Refutable

However, suppose that for every formula φ there is some formula ψ taken from a more restricted class of formulas C, such that " is either refutable or satisfiable" → " is either refutable or satisfiable".

Now if is refutable for some n, it follows that φ is refutable.

Because of the two meanings of the word undecidable, the term independent is sometimes used instead of undecidable for the "neither provable nor refutable" sense.

If every formula in R of degree k is either refutable or satisfiable, then so is every formula in R of degree k+1.

If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬φ is not satisfiable in any structure and therefore refutable; then ¬¬φ is provable and then so is φ, thus Theorem 1 holds.

In classical logic, the negation of a statement asserts that the statement is false; to an intuitionist, it means the statement is refutable citation (e.

Let us call the class of all such formulas R. We are faced with proving that every formula in R is either refutable or satisfiable.

Now is a formula of degree k and therefore by assumption either refutable or satisfiable.

Then, once this claim (expressed in the previous sentence) is proved, it will suffice to prove " is either refutable or satisfiable" only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ (i.

We have proved that φ is either satisfiable or refutable, and this concludes the proof of the Lemma.

We immediately restate it in a form more convenient for our purposes: Theorem 2. Every formula is either refutable or satisfiable in some structure.