proof-by-contradiction-1 (math concept)
See also
python declaration class | python inclusion class | python sample
Definition
proof-by-contradiction-1 is the inference-rule:
\[\left( \boldsymbol{\mathcal{H}} \: \text{assume} \: \neg \boldsymbol{P}, \mathit{Inc}\left( \boldsymbol{\mathcal{H}} \right) \right) \vdash \boldsymbol{P}\]
Where:
\(\boldsymbol{\mathcal{H}}\) is an hypothesis
\(\boldsymbol{P}\) is a propositional formula
\(\mathit{Inc}\) is the inconsistency function
In straightforward language, if from the hypothesis that P is false, it follows that the hypothesis is inconsistent, it follows that P is true (or alternatively that the parent theory is itself inconsistent).
Note
Note the possibility that the base theory-derivation from which the hypothesis is elaborated is inconsistent.
Synonyms
reductio ad absurdum [Bau10]
Sources
[Bau10]