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The text I am reading gives this problem:

Express the following using logic symbols.

The cat is out of the bag only when the contestant is bald.

$D$ is: The cat is out of the bag, and $P$ is: The contestant is bald, thus $D$ only when $P$.

I thought of "only when" as similar to "only if" and answered $D \iff P$. The text gives the answer as $D \implies P$.

That then is:

If the cat is out of the bag, then the contestant is bald, and I can not see that "if-then" has the same meaning as "only when".

As an absolute amateur at this, and assuming that the text is not in error, I have to look here for guidance. The logic text I am reading makes it very clear that connections do not imply causality or sequence in time. "Only when" does not imply a sequence, but does seem to separate the times when the contestant is bald and when, another, contestant is not. Please excuse me if I am not making sense about this.

How do I interpret "only when"?

Mars Plastic
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  • I agree with the text. It could be the contestant is bald and the cat is not out of the bag; cf. this question – J. W. Tanner Mar 12 '20 at 20:51
  • When you see "if and only if" you can use $\iff$. Otherwise, the converse of the statement may not be true and you cannot use $\iff$, only $\implies$. – Andrew Chin Mar 12 '20 at 20:54
  • On further reading, it would seem that "only when" is not like "only if" and I do not know if "only if" is in any way like if-only if. It sounds less silly to me, but maybe the question I am asking is more like: Of the logical connectors, if-then, if-only if, and, or, etc., which one is the same as "only when"? –  Mar 12 '20 at 21:36
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    Effectively, "only when" is equivalent to "only if". – Andrew Chin Mar 12 '20 at 21:38
  • @AndrewChin, I just wrote that I thought it was not, but I am most likely wrong. As is probably evident from my questions, I am very new to this. –  Mar 12 '20 at 21:46
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    If I say "$x$ is a multiple of $6$ only when it's a multiple of $2$," you can't conclude from something being a multiple of $2$ that it's a multiple of $6$; that's why you can't translate it as $\Leftrightarrow$. All I've claimed is that if something is not a multiple of $2$, then it is not a multiple of $6$. – Malice Vidrine Mar 12 '20 at 21:47
  • !!!! Now I understand. –  Mar 12 '20 at 21:58
  • At least I think I understand. –  Mar 12 '20 at 22:12
  • No, I do not. If the two statements are independent, then all that matters is the truth value of the statements. In the example by @MaliceVidrine, there is a dependence from one to the other. I will try to resolve that in my thinking. –  Mar 13 '20 at 14:51
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    @GeoGraphy - I was using a natural example. That's the nature of truth functional logic that the truth values are the only thing that matter. The apparent conceptual dependence doesn't do any logical work here. – Malice Vidrine Mar 13 '20 at 19:06

2 Answers2

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We read $D\implies P$ as "if $D$ then $P$," but what the notation really means is "either $D$ is false or $P$ is true or both."

Given that $D \implies P,$ is it possible that $D$ is true when $P$ is false? No, because then neither of the two halves of the "either or" version of the statement is true: it is not true that $D$ is false and it is not true that $P$ is true.

So the statement $D\implies P$ tells us that the only circumstances under which $D$ can be true are when $P$ also is true. "$D$ true and $P$ false" is ruled out. Hence $D$ is true only when $P$ is true.

Another way to think of this is to recall that we read $D \iff P$ as "$D$ if and only if $P.$"

Now, "$D$ if $P$" is the same as "if $P$ then $D$", that is, it can be written $P \implies D$ or $D \impliedby P$.

So the "if" half of the "if and only if" gives us the $D \impliedby P$ direction of the arrow in $D \iff P$. The other direction of the arrow, $D \implies P$, is given by the "only if". That is, "$D$ only if $P$" is a legitimate way to read $D\implies P$.

David K
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  • Your "We read D⟹P as "if D then P," but what the notation really means is "either D is false or P is true or both," points to exactly the problem I think I am struggling with. Language, rather than ???? For example, does "or both" mean "D is false and P is true", or D is true and P is True, or (hopefully not) P is both true and false? –  Mar 13 '20 at 14:29
  • "Or both" is just a way to emphasis that when we say "A or B" we don't mean exclusive or. So "D is false" is one of the things that might be true. "P is true" is another thing that might be true. And it is possible (though not necessary) that both are true, that is, D is false and P is true. If you understand the mathematical "or", then it is not really necessary to say "or both" because that is already part of the meaning of "or". But sometimes we say it anyway, and that is what I did. – David K Mar 14 '20 at 02:01
  • When someone says "both", they do not mean, "When I said 'true' I really meant true and false." As you noticed, that is not a reasonable statement. – David K Mar 14 '20 at 02:05
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"$D$ only when $P$" means exactly that "not $D$ when not $P$".   That is $\neg P\to\neg D$.

If it is $D$ only when it is $P$, then it must be not $D$ when it is not $P$.

If it is not $D$ when it is not $P$, then it can be $D$ only when it is $P$.

"$D$ only when $P$" also means exactly that "$P$ when $D$".   That is $D\to P$.

If it is $D$ only when it is $P$, then it must be $P$ when it is $D$.

If it is $P$ when it is $D$, then it must be $D$ only when it is $P$.

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Graham Kemp
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