Logical systems that are complete but not sound I was wondering, are there any commonly used logics(with both notions of deductions and of semantics) that are complete but not sound? I'm looking for an example that has actually proven useful to logicians. Thank you for your wisdom!
Sincerely,
VIen
 A: If a logical system is not sound, this means that there is a mis-match between the deductive system and the semantics (the models used when defining soundness).  A typical example would be that, if we use Kripke structures as our semantics (so the appropriate logic would be intuitionistic), then classical logic is unsound.  As far as I can see, such a phenomenon is not "useful" but just a mistake.  (Well, I suppose it could be useful for showing that someone made a mistake.)
A: "Complete" means that every true formula is derivable. "Sound" means that every derivable formula is true. Thus a system in which every formula is derivable would be complete (since the true formulas are a subset of all formulas), but it would not be sound as long as there is at least one formula that is not true. 
For example, propositional logic with "p ∧ ¬p" as an added axiom would be complete but not sound, as "p ∧ ¬p" is not true, but anything can be derived from it.
A: For anyone that comes across this:
A good example in an exam would be the calculus consisting only of the rule:
$\emptyset \vdash_{R_{1}} F$
Any rule can be derived from it (making it complete), because F can be arbitrarily chosen. It is clearly not sound, since every formula can be derived from it (including false ones). It must follow for the calculus to be sound that every formula derived from some arbitrary set $\mathbf{M}$ under the rule $R_{1}$ must be a tautology:
$M \vdash_{K}F \implies M\models F$ (Definition of soundness for a calculus K, where M is some arbitrary set and F is some formula.)
In this case we have $K = \{R_{1}\}$. And therefore:
$\emptyset  \vdash_{R_{1}}F \implies \emptyset \models F \implies F \ is \ a \ tautology$
Since not all formulae derived from the rule are tautologies the calculus cannot be sound.
