We create a simple contradiction in propositional logic and test it with two sound inferences: LK and “Modus Ponens alone”. The former handily identifies the issue, but the latter misses it and characterizes this minimal theory as... more
We create a simple contradiction in propositional logic and test it with two sound inferences: LK and “Modus Ponens alone”. The former handily identifies the issue, but the latter misses it and characterizes this minimal theory as consistent – setting the expectation that it is satisfiable, per a fundamental property. This counter-example challenges canonical definitions of consistency and completeness, proving that the choice of inference matters and it needs to be mentioned explicitly in the formulation of properties, unlike common practice today.
