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just want to know is it "good" to use syntax and semantics together in a formal prove in mathematics.

With Completeness theorm,syntax of first order logic is equal in value to semantics. However,in Zhongwang Lu's mathematical logic towards computer science (the book is written in Chinese and the title is translated by myself),Lu said that many people (in Chinese) mixed up syntax with semantics,and one of the purpose of his writing the book is to correct this trend. So Is it good to mix syntax with semantics in mathematical logic?

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    $\begingroup$ What does it mean, according to the author, "to mix syntax with semantics in mathematical logic?" Theorems like the Completeness one are exactly about the link between the derivability feature property of the calculus (a syntactical concept) and the validity of a formula (a semantical concept). $\endgroup$ – Mauro ALLEGRANZA Jan 26 '20 at 12:39
  • $\begingroup$ maybe the division of syntax and semantics could help somewhere in our understanding(especially in the case of the prove of Godel incompleteness, given that "representative" used in it is syntactic). $\endgroup$ – AnduinWilde Jul 9 '20 at 6:59
  • $\begingroup$ Agreed - in formal systems the distinction between syntax and semantics is clear and useful. On top of this we have results connecting the two. My concern is about the author alluding to "bad mixing" of syntax with semantic. $\endgroup$ – Mauro ALLEGRANZA Jul 9 '20 at 9:00
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Once you've established good relations between syntax and semantics (e.g. variants of the completeness theorem), it is completely fine to mix them up, as long as you have the confidence to seperate them any time. This is however not the case with many beginners in logic. Therefore for pedagogical reasons you should not do this.

On the other hand, when such relations are not yet established, or cannot be well established, then it is absolutely unacceptable to mix them up.

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In many low-level foundations of mathematics introductions there is a lack of precision if it is about completeness and soundness. For example, there is a quite popular "mixing up syntax with semantics"-confusion over the definition of completeness. Some authors use the notion of "completeness" for both:

  • semantic completeness: For any tautology in the system, there is a formal proof. $(\models \phi )\Rightarrow( \vdash \phi)$.

  • syntactical completeness: (or negation completeness) For any sentence $\phi$, either $\phi$ or $\lnot \phi$ is provable in the system. $(\not \vdash \lnot \phi) \Rightarrow (\vdash \phi)$.

These notions are not equivalent, however. Negation completeness is stronger than semantic completeness. Also, at least in german, there is another very similar confusion, since some authors use correctness also for consistency, but of course these notions aren't equivalent as well. Maybe this is what the chinese author means if he says people mix up syntax with semantics.

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  • $\begingroup$ I think he maybe what Lu considers is semantics completeness and I guess he doesn't like that because it's hard to implement in the computer? I just want to know is it "good" to use syntax and semantics together in a formal prove in mathematics. $\endgroup$ – AnduinWilde Jan 27 '20 at 11:06

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