This type of response poses a problem of consistency. A hypothesis is consistent if it does not result in a logical contradiction.
What is consistency ?
- A consistent theory in traditional deductive logic is one that does not result in a logical contradiction. Both semantic and syntactic definitions of the absence of contradiction are possible.
- The term satisfiable is employed in modern mathematical logic in place of this concept in classical Aristotelian logic.
- A consistency proof is a mathematical demonstration of the consistency of a given theory. As part of Hilbert's programme, the early development of mathematical proof theory was motivated by the need to offer finite consistency proofs for all of mathematics.
- The incompleteness theorems, which demonstrated that sufficiently robust proof theories cannot prove their own consistency, had a significant impact on Hilbert's programme.
- Model theory can be used to demonstrate consistency, however it is frequently done in a syntactically-only manner without referencing any models of the logic. Since there is no general contradiction and no cut-free proof of falsity, the cut-elimination implies that the calculus is consistent.
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