John Rosson

Thursday February 15, 2007

Survey of Mathematical IdeasMath 100

Chapter 3, Logic

The Lady and the Tiger

p - Lady in room 1q - Tiger in room 1r - Lady in room 2s - Tiger in room 21 2

ps (pr) (qs)

Given information:

• One sign is true the other false

•The lady and tiger do not share a room

Argument: If the first sign were true, both p and s would have to be true. This would mean that both disjunctions in the second sign would have to be true and so the sign would be true. But this contradicts the first piece of information, so the first sign has to be false. Since they do not share a room the lady has to be in 2 ( r ) and the tiger in 1 (q).

Introduction to Logic

1. Statements and Quantifiers

2. Truth Tables and Equivalent Statements

3. The Conditional

4. More on the Conditional

5. (3.6) Analyzing Arguments using Truth Tables

The Conditional

Conditional

p q p q

T T T

T F F

F T T

F F T

The conditional statement is false only when the first statement, called the antecedent, is true and the second statement, called the consequent, is false.

Recall the truth table for the conditional statement.

A conditional statement is always true when the antecedent is false and always true when the consequent is true.

Calculating Truth Tables

( p q) q

p q( p q) q

T T F T T T F F T

T F F T T F T T F

F T T F T T F F T

F F T F F F T T F

Calculating truth tables involving the conditional is not difficult. All we have to remember is that the conditional is false only when the antecedent is true and the consequent is false.

Calculating Truth Tables

p q p q ~ p q

T T T T T F T T T

T F T F F F T F F

F T F T T T F T T

F F F T F T F T F

Notice that the conditional statement p q is equivalent to the statement ~pq. Since the conditional

can be interpreted as implication, this equivalence can be interpreted as follows. The claim that “p implies q” has the same logical meaning as “either p is false or q is true”.

For example, let p be the statement that “the number n is evenly divisible by 4” and let q be the statement that “ the number n is evenly divisible by 2”. Now, p implies q since any number divisible by 4 is divisible by 2. It is also valid to say that either a number is not divisible by 4 or it is divisible by 2.

This equivalence also means that any statement containing a conditional () may be logically replaced by one using only “not” (~) and “or” ().

Calculating Truth Tables

p q ~ (p q) p ~ q

T T F T T T T F F T

T F T T F F T T T F

F T F F T T F F F T

F F F F T F F F T F

Notice that the the negation of the conditional ~( p q) is equivalent to the statement p ~q.

Since the conditional can be interpreted as implication, this equivalence can be interpreted as follows. The claim that “p does not imply q” has the same logical meaning as “p is true and q is false”.

For example, let p be the statement that “the number n is evenly divisible by 2” and let q be the statement that “ the number n is evenly divisible by 4”. Now, p does not imply q since the number 6 is divisible by 2 ( so p is true) and 6 is not divisible by 4 (so q is false).

Calculating Truth Tables

((p q) p) q

p q ((p q) p) q

T T T T T T T T T

T F T F F F T T F

F T F T T F F T T

F F F T F F F T F

We also get the following tautology. This says that: if p implies q and p is true then q has to be true also.

mortal. is Socrates

man. a is Socrates

mortal. aremen All Assumptions.

Conclusion.

Rule of logic. Modus ponens

But where do the rules of logic come from?

Recall:

The argument would go like this: “All men are mortal” can be express as “For all x, if x is a man then x is mortal”. Specializing (another logical rule) x to Socrates, we have “If Socrates is a man then Socrates is mortal”. The first line becomes

“Socrates is a man” “Socrates is mortal” . So this tautology is the basis of the logical rule modus ponens.

Most of mathematics and much of Artificial Intelligence (AI) is founded on this tautology.

Conditional

The conditional statement is the basic form of deductive reasoning. It has a direction, from antecedent to consequent. Since it is so important, the conditional has many synonyms.

Synonyms for pq.

If p, then q. p is a sufficient condition for q.

If p, q. q is a necessary condition for p.

p implies q. All p’s are q’s.

p only if q. q if p.

Relative FormsDirect

p q p q

T T T

T F F

F T T

F F T

Converse

p q q p

T T T

T F T

F T F

F F T

Inverse

p q ~p ~q

T T T

T F T

F T F

F F T

Contrapositive

p q ~q ~p

T T T

T F F

F T T

F F T

Note that the direct and contrapositive statements are equivalent as are the converse and the inverse.

Note that this is not the negation of the direct statement.

Relative Forms

Let p be the statement “you build it” and let q be the statement “they will come”.

Direct Statement: p q, “If you build it, then they will come.”

Converse : q p, “If they do come, then you did build it.”

Inverse : ~p ~q, “If you do not build it, then they will not come.”

Contrapositive: ~q ~p, “If they do not come, then you did not build it.”

BiconditionalConsider the truth table for the conjunction of a conditional with its converse.

p q (p q) (q p)

T T T T T

T F F F T

F T T F F

F F T T T

This statement claims that it is true both that p implies q and conversely that q implies p.

This relationship between p and q is important enough to get its own symbol called the biconditional (conditional in both directions).

Biconditional

p q p q

T T T

T F F

F T F

F F T

In words, p q means “p is true if and only if q is true.” The statement p q is true precisely when p and q have the same truth values. If p and q are equivalent statements then p q is a tautology.

Biconditional

p q ~ (p q) (p ~ q)

T T F T T F F

T F T F T T T

F T F T T F F

F F F T T F T

Consider the following example.

This statement is a tautology because the two terms of the biconditional are equivalent (have the same true table).

It is true that “a natural number is even if and only if it is divisible by 2.” It is false that “a natural number is even if and only if it is divisible by 4.”

Assignments 3.5, 3.6, 4.1, 4.2, 4.3

Read Section 3.5, 3.6 Due February 20

Exercises p. 145 1-23 odd, 27,29,47

Test 1 over Chapters 1, 2, 3

Thursday, February 22

Read 4.1, 4.2, 4.3 Due February 27

Exercises p. 176 1-4, 5,11,19, 23, 35, 47