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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?

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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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  • $\begingroup$ You should add parentheses to make statement 2 and 3 less ambiguous. $\endgroup$ – MM8 Feb 17 '17 at 17:20
  • $\begingroup$ You are right, thanks for point it out $\endgroup$ – John Feb 17 '17 at 17:21
  • $\begingroup$ Statement three should be $$\lnot (m \land s)\rightarrow \lnot e \equiv (\lnot m \lor \lnot s)\rightarrow \lnot e$$ $\endgroup$ – Namaste Feb 17 '17 at 17:24
  • $\begingroup$ Agreed, "If you don't use magic mushrooms and spider legs" is negating the conjunction, not the individual statements. $\endgroup$ – MM8 Feb 17 '17 at 17:25
  • $\begingroup$ Well, @Timon, after our comments, John corrected his post, but never acknowledged the error. $\endgroup$ – Namaste Feb 17 '17 at 17:27

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