since $F$ wasn't around, you need to prove $E$.

How are you going to get $E$? It only appears the third conditional premiss's consequent, $A \lor E$, so somehow you are going to need to prove the antecedent of the conditional $B$, and then (probably) also prove $\neg A$ so you can use disjunctive syllogism to extract $E$. And yippee, there's a $B$ and a $\neg A$ around in the other premisses. Let's see if we can get at them!

Starting from the beginning then, the first box writes itself. And then it was obvious that the only thing to do with the original second premiss is use modus ponens which gets us to the second box and then the third box wrote itself.

Pause for thought. $D$ isn't going to be useful, but (b) is what we want, but the only way we can use it right now is to put it together with (a) and use disjunctive syllogism to get $B$.

And now we are motoring ....

  • $\begingroup$ I really appreciate the help, especially your explanation. $\endgroup$
    – Gary Choi
    Feb 10, 2014 at 15:31

Starting with the premises, how much can you figure out about the truth values of each of $A-E$? By conjunction elimination on the first premise, you know $\neg C$, so $C$ is false. See what other truth values you can pin down.

  • $\begingroup$ What do you mean disjunction elimination on the first premise? The first premise is a conjunct isn't it? $\endgroup$
    – Gary Choi
    Feb 10, 2014 at 6:40
  • $\begingroup$ Yes thanks, fixed. $\endgroup$ Feb 10, 2014 at 7:09

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