If the domain contains only finitely many objects, substitutional invariance can be satisfied by arguments whose validity depends on contingent cardinality facts, not logical form.

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Substitutional invariance(discussed as potentially insufficient for determining formal validity): The idea that a logical rule holds true no matter what terms you swap in and out—if it works with 'cats' and 'animals,' it should work with any other pair of terms following the same pattern.
cardinality(Central to comparing infinite sets and establishing that no universal set exists.): A measure of the size of a set, indicating the number of elements it contains.
contingent(De Interpretatione 12–13): Equated with 'possible'; on the two-sided interpretation, contingency excludes necessity (possibility implies non-necessity).
domain(Both f1 and f2 have the reals as their domain): The set of input values over which a function is defined.
finitely many(in logic): A limited, countable number of things—you could theoretically count them all and reach an end, unlike infinite things.
logical form(Used to characterize logical consequence): The way that a sentence is built up from the logical particles.
validity(Formal logic; distinguished from syntactic deducibility): The model-theoretic counterpart to deducibility; an argument is valid if its conclusion is true under every interpretation in which its premises are true

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