∀ a1,a2∈A a1RAa2 ⇔ f(a1)RBf(a2)
In such a case the structures are said to be isomorphic.
An isomorphism between two first-order models
B which have universes of discourse
is a bijection
g:A→B such that
for all relations
and all functions
f, and all constants
g(fA(a1,...,an)) = fB(g(a1),...,g(an))
rA(a1,...,an) ⇔ rB(g(a1),...,g(an))
cA is the interpretation for
fA is the interpretation for
rA is the interpretation for
A. Likewise for
A≅B if models
B are isomorphic.
Isomorphic models are first-order indistinguishable, so if
Doets, Kees. From Logic to Logic Programming. MIT Press, 1994.
Copyright © 2014 Barry Watson. All rights reserved.