# Dictionary Definition

entailment n : something that is inferred
(deduced or entailed or implied); "his resignation had political
implications" [syn: deduction, implication]

# User Contributed Dictionary

## English

### Noun

- The act of entailing, the state of being entailed, or something that is entailed

# Extensive Definition

In logic, entailment (or logical
implication) is a relation between sentences of a formal
language, such that if A is a set of sentences and B is a sentence
then A entails B just in case B is true in every interpretation in
which all members of A are true, (i.e. in every model of A). This
is the case just in case the conjunction of all the members of A
and the negation of B is inconsistent, or in other words if C is
the conditional between the conjunction of all members of A as
antecedent and B as the consequent, then C is logically valid just
in case A entails B. Another way of stating this is to say that the
class K of models of A is a (possibly improper) subclass of the
class K' of models of some subset B' of B—i.e.
K⊆K'. Alternatively, we can say that A entails B if and
only if, for every subset B' of B, the class of models of A is a
subclass of the union of the classes of each B'.

In symbols,

- A \models B

states that the set A of sentences entails the
set B of sentences. Notice that entailment is a semantic relation.
Often it is stated less generally for B a single formula rather
than a set of formulæ. In our definition, this is equivalent to the
case when B is a singleton consisting of a sole formula.

Example 1. Let the set A of sentences include
'All horses are animals' and 'All stallions are horses', and the
set B of sentences include 'All stallions are animals'. Then
A\models B, i.e. A entails B, holds.

Example 2. Put A = \ and B = \ . Then A does not
entail B, since the empty model
is a model of A, but it is not a model of B - i.e. it is not the
case that all models of A are models of B.

In Venn diagram
form, A\models B looks like this:

If \varnothing \models X for X=\ a non-empty
finite set of formulae with n>1, we say that the disjunction
\phi_1\lor\dots\lor\phi_n is valid. In particular, if X=\ is a
singleton, then φ is said to be valid. If X is an infinite set of
first-order formulae, then there is some finite subset X' of X such
that the disjunction of the members of X' is valid. This is a
consequence of the compactness
property of first-order languages.

## Relationship between entailment and deduction

Ideally, entailment and deduction would be extensionally equivalent. However, this is not always the case. In such a case, it is useful to break the equivalence down into its two parts:A deductive
system S is complete for a language L if and only if A
\models_L X implies A \vdash_S X: that is, if all valid arguments are deducible (or
provable), where \vdash_S denotes the deducibility relation for the
system S.

A deductive system S is sound for a language L if and
only if A \vdash_S X implies A \models_L X: that is, if no invalid
arguments are provable.

Many introductory textbooks (e.g. Mendelson's
"Introduction to Mathematical Logic") that introduce first-order
logic, include a complete and sound inference system for the
first-order logic. In contrast, second-order
logic - which allows quantification over predicates - does not
have a complete and sound inference system with respect to a full
Henkin (or standard) semantics.

A related topic that sometimes causes confusion
is
Gödel's incompleteness theorem, which states that there are
sentences of certain theories that cannot be proved by the
underlying deductive system for the theory, even though such
sentences are true in the standard interpretation of the theory.
This holds even if the underlying deductive system is complete in
the above sense. It is a consequence of the existence of
nonstandard interpretations of theories.

## Relationship with material implication

In many cases, entailment corresponds to material implication (denoted by \supset) in the following way. In classical logic, A\models B if and only if there are some finite subsets \ of A and \ of B such that \varnothing\models A_1\land\dots\land A_n\supset B_1\lor\dots\lor B_m. There is also the deduction theorem that holds in classical logic.## See also

entailment in Chinese: 蕴涵

entailment in Swedish: medför
(logik)