Entailment:
In logic: A set of
statements {A1 ... An} entails
a statement C iff there is an accepted
rule of inference that
implies that
some of the statements in the set implies conclusion C. 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, with {A1 ... An} a set of premisses and C a conclusion:
{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. Note that this then is a truth: The truth that, on the given premisses, the probability of C equals x. Example: If this is a fair coin that is properly tosses, the probability of its falling heads equals 1/2.
Using conditional probability, this must satisfy
 pr(CA1 & .. & 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.
