The negation of a satisfiable sentence is unsatisfiable. True or False?

Question

Can you provide me please with an answer and justification by using counter example. Thanks for your support.

Answer 1

False.

Simple example:

Consider $\exists x \: P(x)$. This sentence is satisfiable, because we can consuct an interpretation that makes this true. E.g. Let the domain be natual numbers and let $P(x)$ be 'x is even'

OK, the negation is $\neg \exists x\: P(x)$. That sentence is satisfiable too, because once again we can come up with an interpretation that sets this sentence to true. E.g. Let the domain be politicians, and let $P(x)$ be 'x is honest' :)

The point is: to be satisfable there needsd to be some interpretation that sets the sentence to true. Since we can consider completely different interpretations for a sentence and its negation, both sentences can be satisfiable.

Answer 2

If we assume a consistent theory (otherwise, EVERY sentence is satisfiable!) , we must distinguish between satisfiable and provable.

• A sentence is provable if it is true in every possible model.

• A sentence is satisfisable if it is true in at least one model.

If a sentence is only satisfiable , it can be false in some model. In this case, the negation is true in this model, hence satisfiable. Hence, we cannot conclude that the negation is non-satisfiable.

If the sentence is provable, then the negation must be false, hence it is not satisfiable.