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In a propositional logic system for example, it is sound iff $\Gamma \vdash \varphi \implies \Gamma \vDash \varphi$ and complete iff $\Gamma \vDash \varphi \implies \Gamma \vdash \varphi$ (I don't know if this definition also happens to be the same in higher order logics).

Anyhow, what then does it mean for a system to be "consistent"? Is this the same as a logic system being complete or sound, or does it mean something else? Is it a syntactic claim? A semantic one?

user525966
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    I think you mean $\Rightarrow$ rather than $\Leftrightarrow$ – Carl Mummert Nov 07 '18 at 15:49
  • And "system" is a dangerous word to use here: a system of axioms is a very different thing from a proof system - the former is just a set of sentences (the "$\Gamma$") while the latter is a method for deriving sentences from sets of sentences (the "$\vdash$"). Conflating the two is an easy way to get quickly lost. – Noah Schweber Nov 07 '18 at 16:04
  • Yes I meant $\implies$, fixed, thanks – user525966 Nov 07 '18 at 16:09
  • @NoahSchweber I don't think I said "system of axioms" anywhere though? I'm referring to the relationship between syntax and semantics which I assume belongs to the overall "logic system" or "logical calculus"? Or is this wrong? – user525966 Nov 07 '18 at 16:10
  • @user525966 My point is that I've seen the phrase "logical system" (or its relatives) used to refer to multiple different things - the proof system on its own, or the semantics on its own, or the proof system together with the semantics, or even the proof system + the semantics + a particular theory. I'm just saying that using the term "system" in this context (as in "propositional logic system"), until one is very comfortable with the basics, is in my opinion a risky thing to do since it can lead to confusion if you see others use the term differently. – Noah Schweber Nov 07 '18 at 16:18
  • @NoahSchweber Gotcha. What would be the more "technically correct" way to word things instead for what I am trying to say? – user525966 Nov 07 '18 at 16:23
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    @user525966 To be super fussy: "A proof system $\vdash$ for propositional logic is complete, for the usual semantics $\models$, if for every theory $\Gamma$ and every sentence $\varphi$ we have $$\Gamma\models\varphi\implies\Gamma\vdash\varphi$$ (and same but with "$\implies$" switched around for sound)." – Noah Schweber Nov 07 '18 at 16:25
  • @NoahSchweber All of this is relative to some fixed semantic system then? It's conceivable to make a different semantic system behind $\vDash$ where the proof system is no longer complete or sound? – user525966 Nov 07 '18 at 16:43
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    @user525966 Yup, that's correct! And sometimes we start with an alternate semantics and try to find a sound complete proof system for it, and other times we start with an alternate proof system and try to find a semantics with respect to which it's sound and complete. – Noah Schweber Nov 07 '18 at 16:48

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A set of sentences or a theory is consistent if it does not contain a contradiction.

Consistency can be defined in either semantic or syntactic terms. The semantic definition states that the set is consistent if and only if it has a model, i.e., there exists an interpretation under which all formulas in the set are true.

In this case, we say also that the set is satisfiable

In the context of mathematical logic, where we consider a proof system and the corresponding relation of derivability ($\vdash$), we say that

a set $\Gamma$ of sentences is consistent if and only if there is no formula $\varphi$ such that both $\Gamma \vdash \varphi$ and $\Gamma \vdash \lnot \varphi$.

If $\bot$ is part of the language, the above amounts to saying that $\Gamma \nvdash \bot$.

To say that a proof system is sound means that only true formulas can be derived with it.

Thus, soundness implies consistency, because $\bot$ is not true.

Regarding completeness, we have that, in classical logic, where Ex falso holds, an inconsistent proof system is trivially complete : being inconsistent, it proves every formula, and thus also the valid ones.

Mauro ALLEGRANZA
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    I have never seen a good discussion, but there is an interesting thing with proof systems. It is possible that a proof system only proves logically valid formulas (so the system is sound when there are no hypotheses) but from some hypotheses it is possible to prove formulas that are not logically implied by those hypotheses (so the proof system is not sound for proofs with hypotheses). One example is a Hilbert style proof system for propositional logic augmented with a new rule of inference $P \vdash \bot$. The system is sound for formulas with no hypotheses because the new rule can't be used. – Carl Mummert Nov 07 '18 at 15:51
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    Note for the OP (and possibly others) that in a system of the type @CarlMummert describes, the deduction theorem - a principle we often take for granted - must fail. The occasional failure of the deduction theorem in interesting logics is a strange and interesting topic; see the discussion at this mathoverflow question, and in particular the linked paper by Blok and Pigozzi. – Noah Schweber Nov 07 '18 at 16:02
  • Re: the last paragraph: first, by "system" I think you mean "theory," and I think it's a good idea to avoid the overloaded term "system" in this context. It's also worth pointing out that some authors (this is e.g. the by-far-majority use in model theory, in my experience) use "complete" to mean "complete and consistent." – Noah Schweber Nov 07 '18 at 16:07
  • @user525966 We're allowed to use it, but we might never get the opportunity. That is, we're told "If you can derive $P$, then I'll go ahead and let you derive $\perp$ too" - but if we start out with a theory that can't derive $P$, we never get to invoke this clearly bonkers rule. – Noah Schweber Nov 07 '18 at 16:49