In logic, the concept where one kinds of rule leads to another is called entailment (also called logical consequence). For example, “If A is true, B must follow. A necessarily leads to B.”

A valid logical argument is the one in which the conclusion is entailed by premises.

Note

Entailment is not the same as logical equivalence. For example, logically entails , but does not logically entail .

Relation with Conditional

if and only if is a tautology.

Syntactic and Semantic Consequence

The symbol for entailment are or , which represent semantic consequences. There is also a separate symbol to represent syntactic consequence (or provability). Their difference is subtle:

  • syntactic consequence says: sentence is provable from the set of assumptions without knowing whether is true.
  • semantic consequences says: sentence is true in all interpretations of .

If a system is both sound and complete then the semantic consequences and syntactic consequence are equivalents. There are theorems regarding the equivalence between the two concepts both for propositional logic and first-order logic (see Gödel’s completeness theorem), but for higher-order logic this may not be true. 1

Footnotes

  1. logic - Semantic vs syntactic consequence - Philosophy Stack Exchange