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I have the following question:

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And I have drawn up the truth table below:

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My question is I see there is one truth value that both the premise and conclusion have in common, but does that mean that the premise does or does not entail the conclusion? Thanks for any help!

user3371750
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  • Your truth table is wrong. The premise is true in more than one case. – Git Gud Oct 01 '14 at 14:56
  • $warm(Susan) \lor warm(Mary)$ is a disjunction; thus it is false only in the case $F-F$. You have to fix rows 6 and 8. – Mauro ALLEGRANZA Oct 01 '14 at 15:22
  • The conclusion is entailed by the premise when in every row where the premise has value T also the conclusion has T. – Mauro ALLEGRANZA Oct 01 '14 at 15:24
  • It is worst than before : the structure of the t-t must be : David/Susan/Mary/Susan or Mary/David and (Susan or Mary). We have parentheses in the formula ! Thus, when warm(David) is F the final column must have F, because it is a conjunction. – Mauro ALLEGRANZA Oct 01 '14 at 15:28
  • Thank you for all your comments, I have updated my truth table to the values I think it should be now. As the columns 'David and 'Premise' are different, the I understand the answer should be that 'Premise does not entail conclusion'? – user3371750 Oct 01 '14 at 15:34
  • NO. The def is [see above] : "The conclusion is entailed by the premise when in every row where the premise has value T also the conclusion has T". The premise has T only in the last three rows; in these rows warm(David) (the conclusion) has T. Thus : "Premise does entail conclusion". – Mauro ALLEGRANZA Oct 01 '14 at 15:53
  • Ah I see, thanks so much for all your help Mauro! – user3371750 Oct 01 '14 at 15:55
  • You have also to use your "intuition"; if the premise is a conjunction, for sure both concjuncts "individually" are entailed by the premise. If my coffee is black and hot, it is for sure hot. The same approach may help you with your other question: if my coffee is not black, why this must entail that it is hot ? – Mauro ALLEGRANZA Oct 01 '14 at 15:59

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My question is I see there is one truth value that both the premise and conclusion have in common, but does that mean that the premise does or does not entail the conclusion?

No, we cannot conclude this. I'm afraid of confusing you with formalism.. but I better explain it in details. In usual terminology, entailment in the propositional calculus is defined this way: let $\Gamma$ be a set of formulae and $\phi$ a formula. We say that $\Gamma$ entails $\phi$ iff for any valuation $v$, if $v(\psi)=1, \psi \in \Gamma$ then $v(\phi)=1$ (Roughly speaking, whenever all formulae in $\Gamma$ are true, $\phi$ must be true). This is why it's always better, when working with truth tables, to just check the lines all premises are true and see if in any of those lines the conclusion is false. If it's not the case, we say the premises entail the conclusion.

Bruno Bentzen
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