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.


  • $s =$ 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?


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

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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.