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Topic: Exclusive and Inclusive Definitions for Quadrilaterals (revisited, again)
Replies: 4   Last Post: Jun 10, 2006 1:26 AM

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 Michael Weiss Posts: 5 From: Ann Arbor Registered: 4/5/06
Exclusive and Inclusive Definitions for Quadrilaterals (revisited, again)
Posted: Apr 5, 2006 3:12 PM

Pardon the thread necromancy; I know that this topic has been beaten to death in the past, but it's just barely possible that I have something new to contribute on the subject.

To review the two sides of the argument: In general, conversational usage, people tend to use exclusive language (e.g., when we say "kite" we mean "non-rhomboid kite", when we say "trapezoid" we mean "non-parallelogrammic trapezoid', etc. If you go to a furniture store shopping for a rectangular table they're not going to show you squares.

On the other hand, mathematically-savvy people generally prefer inclusive language, on "inheritance" grounds: any property that can be proven to be true for objects of a general category is automatically inherited by all objects of particular subcategories. So, for example, if we prove that "the diagonals of a kite are congruent", then we automatically know that the diagonals of a rhombus are congruent (because a rhombus is a special kite) and also that the diagonals of a square are congruent (because the square is a special rhombus).

I hereby present an argument for an *exclusive* definition of trapezoid, one that excludes parallelograms, based on inheritance grounds.

Part of the context of my argument is my presumption that definitions are chosen (at least in part) for their instrumental value, insofar as that certain theorems become true or false according to what definitions we choose. Lakatos (1976) documents, for example, how the definition of "polyhedron" evolved in response to various proofs and refutations of Euler's theorem.

So one way of deciding which definition is the better one is to ask: are there certain theorems that we would like to be true, but are false under one definition or the other?

Consider the following theorem:

THM. If ABCD is a trapezoid, with parallel bases AB and CD, then the following conditions are equivalent:
(a) The two legs BC and DA are congruent.
(b) The two base angles A and B are congruent.
(c) The two base angles C and D are congruent.
(d) The two diagonals AC and BD are congruent.
If any of these four conditions are met, then ABCD is called an isosceles trapezoid.

Now, this is a nice theorem; it amalgamates two lesser theorems (the "Base Angles Theorem for Trapezoids" and the "Diagonals of an Isosceles Trapezoid" theorem) into a comprehensive description of the useful properties of isosceles trapezoids. I think most of us would agree that we'd like such a theorem to be true.

But please note that it is only true if you insist on an exclusive definition of trapezoid, because every parallelogram satisfies (a) but will not satisfy (b,c,d) unless it is a rectangle.

The ironic thing is that rectangles do satisfy all four of the above conditions. So ideally we would like to define "trapezoid" in such a way that it includes all rectangles, but excludes all non-rectangular parallelograms. Could such a definition be written?

(Of course, I know you could define trapezoid as "a quadrilateral with at least one pair of parallel sides, that is not a parallelogram, unless the parallelogram is a rectangle" -- but that seems rather inelegant.)