Logical consequence
of a given set of premises
A proposition that is true for any interpretation of the non-logical symbols (that is, the names (cf. Name) of objects, functions, predicates) for which the premises are true. If a proposition is a logical consequence of a set of propositions
, one says that
logically implies
, or that
follows logically from
.
If is a set of statements of a formalized first-order logico-mathematical language (cf. Logico-mathematical calculus) and
is a proposition of this language, then the relation "A is a logical consequence of G" means that any model for
is a model for
. This relation is denoted by
. The Gödel completeness theorem of classical predicate calculus implies that the relation
coincides with the relation
, that is,
if and only if
is deducible from
by the methods of classical predicate calculus.
References
[1] | H. Rasiowa, "The mathematics of metamathematics" , Polska Akad. Nauk (1963) |
[2] | K. Gödel, "Die Vollständigkeit der Axiome des logischen Funktionenkalküls" Monatsh. Math. Phys. , 37 (1930) pp. 349–360 |
Comments
The phrase "semantic entailmentsemantic entailment" is sometimes used instead of "logical consequence" ; thus, the expression is read as "G semantically entails A" . The expression
is similarly read as "G syntactically entails A" .
References
[a1] | P.T. Johnstone, "Notes on logic and set theory" , Cambridge Univ. Press (1987) |
[a2] | A. Grzegorczyk, "An outline of mathematical logic" , Reidel (1974) |
Logical consequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logical_consequence&oldid=31698