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My background knowledge: My (poor) understanding of Tarskian semantics is that we are given both:

  • a theory/formal language of first-order logic to be used as the object theory, and
  • a "larger" first-order set theory (usually ZFC or ZFC+relevant inaccessible cardinal axioms) to be used as the metatheory. ["first-order set theory" = theory/formal language of first-order logic whose variables "are" or are called "sets".]

This then allows us to define a structure (in the sense of model theory), where the signature of the structure corresponds to/is the object theory, and the domain of the structure is a set (i.e. variable) of the metatheory. And in what follows below I assume that the domain and the interpretation function have been chosen such that the requirements/axioms of the object theory are actually satisfied by the given structure, i.e. that it is a model of the object theory.

In particular the interpretation function has to interpret not only the variables of the object theory/signature, but also the predicates of the object theory/signature. The interpretation function sends variables to elements of the domain, and the interpretation function sends predicates to subsets of (Cartesian products of) the domain.

Question:

  1. By interpreting predicates as subsets of the domain, doesn't this construction "implement"/enable second-order logic in the object theory?

    In other words, doesn't this construction enable "internal second-order quantification" i.e. quantification over the predicates of the object theory/signature, using exclusively "external first-order quantification" i.e. quantification over the variables of the metatheory?

Both elements of the domain, and subsets of (Cartesian products of) the domain, are sets and thus variables of the (first-order) metatheory, at least as far as I understand.

The Plato encyclopedia article on "first-order model theory" seems to mention most of these definitions, but does not mention second- or higher-order logic at all. My question is maybe similar to what e.g. these answers/comments [1][2][3][4][5][6] to related questions might be implying, but I'm really unsure either way.

hasManyStupidQuestions
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    Yes, in first-order set theory we can quantify over sets (such as the subsets of the domain of a model) and make statements and deductions that would require second-order logic if the domain of discourse included not those sets but only their elements. – Karl Jan 03 '23 at 16:52
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    But a given first-order language only includes a fixed set of relation symbols and does not have second-order quantifiers (i.e. quantifiers "over all possible predicates"), so second-order reasoning is not possible in the object language. In the metatheory we can do whatever we want, like quantify over formulas or symbols in the object language or subsets of models, but then we're reasoning about the language, not in it. – Karl Jan 03 '23 at 17:09
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    @Karl You're right that the first-order object language remains fixed, so any "second order reasoning" definitely can't be internal to / in the original first-order object language. But is there not a straightforward syntactic extension of the original first-order object language to a new second-order ("object") language, just by adding second-order quantifiers? So maybe the question is whether the Tarskian semantics provides an interpretation/semantics not only for the original first-order object language but also for its new second-order extension? Even though the metatheory is first-order? – hasManyStupidQuestions Jan 03 '23 at 23:24
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    Yes, we can define and interpret second-order logic using (first-order) set theory. Set theory is powerful because it enables higher-order reasoning (about the elements of any given set, but not about the whole universe of sets). So working with (for example) natural numbers in set theory, where $\Bbb N$ and its power sets, functions, etc. are objects in the domain of discourse, is very different from working inside a first- or second-order theory whose domain of discourse is just the natural numbers. – Karl Jan 04 '23 at 00:49

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