What's the difference between i.e. and e.g.?
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.