Henkin semantics for second-order logic permits non-standard interpretations of quantifiers over relations, under which second-order logic is complete and its Löwenheim-Skolem theorems hold (Henkin 1950).
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Meanings assigned to logical symbols that differ from the conventional or expected way of understanding them.
Relations
A relation is a way of describing how two or more things are connected or related to each other. For example, "is the parent of," "is taller than," or "is friends with" are all relations that show how people or objects link together. In everyday terms, it's simply a pattern or connection that shows how one thing depends on, compares to, or associates with another.
Second-order logic(as used in mathematical logic)
A formal system that goes beyond basic logic by allowing you to quantify over (talk about) properties and relations themselves, not just individual objects.
quantifiers(the logical form Russell said descriptions should take)
Words like 'all,' 'some,' and 'none' that express how many things we're talking about; Russell argued that phrases like 'the king of France' should be understood using these quantity-words rather than as simple names.
semantics(Distinguished from metasemantics and pragmatics in Kaplan 1989)
The domain that concerns the facts about what meanings words or phrases have.