Proposition

-> true/false do not merely "corresponds to reality."

• In Intuitionistic/Constructive logic

  • true means "has a proof".

  • false means "has a refutation"

    -> true/false do not merely "corresponds to reality."

  • Classical logic treats propositions as having truth values in some Platonic realm.

  • Intuitionistic/Constructive logic rejects this: a proposition is a problem to be solved. Asserting $P$ means "I have a construction that demonstrates $P$."

    -> Kolmogorov called Proposition Aufgaben (tasks) - a proposition is something you do, not something that is.

• Propositions vs. Booleans

  In the context of Intuitionistic/Constructive logic.

  • A proposition is not a boolean.

  • Conflating them is called Boolean blindness.

    | | Proposition | Boolean | | --- | --- | --- | | What it is | An assertion with proof/refutation | A data value: true or false | | "True" means | Has a proof | Equals the value true | | "False" means | Has a refutation | Equals the value false | | Computable? | Not in general | Always-it's just a bit |

    Example. Consider equality of functions $f, g : \mathbb{N} \to \mathbb{N}$.

  • The proposition $f = g$ is legitimate - you can prove it by showing they agree on all inputs.

  • The boolean $f == g$ is not computable-you would have to check infinitely many inputs.

    -> The proposition exists and can be reasoned about.

    -> The boolean cannot be computed.

    -> Conflating them causes confusion.