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THE LOGIC OF CAUSATION

© Avi Sion, 1999, 2000; 2003, 2005; 2008, 2010  All rights reserved.

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THE LOGIC OF CAUSATION

Phase One: Macroanalysis

Chapter 7 – Reduction of Positive Moods.

1. Reduction.

2. Reductions in Figure 1.

3. Reductions in Figure 2.

4. Reductions in Figure 3.

Section 2. Reductions in Figure 1.

First, note that ten moods in subfigure 1d have inconsistent premises. Specifically, if the minor premise (which has form P(S)Q) involves a strong determination, then it conflicts with the weak determination(s) of the major premise (which has form Q(P)R).

For if the minor concerns complete causation, clause (i) of which means that (P + notQ) is impossible – it is incompatible with the major, which implies (P + notQ) is possible, whether it concerns partial causation (see clause (ii) of that) or contingent causation (see clause (iii) of that). Similarly, if the minor concerns necessary causation, clause (i) of which means that (notP + Q) is impossible – it is incompatible with the major, which implies (notP + Q) is possible, whether it concerns partial causation (see clause (iii) of that) or contingent causation (see clause (ii) of that).

Summary of figure 1.

· 30 valid moods:

111-118, 121, 124-125, 128, 131, 134, 136-137, 141, 144, 147-148, 151-152, 155, 161, 163, 166, 171, 174, 181, 184.

· 24 moods without conclusion (nil):

126-127, 135, 138, 145-146, 153-154, 156-158, 162, 164-165, 167-168, 175-178, 185-188.

· 10 impossible moods (**):

122-123, 132-133, 142-143, 172-173, 182-183.

Total of moods = 30 valid and 34 invalid = 64.

Table 7.3.Sources of validity or invalidity in figure 1.

This table may be read as follows:

yes = element of conclusion (m, n, p or q) are implied by the given premises.

no = element of conclusion (m, n, p or q) are not implied (which does not mean denied) by the given premises.

by = by any sort of reduction to (number of mood used) or MA (matricial analysis).

Elements of conclusions for which matricial analysis is required are shaded.

since = for given premises, if an element of conclusion is valid (yes), then its contrary element is invalid (no).

** = incompatible premises.

nil = no valid conclusion.

Ref.

Mood #

Elements of conclusion implied?

§1

111

m

n

p

q

major

mn

yes

yes

no

no

minor

mn

by

by

since

since

concl.

mn

155

166

m

n

§2

112

m

n

p

q

major

mn

yes

no

no

yes

minor

mq

by

since

since

by

concl.

mq

155

q

m

118

§2

113

m

n

p

q

major

mn

no

yes

yes

no

minor

np

since

by

by

since

concl.

np

p

166

117

n

§6

114

m

n

p

q

major

mn

no

no

yes

yes

minor

pq

since

since

by

by

concl.

pq

p

q

117

118

§11

115

m

n

p

q

major

mn

yes

no

no

no

minor

m

by

by

since

by

concl.

m

155

112

m

111

§11

116

m

n

p

q

major

mn

no

yes

no

no

minor

n

by

by

by

since

concl.

n

113

166

111

n

§17

117

m

n

p

q

major

mn

no

no

yes

no

minor

p

since

by

by

by

concl.

p

p

114

MA

113

§17

118

m

n

p

q

major

mn

no

no

no

yes

minor

q

by

since

by

by

concl.

q

114

q

112

MA

Ref.

Mood #

Elements of conclusion implied?

§3

121

m

n

p

q

major

mq

yes

no

no

yes

minor

mn

by

since

since

by

concl.

mq

155

q

m

181

§4

122

m

n

p

q

major

mq

q of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§5

123

m

n

p

q

major

mq

q of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§8

124

m

n

p

q

major

mq

no

no

no

yes

minor

pq

by

since

by

by

concl.

q

MA

q

MA

128,184

§13

125

m

n

p

q

major

mq

yes

no

no

no

minor

m

by

by

since

by

concl.

m

155

121

m

MA

§14

126

m

n

p

q

major

mq

no

no

no

no

minor

n

by

by

by

by

concl.

nil

MA

121

121

MA

§19

127

m

n

p

q

major

mq

no

no

no

no

minor

p

by

by

by

by

concl.

nil

124

124

124

MA

§20

128

m

n

p

q

major

mq

no

no

no

yes

minor

q

by

since

by

by

concl.

q

124

q

124

MA

Ref.

Mood #

Elements of conclusion implied?

§3

131

m

n

p

q

major

np

no

yes

yes

no

minor

mn

since

by

by

since

concl.

np

p

166

171

n

§5

132

m

n

p

q

major

np

p of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§4

133

m

n

p

q

major

np

p of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§8

134

m

n

p

q

major

np

no

no

yes

no

minor

pq

since

by

by

by

concl.

p

p

MA

137,174

MA

§14

135

m

n

p

q

major

np

no

no

no

no

minor

m

by

by

by

by

concl.

nil

131

MA

MA

131

§13

136

m

n

p

q

major

np

no

yes

no

no

minor

n

by

by

by

since

concl.

n

131

166

MA

n

§20

137

m

n

p

q

major

np

no

no

yes

no

minor

p

since

by

by

by

concl.

p

p

134

MA

134

§19

138

m

n

p

q

major

np

no

no

no

no

minor

q

by

by

by

by

concl.

nil

134

134

MA

134

Ref.

Mood #

Elements of conclusion implied?

§7

141

m

n

p

q

major

pq

no

no

yes

yes

minor

mn

since

since

by

by

concl.

pq

p

q

171

181

§9

142

m

n

p

q

major

pq

p, q of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§9

143

m

n

p

q

major

pq

p, q of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§10

144

m

n

p

q

major

pq

no

no

yes

yes

minor

pq

since

since

by

by

concl.

pq

p

q

147,174

148,184

§23

145

m

n

p

q

major

pq

no

no

no

no

minor

m

by

by

by

by

concl.

nil

141

141

MA

MA

§23

146

m

n

p

q

major

pq

no

no

no

no

minor

n

by

by

by

by

concl.

nil

141

141

MA

MA

§25

147

m

n

p

q

major

pq

no

no

yes

no

minor

p

since

by

by

by

concl.

p

p

144

MA

MA

§25

148

m

n

p

q

major

pq

no

no

no

yes

minor

q

by

since

by

by

concl.

q

144

q

MA

MA

Ref.

Mood #

Elements of conclusion implied?

§12

151

m

n

p

q

major

m

yes

no

no

no

minor

mn

by

by

since

by

concl.

m

155

121

m

111

§15

152

m

n

p

q

major

m

yes

no

no

no

minor

mq

by

by

since

by

concl.

m

155

112

m

MA

§16

153

m

n

p

q

major

m

no

no

no

no

minor

np

by

by

by

by

concl.

nil

113

MA

MA

113

§24

154

m

n

p

q

major

m

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

114

114

124

MA

§27

155

m

n

p

q

major

m

yes

no

no

no

minor

m

by

by

since

by

concl.

m

MA

112

m

111

§28

156

m

n

p

q

major

m

no

no

no

no

minor

n

by

by

by

by

concl.

nil

113

121

111

111

§29

157

m

n

p

q

major

m

no

no

no

no

minor

p

by

by

by

by

concl.

nil

113

114

124

113

§30

158

m

n

p

q

major

m

no

no

no

no

minor

q

by

by

by

by

concl.

nil

114

112

112

152

Ref.

Mood #

Elements of conclusion implied?

§12

161

m

n

p

q

major

n

no

yes

no

no

minor

mn

by

by

by

since

concl.

n

131

166

111

n

§16

162

m

n

p

q

major

n

no

no

no

no

minor

mq

by

by

by

by

concl.

nil

MA

112

112

MA

§15

163

m

n

p

q

major

n

no

yes

no

no

minor

np

by

by

by

since

concl.

n

113

166

MA

n

§24

164

m

n

p

q

major

n

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

114

114

MA

134

§28

165

m

n

p

q

major

n

no

no

no

no

minor

m

by

by

by

by

concl.

nil

131

112

111

111

§27

166

m

n

p

q

major

n

no

yes

no

no

minor

n

by

by

by

since

concl.

n

113

MA

111

n

§30

167

m

n

p

q

major

n

no

no

no

no

minor

p

by

by

by

by

concl.

nil

113

114

163

113

§29

168

m

n

p

q

major

n

no

no

no

no

minor

q

by

by

by

by

concl.

nil

114

112

112

134

Ref.

Mood #

Elements of conclusion implied?

§18

171

m

n

p

q

major

p

no

no

yes

no

minor

mn

since

by

by

by

concl.

p

p

141

MA

131

§21

172

m

n

p

q

major

p

p of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§22

173

m

n

p

q

major

p

p of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§26

174

m

n

p

q

major

p

no

no

yes

no

minor

pq

since

by

by

by

concl.

p

p

134

MA

134

§31

175

m

n

p

q

major

p

no

no

no

no

minor

m

by

by

by

by

concl.

nil

131

135

135

131

§32

176

m

n

p

q

major

p

no

no

no

no

minor

n

by

by

by

by

concl.

nil

131

141

136

131

§33

177

m

n

p

q

major

p

no

no

no

no

minor

p

by

by

by

by

concl.

nil

134

134

MA

134

§34

178

m

n

p

q

major

p

no

no

no

no

minor

q

by

by

by

by

concl.

nil

134

134

138

134

Ref.

Mood #

Elements of conclusion implied?

§18

181

m

n

p

q

major

q

no

no

no

yes

minor

mn

by

since

by

by

concl.

q

141

q

121

MA

§22

182

m

n

p

q

major

q

q of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§21

183

m

n

p

q

major

q

q of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§26

184

m

n

p

q

major

q

no

no

no

yes

minor

pq

by

since

by

by

concl.

q

124

q

124

MA

§32

185

m

n

p

q

major

q

no

no

no

no

minor

m

by

by

by

by

concl.

nil

141

121

121

125

§31

186

m

n

p

q

major

q

no

no

no

no

minor

n

by

by

by

by

concl.

nil

126

121

121

126

§34

187

m

n

p

q

major

q

no

no

no

no

minor

p

by

by

by

by

concl.

nil

124

124

124

127

§33

188

m

n

p

q

major

q

no

no

no

no

minor

q

by

by

by

by

concl.

nil

124

124

124

MA

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2016-06-13T10:29:31+00:00