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Is it too pedantic to ask, why in the definition of a structure in model theory sets are assigned to the relational symbols $P, R, ...$ of a language and not to corresponding formulas $Px, Rxy, ...$ (modulo choice of variables)? It would seem to me more consistent with the interpretation of arbitrary open formulas inside model theory and compared to set theory where by the comprehension axiom sets are assigned to open formulas, not to symbols.

Is it just a notational abbreviation - to name a set by $P$ instead of $Px$ - or is there something deeper behind it? If it's an abbreviation: Why is this so seldom (if ever) made explicit?

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    How would you then define the meaning of P(f(x))?2011-06-27
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    What does this have to do with my question?2011-06-27
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    You suggested to change somehow the definition of a structure claiming that it will make matters more consistent. I tried to imagine how will this affect other basic definitions. I think that in defining the semantics of formulas (and by that I mean the relation ${\cal A} \models \phi$), you'll either need to define some complicated operations on sets or, at least implicitly, state that relations are assigned to relation symbols (not formulas). I don't think this is more consistent and wondered if you have a better definition.2011-06-27
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    @Levon: I didn't want to change the definition of a structure but asked for an alternative *reading* of the standard definition: wether the assignment of sets to symbols couldn't be understood as the assignment of sets to (equivalence classes of) formulas (just *represented* by symbols).2011-06-27
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    @Hans: There is a natural one-to-one correspondence between relation symbols and their corresponding formulas (modulo choice of variables). So from formalist point of view, there is no difference whether you assign something to the former or to the later. It is like talking about functions and their graphs. I interpreted your question as an aesthetic question of preferring one over the other. If this is not the case, I am not sure I understand your question, at least the first part.2011-06-27
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    @Hans Stricker: A function symbol is interpreted as a real honest to goodness *function*, a relation symbol as a relation. Why complicate things by bringing in possible valuations on the collection of symbols for variables? (I don't think that such valuations as a technical device are a great idea anyway.)2011-06-27

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