Definition·D095
Model of a formula or theory
A structure models a formula when every assignment satisfies it, and models a set of formulas when it models each one.
For an L-structure
,
, and
:
models
(written
) and
models
(written
) are characterized by:
In words
A structure models a formula exactly when every assignment into it satisfies the formula. A structure models a set of formulas exactly when it models every formula in the set.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 · A09 (computed from the citation graph, not asserted).
Remarks
For a sentence
,
does not depend on
(see D094's notes), so "every assignment satisfies it" and "some assignment satisfies it" coincide;
then agrees with the usual reading "
is true in
". A set of formulas
intended to be modeled this way is called a theory.
for
holds vacuously, for every
(F09, an implication with no antecedent-true case). Modeling is the semantic half of logical consequence, defined next.