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.