I am following a course in intuitionistic mathematics and I have been given an exercise about intuitionistic propositional logic.
The problem
Find four propositions $X_0, X_1, X_2, X_3$, such that the following hold:
(i) For each $i < 4$, the basic formulas of $X_i$ are among $A, B, \bot$
(ii) For each $i < 4$, $X_i \vdash_{NK} A\lor B$ and $A\lor B \vdash_{NK} X_i$
(iii) For each $i<4$, for each $j < 4$, if $i \neq j $ then $X_i \not\vdash_{NI} X_j$
Here $NK$ denotes the natural deduction scheme in classical logic and $NI$ the natural deduction scheme in intuitionistic logic.
Attemps
I am at a point where I have four propositions but recently I have found that (iii) does not hold.
- $X_0 = ((\neg\neg A \to A) \land A) \lor ((\neg\neg B\to B) \land B)$
- $X_1 = \neg A \to B$
- $X_2 = \neg B \to A$
- $X_3 = \neg\neg A \lor \neg\neg B$
Here it can be shown that $X_0 \vdash_{NI} X_3$ and for all the other combinations, I can find Kripke models to show the statements are intuitionistically not valid. Besides these four, there were other candidates that are classically equivalent to $A\lor B$ which were considered such as $A\lor B, \neg\neg(A\lor B)$ and $((A\to B)\to B) \land ((B\to A) \to A)$. Yet combining these have failed to give a satisfying result.
My question therefore is, are there other propositions that are classically equivalent to $A\lor B$ which I have not considered and cannot intuitionistically be proven assuming that proposition?
