I have just thought about an interesting problem in classical propositional logic to propose at this site, but lost my notes... I do recall, however, that the problem involved a particular contingent sentence with three atomic variables, $p$, $q$ and $r$. This first sentence contained exactly two occurrences of connectives, both binary. I am positive that at least one of those connectives was an implication having one of the three aforementioned variables as antecedent.
Question 1. Assuming that the previous information is sufficient to semantically characterize the forgotten sentence, can you determine what was its other connective, as well as the possible logical forms that my sentence could have had?
Now, another thing I do recall is that the problem that I had created consisted in checking that the first sentence above is semantically equivalent to a second sentence, written with the same variables and containing the same connectives as the first sentence, but with one extra implication.
Question 2. What could have been this second sentence? Justify.
The following extra considerations should be unnecessary to solve the problem:
Word of warning.
While one must not disconsider any information from the above statement when looking for a solution, one should also not read more than what is written. In particular:
(1) Nowhere in the statement of the problem is to be found a claim about the first (or the second) sentence being "syntactically unique". In fact, there is obviously no way, for instance, of avoiding permutation of the variables in a given solution. But there might clearly be many other structurally different sentences (belonging to the same equivalence class!) that solve Question 1. The task is to identify such sentences (preferably without just guessing).
(2) In the statement of the problem, a sentence is supposed to be "semantically characterized" when it is characterized up to logical equivalence --- so, again, you had better think first about what is happening at the level of the quotient algebra, and consider next the syntactic restrictions imposed by the statement of the problem.
The assumption made in Question 1 is integral part of the statement, and in fact essential to the existence of a well-determined solution!
logic, because of the syntactical aspects of the problem. Boolean Algebras are part of
math, anyway. I believe the problem is not easy to solve, but might be worth a try. $\endgroup$