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In the first order logic system, the definition of $\models$ is on p32~33 of §4. The Consequence Relation of III Semantics of First-Order Languages of Ebbinghaus' Mathematical Logic (It is very dreadful to transcribe every time, so please allow me use screenshot only. If it is not allowed, I would furthermore remove screenshot. So I guess a screenshot is better than none):

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Is it correct that the definition of $\models$ is entirely based on something similar to "the first order logic system" for sets, which consists of

  • "=" between elements,
  • "holds for" between relations and elements in product spaces,
  • propositional connectives: "not", "and", "or", "if ... then...", "if and only if"
  • quantifiers: "for all ... $\in$ ..., ...", "there is an ... $\in$ ..., such that ..."

?

I know I haven't read some parts of the book. I was wondering about the chicken and egg problem or the circular problem: How is that "first order logic system" for sets defined, if $\models$ of the first order logic system itself is defined in terms of it?

Thanks.

Tim
  • 49,162
  • Not exactly; it is described in "usual" mathematical language that uses set language (see a standard algebra textbook). The concept of interpretation is fairly simple: a domain of elements with relations (properties, etc.) and operations (functions) defined on it. The first definition of logical consequence was due to Aristotle and it did not use set language. – Mauro ALLEGRANZA Aug 28 '20 at 15:39
  • If you want to formalize the meta-theory, a "certain amount" of set theory will be enough. – Mauro ALLEGRANZA Aug 28 '20 at 15:43
  • Maybe useful this post – Mauro ALLEGRANZA Aug 28 '20 at 15:51
  • As well as this one with link to this MO post – Mauro ALLEGRANZA Aug 28 '20 at 15:57
  • Could you be more specific for my questions? – Tim Aug 28 '20 at 23:13
  • What I mean is that the rigorous definition of the relation $\vDash$ is rigorous enough - as for "usual" mathematics - as written, without necessarily formalizing it into a first order set theory. Thus, we can develop logic as a mathematical discipline in itself (the mathematical theory of formal deduction) and then use it to formalize mathematical theories, like e.g. arithmetic, group theory and set theory. – Mauro ALLEGRANZA Aug 29 '20 at 15:34
  • The "chicken-and-egg" issue is quite useless... in fact, formal logic starts at least with Aristotle, without set theory. Thus, at least from an historical point of view, formal logic is "prior to" set theory. – Mauro ALLEGRANZA Aug 29 '20 at 15:35

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