here's an example of intuitionism
Dec. 12th, 2021 02:56 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
juan_gandhiIn Boolean logic,
In intuitionism you can only prove that
Example? Use the semantics trick from Wikipedia.
Namely, take the partial order of open subsets of the set of real numbers between 0 and 1,
Now take
Actually,
And their disjunction,
Q.E.D. We have an example where this equivalence from Boolean logic,
If somebody can show me an example from a finite lattice, that would be very cool. The example I had in my book is just wrong.
¬(A∧B) ≡ (¬A∨¬B); in intuitionism - no so fast. In intuitionism you can only prove that
(¬A∨¬B) ⊢ ¬(A∧B) , but not the opposite way.Example? Use the semantics trick from Wikipedia.
Namely, take the partial order of open subsets of the set of real numbers between 0 and 1,
[0,1]. It is a Heyting algebra (for an obvious reason not worth discussing here. Negation is defined like this: take a set complement, and take its interior (the largest open subset). So, e.g. for [(a,b)], the complement is [0,a)∪(b,1]Now take
A=[0,0.5) and B=(0.5,1]. These two sets don't intersect, so their conjunction is ⊥, and ¬(A∧B) is ⊤.Actually,
A = ¬B and B = ¬A.And their disjunction,
A∨B, is [0,0.5)∪(0.5,1], which is not ⊤.Q.E.D. We have an example where this equivalence from Boolean logic,
¬(A∧B) ≡ (¬A∨¬B), does not hold. Profit!If somebody can show me an example from a finite lattice, that would be very cool. The example I had in my book is just wrong.
