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Scalar comparison of equality indicates “at least equal”
In the absence of indications to the contrary, a scalar [= ‘clever’ in [i] below] comparison of equality is interpreted as “at least equal”, not “exactly equal”:

i) Jill is as clever as Liz. [Jill may be cleverer]
ii) Jill isn’t as clever as Liz [Jill must be less clever]

Example [i] is consistent with Jill being cleverer than Liz: we can say Jill’s as clever as Liz, somewhat more so in fact. [n] Scalar equality therefore normally exclude only the relation of inferiority: it gives a lower bound. . . . . .

[n] This can indeed apply also to equal itself, as in Kim is the equal of Pat when it comes to solving crossword puzzles (which is consistent with Kim being better) or We hope to equal last year’s profit (consistent with bettering it). (CGEL, p.1100~1101)

  1. as ~ as:
    Does the phrase “at least equal” refer to as ~ as construction or one of *as*s?
    (I want to know whether the meaning comes from the structure or the lexical meaning.)

  2. equal:
    I don’t find any lexical source that equal implies being better. Is there any dictionary saying the meaning?

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    You're mistakenly assuming "Jill is as clever as Liz" means the same as "Jill is at least as clever as Liz". In actual fact, if people really have that second meaning in mind, they'll normally either include at least, or they'll say something like "Jill is as clever as Liz - if not cleverer" (or maybe more clever, since that particular adjective doesn't sit well with the -er suffix).
    – FumbleFingers
    Commented Sep 1, 2013 at 14:55
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    Yes, I'm with FumbleFingers on this one. Despite the text you've quoted, I would naturally assume that "Jill is as clever as Liz" means that the two are equally clever, or quite nearly so. If I meant something different, I'd add qualifiers.
    – WendiKidd
    Commented Sep 1, 2013 at 15:55

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The key is in the negative statement: "Jill isn't as clever as Liz." 100% of native speakers will interpret this to mean "Jill is less clever than Liz." No one will think it could mean that Jill is much, much cleverer than Liz, unless the statement is meant as a joke.

Back to the positive statement, then, "Jill is as clever as Liz." Yes, this leaves room for Jill to be much cleverer than Liz. I could nod agreement and say "At least." If Jill were far cleverer than Liz, though, it would feel too understated and I might shake my head in disagreement.

The same thing happens with "equal." If I say "You are not my equal," that means "You are my inferior." (Barring weird cases where I'm explaining politeness rules or something.) But if I say "You are my equal" then I don't think anyone would interpret that to mean other than what it literally says.

This is all the domain of pragmatics by the way, not syntax or semantics, so you won't find it in a dictionary. Pragmatics studies (among other things) the implications we make by the things we don't say. For example, when we say "A or B" we usually mean "but not both." If both were an option, we'd have said "A and B" to begin with.

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