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.


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 $R \to Q$.
  • $\begingroup$ I would be very happy to see other examples. I find this an extremely interesting question! $\endgroup$ Aug 1 '14 at 0:26
  • $\begingroup$ Nice answer, +1 $\endgroup$
    – rogerl
    Aug 1 '14 at 1:13
  • $\begingroup$ What is the actual, mathematical converse of $P \implies (Q \implies R)$? $\endgroup$ Sep 9 '16 at 2:09
  • $\begingroup$ In an intro logic class, the converse of $A \to B$ is $B \to A$. But mathematicians use the word 'converse' in a more general way. @Kurt Mueller $\endgroup$ Sep 9 '16 at 13:55

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