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TOPIC 8
AN INTRODUCTION TO SYMBOLIC LOGIC
One of the major topics for the study of geometry in high school is to do some deductive
reasoning. This requires us to know about true statements that concern the attributes of
geometric shapes and their relationships with each other. There are statements that are TRUE
and others which are FALSE. We will learn how to verify that some statements are true, and we
will be able to make successful arguments about their truth.
Assume that these statements are true.
Statement #1: Shape A is a NIM. Shape C is not a NIM. If a shape is not a NIM, then it is a
ROF. Therefore, shape C is a ROF.
Is the last sentence a true conclusion?
Statement #2: Sally is going to study if it does not rain. People play tennis when the weather is
good. Sally is studying. There the weather is not good.
Is the last sentence a true conclusion?
Statement #3: If lines are parallel, then they do not intersect. If lines do not intersect, then they
do not form angles. If lines do not form angles, then there is no need to have a protractor.
Therefore, if lines are parallel, then there is no need for a protractor.
Is the last sentence a true conclusion?
Statement #4: All zebras are mammals. If an animal is a zebra, then it has black and white
stripes. Some flags and zebras have black and white stripes. Therefore, some flags are
mammals.
Is the last sentence a true conclusion?
The statements above involve the use of the logical connectives NOT, AND, OR, IF…THEN,
and (eventually) IFF (“if and only if”). The goal now is to learn how these logical connectives
work with TRUE and FALSE values.
TOPIC 8: Introduction to Symbolic Logic
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Types of Logical Statements
Examples
Statements
Negation of a Statement
Conjunction of two
Statements
Disjunction of two
Statements
Conditional Statement
Biconditional Statement
It is raining.
The sun is not shining.
Symbolic Form
TOPIC 8: Introduction to Symbolic Logic
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Suppose that the letters P, Q, and R stands for a sentence which could be either true or false. So
P might represent “Today is Tuesday.” Maybe Q stands for “There is pizza for lunch.” And R
is the symbol for “I will eat outside.”
Of course, P, Q, and R could stand for geometric sentences, too. Perhaps P is “Lines are
parallel” with Q being “Alternate interior angles are congruent” and R is “a pair of
supplementary angles are formed.”
There are FIVE LOGICAL CONNECTIVES: NOT, AND, OR, IF…THEN, and IF and only IF.
The symbols in logic are as follows:
P
Q
Symbols:
not P
 P
P and Q
P Q
P or Q
P Q
If P, then Q
PQ
P iff Q
P Q
Translate each conditional sentence from English into symbolic form. Use the letter
substitutions below.
P: Kermit is green with envy.
Q: Miss Piggy squealed.
R: Gonzo came from outer space.
S: The acrobats juggle flying fish.
Translate the symbols into an English sentence.
1.
2.
PR
If Kermit is green with envy, then Gonzo came from outer space.
S ~ Q
If the acrobats juggle flying fish, then Miss Piggy does not squeal.
3.
SR
4.
~ P  ~ S If Kermit is not green with envy, then the acrobats do not juggle flying fish.
5.
6.
The acrobats juggle flying fish and Gonzo came from outer space.
~  S  R  It is not the case that the acrobats juggle flying fish or Gonzo came from
outer space.
P  R Kermit is green with envy iff Gonzo came from outer space.
TOPIC 8: Introduction to Symbolic Logic
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Determining if a statement is True or False
Referring to the statement in part II, let’s assume that statements P and S are TRUE and
statements R and Q are FALSE.
Write each statement in symbolic form and determine whether it is true or false.
7.
Kermit is not green with envy. (not P)
8.
Miss Piggy did not squeal. (not Q)
9.
Kermit is green with envy and the acrobats juggle flying fish. (P and S)
10. Gonzo did not come from outer space and Kermit is not green with envy. (not R and not P)
11. Miss Piggy squealed and the acrobats juggled flying fish. (Q and S)
12. Miss Piggy squealed or Kermit is green with envy. (Q or P)
13. Gonzo did not come from outer space or the acrobats juggle flying fish. (not R or S)
14. Miss Piggy squealed or Gonzo came from outer space. (Q or R)
15. If Kermit is green with envy, then the acrobats are juggling flying fish. (PS)
16. If Miss Piggy squealed, then Gonzo came from outer space. (Q  R)
17. If the acrobats are not juggling flying fish, then Kermit is not green with envy.
(not S not P)
18. If Gonzo did not come from outer space, then Kermit is not green with envy. (not R  not
P)
TOPIC 8: Introduction to Symbolic Logic
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Translate each of these English statements into logical language, using the symbols for NOT,
AND OR, and IF THEN. Do not worry about the truth of the statements here.
19.
P represents “it is snowing” and Q represents “school is closed. What does
P   Q state?
20.
If P represents “x is an odd number” and Q represents “x2 is an odd number, what does
 P  Q represent?
21.
If P represents “I am a sophomore” and Q represents “I take geometry.” Let R represent
“Classes meet for 60 minutes.” Write the following in symbolic logical notation: “If I
am a sophomore, then I take geometry, or if I am not a sophomore and my classes do not
meet for 60 minutes, then I am not taking geometry.”
We will build a truth table for each of the connectives.
P
Q
not P P and Q
P or Q
If P, then Q P iff Q
 P
P Q
P Q
PQ
P Q
Symbols:
______________________________________________________________
T
T
F
F
T
F
T
F
F
F
T
T
T
F
F
F
T
T
T
F
T
F
T
T
T
F
F
T
We obviously need some examples to see that these truth tables are valid, but we should accept
that these are the postulates (“true without needing proof”) of symbolic logic. We should be
convinced that this foundation of the more advanced statements can be trusted.
TOPIC 8: Introduction to Symbolic Logic
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Not P
P
Consider this Venn diagram for the
negation of P.
Use this Venn
diagram for:
P
Q
P and Q
P or Q
Q
Test the truth table for
If P, then Q
(which can be written P  Q)
and
If P, then Q and if Q, then P
(which is P if and only if Q)
P
TOPIC 8: Introduction to Symbolic Logic
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Here are some examples of TRUE and FALSE statements with these connectives. See if you are
convinced about the truth of the logical connectives:
22.
(Tuesday is the day after Monday) AND (Bryn Mawr sophomores must be girls.)
23.
(
24.
(I am eating lunch at 11:30 AM) OR (I am doing homework at 11:30 AM)
25.
If 2x = 12, then x = 6
26.
If 4 > 1, then 4 + 2 > 1 + 2
27.
If Wednesday is the name of a month, then triangles have three sides.
28.
It is not the case that Mr. Stephens is not a science teacher.
29.
ABC if and only if AB + BC = AC
a2
> 2 for all “a”) AND ( 4 + 5 = 9)
a2  1
30. 4 > 1 if and only if 4 + 2 > 1 + 2
These are the basic connectives of symbolic logic. From them, we can make combinations to
represent more complicated logical statements. We want to be able to make a truth table for the
complex statements to know what the logical outcomes are. Eventually, we are most interested
in those statements that are TRUE.
To build the truth table for (P  Q)  (P  Q), you work with each basic connective
separately, combining them until the entire statement is involved.
P
Q
T
T
F
F
T
F
T
F
Do:
P Q
PQ
(P  Q)  (P  Q)
first
second
third
TOPIC 8: Introduction to Symbolic Logic
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The truth values in the column marked “third” is done by using the OR connective with columns
marked “first” and “second”.
Exercises: Build truth tables for these.
31.
~P→~Q
32.
P→(P∨Q)
33.
~(P∨Q)
34.
~P∧~Q
35.
(P∧Q)→R
36.
~(P∧Q)↔(~P ⋁~Q)
37.
(P  Q)   P
TOPIC 8: Introduction to Symbolic Logic
38.
 (P  Q)  (P   Q)
39.
[  Q  (P  Q) ]   P
40.
Q  ( P  Q)  P
41.
(  P  Q) [ (P  Q)  Q ]
page 9
TOPIC 8: Introduction to Symbolic Logic
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Two statements are logically equivalent if they have the same truth values (that is, they have
identical truth tables.)
42.
If P represents “I eat pizza” and Q represents “I am hungry.” Which of the following
statements is logically equivalent to  (P  Q)? (Use the truth table you just wrote
above.)
I do not eat pizza and I am hungry.
If I do not eat pizza, then I am not hungry.
I do not eat pizza and I am not hungry.
I eat pizza and I am not hungry.
A compound statement that is always true is called a tautology.
Write out the truth tables for each of the next logical statements to determine if any of them are
tautologies.
Example:
Look at the truth table of (p  q)p
p
q
p  q (p  q)p
_________________________________
T
T
F
F
T
F
T
F
T
F
F
F
T
T
T
T
Since the last column indicates that (p  q)p is always true, then this logical statement is called
a tautology.
TOPIC 8: Introduction to Symbolic Logic
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Exercises.
43.
44.
a)
Write out the truth table for  (pq)  (p   q)
b)
Is this statement a tautology?
c)
Let P represent “I have a ticket” and Q represent “I see the show”. Which
sentence is logically equivalent to  (pq) ?
a)
Write out the truth table for [p  (p q)] q
b)
Is this statement a tautology?
TOPIC 8: Introduction to Symbolic Logic
45.
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a)
Write out the truth table for (  P Q)  (P  Q )
b)
Is this statement a tautology?
c)
In (  P q)  (P  Q), let P represent “we do not pollute the water” and Q
represent “The fish will die”. Which statement is logically equivalent to “If we
pollute the water, the fish will die”?
Let’s look particularly at statements which are called
conditionals.
A conditional has two parts: The hypothesis is the part which follows the word IF, and
the conclusion is the part that follows the word THEN. In P  Q, P is the hypothesis and Q is
the conclusion.
We spend a lot of time looking at conditionals because all of our definitions, postulates,
and theorems can be written in the form of a conditional (or variations of conditionals).
Examples: Identify the hypothesis and conclusion.
46.
If an object is a rose, then it is a flower.
47.
If it snows, then the game will be canceled.
Disguised conditionals
Rewrite each of these in if-then form.
48.
All Eskimos like pie.
49.
My eyes get tired when I read without my glasses.
TOPIC 8: Introduction to Symbolic Logic
50.
page 13
Elmo giggles only if he is tickled.
Variations of a conditional
PQ
Converse of the conditional is:
Inverse of the conditional is:
Contrapositive of the conditional is :
51.
Write the three variations of this conditional:
If it is a rose, then it is a flower.
Converse:
Inverse:
Contrapositive:
How do the variations compare with the original conditional? That is, which are EQUIVALENT
to the original statement? Complete this truth table to find out.
52.
ORIGINAL CONVERSE INVERSE
P
~P
Q
T
T
F
F
T
F
T
F
Conclusion:
PQ
~Q
F
F
T
T
F
T
F
T
T
F
T
T
QP
T
T
F
T
~ P~Q
T
T
F
T
CONTAPOSITIVE
~ Q ~ P
T
F
T
T
TOPIC 8: Introduction to Symbolic Logic
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Biconditional Statements
Example: Two lines are perpendicular if and only if they intersect to form right angles.
Symbols:
Truth Value:
Every biconditional means two conditionals:
(1) If two lines are perpendicular, then they intersect to form right angles
(2) If two lines intersect to form right angles, then they are perpendicular.
How are biconditionals used?
Write the two statements implied by the biconditional.
A student is good at geometry proofs if and only if she is logical.
If a student is good at geometry proofs, then she is logical.
If a student is logical, then she is good at geometry proofs.
ONLY IF
This phrase is complicated to decipher. So read through this to see if it makes sense to you.
ONLY IF comes as half of IF AND ONLY IF.
In the phrase A if and only if B, there are two if then statements.
 One of them is if A, then B
 The other is If B then A
 (Technically A  B is logically equivalent to (A  B)  (B  A) )
 So in “A if and only if B”, we can rewrite this as A if B and A only if B)
This is which is which:
A if B
(B  A)
and

Notice that this
is If B, then A
(The “if” directly
precedes the B)
A only if B
(A  B)
This one is
If A, then B
(The “only if” points
the “if” back to A,
not to the B that it directly
precedes.
TOPIC 8: Introduction to Symbolic Logic
page 15
Logical Arguments
We are going to put together various facts and premises (in either algebra or geometry, for our
purposes), using the logical connectives that we have studied. We are going to learn how to use
these abstract logical properties in order to build arguments and establish the truth of algebraic
and geometric conditionals.
52.
A logical argument consists of ____________________ and a _____________________.
53.
Each premise is a statement that is given and is accepted as ___________________.
54.
The conclusion is a statement arrived at through __________________________ from
the given ___________________.
55.
The argument is valid if the conclusion has been arrived at using accepted
________________________________________________.
Five Patterns for Valid Arguments
1. Law of Detachment (Modus Ponens )
Example:
Premises If Suzie’s parents go out to dinner, then she will stay home with a baby-sitter.
On Friday night, Suzie’s Mom and Dad are going out to dinner to celebrate Mom’s birthday.
Conclusion:
Pattern:
Truth table: Build the truth table for P Q and look only at any rows when P is true.
TOPIC 8: Introduction to Symbolic Logic
page 16
2. Law of Syllogism
Example:
Premises: If Charlie Brown forgets to feed Snoopy, then Snoopy will run away from home.
If Snoopy runs away from home, Woodstock will fall out of his nest.
Conclusion:
Pattern:
Truth table: Build the truth table for P Q , Q  R and P  R look only at any rows when
P Q , Q  R are both true.
3. Law of the Contrapositive (Modus Tollens)
Example:
Premises: If it is raining, then I will get wet when I go outside.
Yesterday, when I went outside, I did not get wet.
Conclusion:
Pattern:
4. Law of Disjunction
Example:
Premises: Mr. Collins disagrees with Lady Catherine DeBourgh or Elizabeth refuses to
dance with Mr. Darcy
TOPIC 8: Introduction to Symbolic Logic
page 17
Mr. Collins agrees with Lady DeBourgh.
Conclusion:
Pattern:
Truth table: Build the truth table for P  Q. Look only at any rows when P is false.
5. Law of Simplification
Example:
Premises: Captain Jack gets the Pearl back and Will marries Elizabeth.
Conclusion:
Pattern:
Truth table: Build the truth table for P  Q. Look only at any rows when P and Q are each
true.
TOPIC 8: Introduction to Symbolic Logic
page 18
Recognizing valid arguments.
Determine whether or not each logical argument is valid. If it is, give the name of the
pattern. This is the beginning to the idea of PROOF.
56.
RS
R
S
PS
57.
~S
~ P
~RS
58.
59.
R
~ S
~RS
S T
 R ~ T
RS
60.
R
S
Practice making valid conclusions
Make a valid conclusion, if possible. Use the given statement and each of the four extra
premises, one at a time. Make a conclusion for part “a, one for part “b”, one for part “c”,
and one for part “d”. Name the reasoning pattern that validates your conclusion.
61.
Given: All marathoners have stamina.
a) Nick is a marathoner.
b) Heidi has stamina.
c) Missy does not have stamina.
d) Arlo is not a marathoner.
62.
Given: All squares are rhombuses.
a) ABCD is a square.
b) EFGH is not a rhombus.
c) JKLM is a rhombus.
d) NOPQ is not a square.
Arranging a valid argument
The following statements are true:
 Snoopy will bark or Woodstock will fly away.
 If Snoopy barks, then Lucy will hold the football.
 If Woodstock flies away, then Charlie Brown will not kick the football.
 Lucy will not hold the football.
Use valid reasoning to prove: Charlie Brown will not kick the football.
Symbolic Logic
page 20
Proofs with laws of logical reasoning
Write down an example for each of these.
63.
syllogism
64.
law of detachment (modus ponens)
65.
law of the contrapositive (modus tollens)
66.
law of disjunction
67.
law of simplification (conjunction)
Provide reasons (from the laws of reasoning above) for each step of the logical proof.
68. Given:
Prove:
C C  A A  D
F
C
C  A
A
AD
D
F  D
D  F
F
F  D
Symbolic Logic
69.
page 21
Put the following statements in an order which would constitute a proof.
A  B C  D
A C
70.
BC
A
E
D
E D
C
Put the following statements in an order which would constitute a proof.
H
P

G
P
 Q
P
H
G Q
Q