Information About ™Entailment |
| CATEGORIES ABOUT ENTAILMENT | |
| logic | |
| SHOPPER'S DELIGHT | |
|
Implication or '''entailment''' is used in Propositional Logic and Predicate Logic to describe a relationship between two sentences or sets of sentences. SEMANTIC IMPLICATION (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 for a formula ''X'' then ''X'' is said to be "valid" or " Tautological ". LOGICAL IMPLICATION (in HTML ''A'' ├ ''B'', in LaTeX A dash 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 Calculus ), 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 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 arguments are provable. A deductive system S is Sound for a language '''L''' if and only if implies : that is, if no invalid arguments are provable. RELATIONSHIP WITH MATERIAL IMPLICATION In many cases, entailment corresponds to s. |