proof-by-contradiction-2 (math concept)
See also
python declaration class | python inclusion class | python sample
Definition
proof-by-contradiction-2 is the inference-rule:
\[\left( \boldsymbol{\mathcal{H}} \: \textit{assume} \: \boldsymbol{x} \neq \boldsymbol{y}, \; Inc\left( \boldsymbol{\mathcal{H}} \right) \right) \vdash \boldsymbol{x} = \boldsymbol{y}\]
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 x is not equal to y, it follows that the hypothesis is inconsistent, it follows that x = y (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]