The term entailment (and the verb entails) seem to have been introduced in this sense by G.E. Moore. Note that this entailment is as valid or invalid as the presumed rule of inference (supposing it has been used correctly, with correct substitutions etc.) And also note that a rule of inference need not be deductively valid to be somehow useful.

In general, "|-" tends to be used for entailment (also read as 'follows from'), like so:

{A1 ... An} |- C

As regards deductive logic, every entailment must correspond to a valid implication to be valid, and goes beyond this by adding the permission to add the conclusion of the rule of inference to the proof in which the entailment figures.

Thus, as one can use "Ø |- Q" for "|- Q", in the sense that Q is provable itself iff Q is provable from no assumptions at all, the above entailment must satisfy

|- A1 & ... & An --> C

to be deductively valid. (Hence, in fact here a common trick is used to have "|-" in between a set of statements and a statement, and in front of a statement. This is perfectly good set theory, and relies on the fact that both {A1 ... An} and Ø are sets.)

Note that the above also holds for probabilistic statements, which conform in general principle to

{A1 ... An} |- pr(C)=x

i.e. the premisses entail that - it is true that, if the premisses are true - the probability of C equals x.

Using conditional probability, this must satisfy

|- pr(C|A1 & .. & An)=x V pr(A1 & .. & An)=0

This and the previous statement, accordingly, must be true - which requires that one has a theory of probability that enables this, and also that one has some assumptions about or involving probabilities, at least if 0<x<1.