An implication may be either an entailment or a conditional
statement; these are closely related but not quite the same
thing. A conditional statement is the result of a binary
operation on propositions within a given context in a given
logic. If and are propositions in some context, then so is the
conditional statement , at least if the logic has a notion of
conditional. Notice that , , and are all statements in the
object language (the language that we are talking about),
whereas the hypothetical judgements and are statements in the
metalanguage (the language that we are using to talk about the
object language). Depending on what logic one is using, might be
anything, but it's probably not fair to consider it a
conditional statement unless it is related to entailment as
follows: You can think of entailment as being an external hom
(taking values in the poset of truth values) and the conditional
as being an internal hom (taking values in the poset of
propositions). In particular, we expect these to be relate