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As I understand a set of sentences $\Phi$ in first-order logic is consistent iff for all $\psi$ either $\Phi \vdash \psi$ or $\Phi \vdash \neg \psi$ is false.

On the other hand $\Phi$ is absolute consistent iff there exists a sentence $\psi$ such that $\Phi \vdash \psi$ is false.

That consistency implies absolute consistency seems self explanatory.

Since $\bot$ is usually defined by the definition schema $\forall \psi:\bot \implies \psi$ and negation is defined by $\forall \psi:\neg \psi \iff (\psi \implies \bot)$.

Question: does this mean that in first-order logic consistency and absolute consistency are equivalent?

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  • $\begingroup$ NO; a set $\Gamma$ of formulas is consistent iff there is no $\psi$ such that $\Gamma \vdash \psi$ and $\Gamma \vdash \lnot \psi$. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Feb 4, 2018 at 10:59
  • $\begingroup$ I read it on en.m.wikipedia.org/wiki/Consistency#Definition (I was searching for the notion of consistency defined without reference to negation) $\endgroup$
    – Adam
    Commented Feb 4, 2018 at 11:16
  • $\begingroup$ Check the def; you have copied them wrongly. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Feb 4, 2018 at 12:37
  • $\begingroup$ I don't see the difference, the definition of consistency is stated in natural language where the negation of an universal quantification is equivalent to the existence of a (possibly non-constructible) counterexample, is it not? $\endgroup$
    – Adam
    Commented Feb 4, 2018 at 13:12
  • $\begingroup$ We have a set of sentence $\Phi$: what does it mean to negate it ? $\endgroup$
    – Mauro ALLEGRANZA
    Commented Feb 4, 2018 at 13:16

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Yes, in first order logic these concepts are equivalent. I will use $T$ to stand for a set of sentences, rather than $\phi$, which should stand for a single sentence.

If $T$ proves both $\psi$ and $\lnot \psi$ for some formula $\psi$, then $T$ proves $\rho$ for all formulae $\rho$. So if $T$ is not consistent it is not absolutely consistent.

On the other hand, it is immediate that if $T$ is not absolutely consistent then it is not consistent.

The difference between the different forms of consistency only happens in logics other than first order logic:

  • In complete generality, a logic might not have a negation operation. In a logic like that, we can talk about absolute consistency but not about simple consistency.

  • In paraconsistent logics, it is possible that the principle of explosion does not apply. A theory can sometimes prove $\phi \land \lnot \phi$ for a sentence $\phi$, but not prove $\psi$ for some other sentence $\psi$. One motivation for studying paraconsistent logics is to handle exactly this situation, when we may be presented with contradictory assertions in one area, but we do not want to apply those to other areas.

There is also semantic consistency: a theory is semantically consistent if and only if it has a model. In first order logic, semantic consistency is equivalent to consistency. But, for example, in second order logic with standard semantics a theory can be syntactically consistent but not semantically consistent.

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