# In what order are chained implications evaluated? (i.e. $a \implies b \implies c$)

Implication does not appear to be associative:

a  b  c | (a -> b) -> c | a -> (b -> c)
F  T  F | F             | T
F  F  F | F             | T


Is $a \implies b \implies c$ evaluated as $a \implies (b \implies c)$ or $(a \implies b) \implies c$?

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Informally, I would take $a\implies b \implies c\;$ to mean $$(a\implies b) \wedge (b\implies c).$$ –  anon Jan 31 '12 at 3:49
@anon That's definitely different than taking a->b->c to mean (a->(b->c)). –  Doug Spoonwood Jan 31 '12 at 4:02
@Doug: Yes, they're obviously different. What I posted would be highly ill-advised notation in any formal logic setting, but if instead someone comes across it elsewhere in mathematics it might actually be intended to be an abbreviation of what I wrote. –  anon Jan 31 '12 at 4:26

In computer science, the convention that $a \to b \to c$ means $a \to (b\to c)$ is near universal -- because, through the Curry-Howard isomorphism, such formulas correspond to useful curried types in functional languages.

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The usual convention associates to the right, so $a \rightarrow b \rightarrow c$ is the same as $(a \land b) \rightarrow c$. But it's much better to put parentheses around.

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What if the language doesn't have logical conjunction "^"? I mean I can see (a∧b)→c and a->(b->c) as logically equivalent, but is that all you mean by "the same"? And would you classify all logical theorems as "the same"? –  Doug Spoonwood Jan 31 '12 at 4:01

It really depends on the author, of course.

That said, Carl Mummert, who researches in mathematical logic, indicates that he uses a->b->c to mean (a->(b->c)) in a comment in another thread. So, as others have kindly pointed out, if you're reading a logic text, I'd think it probable that you'd encounter that convention more often than anything else.

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My understanding is that one reads from the left, so $a\rightarrow(b\rightarrow c)$. –  André Nicolas Jan 31 '12 at 4:56
As explained by @anon, the guess in your last sentence should probably be qualified by if you are in logic. For example, it does not correspond to my experience. –  Did Jan 31 '12 at 6:51
Thank you for the mention of my comment. The answer by Henning Makholm has the explanation behind this convention. The answer by Yuval Filmus alludes to the same thing. –  Carl Mummert Feb 1 '12 at 3:19