Title Logic Made Easy (2004) 3.8 MB 260
```                            Cover
Contents
Introduction: Logic Is Rare
The Mistakes We Make
Logic Should Be Everywhere
How History Can Help
1  Proof
Disproof
2  All
All S are P
Vice Versa
Familiarity—Help or Hindrance?
Clarity or Brevity?
3  A NOT Tangles Everything Up
The Trouble with Not
Scope of the Negative
A and E Propositions
When No Means Yes —The "Negative Pregnant" and Double Negative
4  SOME Is Part or All of ALL
Some Is Existential
Some Are; Some Are Not
A, E, I, and O
5  Syllogisms
Sorites, or Heap
Atmosphere of the "Sillygism"
Knowledge Interferes with Logic
Truth Interferes with Logic
6  When Things Are IFfy
The Converse of the Conditional
Causation
The Contrapositive Conditional
7  Syllogisms Involving IF, AND, and OR
Disjunction, an "Or" Statement
Conjunction, an "And" Statement
Hypothetical Syllogisms
Common Fallacies
Diagramming Conditional Syllogisms
8  Series Syllogisms
9  Symbols That Express Our Thoughts
Leibniz's Dream Comes True: Boolean Logic
10  Logic Machines and Truth Tables
Reasoning Machines
Truth Tables
True, False, and Maybe
11  Fuzzy Logic, Fallacies, and Paradoxes
Shaggy Logic
Fallacies
12  Common Logic and Language
13  Thinking Well—Together
Theories of Reasoning
Notes
Introduction
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
References
Acknowledgments
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Y
Z
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Document Text Contents
Page 130

126 LOGIC MADE E A S Y

liest attempts at creating machine intelligence. In 1956, artificial

intelligence pioneers implemented modus ponens in their pro-

gram The Logic Theorist, a program designed to make logical

conclusions. Given an initial list of premises (true propositions),

the program instructed the computer to look through the list

for a premise of the form "if p then q and a premise p. Once
these premises were found, the logical consequent q was
deduced as true and could therefore be added to the list of true

premises. By searching for matches in this way, the program

used modus ponens to expand its list of true propositions.

Armed with modus ponens and some substitution and simplifi-

cation rules, The Logical Theorist was able to prove an impres-

sive number of mathematical theorems.12

Although modus ponens seems like a very simplistic form of

deduction, we can use this structure to form elaborate argu-

ments. Consider the following statement: "If you clean up your

room and take out the trash, then we can go to the movies and

buy popcorn." What do you have a right to expect should you

clean up your room and take out the trash? You have a perfect

right to expect that we will both go to the movies and buy

popcorn. This statement is a conditional of the form: If p and
q, then r and s where "p and q is the antecedent and "r and s" is
the consequent. Utilizing modus ponens, the deduction looks

like:

Ifp and q, then r and s.

p and q.

Therefore, r and s.

Another more elaborate form of modus ponens can be

employed by utilizing the law of the excluded middle. One of

the premises, the assertion of the antecedent, is often implied.

Page 131

SYLLOGISMS INVOLVING IF, AND, AND OR ny

The following example is familiar to anyone who has completed

a United States income tax form:

If you itemize your deductions, then you enter the

amount from Schedule A on line 36.

If you do not itemize your deductions, then you enter

your standard deduction on line 36.

Therefore, either you will enter the amount from

Schedule A on line 36 or you will enter your standard

deduction on line 36.

Symbolically this syllogism is similar to any syllogism of the

form:

If p then q.

If not-p then r.

Therefore, q or r.

The unstated premise is "Either p or not-p"—the law of the

excluded middle—in this case, "Either you itemize your deduc-

tions or you do not itemize your deductions ."When it is inserted

mentally, we know that one or the other of the antecedents is

true and therefore one or the other of the consequents must be

true.

Some conditionals are relatively easy for individuals to evalu-

ate even when they require the reasoner to envision a large

number of scenarios. Most adults would easily negotiate the fol-

lowing: "If your lottery number is 40 or 13 or 5 2 or 33 or 19,

then you win \$100." Under some circumstances, we seem to

have a singular ability to focus on the pertinent information.

The second valid inference schema of the Stoics was given as:

If the first, then the second; but not the second; therefore not the first. As

Page 259

I N D E X 255

Sophists, 20,73, 177-78
law of the excluded middle in, 30—31

sorites, 85-S7, 86, 57,89, 188
definition of, 85

spatial inclusion, 143
spatial relations, 195, 206
split-brain research, 206
Square of Opposition diagram, 65—66, 66
Stanhope, Charles, Earl, 93-94, 160-61, 222n
Staudenmayer, Herman, 106—7
stepped drum calculator, 148, 149, 228n
Stoics, 20 -21 , 31, 33, 39, 45, 62, 94-95, 114-15 ,

118-36,138,155

exclusive "or" used by, 119—20, 130
founder of, 123
metaphysical argument of, 118—19
propositional reasoning of, 119, 134, 136, 144

logisms, Stoic
Student's Oxford Aristotle,The (Ross, trans.), 77, 22 In
summation symbol, 157n
syllogism machines, 160—65
syllogisms, 72, 73-95, 117, 118-36, 160,

168-69, 194,206,207
Aristotle's, 73-74, 76, 94-95, 117, 118, 124,

125,207

Buddhist, 206-7
definition of, 74
diagrams of, 80-87, 81, 82, 83, 84, 86, 81, 94
figures of, 75-85, 222n

five-step, of Naiyâyiks, 128—29
invalid, 80, 81, 83-84, 89-90, 180
"is" in, 196
middle term of, 75, 222n
mnemonic devices for, 76, 77, 79-80, 81, 93
mnemonic verses for, 79-80, 81, 93
moods of, 74-85, 88, 223n
numerically definite quantifiers in, 84-85
possible number of, 78, 79
reasoning mistakes and, 88-91, 180, 183-85, 186
reduced statements and, 78-79, 80
rules forjudging validity of, 80, 88
"some" statements in, 82—85
sorites as, 85-87, 86, 89, 188
subject and predicate of, 75, 77, 81—82
valid, 74, 76-87, 89, 186
variables as terms in, 76

logisms
Sylvie and Bruno (Carroll), 73
symbolic logic, 2 2 - 2 4 , 144, 145-59, 165, 175

of De Morgan, 149-51, 154-57
of Leibniz, 145-49, 150
of Peirce, 157

Symbolic Logic (Carroll), 23
symbolic notation, 157-59, 163, 170-71
Syntagma Logicum, orThe Divine Logike (Granger), 193

Tarski, Alfred, 168
temporal relations, 143, 195

in conditional propositions, 96, 112, 113—14
terminology, 59, 91-95

of Lever, 92-93
of Stanhope, 93-94

Tests at a Glance (EducationalTesting Service), 69—70
Thaïes, 3 1 - 3 2 , 3 9
Thinking and Deciding (Baron), 64
THOG problem, 1 2 0 - 2 1 , 1 2 7 , 2 1 4
Thornquist, Bruce, 186
threats, 96, 133

Through the Looking Glass (Carroll), 96
Titus, Letter of Paul to, 188-89
Topics (Aristotle), 30, 40, 176
truth:

interference of, 90—91
validity confused with, 184, 204

truth degrees, 173-75, 176, 177, 188
truth tables, 165-68, 229n
truth values, 67,85, 102, 113, 166, 168-72, 220n

in Boolean algebra, 154, 173
conditional propositional and, 169—70, 169, 188
in fuzzy logic, 173—75
infinite, 188
of many valued systems, 168—71, 173
mechanical devices and, 171—72
negations and, 169-70, 170, 171
"possible" as, 168-70
and violation of existential presuppositions, 170

Uber das Unendliche (Hilbert), 157
undistributed middle, fallacy of, 180
union, Boolean, 153
universally characteristic language, 145—47, 149
universal propositions, 35—36, 41—42, 51 , 59,

59-61 ,64 ,65 ,88 ,92 ,116-17 ,181 ,194

propositions
universe class, Boolean, 152
universe of discourse, 87, 149, 150, 152, 174-75

vague concepts, 174, 177, 188
Venn, John, 46, 66, 82, 85-87, 162

Page 260

256 I N D E X

Venn, John (continued)
logical-diagram machine of, 162—63

Venn diagrams, 46^-7, 47, 51 , 220n
of conditional syllogisms, 134—36, 134, 135, 136
jigsaw puzzle version of, 162
of modus ponens, 134-35, 134, 135
of negation, 60, 60, 61
of particular propositions, 66, 66, 67, 68, 68
of sorites, 85-87, 86
of valid syllogisms, 80, 82-84, 82, 83, 84

visualization, 57, 142^4-3, 209
Volapuk, 147

Wason, Peter C , 15, 55, 57, 68, 91, 98-101,
113-14, 144, 219n

THOG problem and, 120-21

Wason Selection Task, 99-101, 100, 103-5, 104,
214