Semantic consistency of a theory T is defined by there being a model M in which all theorems of T are true. Syntactic consistency of a theory T is defined by there being no formulas A such that both A and it's negation ~A are provable. So let's assume semantic consistency and then assume syntactic inconsistency. Since T is inconsistent, both A and its negation ~A are provable. But since it's also semantically consistent, there must be a model where all of its theorems are true. So there must be a model M where both A and ~A is true, which is not possible. So semantic consistency implies syntactic consistency.

Is this conclusion right or am I severely misunderstanding something? I thought that semantic consistency only implied syntactic consistency under the additional assumption of soundness, but soundness was not assumed here. Is this conclusion a consequence of defining semantic consistency too broadly? Should it be limited to there being a model where the axioms are true?

  • $\begingroup$ @NoahSchweber how is that soundness? That is the definition of semantic consistency we are working with. It isn't an additional assumption. If we define semantic consistency to mean that all theorems are true in one model we don't have to assume that all theorems are true in every model to get that ~A and A are true in M. $\endgroup$
    – Anon
    Sep 21 at 19:10
  • $\begingroup$ Ah, sorry, I misread your definition. Yes, if you phrase semantic consistency this way, then you have what you need; but the "assumptions-free" version of semantic consistency is "the elements of $T$ are true in some model," which doesn't bake in the tameness of the deductive apparatus. $\endgroup$ Sep 21 at 19:12
  • $\begingroup$ @NoahSchweber thank you, but what do you take to be the elements of T to not assume soundness? Is it just the axioms? $\endgroup$
    – Anon
    Sep 21 at 19:14
  • $\begingroup$ Yes, exactly. (Usually by "theory" we mean simply "set of sentences," especially when working at this level of generality.) $\endgroup$ Sep 21 at 19:15
  • $\begingroup$ @NoahSchweber awesome that makes perfect sense then. Thank you! $\endgroup$
    – Anon
    Sep 21 at 19:15


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