Complicating the Converse

(An archive question of the week)

Usually when we discuss converses (and inverses and contrapositives) we use clear, idealized examples. But statements in real life — even in real math — are not quite so straightforward. The difficulty is not merely in the language, but in the complexity of our statements. A question in the beginning of 2017 brought out some interesting issues.

Context: Partial converses?

Before I go to the question itself, I have to show the context, so some comments in the exchange will make more sense. Navneet first asked a question that didn’t get archived:

My book proves this statement- 

"If the midpoint of a chord is joined to the centre of the circle, then the line passes through the midpoint of the corresponding minor arc."
I understood the proof. But I have a confusion regarding its converse. What should be the converse of this theorem?

The converse of- 
  If p, then q, 
where p and q are sentences, is
  If q, then p.
So, I think the converse of this should be-
"If a line passes through the midpoint of the minor arc, then the line should join the midpoint of the corresponding chord to the centre of the circle"!!!
Please visit the link- http://www.themathpage.com/abooki/logic.htm .
Under the converse category you will find this sentence-
"If a statement has two hypotheses -- If a and b, then c -- then a partial converse is:  If a and c, then b."
Do you think that the theorem stated in the beginning has two hypotheses?

I was obligated to answer this, because it was on a thread I “owned”; but I was not in a position to say much. I replied briefly:

I am on vacation and do not have good equipment, so I can't give long answers, but I will do what I can.

You are right that the converse of this theorem as stated is not reasonable, and of course is not true. The converse of a theorem is not in general true, so this is not a problem. If you were assigned to state the converse and whether it is true, you could state it more or less as you did.

But if your goal is simply to find a converse that makes sense and try to prove it, then a "partial converse" like what you stated is appropriate. In effect, you are restating the theorem as something like, "Given that a line passes through the center of a circle, if it passes through the midpoint of a chord, then it passes through the midpoint of the arc."

The point here is that we can’t just mechanically swap two halves of a sentence to make the converse. The statement here consists of more than just “if p then q“. Navneet’s initial attempt took p as “the midpoint of a chord is joined to the centre of the circle”, and q as “the line passes through the midpoint of the corresponding minor arc”. Merely swapping these produces nonsense. My suggestion was to rewrite the original in three parts: “Given c, if p then q“, so that the converse would be “Given c, if q then p“. Specifically, “Given that a line passes through the center of a circle, if it passes through the midpoint of a chord, then it passes through the midpoint of the arc.”

The converse, then, taken mechanically, would be, “Given that a line passes through the center of a circle, if it passes through the midpoint of the arc, then it passes through the midpoint of a chord.” This makes sense, and it does sound like the “partial converse” idea he had discovered — that was a good insight. But I don’t find that to be a common concept.

Dueling converses

Now, before I made that response, Navneet had already asked another very similar question, on a new thread; but this time his textbook had stated a converse. This one was archived, because it got a fuller answer:

Converses in Construction

According to my text, the midpoint theorem states:

   "If the midpoints of any two sides of a
   triangle are joined, then the line is
   parallel to the third side."

My book goes on to phrase the converse of this theorem like this:

   "The line drawn through the midpoint of
   one side of a triangle and parallel to
   another side bisects the third side."

Do you feel that this is the converse of the midpoint theorem?

We know that the converse of "if p, then q" is "if q, then p." So I think that the converse of the midpoint theorem should be this:

   "If a line is parallel to a side of a
   triangle, then the line should join
   the midpoints of the other two sides."

Now clearly, this converse is false. But am I wrong somewhere? If yes, please correct me and explain in detail.

Note that the book’s converse is not written in if-then form, and also that it does not have the same meaning as Navneet’s.

Doctor Rick replied (a couple hours after my reply), expanding on what I had said:

Hi, Navneet.

Let me borrow an idea from Doctor Peterson, who responded to another, related question of yours separately just now:

   "If your goal is simply to find a
   converse that makes sense and try to
   prove it, then a 'partial converse'
   like what you stated is appropriate."

There is some latitude in what we call the converse of a theorem. We could take your statement of what a converse is, and add a bit to it to express how the concept is really applied to theorems:

  Theorem: In context C, if p, then q.

  Converse of the theorem: In context C, if q, then p.

There is always a context, and this context is assumed to be the same for both the theorem and its converse. The latitude I spoke of consists in what else we reasonably choose to include as part of the context.

We can divide a statement into context, hypothesis, and conclusion somewhat arbitrarily, so things may not be as clear as the usual explanation of converses suggests. Combining this with the fact that, as in the book’s converse, not all theorems are stated explicitly as “if … then …”, things can look pretty confusing.

At minimum, a theorem in Euclidean geometry brings with it the entire context of Euclidean geometry; that is, the postulates and definitions of Euclidean geometry. So, in this case, we could restate the theorem as:

     Context: In Euclidean geometry, given a
              triangle and a line passing through
              the midpoint of one side and a point 
              on another side of the triangle, ...
   Condition: If the second point is the midpoint
              of its side ...
  Conclusion: Then the line is parallel to the
              third side of the triangle.

And the converse of the theorem in this form is:

     Context: In Euclidean geometry, given a
              triangle and a line passing through
              the midpoint of one side and a point 
              on another side of the triangle, ...
   Condition: If the line is parallel to the third
              side of the triangle ...
  Conclusion: Then the second point is the
              midpoint of its side.

That's what your book did. Why? Because the goal was to write a "converse theorem" that is TRUE! There is no point in writing a converse that is not a valid theorem, because we can't do anything with it. Therefore, the author chose, quite reasonably, to include enough in the "context" to make the converse a valid theorem.

And the author then chose to reword it in a way that looks quite different from the original, but is still a valid converse: “The line drawn through the midpoint of one side of a triangle and parallel to another side bisects the third side.”

Clarifications

Navneet wrote back,

Thanks! Your explanation was so easy and precise.

Now, I want a little more help.

(1) I want to understand what you meant to convey in these lines, with stress to the meaning of "latitude":

   "The latitude I spoke of consists in what
   else we reasonably choose to include as
   part of the context."

(2) Why did Doctor Peterson say this?

   "If your goal is simply to find a converse
   that makes sense and try to prove it, then
   a 'partial converse' like what you stated
   is appropriate."

I think that a "partial converse" is valid for a theorem only if it has two hypotheses. If in the beginning only he said that the given theorem does not have two hypotheses, then why does he go on to say that a "partial converse" is appropriate for this theorem? Is the meaning of "partial converse" different here?

Doctor Rick responded first to the question about “latitude”, which is not a familiar word to many students:

To your first question, Dictionary.com offers this definition, which is the sense of "latitude" that I had in mind:

   freedom from narrow restrictions;
   freedom of action, opinion, etc.:
   "He allowed his children a fair amount of latitude."

So I was saying that we have some freedom to decide what we call the converse of a theorem, because we have some freedom to choose what we consider to be part of the context and what is part of the conditions of the theorem.

Then he explained (correctly — that’s what twin brothers are for) what I meant by the quoted line:

To your second question, I have to go back and look at the context of what my brother Doctor Peterson said.... I see that you introduced the concept of a partial converse with him; and that he told you we'd prefer not to use that idea -- it isn't something we've seen. 

But in terms of the statement you quoted,

   If a statement has two hypotheses -- "If a and b, then c" -- then a
   partial converse is: "If a and c, then b."

We need to do the same sort of thing I did in my last response: REWRITE the theorem so that it *does* have two hypotheses. I rewrote it this way:

     Context: In Euclidean geometry, given a
              triangle and a line passing
              through the midpoint of one side
              and a point on another side of
              the triangle, ...
   Condition: If the second point is the
              midpoint of its side ...
  Conclusion: Then the line is parallel to the
              third side of the triangle.

We could just as well say this:
  Hypothesis 1: In Euclidean geometry, given a
                triangle and a line passing
                through the midpoint of one side
                and a point on another side of
                the triangle, ...
  Hypothesis 2: If the second point is the
                midpoint of its side ...
    Conclusion: Then the line is parallel to the
                third side of the triangle.

Now the "partial converse" would be: "If [Hypothesis 1] and [Conclusion], then [Hypothesis 2]." That's the same as my "If [Context] and [Conclusion] then [Condition]."

The two hypotheses in themathpages‘ concept of a partial converse are identical to the “context” and “condition” (hypothesis) in Doctor Rick’s explanation.

We did not have a context for your question in the beginning. (Context seems to be coming up a lot, doesn't it?) It seemed possible that you were asking about the converse because you were ASKED to state a converse for this theorem, without regard as to whether it would be a valid theorem. In that context, your converse could be considered correct. By taking the theorem exactly as stated, we produce a converse that is NOT a theorem -- which would be OK in this context.

But Doctor Peterson also said, "if your goal is simply to find a converse that makes sense and try to prove it, then a 'partial converse' like what you stated is appropriate." Now that you have told us that your *book* gives the converse, we see that the goal was to write a *valid* converse theorem. We needed the extra "latitude" to do that, but it's appropriate in this context -- and much more useful.

In other words, you do better math when you are trying to accomplish something interesting and useful, not just following rules. And textbook questions that ask you to “write the converse” of a statement may be  missing the point of writing converses.

Another example

Navneet, being a good and thoughtful student, wanted to practice these ideas, so he tried applying it to a simpler example:

Thanks!

Given this theorem:

  "Angles opposite to equal sides of a triangle are equal."

Then, as we know, its converse should be this:

  "Sides opposite to equal angles of a triangle are equal."

Should we separate the theorem here into context, condition, and conclusion for finding its converse?

I *GUESS* it should be separated like this:

    Context: In Euclidean geometry, given a triangle....
  Condition: If any two angles are equal....
 Conclusion: Then the sides opposite to the equal angles are equal.

If I have not done right, then please correct me.

Lastly, wouldn't you agree that if we have the freedom to choose what we put in the part of context and conclusion, then we can create more than one converse to any given theorem?

Doctor Rick answered:

This theorem is simpler than the others; I think the converse is reasonably clear, so you don't need to belabor it. If you want to do this ... then yes, that is a reasonable separation -- though when we switch the condition and conclusion here, we'd need to change the wording somewhat.

In proving such a theorem, you would give specific names to the entities mentioned in the statement of the theorem. The situation is clearer once we've done that. 

For this theorem, we'd have

    Context: Euclidean geometry; a given triangle ABC.
  Condition: Angles A and B of triangle ABC are equal.
 Conclusion: Sides BC and AC of triangle ABC are equal.

Now, switching the condition and conclusion yields a perfectly understandable converse (which happens to be a valid theorem):

    Context: Euclidean geometry; a given triangle ABC.
  Condition: Sides BC and AC of triangle ABC are equal.
 Conclusion: Angles A and B of triangle ABC are equal.

To your last question, about the multiplicity of converses for any given theorem: yes, that's the point Doctor Peterson and I have been making. In general, we'd only be interested in a converse that can be proved to be a valid theorem.

I suppose it's possible that one theorem could have more than one valid converse theorem, but I'm not going to start hunting for one. It just isn't important to me whether that ever happens or not.