Is it possible for a set of well-formed formulas to satisfy a formula and its negation?

I know intuitively this is impossible, however I think I tied myself into a knot using the Soundness theorem and inconsistency of deductions.

We know that if a set of well formed formulas can deduce a formula and its negation, that set is called inconsistent.

We also know that if a set of well formed formulas can deduce a formula it also tautologically implies that formula, by the Soundness theorem.

Does this mean that if a set is inconsistent it then can tautologically imply a formula and its negation?

Solution 1:

The title of your question:

Is it possible for a set of well-formed formulas to satisfy a formula and its negation?

makes little sense: formulas don't satisfy formulas. Rather, models or structures satisfy formulas.

You also write:

... and inconsistency of deductions.

Again, this not at all meaningful. I have no idea what an 'inconsistent deduction' is. What you can have, is an inconsistent set of formulas.

The rest of your post is all correct, and so yes, an inconsistent set of formulas implies any formula and its negation. But again, this does not mean that that set of statements satisfies that formula and its negation. In fact, nothing will satisfy that formula and its negation, just as nothing will satisfy that inconsistent set of formulas in the first place.

Solution 2:

Soundness means that whenever $\phi$ is derivable from $\Gamma$, then for any model of the theory that satisfies $\Gamma$, that model satisfies $\phi$.

Consistency of $\Gamma$ means that $\bot$ is not derivable from $\Gamma$.

So if $\Gamma$ is inconsistent, that just means that no model satisfies $\Gamma$.

Consistency is a property of the theory $\Gamma$, whereas soundness is a property of the logical system.

Solution 3:

Per wikipedea reference:

Soundness of a deductive system is the property that any sentence that is provable in that deductive system is also true on all interpretations or structures of the semantic theory for the language upon which that theory is based.

While consistency is usually a syntactic property of a set of wffs with a deductive system and like you said it means for any formula $p$ this set cannot derive both $p$ and $\lnot p ~(\bot)$. Thus soundness implies consistency (see a post in this site and there're also many other similar ones you can search online), but surprisingly consistency doesn't imply soundness in general as the famous $PA \cup \lnot Con(PA)$ theory is consistent (per Gödel's 2nd incompleteness theorem) but unsound (you can also easily search online for many similar posts).

With this background regarding your question, if a set is inconsistent then it must be unsound, meaning it implies an empty model of your set of wffs with all other formulas it can deduce (expressed by the language of a formula and its negation) as emphasized in my linked post...