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Can a model of a sentence in predicate logic contain a function that can map to a member not contained in the domain of that model?

Example. Is this interpretation correct : domain= $\{1,2\}$ ; sentence=$'(∃x)(∃y)(P(f(xy)))'$ ; $f(xy) =$ the the product of $x$ and $y$ ; $P(x)=x$ is positive. The sentence seems true, but on an assignment $(x=2$ and $y=2)$, the function maps to $4$, which is not contained in the domain.

EDIT : Second question : How would we qualify this model? True, false, neither of those?

Thrasymaque
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    Not a model. The interpretation of $f$ must be a dyadic function from the domain to itself. – Fabio Somenzi May 10 '17 at 19:42
  • Additionally Your statement, at the start is not valid because we know nothing about the domain of $y$, so when you write $'(∃x)(P(f(xy)))'$, we assume there exists an x in D. By y remains a free variable. – amWhy May 10 '17 at 19:50
  • You are right, I will edit. – Thrasymaque May 10 '17 at 19:50
  • Note that indeed, $$\exists x\exists y(P(xy))$$ Take $x=1, y=1$, e.g., you need only prove here that indeed, there is such a pair $x, x \in D$ such that $xy>0$ is true. Now perhaps you mean to define $f(x, y) = xy$? – amWhy May 10 '17 at 19:56

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The simple answer is No; interpretations of function symbols must be functions that map objects from the domain to the domain, so you can't map to anything outside the domain.

Also, your sentence should be $\exists x \exists y P(f(x,y))$

Bram28
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