Lets formulate the incidence lemma as follows. We have a possibly infinite set of variables X and the domain of discourse U. Lets define an interpretation of the variables X in the domain U as a function:
$\sigma$ : X $\mapsto$ U
Let $\sigma_1$ and $\sigma_2$ be two interpretations of the variables. Now if I have a formula A and also some interpretation of the function symbols, constants and relation symbols, I can say:
Coincidence Lemma: If $\sigma_1(x)$ = $\sigma_2(x)$ for all x occuring free in A, then A has the same truth value in both interpretations $\sigma_1$ and $\sigma_2$.
Are there any logics that violate this lemma? Any logics that violate this lemma if we correspondingly look at the function symbols or constants or relation symbols occuring in the formula A?
P.S.: I have loosely translated the name of the lemma
from here:
Lemma 1.13: Koinzidenzlemma
Beweise und Programme. Anmerkungen zu
Heytings Formalisierung der intuitionistischen Logik
Helmut Schwichtenberg
See also:
http://de.wikipedia.org/wiki/Koinzidenzlemma