Do completeness and correctness theorems apply to inconsistent sets in predicate logic?

Question

Simple question, the way I understand the theorems, the result is that the completeness and correctness theorems apply to any set.

However, consider the set of formulas:

$\phi, \neg \phi$

Or any other inconsistent set. Then it is commonly known that any formula can be derived from the set. However, it doesn't seem like any formula would be the logical consequence of the above set, since the formulas are not even true in any interpretation. So is any formula in fact the logical consequence of the above set?

Answer 1

is any formula the logical consequence of an inconsistent set [of formulas] ?

YES

The definition of logical consequence is :

Let $\Gamma$ be a set of formulas and $\varphi$ a formula. Then $\Gamma$ logically implies $\varphi$ (and $\varphi$ is a logical consequence of $\Gamma$), written $\Gamma \vDash \varphi$, iff every interpretation that satisfies every member of $\Gamma$ also satisfies $\varphi$.

Thus, the "logical form" of the definition is :

for all interpretation $I$, if $\text { Satisfy }(\Gamma, I)$, then $\text { Satisfy }(\varphi, I)$.

If $\Gamma$ is inconsistent, then $\text { Satisfy }(\Gamma, I)$ is false for every $I$; thus, the conditional: if $\text { Satisfy }(\Gamma, I)$, then $\text { Satisfy }(\varphi, I)$, is vacuously true for every $I$.

And this, in turn, holds for a formula $\varphi$ whatever.

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How completeness applies to this case ?

If $\Gamma = \{ \phi, \lnot \phi \}$ we can apply the rules of the calculus to derive a formula $\psi$ whatever.

With e.g. Natural Deduction, we have :

1) $\phi$ --- premise

2) $\lnot \phi$ --- premise

3) $\bot$ --- from 1) and 2) by $\lnot$-E

4) $\psi$ --- from 3) by $\bot$-E.

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How correctness applies to this case ?

Correctness (or soundness) means that the rules applied to (a set of) true premises produce a true conclusion.

But in an inconsistent set of premises not all formulas are true; thus, the definition does not licence us to assert that in this case the conclusion must be true.