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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 3. Reductions in Figure 2.

First, note that six moods in subfigure 2d have inconsistent premises. Specifically, if the minor premise (which has form P(S)Q) involves a strong determination, then it conflicts with any weak determination of same polarity in the major premise (which has form R(P)Q).

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 when it concerns partial causation (see clause (ii) 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 when it concerns contingent causation (see clause (ii) of that).

Additionally, we may directly reduce a number of moods in figure 2 to figure 1, by converting the major premise. This is feasible when the major premise involves only strong causation; i.e. subfigures 2a and 2b are thus reducible respectively to subfigures 1a and 1b. This is not feasible when the major premise involves weak causation, since its conversion results in negation of the complement; which means that subfigures 2c and 2d have to be evaluated relatively independently (i.e. within the same figure, even if possibly through some moods reduced to figure 1).

Summary of figure 2.

· 18 valid moods:

211-218, 221, 226, 231, 235, 251, 253, 256, 261-262, 265.

· 40 moods without conclusion (nil):

222, 224-225, 227-228, 233-234, 236-238, 241, 244-248, 252, 254-255, 257-258, 263-264, 266-268, 271, 273-278, 281-282, 284-288.

· 6 impossible moods (**):

223, 232, 242-243, 272, 283.

Total of moods = 18 valid and 46 invalid = 64.

Table 7.4.Sources of validity or invalidity in figure 2.

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

211

m

n

p

q

major

mn

yes

yes

no

no

minor

mn

by

by

since

since

concl.

mn

111

111

m

n

§2

212

m

n

p

q

major

mn

yes

no

no

yes

minor

mq

by

since

since

by

concl.

mq

112

q

m

112

§2

213

m

n

p

q

major

mn

no

yes

yes

no

minor

np

since

by

by

since

concl.

np

p

113

113

n

§6

214

m

n

p

q

major

mn

no

no

yes

yes

minor

pq

since

since

by

by

concl.

pq

p

q

114

114

§11

215

m

n

p

q

major

mn

yes

no

no

no

minor

m

by

by

since

by

concl.

m

115

115

m

115

§11

216

m

n

p

q

major

mn

no

yes

no

no

minor

n

by

by

by

since

concl.

n

116

116

116

n

§17

217

m

n

p

q

major

mn

no

no

yes

no

minor

p

since

by

by

by

concl.

p

p

117

117

117

§17

218

m

n

p

q

major

mn

no

no

no

yes

minor

q

by

since

by

by

concl.

q

118

q

118

118

Ref.

Mood #

Elements of conclusion implied?

§3

221

m

n

p

q

major

mq

no

yes

no

no

minor

mn

by

by

by

since

concl.

n

MA

256

MA

n

§4

222

m

n

p

q

major

mq

no

no

no

no

minor

mq

by

by

by

by

concl.

nil

MA

MA

MA

MA

§5

223

m

n

p

q

major

mq

q of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§8

224

m

n

p

q

major

mq

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

MA

MA

MA

MA

§13

225

m

n

p

q

major

mq

no

no

no

no

minor

m

by

by

by

by

concl.

nil

221

222

221

221

§14

226

m

n

p

q

major

mq

no

yes

no

no

minor

n

by

by

by

since

concl.

n

221

256

221

n

§19

227

m

n

p

q

major

mq

no

no

no

no

minor

p

by

by

by

by

concl.

nil

224

224

224

224

§20

228

m

n

p

q

major

mq

no

no

no

no

minor

q

by

by

by

by

concl.

nil

222

222

222

222

Ref.

Mood #

Elements of conclusion implied?

§3

231

m

n

p

q

major

np

yes

no

no

no

minor

mn

by

by

since

by

concl.

m

265

MA

m

MA

§5

232

m

n

p

q

major

np

p of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§4

233

m

n

p

q

major

np

no

no

no

no

minor

np

by

by

by

by

concl.

nil

MA

MA

MA

MA

§8

234

m

n

p

q

major

np

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

MA

MA

MA

MA

§14

235

m

n

p

q

major

np

yes

no

no

no

minor

m

by

by

since

by

concl.

m

265

231

m

231

§13

236

m

n

p

q

major

np

no

no

no

no

minor

n

by

by

by

by

concl.

nil

233

231

231

231

§20

237

m

n

p

q

major

np

no

no

no

no

minor

p

by

by

by

by

concl.

nil

233

233

233

233

§19

238

m

n

p

q

major

np

no

no

no

no

minor

q

by

by

by

by

concl.

nil

234

234

234

234

Ref.

Mood #

Elements of conclusion implied?

§7

241

m

n

p

q

major

pq

no

no

no

no

minor

mn

by

by

by

by

concl.

nil

MA

MA

MA

MA

§9

242

m

n

p

q

major

pq

p of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§9

243

m

n

p

q

major

pq

q of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§10

244

m

n

p

q

major

pq

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

MA

MA

MA

MA

§23

245

m

n

p

q

major

pq

no

no

no

no

minor

m

by

by

by

by

concl.

nil

241

241

241

241

§23

246

m

n

p

q

major

pq

no

no

no

no

minor

n

by

by

by

by

concl.

nil

241

241

241

241

§25

247

m

n

p

q

major

pq

no

no

no

no

minor

p

by

by

by

by

concl.

nil

244

244

244

244

§25

248

m

n

p

q

major

pq

no

no

no

no

minor

q

by

by

by

by

concl.

nil

244

244

244

244

Ref.

Mood #

Elements of conclusion implied?

§12

251

m

n

p

q

major

m

no

yes

no

no

minor

mn

by

by

by

since

concl.

n

161

161

161

n

§15

252

m

n

p

q

major

m

no

no

no

no

minor

mq

by

by

by

by

concl.

nil

162

162

162

162

§16

253

m

n

p

q

major

m

no

yes

no

no

minor

np

by

by

by

since

concl.

n

163

163

163

n

§24

254

m

n

p

q

major

m

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

164

164

164

164

§27

255

m

n

p

q

major

m

no

no

no

no

minor

m

by

by

by

by

concl.

nil

165

165

165

165

§28

256

m

n

p

q

major

m

no

yes

no

no

minor

n

by

by

by

since

concl.

n

166

166

166

n

§29

257

m

n

p

q

major

m

no

no

no

no

minor

p

by

by

by

by

concl.

nil

167

167

167

167

§30

258

m

n

p

q

major

m

no

no

no

no

minor

q

by

by

by

by

concl.

nil

168

168

168

168

Ref.

Mood #

Elements of conclusion implied?

§12

261

m

n

p

q

major

n

yes

no

no

no

minor

mn

by

by

since

by

concl.

m

151

151

m

151

§16

262

m

n

p

q

major

n

yes

no

no

no

minor

mq

by

by

since

by

concl.

m

152

152

m

152

§15

263

m

n

p

q

major

n

no

no

no

no

minor

np

by

by

by

by

concl.

nil

153

153

153

153

§24

264

m

n

p

q

major

n

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

154

154

154

154

§28

265

m

n

p

q

major

n

yes

no

no

no

minor

m

by

by

since

by

concl.

m

155

155

m

155

§27

266

m

n

p

q

major

n

no

no

no

no

minor

n

by

by

by

by

concl.

nil

156

156

156

156

§30

267

m

n

p

q

major

n

no

no

no

no

minor

p

by

by

by

by

concl.

nil

157

157

157

157

§29

268

m

n

p

q

major

n

no

no

no

no

minor

q

by

by

by

by

concl.

nil

158

158

158

158

Ref.

Mood #

Elements of conclusion implied?

§18

271

m

n

p

q

major

p

no

no

no

no

minor

mn

by

by

by

by

concl.

nil

241

231

231

231

§21

272

m

n

p

q

major

p

p of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§22

273

m

n

p

q

major

p

no

no

no

no

minor

np

by

by

by

by

concl.

nil

233

233

233

233

§26

274

m

n

p

q

major

p

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

234

234

234

234

§31

275

m

n

p

q

major

p

no

no

no

no

minor

m

by

by

by

by

concl.

nil

241

231

231

231

§32

276

m

n

p

q

major

p

no

no

no

no

minor

n

by

by

by

by

concl.

nil

233

231

231

231

§33

277

m

n

p

q

major

p

no

no

no

no

minor

p

by

by

by

by

concl.

nil

233

233

233

233

§34

278

m

n

p

q

major

p

no

no

no

no

minor

q

by

by

by

by

concl.

nil

234

234

234

234

Ref.

Mood #

Elements of conclusion implied?

§18

281

m

n

p

q

major

q

no

no

no

no

minor

mn

by

by

by

by

concl.

nil

221

241

221

221

§22

282

m

n

p

q

major

q

no

no

no

no

minor

mq

by

by

by

by

concl.

nil

222

222

222

222

§21

283

m

n

p

q

major

q

q of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§26

284

m

n

p

q

major

q

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

224

224

224

224

§32

285

m

n

p

q

major

q

no

no

no

no

minor

m

by

by

by

by

concl.

nil

221

222

221

221

§31

286

m

n

p

q

major

q

no

no

no

no

minor

n

by

by

by

by

concl.

nil

221

241

221

221

§34

287

m

n

p

q

major

q

no

no

no

no

minor

p

by

by

by

by

concl.

nil

224

224

224

224

§33

288

m

n

p

q

major

q

no

no

no

no

minor

q

by

by

by

by

concl.

nil

222

222

222

222

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2016-06-13T10:29:07+00:00