# Logical systems that are only weakly sound / weakly complete?

I understand the basic concepts of soundness and completeness in logical systems. However, when I took a look at the Wikipedia pages on both concepts, I saw that it listed a seperate strong form of each.

\begin{align} & \text{Soundness: if } \vdash \phi, \text{then} \vDash \phi \\ & \text{Strong Soundness: if } \Gamma \vdash \phi, \text{then } \Gamma \vDash \phi \\ & \\ & \text{(semantic) Completeness: if } \vDash \phi, \text{then} \vdash \phi \\ & \text{Strong Completeness: if } \Gamma \vDash \phi, \text{then } \Gamma \vdash \phi \\ & \text{(where } \phi \text{ is a sentence, and } \Gamma \text{ is a set of sentences)} \end{align}

This piqued my interest as I had not heard of this notion before. Looking at the definitions, I can see that the strong forms imply the weak forms, because $$\Gamma$$ can be the empty set.

What I noticed is that in almost all cases, the weak forms would appear to imply the strong forms, because it seems to hold (at least in 2-valued logics) that $$\vDash A \to B$$ is equivalent to $$A \vDash B$$ and $$\vdash A \to B$$ is equivalent to $$A \vdash B$$, for any A and B.

I know it must be possible for a logic to be weakly sound but not strongly sound, or weakly complete but not strongly complete, but are there any specific examples of logics that exhibit this property?

EDIT: @Mark Savings points out that weak soundness does not imply strong soundness when $$\Gamma$$ is infinite; I thus refine my question to ask if there are any examples of finite logics that are weakly sound/complete, but not strongly sound/complete.

• Your argument about weak soundness implying strong soundness doesn’t seem to be valid. It works only when $\Gamma$ is finite. Dec 19, 2022 at 0:41
• @MarkSaving Good catch. I will edit my question to specifically exclude infinitely long proofs.
– Nico
Dec 19, 2022 at 0:56

For the soundness part, strong and weak are of course the same for finitary deduction systems (where every proof rule has only finitely many premisses), since a proof can only use finitely many formulas from the theory $$\Gamma$$.
• GL (K + the Gödel-Löb axiom $$□(□p \rightarrow p) \rightarrow □p$$)
• Grz (K + $$◻(◻(p \rightarrow ◻p) \rightarrow p) \rightarrow p$$).
The proof idea is as follows: Construct an infinite $$\Gamma$$ of which every finite subset is consistent with GL (resp. Grz) (construct a suitable frame), but the whole $$\Gamma$$ is not valid on any frame of GL (resp. Grz). Since every finite subset is consistent (and the logic is finitary), $$\Gamma \nvdash \bot$$. Since $$\Gamma$$ is not valid on any frame of GL, $$\Gamma \vDash \bot$$. (The full proof should be most introductory modal logic books, e.g. Blackburn et al. Thm 4.43.)