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→Reports of my counter-intuitiveness are greatly exaggerated: fix typos
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My guess as to what's going on here — why the classical and intuitional reasoners appear to be talking past each other on this score — is that they are really talking about two different domains of mathematical objects. That is, the variables <math>p, q\!</math> range over <math>\mathbb{B}</math> in the clasical reading while they range over a space of propositions, say, <math>X \to \mathbb{B}</math> in the intuitional reading of the formulas. Just my initial guess.
My guess as to what's going on here — why the classical and intuitional reasoners appear to be talking past each other on this score — is that they are really talking about two different domains of mathematical objects. That is, the variables <math>p, q\!</math> range over <math>\mathbb{B}</math> in the classical reading while they range over a space of propositions, say, <math>X \to \mathbb{B}</math> in the intuitional reading of the formulas. Just my initial guess.