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I am studying for my exams and I am facing some issues with the "at least" statement in propositional logic.

Mary wants to mix a magic potion. Here is the recipe.

(1): You need at least one of the following ingredients: spider legs, eyes of a toad, magic mushrooms.

(2): If you are using magic mushrooms, then you can not use the other two ingredients.

(3): If you don't use magic mushrooms and spider legs, then you aren't allowed to use eyes of a toad.

Let

  • $s =$ spider legs
  • $m =$ magic mushrooms
  • $e =$ eyes of a toad

My solution would be something like this, but I am not sure about the first one. Feedback would be nice

  • (1) $= (m \vee e) \wedge (m \vee s) \wedge (e \vee s)$
  • (2) $= (\neg s \wedge \neg e) \rightarrow m$
  • (3) $= (\neg m \wedge \neg s) \rightarrow (\neg e)$

Is the first equation correct?

2 Answers 2

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$(1)\;\text{ should be }\;s\vee e\vee m$

$$s \lor e \lor m\;\text{ asserts that at least one (or perhaps two, maybe even all three) of }\;s, \,e,\, m \;\text{ holds.}$$ $(2)$ We have $\quad m\rightarrow \lnot( s \lor e) \equiv m\rightarrow (\neg s\wedge\neg e)$

$\quad$ This means that "If we use magic mushrooms, then we cannot use (spider legs or eyes of a toad)", or equivalently, "If we use magic mushrooms, then we cannot use spider legs, and we cannot use eyes of a toad."

$(3)$ We can express this statement into propositional logic notation: $\;\;\lnot(m \land s)\rightarrow \lnot e \equiv (m\land s) \lor \lnot e$

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  1. $s\vee e\vee m$
  2. $m\rightarrow (\neg s\wedge\neg e)$
  3. $\neg(m\wedge s)\rightarrow\neg e $
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    You should add parentheses to make statement 2 and 3 less ambiguous. – 2017-02-17
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    You are right, thanks for point it out – 2017-02-17
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    Statement three should be $$\lnot (m \land s)\rightarrow \lnot e \equiv (\lnot m \lor \lnot s)\rightarrow \lnot e$$ – 2017-02-17
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    Agreed, "If you don't use magic mushrooms and spider legs" is negating the conjunction, not the individual statements. – 2017-02-17
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    Well, @Timon, after our comments, John corrected his post, but never acknowledged the error. – 2017-02-17
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    I thought about it for a while. I interpreted the sentence as "if you don't use use magic mushrooms and you don't use spider legs...". You interpreted it as "if you don't use both magic mushrooms and spider legs...". But I agree that your intepretation is probably more correct. I edited the answer – 2017-02-17
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    @amWhy I was writing the comment to explain, just a second! – 2017-02-17