# Counterexample in propositional logic

There is this lemma: Let $\Sigma\subset \textrm{Prop}(A)$ and $p, q \in \textrm{Prop}(A)$. Then $\Sigma\models p \implies \Sigma\models p\vee q$. I can't figure out a counterexample for the opposite implication ($\textrm{Prop(A)}$ denotes the set of propositions and $A$ is a set of propositional atoms.

Thanks for help.

-pizet

Let $q=\lnot p$.${}{}{}{}{}{}{}{}$

• I think this is the smallest complete answer, I have seen in the site, Andre. However, that "Let" could be also ommitted. ;-) +1 Apr 13, 2013 at 5:21
• There have been shorter answers, a famous one by did, for example. About the suggestion to let go of let, I think one is not supposed to start a sentence with math symbols. Apr 13, 2013 at 5:25
• Thanks. Yes. you are right. We cannot start such that in Maths at least. Apr 13, 2013 at 5:27
• @BabakS. There is math.stackexchange.com/a/74383/462 and math.stackexchange.com/a/180649/462 (and look at the comments on that one.) Apr 13, 2013 at 15:24

$${\bf \Sigma \models p \lor q \overset{?}\implies \Sigma \models p }\tag{{\bf converse}}$$

What if $\;p\;$ is false and $\;q\;$ is true?: Suppose, e.g., $$\bf \text{ Suppose}\;\;\; q \;= \;\lnot p$$

$\quad$ Then your stated lemma: $\;\Sigma\models p \implies \Sigma\models p\vee q\;$ certainly holds.

But its converse (highlighted) certainly fails to hold.

• Thank you very much, it's quite clear. Apr 12, 2013 at 23:07
• @amWhy: Yes, very clear amd it is nice to get that feedback from OP! +1 Apr 13, 2013 at 2:29