How can I show that the Gödel-Dummett logic has the Scroggs' Property? Gödel-Dummett logic is an extension of the intuitionist logic (IPL) by the following axiom:
$$
(p → q) ∨ (q → p)
$$
A logic has the Scroggs' Property if it isn't characterized by any finite logical model/matrix, although every proper extension of it has a finite characteristic model/matrix.
[By proper extension I mean that the theory of the original logic is a proper-subset of the theory of the extended one]
How can I show that the Gödel-Dummett logic has the Scroggs' Property?
 A: The simplest finite model of your axiom $\bf L$ is the two-element boolean algebra. More interestingly, there are the models $G_n$ (say) of intuitionistic logic used by Gödel to show that intuitionistic propositional logic has infinitely many truth values. The Heyting algebra $G_n$ has $n$ elements, linearly ordered by logical strength, with $a \lor b = \max(a, b)$ and $a \land b = \min(a, b)$. So the two-element boolean algebra is $G_2$. The existence of the $G_n$ shows that the your axiom schema $\bf L$ has arbitrarily large finite models.
What you call Tautologies($\bf L$) (which I would just call the theory of $\bf L$) is not the same as the theory of $M$ (your Tautologies($M$)) for any finite model $M$: in a model with $n$ elements, a formula $\phi_n$ over propositional variables $p_1, \ldots, p_{n+1}$ asserting that at least two of $p_i$ and $p_j$ are equivalent holds, but $\phi_n$ does not hold in $G_{n+1}$ and hence is not in the theory of $\bf L$ (since $\bf L$ is sound for linearly ordered Heyting algebras).
