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P l a n e G e o m e t r y
An Adventure in Language and Logic
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INTRODUCTION TO LOGIC
Hypothesis and conclusion:
Necessary and sufficient
The relationship between truth and validity
LOGIC, WE COULD SAY, is the study of If-then sentences.
If a number is divisible by 10, then it is divisible by 2.
The clause introduced by If --
A number is divisible by 10
-- is called the hypothesis. It is what we are given, or what we may assume.
The clause introduced by then --
It is divisible by 2
-- 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; or necessary for that hypothesis to happen or be true.
Because if a number is divisible by 10, it necessarily follows that it will be divisible by 2. (Equivalently, it is necessary for a number to be divisible by 2, if the number is to be divisible by 10. See further in the contrapositive form.)
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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Do the problem yourself first!
The hypothesis is "Two numbers are even." It is the sufficient condition.
The conclusion is "Their sum is even." It is the necessary condition.
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?
The hypothesis is "A number is divisible by 6." It is that clause that follows "if." It is the sufficient condition.
The conclusion is "The number is divisible by 3." It is 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?
Yes, because the statement is true.
Problem 4. If a number is a multiple of 5, then it ends in 0.
Is the hypothesis a sufficient condition for that conclusion?
No, because the statement is false. 15 is a multiple of 5, but 15 does not end in 0.
Problem 5. If a number is prime, then it is odd.
Is the conclusion a necessary condition of that hypothesis?
No, because the statement is false. 2 is a prime, and it is not odd.
Problem 6. If two numbers are even, then their product is even.
Is the conclusion a necessary condition of that hypothesis?
Yes, because the statement is true.
Problem 7. Express each of these as an If-then sentence.
a) a is a sufficient condition for b.
If a, then b.
b) a is a necessary condition of b.
If b, then a.
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.
If a triangle is isosceles, then the angles at the base are equal.
The hypothesis is A triangle is isosceles. The conclusion is The angles at the base are equal.
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.
If a number is a multiple of 5, then it ends in 5.
False. 20, for example, is a multiple of 5.
b) If a number is a multiple of 10, then it ends in 0.
If a number ends in 0, then it is a multiple of 10.
True. All numbers that end in 0 are multiples of 10.
Problem 10. State the converse of All right angles are equal.
All equal angles are right angles.
Which is false.
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"
When 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"?
If a number is a multiple of 10, then it ends in 0;
and conversely, if a number ends in 0, then it is a multiple of 10.
Problem 12. What does it mean to say, "p is both necessary and sufficient for q."
If p, then q is true; and conversely, If q, then p is true. That is, p if and only if 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.
True. Because if a number is divisible by 6, then it is divisible by both 2 and 3. And conversely.
b) A number is a multiple of 9 if and only if it is a multiple of 3.
False. It is true that if a number is a multiple of 9, then it is a multiple of 3. (All multiples of 9 are multiples of 3.)
But the converse is false.
c) A fraction is in lowest terms if and only if the numerator and
c) denominator have no common divisors except 1.
True. Because if a fraction is in lowest terms, then the numerator and denominator have no common divisors except 1. And conversely.
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
is
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.
"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."
That sentence 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.
If a number is not even, then it does not end in 6.
b) If two lines are parallel, then they do not meet.
If two lines meet, then they are not parallel.
c) If q, then not-p.
If p, then not-q.
*
The contrapositive illustrates the meaning of a necessary condition. Again, it is necessary for a number to be divisible by 2, if the number is to be divisible by 10. That is,
If a number is not divisible by 2, then it is not divisible by 10.
Equivalently,
If a number is divisible by 10, then it is divisible by 2.
In that form, you see explicitly that the necessary condition is introduced by then.
Problem 15. A necessary condition for voting is that you be 18.
a) Express that as a contrapositive.
If you are not 18, then you may not vote.
a) Express that in simple If-then form.
If you may vote, then you're 18.
The remaining variation of an If-then sentence is called the inverse. The inverse of
If a, then b
is
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 16. If a triangle is isosceles, then the base angles are equal.
a) State the inverse.
If a triangle is not isosceles, then the base angles are not equal.
b) State the converse.
If the base angles of a triangle are equal, then the triangle is isosceles.
c) State the contrapositive.
If the base angles of a triangle are not equal, then the triangle is not isosceles.
d) Of the three statements a), b), c), which ones mean the same, that is,
d) which are logically equivalent?
a) and b). Sentence c) is equivalent to the original.
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 17. 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 18. 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).
The relationship between truth and validity
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 contained within, or 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 19. 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 true and valid, and between true and logical. Being valid or logical depends only on the form. We can judge truth only on the meaning.
Here is another example of this:
c) All lies are true.
d) 1 + 1 = 3 is a lie.
d) Therefore, 1 + 1 = 3 is true.
d) All multiples of 5 are even.
c) 8 is a multiple of 5.
d) 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, or even the meaningfullness, of its hypothesis. This is fundamental to all argument, all explanations founded on logic, and especially to those theories of what is real, called science.
In science, the conclusions are what is actually observed. The hypotheses are the theoretical or mathematical explanations for them. It is meaningless, then, to speak of a scientific theory as being "proved." We do not "prove" a hypothesis. We adopt a hypothesis, which typically is not observable, in order to explain what we can observe. Observations continue to test a hypothesis and are evidence for its truth. A hypothesis will be proved false, however, when it leads to a false conclusion. See Problem 20c, below.
e) All multiples of 10 are even.
e) 80 is even.
e Therefore, No valid conclusion. This is not the form of a syllogism.
| f) | All pretty girls are numbers. A point is a pretty girl. Therefore, A point is a number. |
| This is verbal nonsense. Nevertheless, that formal structure is a valid argument. | |
| This example illustrates that words -- girls, numbers, point -- that have a customary meaning, in logic need not have that meaning, or any meaning for that matter (unless they are defined to have one). They are nothing but the formal elements of the argument, playing the same roles as A, B and x. | |
Problem 20. 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) 19b) and d).
c) In a valid argument, if the conclusion is false, then the hypothesis
b) must be false. Compare Problems 19b) and c).
c) It is this principle that leads to a method called proof by
c) contradiction. We assume that what we want to prove is false —
b) 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.
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