I have a feeling it’s not, because ¬¬P → P is not provable. If it is, I’m not sure what kind of reductio I’d need to negate ¬(¬¬P → P). I believe a textbook somewhere said it was provable in intuitionistic logic, so am I missing something or is the textbook wrong?
I’ve heard that some axioms, such as “all functions are continuous” or “all functions are computable”, are compatible with intuitionistic type theories but not their classical equivalents. But if they aren’t compatible with LEM, shouldn’t that mean they prove not LEM? But not LEM means not (A or not A) which in particular implies not […]
I am trying to understand what’s wrong with the following logic related to “multiple conditioning.” Why is the probability of [(A given B) given C] not the same as the probability of [A given (B and C)] ? I know it’s not true, but only because numbers disagree. I am having a hard time parsing […]
Several times in my studies, I’ve come across Hilbert-style proof systems for various systems of logic, and when an author says, “Theorem: $\varphi$ is provable in system $\cal H$,” or “Theorem: the following axiomatizations of $\cal H$ are equivalent: …,” I usually just take the author’s word as an oracle instead of actually trying to […]
I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals <something>, … One of my students just rose and asked me: Why do we assume so much in math? Is math really built on assumptions? I couldn’t answer him, […]