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Improvement of Reasoning
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14.1 Definitions
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Was Lincoln an educated man is a question that cannot be properly answered unless we agree upon the meaning of an educated man. Understanding cannot exist and progress cannot be made in any discussion or problem unless the terms involved are properly defined or, by agreement, are to be undefined.
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14.1A Requirements of a Good Definition
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PRINCIPLE
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All terms in a definition must have been previously defined (or be those that, by agreement, are left undefined).
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Thus, if we are to define a regular polygon as an equilateral and equiangular polygon, it is necessary that equilateral, equiangular, and polygon be previously defined.
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The term being defined should be placed in the next larger set or class to which it belongs.
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Thus, the terms polygon, quadrilateral, parallelogram, and rectangle should be defined in that order. Once the term polygon has been defined, the term quadrilateral is then defined as a kind of polygon. Then the term parallelogram is defined as a kind of quadrilateral and, lastly, the term rectangle is defined as a kind of parallelogram. Proper sequence in definition can be understood by using a circle to represent a set of objects. In Fig. 14-1, the set of rectangles is in the next larger set of parallelograms. In turn, the set of parallelograms is in the next larger set of quadrilaterals, and, finally, the set of quadrilaterals is in the next larger set of polygons.
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PRINCIPLE
The term being defined should be distinguished from all other members of its class.
Fig. 14-1
Thus, the definition of a triangle as a polygon with three sides is a good one since it shows how the triangle differs from all other polygons.
PRINCIPLE
The distinguishing characteristics of a defined term should be as few as possible.
Thus, a right triangle should be defined as a triangle having a right angle, and not as a triangle having a right angle and two acute angles.
SOLVED PROBLEMS
Observing proper sequence in definition In which order should the terms in each of the following sets be defined: (a) Englishman, European, Londoner; (b) quadrilateral, square, rectangle, parallelogram.
CHAPTER 14 Improvement of Reasoning
Solutions
(a) European, Englishman, Londoner (b) Quadrilateral, parallelogram, rectangle, square
14.2 Correcting faulty definitions Correct the following definition: A trapezoid is a quadrilateral having two parallel sides.
Solution
The given definition is incomplete. The correct definition is a trapezoid is a quadrilateral having only two parallel sides. This definition distinguishes a trapezoid from a parallelogram.
14.2 Deductive Reasoning in Geometry
The kinds of terms and statements discussed in this section comprise the deductive structure of geometry, which can be visualized as in Fig. 14-2.
Fig. 14-2
14.2A Undefined and Defined Terms
Point, line, and surface are the terms in geometry which are, by agreement, not defined. These undefined terms begin the process of definition in geometry and underlie the definitions of all other geometric terms. Thus, we can define a triangle in terms of a polygon, a polygon in terms of a geometric figure, and a geometric figure as a figure composed of line segments, or parts of lines. However, the process of definition cannot be continued further because the term line is undefined.
14.2B Assumptions
Postulates and axioms are the statements which are not proved in geometry. They are called assumptions because we willingly accept them as true. These assumptions enable us to begin the process of proof in the same way that undefined terms enable us to begin the process of definition. Thus, when we draw a line segment between two points, we justify this by using as a reason the postulate two points determine one and only one straight line. This reason is an assumption since we assume it to be true without requiring further justification.
14.2C Theorems
Theorems are the statements which are proved in geometry. By using definitions and assumptions as reasons, we deduce or prove the basic theorems. As we use each new theorem to prove still more theorems, the process of deduction grows. However, if a new theorem is used to prove a previous one, the logical sequence is violated. For example, the theorem the sum of the measures of the angles of a triangle equals 180 is used to prove that the sum of the measures of the angles of a pentagon is 540 . This, in turn, enables us to prove that each angle of a regular pentagon measures 108 . However, it would be violating logical sequence if we tried to use the last theorem to prove either of the first two.
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