What is the fallacy of defining a square as “a closed-plane figure whose sides are all equal”?

I am determined to prove my professor wrong. Here is a question from a recent exam:

Using the six definitional criteria, evaluate the following definition.

A square is a closed-plane figure whose sides are all equal.

1. obscure
2. circular
3. too narrow
4. unsuitable attribute

I do not believe that the right answer is offered as an option here.

I would argue that the fallacy committed in this definition is neither:

a.) obscure; the language is clear and not overly technical

b.) circular; at no point is the term used in the definition

c.) too narrow; the defined parameters do not exclude any squares

nor

d.) unsuitable attribute; everything here refers to geometry and is suitable in describing a square

I believe that the definitional fallacy here is: "too wide". There are myriad polygons that could fit this definition besides the square. It should, in fact, be made narrower by including the parameter of having four sides or being a quadrilateral.

This was his response to my challenge of the question:

For the reason you say below the definition is too narrow by excluding polygons. 82% of the students put too narrow. I suggest focusing on exam 4-b and getting a higher square [sic] making your concerns moot. Otherwise, make an appointment to see me. You are doing well in the course, Professor X

First of all, the definition DOES NOT exclude polygons such as hexagons and equilateral triangles. Second, his statistic about other students' selections is not strong evidence and basically irrelevant, especially if the correct answer was unavailable. I really think that I am right on this one. Can anyone support my argument?

Here is wikipedia's summary of fallacies of definition for help.

Thanks,

Oscar

• In the same way as the 82% is irrelevant, any 'popular support' you're going to get here is irrelevant as well. Instead, make an appointment to see the professor as he suggested. In general, teachers are very happy to help a student who is keen to learn, and personal contact helps them understand the issues students are having. So, if you take on the attitude I may be wrong (even though you don't think so), you're going to have a more fruitful conversation.
– user2953
Nov 7, 2016 at 22:44
• Thanks for the life advice. Also looking for direct answers, though ;) The popular support is for my own peace of mind. I am more interested in affirmations or arguments against my opinion that seem logical, since my teacher's argument does not make sense to me. As I said, his claim that the definition excludes polygons is false. Nov 7, 2016 at 23:14
• Hi. FWIW, your criricism seems to me correct. And even more: the given definition does not state that the "sides" are straight. They could be e.g. circular arcs. So, not only polygons are let in. Definitely too wide... ask your teacher for a reasoned answer. Nov 7, 2016 at 23:26
• I'm with you. The def is too wide, since it includes pentagons and such. Then again I'm not at the mercy of your professor :-) Nov 7, 2016 at 23:31
• I agree with you. The answer is 'too wide'. The professor's answer does not make sense. The definition is 'closed-plane figure whose sides are all equal'. This covers far MORE polygons than squares, so it's too wide, not too narrow. I recommend telling him, the definition covers more polygons than a square, and therefore is too wide... Maybe give him an example of a narrow definition... say 'a rectangle is a quadrilateral with 4 equal sides and 4 equal angles', so that he sees the contrast. 'Quadrilateral with 4 equal sides and 4 equal angles' covers less than all the rectangles. Nov 7, 2016 at 23:48

I'm not sure what constitutes an answer to this question, exactly, but as a mathematician my professional opinion is that the given definition of a square is too wide. This is essentially a definition of an equilateral polygon, a very broad class of shapes which includes squares as well as rhombuses (diamond-shaped quadrilaterals), regular polygons, and many other types of polygons.

As Ram Tobolski points out, the definition also refers to the "sides" of a figure, which really only makes sense if the figure is a polygon. So of the four answer choices provided, I think "unsuitable attribute" is most correct, since a general plane figure does not have the attribute of "sides".

• I hadn't read the definition properly, and it clearly includes the rhombus - so even if it had said four sides and four straight sides it would be too wide. Nov 11, 2016 at 14:42
• Ah, indeed. So, it should even include "having four interior angles measuring 90 degrees." Nov 11, 2016 at 16:46

I'll try to demonstrate visually my point from the comments. I not only agree with the criticism of the definition by the OP, but I claim that the given definition is even wider, in that it covers not just perfect polygons.

Because the given definition

[A square is] a closed-plane figure whose sides are all equal.

applies to the following figure, which is not a polygon.

This figure is a closed-plane figure with four sides, all equal (trust me, they are equal :)). And it is not a polygon.

I am sorry I have to side with your teacher, but the correct answer is 3.
This conclusion is obtained by understanding the definition of "too narrow." And that is - if a definition excludes other possible members, then the definition is too narrow.
You (correctly) argue that the definition excludes other members (polygons, etc.). Therefore you should have selected the 3rd answer - the definition is too narrow.

• You seem to be confused about what it means for a definition to include or exclude something. Something is "included" in a definition if the thing satisfies the definition, and "exluded" if it does not satisfy the definition. The definition given by the OP includes many polygons that aren't squares, since these polygons satisfy the definition. These polygons shouldn't have been included in the definition, so the definition is too wide. Nov 16, 2016 at 1:34