This article is about logical implication. For the historical legal practice of restricting the descent of property, see fee tail.

Implication or entailment is used in propositional logic and predicate logic to describe a relationship between two sentences or sets of sentences.

Semantic Implication

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$A\; \backslash models\; B$ (in HTML A ⊧ B, in LaTeX A \models B)

states that the set A of sentences semantically entails the set B of sentences.

Formal definition: the set A entails the set B if and only if, in every model in which all sentences in A are true, all sentences in B are also true. In diagram form, it looks like this:

We need the definition of entailment to demand that every model of A must also be a model of B because a formal system like a knowledge base can't possibly know the interpretations which a user might have in mind when they ask whether a set of facts (A) entails a proposition (B).

In pragmatics (linguistics), entailment has a different, but closely related, meaning.

If $\backslash varnothing\; \backslash models\; X$ for a formula X then X is said to be "valid" or "tautological".

Logical Implication

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$A\; \backslash vdash\; B$ (in HTML A |– B, in LaTeX A \vdash B)

states that the set A of sentences logically entails the set B of sentences. It can be read as "B can be proven from A".

Definition: A logically entails B if, by assuming all sentences in A and applying a finite sequence of inference rules to them (for example, those from propositional logic), one can derive all sentences in B.

This is, of course, relative to a specific logic (proof calculus). In cases where multiple logics are under discussion, it may be useful to put a subscript on the ⊢ symbol.

Relationship between Semantic and Logical Implication

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Ideally, semantic implication and logical implication would be equivalent. However, this may not always be feasible. (See Gödel's incompleteness theorem, which states that some languages (such as arithmetic) contain true but unprovable sentences.) 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 $A\; \backslash models\_L\; X$ implies $A\; \backslash vdash\_S\; X$: that is, if all valid arguments are provable.

A deductive system S is sound for a language L if and only if $A\; \backslash vdash\_S\; X$ implies $A\; \backslash models\_L\; X$: that is, if no invalid arguments are provable.

Relationship with Material Implication

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In many cases, entailment corresponds to material implication: that is, $A,\; X\; \backslash models\; Y$ if and only if $A\; \backslash models\; X\; \backslash to\; Y$ . However, this is not true in some many-valued logics.

See also

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• Inference

Logic

蕴涵