-- is called the conclusion. It is the statement that "follows" from the hypothesis. Or, given the hypothesis, it is the statement that we must prove.

When the If-then sentence is true, we say that the hypothesis is a sufficient condition for the conclusion. Thus it is sufficient to know that a number is divisible by 10 -- in order to conclude that it is divisible by 2.

The conclusion is then called a necessary condition of that hypothesis. For, if a number is divisible by 10, it necessarily follows that it will be divisible by 2.

Problem 1. If two numbers are even, then their sum is even.

Name the hypothesis and the conclusion. Which of them is the sufficient condition, and which the necessary condition?

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Note that the hypothesis is a sentence, independent of "If." And the conclusion is a sentence, independent of "then."

Problem 2. A number is divisible by 3 if it is divisible by 6.

Name the hypothesis and the conclusion. Which of them is the sufficient condition, and which the necessary condition?

Problem 3. If a number ends in 0, then it is a multiple of 5.

Is the hypothesis a sufficient condition for that conclusion?

Problem 4. If a number is a multiple of 5, then it ends in 0.

Is the hypothesis a sufficient condition for that conclusion?

Problem 5. If a number is prime, then it is odd.

Is the conclusion a necessary condition of that hypothesis?

Problem 6. If two numbers are even, then their product is even.

Is the conclusion a necessary condition of that hypothesis?

Problem 7. Express each of these as an If-then sentence.

a) a is a sufficient condition for b.

b) a is a necessary condition of b.

Other forms

The hypothesis and conclusion will not always appear in If-then form. For example,

All right angles are equal.

This is equivalent to saying

If these angles are right angles, then they are equal.

Therefore, the hypothesis of All right angles are equal is

These angles are right angles.

And the conclusion is

These angles are equal.

Problem 8. Express the following in If-then form, and so discover its hypothesis and conclusion.

In an isosceles triangle the angles at the base are equal.

The converse

To form the converse of an If-then sentence, exchange the hypothesis and conclusion. The converse of

If p, then q,

where p and q are sentences, is

If q, then p.

Clearly, if an If-then sentence is true, its converse is not necessarily true.

Problem 9. State the converse of each statement, and then decide whether the converse is true. (Note that each statement is true.)

a) If a number ends in 5, then it is a multiple of 5.

b) If a number is a multiple of 10, then it ends in 0.

Problem 10. State the converse of All right angles are equal.

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

"if and only if"

If a statement If a, then b and its converse If b, then a are both true, we say "a if and only if b."

In other words, a is both necessary and sufficient for b.

For example,

A triangle is isosceles if and only if the base angles are equal.

This means

If a triangle is isosceles, then the base angles are equal

and conversely,

If the base angles are equal, then the triangle is isosceles.

Problem 11. What does it mean to say, "A number is a multiple of 10 if and only if it ends in 0"?

Problem 12. What does it mean to say, "p is both necessary and sufficient for q."

Problem 13. Which of these if and only if statements is true. Explain.

a) A number is divisible by 6 if and only if it is divisible by both 2 and 3.

b) A number is a multiple of 9 if and only if it is a multiple of 3.

c) A fraction is in lowest terms if and only if the numerator and
c) denominator have no common divisors except 1.

The contradiction

If a is a statement, then its contradiction (or its negation) is a statement that is equivalent to saying, "It is not true that a." We symbolize the contradiction as Not-a.

For example, the contradiction of

These lines are parallel

These lines are not parallel.

Now, a statement must be either true or false. (That is called the law of the excluded middle. We may even take it to be the definition of a "statement.") And according to what is called the law of non-contradiction: A statement and its contradiction cannot both be true. One of them must be true and the other, false.

The contrapositive

"If you're not in Kansas, then you're not in Salina."

According to that statement, where is Salina?

It must be in Kansas. Because if you're anywhere in the shaded area
-- Not in Kansas -- then according to the statement, you're not in Salina.

In other words, "If you're in Salina, then you're in Kansas." This has the form

If a, then b.

The statement "If you're not in Kansas, then you're not in Salina" has the form

If not-b, then not-a.

We call it the contrapositive of "If a, then b." The hypothesis and conclusion are exchanged and contradicted.

We say that a statement and its contrapositive are "logically equivalent." That is a technical way of saying that they mean the same. They will either both be true or both be false.

Problem 14. State the contrapositive.

a) If a number ends in 6, then it's even.

b) If two lines are parallel, then they do not meet.

c) If q, then not-p.

The inverse

The remaining variation of an If-then sentence is called the inverse. The inverse of

If a, then b

If not-a, then not-b.

We contradict both the hypothesis and the conclusion.

Now the inverse means the same as the converse --

If b, then a

-- because the inverse is the contrapositive of the converse.

Problem 15. If a triangle is isosceles, then the base angles are equal.

a) State the inverse.

b) State the converse.

c) State the contrapositive.

d) Of the three statements a), b), c), which ones mean the same, that is,
d) which are logically equivalent?

The student should realize that to construct the variations of an
If-then sentence, it is not necessary to know what the If-then sentence means! The variations can be constructed purely formally.

Problem 16. State the contrapositive.

a) If p, then q. If not-q, then not-p.

b) If not-q, then not-p. If p, then q.

c) If not-p, then not-q. If q, then p.

d) If q, then p. If not-p, then not-q.

Problem 17. In the previous problem, with respect to sentence a),

What is sentence d) called? The converse.

What is sentence c) called? The inverse.

What is sentence b) called? The contrapositive.

Which of those four sentences mean the same? a) and b). c) and d).

Valid arguments

The classic example of what is called a valid argument is the syllogism:

1. All men are mortal.

2. Socrates is man.

3. Therefore, Socrates is mortal.

Statements 1 and 2 are the hypothesis. Statement 3 is the conclusion. Equivalently,

If all men are mortal, and Socrates is a man,

then Socrates is mortal.

What characterizes this, or any, valid argument is the following:

If the hypothesis is true, then the conclusion must be true.

For if the class of things called "Men" are all members of the class called "Mortals," and Socrates is a member of "Men," then Socrates necessarily must be a member of "Mortals."

The essence of this argument is its form, which is as follows:

1. All A's are B's.

2. x is an A.

3. Therefore, x is a B.

Problem 18. Try to draw a valid conclusion from the following hypotheses. Keep in mind that you may assume only what is stated in the hypothesis.

a) All butchers wear straw hats.

b) My uncle wears a straw hat.

b) Therefore, No valid conclusion. This is not the form of a syllogism. If the second statement were, "My uncle is a butcher," then we could conclude, "My uncle wears a straw hat."
Look at the form above. x must belong to the first named class, not the second.

b) All multiples of 10 are odd.

d) 80 is a multiple of 10.

d) Therefore, 80 is odd.

We see an important point here. This conclusion is valid. But it is not true. Thus we must distinguish between truth and validity. Validity depends only on the form. We can judge truth only on the meaning.

c) All multiples of 5 are even.

c) 8 is a multiple of 5.

c) Therefore, 8 is even.

We see here another point, and an extremely important one:

A false hypothesis can lead to a true conclusion.

In other words, the truth of a conclusion does not guarantee the truth of its hypothesis. This is fundamental to all argument, and especially to that philosophy called science.

d) All multiples of 10 are even.

e) 80 is even.

c) Therefore, No valid conclusion. This is not the form of a syllogism.

Problem 19. Complete the following with either "must be true,"
"must be false," or "may be true or false."

a) In a valid argument, if the hypothesis is true, then the conclusion
a) must be true. This in fact is all that logic can tell us about truth.

b) In a valid argument, if the hypothesis is false, then the conclusion
b) may be true or false. If at least one statement of the hypothesis is
b) false, then we consider the hypothesis to be false. Compare Problems
b) 18b) and c).

c) In a valid argument, if the conclusion is false, then the hypothesis
b) must be false. Compare Problem 18b).

c) It is this principle, in fact, that leads to a method called proof by
c) contradiction. We take as hypothesis that what we want to prove is
b) false — statement a is false. We then show how that hypothesis leads
c) to a false conclusion -- that 80 for example is an odd number. The
c) hypothesis therefore must be false. Statement a is in fact true.

We are now ready to state the first principles of geometry.

First Principles

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