The "converse" of $P\rightarrow(Q\rightarrow R)$

Question

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:

• $(Q\rightarrow R)\rightarrow P$

• $P\rightarrow(R\rightarrow Q)$

• $R\rightarrow(P\wedge Q)$

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.

Answer 1

Here are three in the order of the bullets above. Note that the quantifiers do cause some issues, especially in #1, but I think these meet the spirit of the question. I am not sure they are accessible to a layman, though.

1. If $A$ is a closed set of real numbers then if $(x_n)$ is a Cauchy sequence in $A$ then $\lim x_n$ is in $A$. Converse: If every Cauchy sequence in a set of real numbers $A$ converges to a point in the set $A$, then the set $A$ is closed.

2. If $X$ is a metric space then if $X$ is complete and totally bounded then $X$ is compact. Converse: If $X$ is a metric space then if $X$ is compact then $X$ is complete and totally bounded.

3. If $X$ is an infinite compact metrizable space then if $X$ is perfect and totally disconnected then $X$ is homeomorphic to the Cantor set. Converse: If $X$ is homeomorphic to the Cantor set then $X$ is an infinite compact metrizable space and perfect and totally disconnected.

Now here are three more plain examples:

1. If you are an skilled singer, then if you are given any song, you can sing it well. Converse: If, whenever you are given a song, you can sing it well, then you are a skilled singer.

2. If you work all days except holidays, then if you are not working on a given day then the day is a holiday. Converse: If you work all days except holidays, then if a day is a holiday then you don't work on that day.

3. If someone is in the U.S. Congress and not in the Senate, they are in the House of Representatives. Converse: if someone is in the House of Representatives, they are in the U.S. Congress and not in the Senate.

By the way, here is a fourth option:

• Urysohn's metrization theorem: If $X$ is a second countable topological space, then if $X$ is Hausdorff and regular then $X$ is metrizable. "Converse:" If $X$ is metrizable then $X$ is Hausdorff and regular. Note that this partial "converse" ignores second countability; the "converse" is just $R \to Q$.