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I am a logic neophyte and simpleton studying for an exam in a graduate-level course in elementary symbolic logic. I am trying to symbolize the following categorical statement:

"No artist is a fortune teller, unless he or she reads tarot cards for someone."

Domain: people.

I symbolized the terms as follows:

$Ax = x$ is an artist, $Fx = x$ is a fortune teller, $Rxy= x$ reads tarot cards for $y$.

Here is what I got: $$(∀x)\Big((Ax∧¬(∃y)(Rxy))⟹¬Fx)\Big).$$

Is this right? Have I left something out, misplaced something, or added a superfluous bracket?

ryang
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Zane A.
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3 Answers3

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Looks right to me if you remove the extraneous closing parenthesis at the end. It reads "for all x, if x is an artist and there is no y for whom x reads tarot cards, then x is not a fortune teller". That's arguably equivalent to the original statement.

Karl
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$$(∀x)\Big((Ax∧¬(∃y)(R_xy))⟹¬Fx)\Big).$$ Is this right? Have I left something out, misplaced something, or added a superfluous bracket?

Adding to Karl's answer, a human-friendlier rewrite of your suggested formalisation (minus that extraneous closing parenthesis) is just $$∀x\;\Big(Ax∧¬∃y\,Rxy⟹¬Fx\Big),$$ while logically-equivalent minimalist formalisations are $$∀x∃y\;\Big(Rxy ∨¬Ax ∨ ¬Fx\Big)$$ and $$¬∃x∀y\;\Big(¬Rxy ∧Ax ∧ Fx\Big)$$ and $$∀x∃y\;\Big(Ax ∧ Fx ⟹Rxy\Big).$$

ryang
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    Interesting. I agree that the rewrite looks much better. Thanks (once again), Ryan. – Zane A. May 11 '22 at 20:08
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    @ZaneA. You're quite welcome! Your suggestion right under Carlyle's answer is correct, and is logically equivalent to my final one above. BTW, please enclose mathematics within dollar signs when typing on this website (like how I edited your Question above); this is called mathjax and, for readability, is pretty much insisted on by the community. – ryang May 11 '22 at 20:16
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    Possibly of interest, if time permits: how to translate "unless" and "only if" – ryang May 11 '22 at 20:30
  • OK. I will be sure to do so in future posts. I'll check out the link that you sent me. Thanks so much!

    Edit: re-posted my reply to Carlyle with the dollar signs.

     – Zane A. May 11 '22 at 20:34
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    Nice, thanks! @ZaneA. P.S. My first suggestion above is based on the conventional precedence order for logical connectives. – ryang May 11 '22 at 20:43
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This looks correct, but it can be written more simply as "if someone is an artist and a fortune teller, then they read tarrot cards for someone"

On the predicate level this can be seen in that "P implies Q" is equivalent to "not Q implies not P"

Carlyle
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