# Can the superlative be paraphrased by the determiner 'any'?

If

"John can't solve the simplest puzzle" = "John can't solve any puzzle"

is true, why is

"John can solve the simplest puzzle" = "John can solve any puzzle"

false, but

"I'm surprised that John can solve the simplest puzzle" = "I'm surprised that John can solve any puzzle"

true is?

In other words, are there coherent and standard correlations between the superlative and the determiner any?

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None of those equations actually hold; they need > or < signs. The downward entailment wikipedia article may help, via remarks like: “The proposition “At most two boys ran” entails that “At most two boys ran fast”.” – jwpat7 Sep 3 '12 at 13:12
As far as symbols are concerned, the = signs should probably be &rArr; (double right arrow) for "implies". – Andrew Leach Sep 3 '12 at 13:53
This isn't just English, it's also logic. Consider "John can't sail across the calmest sea", "John can't sail across the narrowest sea", "John can't sail across the bluest sea", and "John can't sail across any sea". The first two would probably be taken to mean "John can't sail across any sea", but they don't logically imply it. The third is a kind of non sequitur. – Peter Shor Sep 3 '12 at 13:57
Because the negation of "John can't solve any puzzle" isn't "John can solve any puzzle", but "John can solve at least one puzzle". – Brian Hooper Sep 3 '12 at 14:42

This is another negative polarity issue.

In the first, "John can't solve the simplest puzzle" implies "John can't solve any puzzle". This is similar to the third, where "I'm surprised that" introduces the negativity.

"I'm surprised that John can solve the simplest puzzle" implies "I'm surprised that John can solve any puzzle."

In the second, "John can solve the simplest puzzle" does not imply that "John can solve any puzzle." There is no negative polarity, and any can't be used in the same way.

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Exactly. Negatives and NPIs are very complex and do not work by equations. See here for lists of NPIs and Negative Triggers, and here for explanations of what they are. – John Lawler Sep 3 '12 at 14:29