As everyone know, even when reading mathematics books, a paragraph written in natural language contains much more information than just its purely logical translation. For my next tutor session, I am planning to demonstrate the differences by showing that what exactly is the converse of a theorem with logical form of $P\rightarrow(Q\rightarrow R)$ is dependent on how it is actually phrased.
When considering a theorem of the form $P\rightarrow(Q\rightarrow R)$, if we consider each letter to be a blackbox, then the 3 possible converses are:
$(Q\rightarrow R)\rightarrow P$
I would like 3 examples of theorem, each of which have to be a theorem (ie. proved) which under the standard translation to logic will have the form $P\rightarrow(Q\rightarrow R)$ such that: it and its proof must be easily understood by layman, it is commonly phrased in a certain way that suggest one of the converse above, that converse is also a theorem but the other 2 converses are false or even meaningless. Obviously, the 3 examples should have different type of correct converse among the 3 listed above. Note that despite saying that the theorem must have the logical form of $P\rightarrow(Q\rightarrow R)$, it certainly can contains free variables: as usual, free variables are implicitly under universal quantification.
Thank you for your help.