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Assume 'S' is a first-order sentence about a subject 'Z'.

When one stipulates a Model for 'S' with a domain 'D' does one always assume that the domain will contain all the objects within the subject 'Z'?

E.g. Let S = 'All lizards like flies'

Then when assigning a model/structure to 'S' do we assume that the domain will contain all the objects which are lizards? Since the sentence is about lizards.

I apologise for the crude example and (I think) poorly structured question, but I hope you understand what I'm getting at.

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  • In pure FOL theories there are no objects, only predicates, lizards and flies are 1-place predicates L(x)="x is a lizard" and F(x)="x is a fly", like is a 2-place predicate P(x,y)="x likes y". Your sentence is ∀x∀y(L(x) & F(y) → P(x,y)). You only get objects when you interpret the theory on a domain, and predicates are then interpreted as relations on them, so L, F and P return true of false when domain objects are substituted into them. The domain need not include all lizards, or all flies, or any of them, it can even be empty.
    – Conifold
    Commented Mar 9, 2023 at 9:39

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imo it strongly depends on a construction of your domain of discourse. It's like a kid asking: "Mom, did I write all the words on this piece of paper?" - just define "all"!

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