##### 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

sorites (heap) paradox, 188

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

see also compound propositions; conditional syl-

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

see also conclusion; premises; specific types of syl-

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

see also Boolean algebra

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

see also "all" statements; A propositions; E

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

village barber paradox, 189—90

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

Whitehead, Alfred North, 157

Wilkins, M. C , 49

William of Ockham, 154-55

William of Shyreswood, 79-80

Wilson.Thomas, 2 1 , 92, 224n

Winkler, Peter, 99, 224n

Woodworth.R. S.,88

Yi Ching, or Book of Changes, 148

yinand yang, 148

Zadeh,Lotfi, 174

ZenoofElea, 32 -33 , 192, 220n

Zwicky.A.M., 115

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

sorites (heap) paradox, 188

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

see also compound propositions; conditional syl-

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

see also conclusion; premises; specific types of syl-

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

see also Boolean algebra

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

see also "all" statements; A propositions; E

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

village barber paradox, 189—90

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

Whitehead, Alfred North, 157

Wilkins, M. C , 49

William of Ockham, 154-55

William of Shyreswood, 79-80

Wilson.Thomas, 2 1 , 92, 224n

Winkler, Peter, 99, 224n

Woodworth.R. S.,88

Yi Ching, or Book of Changes, 148

yinand yang, 148

Zadeh,Lotfi, 174

ZenoofElea, 32 -33 , 192, 220n

Zwicky.A.M., 115