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late 14c., "to enfold, enwrap, entangle" (the classical Latin sense), from Old French emplier, from Latin implicare "involve" (see implication). Meaning "to involve something unstated as a logical consequence" first recorded c.1400; that of "to hint at" from 1580s. Related: Implied; implying. The distinction between imply and infer is in "What do you imply by that remark?" But, "What am I to infer from that remark?"
(=> or a thin right arrow) A binary Boolean function and logical connective. A => B is a true implication unless A is true and B is false. The truth table is
A B | A => B ----+------- F F | T F T | T T F | F T T | T
It is surprising at first that A => B is always true if A is false, but if X => Y then we would expect that (X & Z) => Y for any Z.
If A is actually an expression X & Y then the implication is called a syllogism.