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How to write the following sentence using propositional logic:

Only monkeys properly appreciate the value of art.

Is this good:

Let $M(x)$ be '$x$ is a monkey' and $A(x)$ be '$x$ properly appreciates the value of art', where the domain of $x$ is the set of all animals.

In this case, $\forall x\ A(x) \rightarrow M(x)$ or $\forall x\ A(x) \iff M(x)$?

Clarinetist
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fast-and-last
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  • If only monkeys appreciate art, then if you are not a monkey then you don't appreciate art. For all x not an element of monkey implies x is not an art lover. That should be pretty easy to translate into propositional logic. – Peter Webb Feb 22 '15 at 07:58
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    Your second answer, $\forall x\ A(x) \iff M(x)$, is too strong. It might be translated to "All and only monkeys properly appreciate the value of art". – joeA Feb 22 '15 at 08:25
  • Yes, if my answer wasn't clear enough I apologize--the correct form is $(\forall x)(A(x)\to M(x))$; like @joeA pointed out, the second answer is too strong. – Daniel W. Farlow Feb 22 '15 at 08:31

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You seem right, but you may want to formulate it as follows to reflect the nature of the question: If $x$ properly appreciates art, then $x$ is a monkey AND if $x$ is not a monkey, then $x$ does not appreciate art, but this latter statement is just the contrapositive of the previous statement which takes the form $(\forall x)(A(x)\to M(x))$. Thus, I do not see any issue with your formulation.

Daniel W. Farlow
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