Status of formulas in an unsatisfiable propositional theory

[pallid zealot]

Suppose we have a propositional theory T that is unsatisfiable. Then it has no models. A formula phi is true in T if it is true in every model of T, by definition. Then if there are no models, it is any phi is true in T. Likewise, a formula phi is contradictory in T if it is false in every model of T, by definition. Thus again any phi is contradictory in T.

So is it correct to say that for an unsatisfiable theory T all formulas are both true and contradictory?

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[green geode]

No, that conclusion is not correct. While it is true that an unsatisfiable theory has no models, it does not imply that every formula in that theory is both true and contradictory.

In classical propositional logic, a formula is considered contradictory if it is false in every model of the theory. If a theory is unsatisfiable, it means there are no models where all the formulas are simultaneously true. However, this doesn't mean that every formula is false in every possible model.

In fact, in classical logic, an unsatisfiable theory contains at least one formula that is false in every model, making the theory itself unsatisfiable. However, it doesn't mean that every individual formula in the theory is false in every model. Some formulas may be true in some models, false in others.

Therefore, it would be more accurate to say that in an unsatisfiable theory, there exists at least one formula that is false in every model, making the theory unsatisfiable, rather than stating that every formula is both true and contradictory.

[pallid zealot]

If a theory is unsatisfiable, it means there are no models where all the formulas are simultaneously true.
You mean all the axioms?
However, this doesn't mean that every formula is false in every possible model.
The definition we were given in class is: "a formula is contradictory in T if it is false in every model of T". And a model of T is any model that satisfies all axioms in T. Thus if T is contradictory, there are no models that satisfy all the axioms of T. Thus the set of models of T is empty. Thus any formula is false in every model of T, it's a vacuous truth. I don't get where I am wrong here

[quartz estuary]

the implication is a vacuous truth

x in emptyset implies x | 666 is also a true statement

[pallid zealot]

Yeah that’s what I was thinking as well

Thank you!

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