CRYSTAL Alberta

Deductive Logic - a Description

Mathematics has traditionally been viewed as a study of logical systems—a set of axioms and rules for creating concepts or theorems from the axioms—in which truth is transmitted from premises to conclusions. For any particular system, such as geometry, certain fundamental but unproven assumptions (called axioms or postulates) are accepted as absolutely true and rules of logic are used to deduce new concepts or theorems. These new concepts add to the accepted knowledge and can be used to further expand the store of mathematical knowledge in a particular field.

For example, more than two thousand years ago, Euclid used known properties (axioms and proven postulates) of lines, angles and shapes to prove the Pythagorean theorem—in a right-angle triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

This kind of thinking from accepted truths to specific new concepts is one example of reasoning by deduction, called deductive logic. We can translate this deductive logic into a general form, with a simple example, as follows:

* Where p represents the hypothesis and q represents the conclusion.

Notice that the conclusion or product of this reasoning must be no less certain than the starting hypothesis; that is, if the hypothesis is true then the conclusion must be true. In mathematical proofs, the hypotheses are the axioms (assumed to be true) or previously proven concepts or theorems. This means that mathematics can include absolute proofs and claim to discover new truths based on previous or accepted truths.

Proof by Contradiction

The method described above is sometimes referred to as a direct proof. However, mathematicians very often use deductive reasoning in a proof by contradiction (also known as “reductio ad absurdum”—Latin, meaning reduction to the absurd). In this method, we assume the opposite of what we are trying to prove and then derive a contradiction or error. This then shows that the assumption is false and, therefore, the original hypothesis is true.

A well-known example of proof by contradiction is the proof that the square root of two is irrational—cannot be expressed as a fraction a/b where a and b are non-zero integers with no common factor; that is, as a simple fraction. The proof by contradiction is outlined below.

(1) If the square root of 2 is rational (assuming the opposite of the premise), then

(2) This conclusion can be shown to be incorrect

(3) Therefore, the square root of 2 is irrational.

In general symbols, proof by contradiction follows the outline:

* Where p represents the hypothesis and q represents the conclusion.

A proof by contradiction is considered to be as rigourous (logically valid) as a direct proof.

Some Pitfalls of Deductive Logic

• If one or more of the starting premises is not true, then the conclusion cannot be true even if a valid argument is used.
• If there is an error in logic (an invalid argument), then no reliable conclusion can be made regardless of whether the premises are true.

Creating and Testing Hypotheses (text)

Although mathematics has traditionally been defined in terms of deductive logic based on rigorous proofs, many modern philosophers and mathematics educators argue that this is not the only way mathematical knowledge is created. Furthermore, it is not always necessary or even possible to start with first principles (axioms). Rigorous proofs have their place in mathematics; however, some alternate views of mathematics propose several key points:

• initial assumptions or hypotheses may not be known to be true
• hypotheses or conjectures need to be tested by trying to disprove them with counterexamples—examples that shows the hypothesis to be false
• more/less plausible claims can, in some cases, replace the true/false claims required in deductive logic.

Based on these points, problem solving in mathematics is not restricted to rigid rules (as in deductive logic). Hypotheses can be made from informed guesses based on trial-and-error or from a stated mathematical claim. Hypotheses can also be obtained by making deductions (i.e., “If….then….” statements) based on the implications of known mathematical facts.

Regardless of the origin of the hypothesis, predictions are made and tested by trying to find counterexamples that would falsify the hypothesis. This plan is based on the philosophy of Karl Popper who argued that you can never prove a hypothesis to be true because it is not possible to test every example; however, one counterexample can prove the hypothesis to be false.

Counterexamples serve either to completely reject the hypothesis, or to refine it by restricting or revising definitions and conditions. If no counterexamples can be found, the hypothesis becomes plausible (but certainly not proven no matter how many supporting instances are found). A plausible hypothesis may also be broken down into sub-concepts that are subjected to additional testing and possible refinements.

The process of making and testing hypotheses can be “messy”; that is, jumping between testing, evaluating and revising. Some would argue that mathematics does not develop in a linear, logical fashion.

Example:

For any two integers, x and y, how does the sum of the squares compare with the square of the sum?

Hypothesis: The square of a number increases very rapidly as the size of the number increases, therefore the square of the sum is greater than the sum of the squares; i.e., (x + y)2 > x2 + y2.

Testing:

Revised Hypothesis:

If both x and y are non-zero positive or negative integers, then the square of the sum is greater than the sum of the squares;

i.e., (x + y)2 > x2 + y2.

According to the limited testing, it appears that the revised hypothesis is plausible. Other questions may arise from this hypothesis; for example, does it apply to real numbers? to imaginary numbers? to matrices?

This is not the only deductive reasoning possible. Using concepts from algebra, reasoning by deduction may also be possible to prove the revised hypothesis.

Some Pitfalls of Creating and Testing Hypotheses

• inadequate testing: insufficient effort to find counterexamples
• incomplete/inadequate conditions for revised hypotheses

Reasoning by Deduction - Exercise

1. Reasoning by deduction uses a/an __i__ argument that ensures the conclusion must be __ii__ if the premises are __iii__.
2. The choice that best completes the above statement is

1. Complete the conclusion for the following deductive argument.
2. If an integer is an even number, then its square is also even.
3. 4 is an even number.
4. Therefore, ….
5. One of Euclid’s axioms, considered to be absolutely true, is very commonly used and can be expressed in symbols as
6. If A = B and B = C, then ….
7. A simple and logical completion of this statement is ...
8. Deductive logic can also be presented in the form of premises leading to a logical conclusion. For example,
9. Premise 1: All men are mortal.
10. Premise 2: Aristotle is a man.
11. Conclusion: Aristotle is mortal.
12. Complete the conclusion from the following premises.
13. Premise 1: All plane figures enclosed by straight lines are polygons.
14. Premise 2: A square is plane figure with four straight sides in a closed path.
15. If a valid argument is used and the conclusion is shown to be false, then the premise or starting hypothesis must be
16. A. inconclusive
17. B. probable
18. C. false
19. D. true
20. Euclid, a Greek mathematician, is credited with one of the first proofs by contradiction of the premise, “There are infinitely many prime numbers.”
21. What is the opposite premise; that is, the first step in the proof by contradiction?
22. Consider the following claim: if a2 – b2 = 1, then a and b cannot be positive integers. Complete a proof by contradiction of this claim.
23. Counter-examples do not constitute a proof but they are useful in both falsifying and revising hypotheses. Consider the hypothesis: “All numbers that are not positive are negative.” Identify a counterexample and revise the hypothesis if necessary.
24. Hypothesis:
25. 
26. Test this hypothesis and revise if necessary.

Answers

1. B

2. 4² or 16 is an even number

3. A = C

4. Conclusion: A square is a polygon.

5. C

6. There are a finite number of prime numbers.

7. (1) If a2 – b2 = 1, then a and b are positive integers

(2) a2 - b2 = (a - b)(a + b) = 1

either a - b = 1 and a + b = 1

or a - b = -1 and a + b = -1

solving these equations gives either a = 1 and b = 0 or a = -1 and b = 0

Both solutions contradict the premise in (1).

(3) Therefore, the original claim is true.

8. Zero is neither positive nor negative. All nonzero numbers that are not positive are negative.

9.

Revised hypothesis:

Inductive Reasoning Exercise

1. Reasoning by induction
2. A. uses patterns to create logically valid proofs
3. B. claims to use accepted truths to create new truths
4. C. develops a general conclusion based on general concepts
5. D. develops a general conclusion based on specific observations
6. Inductive reasoning does not create
7. A. tentative hypotheses
8. B. absolute truths
9. C. probable answers
10. D. supporting evidence
11. Choose the best general conclusion that follows from this specific observation:
12. Every time I touch snow, it feels cold.
13. A. All snow is cold.
14. B. Some snow is cold.
15. C. My hands may be cold.
16. D. My hands are made of snow.
17. In the thirteenth century, Italian mathematician Fibonacci presented a series of numbers that is found in many aspects of nature; for example, in the arrangement of the seeds on a sunflower head. These seeds are observed in two sets of spiral rows, where one curves to the left and the other curves to the right. Complete the following pattern of seeds observed in a sunflower seed head: 1, 2, 3, 5, 8, , ,
18. A. 8, 5, 3
19. B. 13, 34, 89
20. C. 13, 21, 34
21. D. 40, 320, 12 800
22. (a) Using the number 5 as your starting number, complete the following steps.
23. Choose a number.
24. Multiply it by 3.
25. Add 6.
26. Divide by 3.
27. Subtract 1.
28. What is the new number that you obtained?
29. (b) Choose another number (pick any number that you wish) and repeat the steps from part (a). What is your starting number and the new number that you obtained?
30. (c) Create a generalization that describes how your new number compares with your starting number.
31. (d) Make a hypothesis and then prove it algebraically by repeating the steps in part (a). (Hint: Remember to simplify your algebraic process.)
32. Use the following table of evidence to answer the questions below.

1. (a) Describe a relation between the lengths of the sides of these triangles.
2. (b) Write a generalization, in words and symbols, describing how the squares of the length of each of the sides are related.
3. (c) What is the name of this relationship?

Answers

1. D
2. B
3. A
4. C
5. (a) 6
6. (b) For example, if the starting number chosen was 7, then the new number will be 8.
7. (c) The new number solution is one more than the original number.
8. (d) Hypothesis: If x = the original number, then the new number will be one more than the original number after completing the following series of steps.
9. Chose a number. x
10. Multiply it by 3. 3x
11. Add 6 . 3x + 6
12. Divide by 3. 3x + 6
13. 3
14. Subtract 1. 3x + 6 - 1
15. 3
16. Simplying: 3 (x + 2) -1 = (x + 2) – 1 = x + 1
17. 3
18. (a) If you square side a and add it to the square of side b, then take the square root of this sum, you get the same number as given for side c.
19. (b) The square of side c is equal to the sum of squares of sides a and b.
20. c2 = a2 + b2
21. (c)Pythagorean Theorem, c2 = a2 + b2