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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 4. Reductions in Figure 3.

First, note that ten moods in subfigure 3d have inconsistent premises. Specifically, if the minor premise (which has form Q(S)P) 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 (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). Similarly, if the minor concerns necessary 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).

Additionally, we may directly reduce a number of moods in figure 3 to figure 1, by converting the minor premise. This is feasible when the minor premise involves only strong causation; i.e. subfigures 3a and 3c are thus reducible respectively to subfigures 1a and 1c. This is not feasible when the minor premise involves weak causation, since its conversion results in negation of the complement; which means that subfigures 3b and 3d 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 3.

· 18 valid moods:

311-313, 315-316, 321, 326, 331, 335, 341, 351, 353, 356, 361-362, 365, 371, 381.

· 36 moods without conclusion (nil):

314, 317-318, 324-325, 327-328, 334, 336-338, 344-348, 352, 354-355, 357-358, 363-364, 366-368, 374-378, 384-388.

· 10 impossible moods (**):

322-323, 332-333, 342-343, 372-373, 382-383.

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

Table 7.5.Sources of validity or invalidity in figure 3.

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

311

m

n

p

q

major

mn

yes

yes

no

no

minor

mn

by

by

since

since

concl.

mn

111

111

m

n

§2

312

m

n

p

q

major

mn

no

yes

no

no

minor

mq

by

by

by

since

concl.

n

MA

365

MA

n

§2

313

m

n

p

q

major

mn

yes

no

no

no

minor

np

by

by

since

by

concl.

m

356

MA

m

MA

§6

314

m

n

p

q

major

mn

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

MA

MA

MA

MA

§11

315

m

n

p

q

major

mn

no

yes

no

no

minor

m

by

by

by

since

concl.

n

116

116

116

n

§11

316

m

n

p

q

major

mn

yes

no

no

no

minor

n

by

by

since

by

concl.

m

115

115

m

115

§17

317

m

n

p

q

major

mn

no

no

no

no

minor

p

by

by

by

by

concl.

nil

314

313

313

313

§17

318

m

n

p

q

major

mn

no

no

no

no

minor

q

by

by

by

by

concl.

nil

312

314

312

312

Ref.

Mood #

Elements of conclusion implied?

§3

321

m

n

p

q

major

mq

yes

no

no

yes

minor

mn

by

since

since

by

concl.

mq

121

q

m

121

§4

322

m

n

p

q

major

mq

q of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§5

323

m

n

p

q

major

mq

q of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§8

324

m

n

p

q

major

mq

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

MA

MA

MA

MA

§13

325

m

n

p

q

major

mq

no

no

no

no

minor

m

by

by

by

by

concl.

nil

126

126

126

126

§14

326

m

n

p

Q

major

mq

yes

no

no

No

minor

n

by

by

since

by

concl.

m

125

125

m

125

§19

327

m

n

p

Q

major

mq

no

no

no

No

minor

p

by

by

by

by

concl.

nil

324

324

324

324

§20

328

m

n

p

Q

major

mq

no

no

no

No

minor

q

by

by

by

by

concl.

nil

324

324

324

324

Ref.

Mood #

Elements of conclusion implied?

§3

331

m

n

p

Q

major

np

no

yes

yes

No

minor

mn

since

by

by

since

concl.

np

p

131

131

N

§5

332

m

n

p

q

major

np

p of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§4

333

m

n

p

q

major

np

p of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§8

334

m

n

p

q

major

np

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

MA

MA

MA

MA

§14

335

m

n

p

q

major

np

no

yes

no

no

minor

m

by

by

by

since

concl.

n

136

136

136

n

§13

336

m

n

p

q

major

np

no

no

no

no

minor

n

by

by

by

by

concl.

nil

135

135

135

135

§20

337

m

n

p

q

major

np

no

no

no

no

minor

p

by

by

by

by

concl.

nil

334

334

334

334

§19

338

m

n

p

q

major

np

no

no

no

no

minor

q

by

by

by

by

concl.

nil

334

334

334

334

Ref.

Mood #

Elements of conclusion implied?

§7

341

m

n

p

q

major

pq

no

no

yes

yes

minor

mn

since

since

by

by

concl.

pq

p

q

141

141

§9

342

m

n

p

q

major

pq

p and q of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§9

343

m

n

p

q

major

pq

p and q of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§10

344

m

n

p

q

major

pq

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

MA

MA

MA

MA

§23

345

m

n

p

q

major

pq

no

no

no

no

minor

m

by

by

by

by

concl.

nil

146

146

146

146

§23

346

m

n

p

q

major

pq

no

no

no

no

minor

n

by

by

by

by

concl.

nil

145

145

145

145

§25

347

m

n

p

q

major

pq

no

no

no

no

minor

p

by

by

by

by

concl.

nil

344

344

344

344

§25

348

m

n

p

q

major

pq

no

no

no

no

minor

q

by

by

by

by

concl.

nil

344

344

344

344

Ref.

Mood #

Elements of conclusion implied?

§12

351

m

n

p

q

major

m

yes

no

no

no

minor

mn

by

by

since

by

concl.

m

151

151

m

151

§15

352

m

n

p

q

major

m

no

no

no

no

minor

mq

by

by

by

by

concl.

nil

312

MA

312

312

§16

353

m

n

p

q

major

m

yes

no

no

no

minor

np

by

by

since

by

concl.

m

356

313

m

313

§24

354

m

n

p

q

major

m

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

314

314

314

314

§27

355

m

n

p

q

major

m

no

no

no

no

minor

m

by

by

by

by

concl.

nil

156

156

156

156

§28

356

m

n

p

q

major

m

yes

no

no

no

minor

n

by

by

since

by

concl.

m

155

155

m

155

§29

357

m

n

p

q

major

m

no

no

no

no

minor

p

by

by

by

by

concl.

nil

314

313

313

313

§30

358

m

n

p

q

major

m

no

no

no

no

minor

q

by

by

by

by

concl.

nil

312

314

312

312

Ref.

Mood #

Elements of conclusion implied?

§12

361

m

n

p

q

major

n

no

yes

no

no

minor

mn

by

by

by

since

concl.

n

161

161

161

n

§16

362

m

n

p

q

major

n

no

yes

no

no

minor

mq

by

by

by

since

concl.

n

312

365

312

n

§15

363

m

n

p

q

major

n

no

no

no

no

minor

np

by

by

by

by

concl.

nil

MA

313

313

313

§24

364

m

n

p

q

major

n

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

314

314

314

314

§28

365

m

n

p

q

major

n

no

yes

no

no

minor

m

by

by

by

since

concl.

n

166

166

166

n

§27

366

m

n

p

q

major

n

no

no

no

no

minor

n

by

by

by

by

concl.

nil

165

165

165

165

§30

367

m

n

p

q

major

n

no

no

no

no

minor

p

by

by

by

by

concl.

nil

314

313

313

313

§29

368

m

n

p

q

major

n

no

no

no

no

minor

q

by

by

by

by

concl.

nil

312

314

312

312

Ref.

Mood #

Elements of conclusion implied?

§18

371

m

n

p

q

major

p

no

no

yes

no

minor

mn

since

by

by

by

concl.

p

p

171

171

171

§21

372

m

n

p

q

major

p

p of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§22

373

m

n

p

q

major

p

p of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§26

374

m

n

p

q

major

p

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

334

334

334

334

§31

375

m

n

p

q

major

p

no

no

no

no

minor

m

by

by

by

by

concl.

nil

176

176

176

176

§32

376

m

n

p

q

major

p

no

no

no

no

minor

n

by

by

by

by

concl.

nil

175

175

175

175

§33

377

m

n

p

q

major

p

no

no

no

no

minor

p

by

by

by

by

concl.

nil

334

334

334

334

§34

378

m

n

p

q

major

p

no

no

no

no

minor

q

by

by

by

by

concl.

nil

334

334

334

334

Ref.

Mood #

Elements of conclusion implied?

§18

381

m

n

p

q

major

q

no

no

no

yes

minor

mn

by

since

by

by

concl.

q

181

q

181

181

§22

382

m

n

p

q

major

q

q of major premise and

minor

mq

m of minor premise

concl.

**

are incompatible

§21

383

m

n

p

q

major

q

q of major premise and

minor

np

n of minor premise

concl.

**

are incompatible

§26

384

m

n

p

q

major

q

no

no

no

no

minor

pq

by

by

by

by

concl.

nil

324

324

324

324

§32

385

m

n

p

q

major

q

no

no

no

no

minor

m

by

by

by

by

concl.

nil

186

186

186

186

§31

386

m

n

p

q

major

q

no

no

no

no

minor

n

by

by

by

by

concl.

nil

185

185

185

185

§34

387

m

n

p

q

major

q

no

no

no

no

minor

p

by

by

by

by

concl.

nil

324

324

324

324

§33

388

m

n

p

q

major

q

no

no

no

no

minor

q

by

by

by

by

concl.

nil

324

324

324

324

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2016-01-21T06:43:08+00:00