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I am confused regarding a propositional logic representation of a sentence.

Please note that this sentence is not realistic:

"A person who is male (M) is smart (S) if he is tall (T), but otherwise is not smart."

Which of the following propositional logic formulas correctly represents the above:

(a) $(M \wedge S) \iff T$

(b) $M \Rightarrow (S \iff T)$

Note that $\Rightarrow$ and $\iff$ represents "only if/implies" and "iff and only if," respectively.

The solution to the problem is (b)

The solution claims that (a) is NOT an option because it asserts, among other things, that all tall people are male, which is not what is asserted. This makes sense to me, I think (not confident, but this is not the main point of this post).

The solution claims that (b) IS a correct representation because it is saying that if a person is male, then they are smart IF AND ONLY IF they are tall. I don't understand that, and based on my current understanding, I don't agree with this because the "IFF" part is saying that if the person is tall, then they are smart, and that is not what the problem statement said. The problem statement only states that a person who is male is smart if he is tall.

So I feel like (b) is not correct, but if you swap out the $\iff$ for $\Rightarrow$ then it becomes correct.

Am I misunderstanding something here?

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In words, we have:

A person who is male (M) is smart (S) if he is tall (T), but otherwise is not smart.

Or equivalently:

A person who is male (M) is smart (S) if he is tall (T), but otherwise if he is not tall, then he is not smart.

Inserted phrase in bold is implicitly understood.

Symbolically, by the contrapositive rule and by the definition of '$\iff$', we have:

$(T\implies S)\land (\neg T \implies \neg S)$

$\equiv \space \space (T \implies S )\land (S \implies T)$

$\equiv \space \space T \iff S$

$\equiv \space \space S \iff T $

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  • $\begingroup$ Yes, I am of the relations, but I'm not sure how this helps. Could you explain further? In the middle expression, $T \Rightarrow S$ is saying that a tall person is smart, right? If so, that's not what the sentence is asserting? $\endgroup$
    – 24n8
    Feb 20 '20 at 3:25
  • $\begingroup$ Hmm, perhaps I am misunderstanding the "but otherwise is not smart" part. Is that what makes the "only if" "IFF?" $\endgroup$
    – 24n8
    Feb 20 '20 at 3:27
  • $\begingroup$ I've thought about this some more, and it still doesn't make sense. At first I thought the "but otherwise is not smart" is manifested in $\lnot S \implies \lnot T$, but when I think about "but otherwise is not smart," I'm interpreting it as saying "if the male person is not tall, then he is not smart." Wouldn't that make it $\lnot T \implies \lnot S$? $\endgroup$
    – 24n8
    Feb 20 '20 at 3:53
  • $\begingroup$ Indeed, @lamanon , the original sentence says "smart if tall but not smart if otherwise" which is $(T\to S)\wedge(\neg T\to\neg S)$. That still equates to $(S\leftrightarrow T)$ $\endgroup$ Feb 20 '20 at 4:10
  • $\begingroup$ @Iamanon "A person who is male (M) is smart (S) if he is tall (T), but otherwise if he is not tall, then he is not smart." (Inserted phrase in bold is implicit.) $\endgroup$ Feb 20 '20 at 4:26
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The solution claims that (b) IS a correct representation because it is saying that if a person is male, then they are smart IF AND ONLY IF they are tall. I don't understand that, and based on my current understanding, I don't agree with this because the "IFF" part is saying that if the person is tall, then they are smart, and that is not what the problem statement said. The problem statement only states that a person who is male is smart if he is tall.

No, there is that "otherwise" clause in the statement.

The sentence says that "A male person is: smart if tall, and not smart if otherwise (ie not tall)."

That claims that "A male person is smart exactly when tall."

$$M\to(S\leftrightarrow T)$$

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  • $\begingroup$ I think I'm struggling with the semantics here for some reason. "smart if tall" is the same as saying "smart only if tall" and "if tall, then smart" right? In propositional logic form, it's $T \implies S$? $\endgroup$
    – 24n8
    Feb 20 '20 at 4:09
  • $\begingroup$ On second thought "smart if tall" $\neq$ "smart only if tall" $\endgroup$
    – 24n8
    Feb 20 '20 at 4:12
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    $\begingroup$ "Smart if tall" is "If tall, then smart", ie: $T\to S$. That is also "Tall only if smart" and "Not tall if not smart" ie: $\neg S\to\neg T$. $\endgroup$ Feb 20 '20 at 4:12
  • $\begingroup$ Got it. If it's "Smart only if tall" then that's manifested in $S \implies T$ and it also means "if smart, then tall," right? I think I struggle with the different ways to phrase things. The "if ___ then " format is more intuitive to me than "____ if _____" or " only if ____" $\endgroup$
    – 24n8
    Feb 20 '20 at 4:16
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    $\begingroup$ Take "If they are tall, then they are smart" as a promise. That promises that they cannot be tall should they not be smart. Thus the promise entails that "if they are not smart, then they are not tall." Et vice versa. $\endgroup$ Feb 20 '20 at 5:07

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