In linguistics, entailment is the relationship between two sentences where the truth of one (A) requires the truth of the other (B). This relationship is generalized below.
As a tool or method to make progress, one might use the entailment concept. For example, if one recognizes that "were A true, B must be true as well", then one might usefully shift attention from some valid position (say A) to another (say B).
The linguistic concept above can be generalized for a set of A's in logic and mathematics; call this set T. Then entailment or logical implication is a logical relation that holds between a set T of propositions and a proposition B, when every model (or interpretation or valuation) of T is also a model of B. In symbols,
which may be read 1) "T entails B", 2) "T implies B", or 3) "B is a (logical) consequence of T". In such an implication, T is called the antecedent, while B is called the consequent.
In other words, (1) holds when the class of models of T is a subset of the class of models of B. Without using the language of models, (1) states that the material conditional formed from the conjunction of all the elements of T and B (i.e. the corresponding conditional) is valid. That is, it is valid that
where the A_{i} are the elements of T. (If T has infinite cardinality then, provided the language of T has the compactness property, some finite subset of T implies B.) The statement in terms of the material conditional holds only in logics that have the semantic equivalent of the deduction theorem (and, as noted earlier, if T is infinite, then the compactness property is also required if the language disallows conjunctions over infinite sets of formulas). Thus, the original statement in terms of models is more general. The weaker truth function material implication, denoted by '→', should not be confused with the stronger logical implication.
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 , i.e. A entails B, holds.
Example 2. Put and . 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, because the type signatures differ.
If for a nonempty finite set of formulae with n > 1, we say that the disjunction is valid. In particular, if X = {φ} is a singleton, then φ is said to be valid. If X is an infinite set of firstorder 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 firstorder languages.
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 implies : that is, if all valid arguments are deducible (or provable), where denotes the deducibility relation for the system S.
A deductive system S is sound for a language L if and only if implies : that is, if no invalid arguments are provable.
Many introductory textbooks (e.g. Mendelson's "Introduction to Mathematical Logic") that introduce firstorder logic, include a complete and sound inference system for the firstorder logic. In contrast, secondorder logic — which allows quantification over predicates — does not have a complete and sound inference system with respect to a full Henkin (or standard) semantics.
In many cases, entailment corresponds to material implication (denoted by ) in the following way. In classical logic, if and only if there are some finite subsets of A and of B such that . There is also the deduction theorem that holds in classical logic.
It is impossible to state rigorously the definition of 'logical implication' as it is understood pretheoretically, but many have taken the Tarskian modeltheoretic account as a replacement for it. Some, e.g. Etchemendy 1990, have argued that they do not coincide, not even if they happen to be coextensional (which Etchemendy believes they are not). This debate has received some recent attention.^{[2]}
The usage of the terms logical implication and material implication varies from field to field and even across different contexts of discussion. One way to minimize the potential confusion is to begin with a focus on the various types of formal objects that are being discussed, of which there are only a few, taking up the variations in language as a secondary matter.
The main formal object under discussion is a logical operation on two logical values, typically the values of two propositions, that produces a value of false only in the case the first operand is true and the second operand is false:
In set theory there is
the same difference between the operation
and the relation
meaning .
On the way from to the difference between logical () and material implication () can be seen in an easy calculation:
The operation can be expressed by and . The relation can be expressed by and the universal quantifier .
A common exercise for an introductory logic text to include is symbolizations. These exercises give a student a sentence or paragraph of text in ordinary language which the student has to translate into the symbolic language. This is done by recognizing the ordinary language equivalents of the logical terms, which usually include the material conditional, disjunction, conjunction, negation, and (frequently) biconditional. More advanced logic books and later chapters of introductory volumes often add identity, Existential quantification, and Universal quantification.
Different phrases used to identify the material conditional in ordinary language include if, only if, given that, provided that, supposing that, implies, even if, and in case. Many of these phrases are indicators of the antecedent, but others indicate the consequent. It is important to identify the "direction of implication" correctly. For example, "A only if B" is captured by the statement
A → B,
but "A, if B" is correctly captured by the statement
B → A
When doing symbolization exercises, it is often required that the student give a scheme of abbreviation that shows which sentences are replaced by which statement letters. For example, an exercise reading "Kermit is a frog only if muppets are animals" yields the solution:
A → B
A—Kermit is a frog.
B—Muppets are animals.
Using the horseshoe "⊃" symbol for implication is falling out of favor due to its conflict with the superset symbol used by the Algebra of sets. A set interpretation of "" is "{x A(x) is true} {x B(x) is true}".
The use of the operator is stipulated by logicians, and, as a result, can yield some unexpected truths. For example, any material conditional statement with a false antecedent is true. So the statement "2 is odd implies Paris is in America" is true. Similarly, any material conditional with a true consequent is true. So the statement, "If pigs fly, then Paris is in France" is true.
These unexpected truths arise because speakers of English (and other natural languages) are tempted to equivocate between the material conditional and the indicative conditional, or other conditional statements, like the counterfactual conditional and the material biconditional. This temptation can be lessened by reading conditional statements without using the words "if" and "then". The most common way to do this is to read A → B as "it is not the case that A and/or it is the case that B" or, more simply, "A is false and/or B is true". (This equivalent statement is captured in logical notation by , using negation and disjunction.)
Implication, when taken as an operation over symbols, has two important properties that ally it to some wellknown relations in mathematical discourse. These are:
One implication of these properties is that the twosided relation, "A → B = T and B → A = T", defines an equivalence over possible inputs.

