Changes

My guess as to what's going on here &mdash; why the classical and intuitional reasoners appear to be talking past each other on this score &mdash; 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.

My guess as to what's going on here &mdash; why the classical and intuitional reasoners appear to be talking past each other on this score &mdash; 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.

On the reading <math>p, q : X \to \mathbb{B},</math> another guess at what's going on here might be the difference between the following two statements:

On the reading <math>P, Q : X \to \mathbb{B},</math> another guess at what's gone awry here might be the difference between the following two statements:

<blockquote>

<blockquote>

<math>(\forall x \in X)(p \Rightarrow q) \lor (q \Rightarrow p)</math>

<math>(\forall x \in X)(Px \Rightarrow Qx) \lor (Qx \Rightarrow Px)</math>

</blockquote>

</blockquote>

<blockquote>

<blockquote>

<math>(\forall x \in X)(p \Rightarrow q) \lor (\forall x \in X)(q \Rightarrow p)</math>

<math>(\forall x \in X)(Px \Rightarrow Qx) \lor (\forall x \in X)(Qx \Rightarrow Px)</math>

</blockquote>

</blockquote>

Or maybe these are really the same guess?

Or maybe these are really the same guess?