Explanation of the ‘If and only if’ statement’s rules

logic

I had never had a problem with understanding 'if and only if' statements until I decided to learn some formal logic. And there I found that:

A compound 'if and only if' statement is true if both of its atomic statements are true or false

rules for iff statements

If I understood correctly the text and the table, there is a problem with (at least) this statement:

Fish can swim if and only if it's August 9.

From the formal logic rule cited above this statement is true, since both statements are true. But intuitively it's false. Since fish can swim regardless of today's date.

Also it's definitely false, if we dissect it in two:

  • if it's August 9, fish can swim — True
  • if fish can swim, it's August 9 — False

Could anyone please explain, what's wrong with my fish example and reasoning? Or if the table at the picture is correct?

Best Answer

Your reasoning is wrong because you are viewing implication as causation rather than merely correlation. All $A \implies B$ means is that if $A$ is true then $B$ is true; it says nothing about why $B$ is true. (In particular, $B$'s truth may have nothing to do with $A$ at all; e.g. "if $3 = 5$ then I am Patrick Stevens" is true, because I am Patrick Stevens; nothing at all to do with the truth or falsity of $3 = 5$.)

"If fish can swim, then it's August 9th" is true as a statement of logic, because it's August 9th. Of course, it's not true as a statement of English, because the mere fact that fish can swim does not cause it to be August 9th. The "if/then" relationship means something different in formal logic to what it means in English.

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