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THE LOGIC OF CAUSATION

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THE LOGIC OF CAUSATION

Phase One: Macroanalysis

Chapter 6 – List of Positive Moods.

1. Valid and Invalid Moods.

2. Moods in Figure 1.

3. Moods in Figure 2.

4. Moods in Figure 3.

1. Valid and Invalid Moods.

As we have seen in the preceding chapter, causative syllogism with both premises affirmative has 64 conceivable moods in each of three figures. In the present chapter, we shall list all these moods, and for each mood specify whether it is valid or invalid, and briefly the basis of this evaluation.

For any positive mood, there are four initially conceivable, putative conclusions, corresponding to the four generic determinations, which we have symbolized as m (for complete causation), n (for necessary causation), p (for partial causation) and q (for contingent causation). However, at most two such conclusions may be valid for any given mood, since the determinations m and p are contrary and n and q are contrary. Thus, there are eight logically possible conclusions for any positive causative syllogism, namely:

mn, mq, np, pq, m, n, p, q.

A putative conclusion is valid – if it logically follows from the given premises, i.e. if its contradictory is logically incompatible with them or any of their implications. A putative conclusion is declared invalid – if it is not valid, for whatever reason; the reason may be that the premises themselves are inconsistent, or that the contradictory of the putative conclusion is compatible with them (in which case the putative conclusion is a non-sequitur), or that the putative conclusion is incompatible with the premises (in which case the putative conclusion is an antinomy and its contradictory is valid).

If one of the eight joint or generic determinations is demonstrably inferable from the premises concerned, the mood is valid. If none of them can be legitimately drawn from the premises, the mood is invalid. Additionally, some moods are invalid at the outset because the premises concerned are in fact incompatible in some respect(s); i.e. at least one clause of each is implicitly denied by at least one clause of the other.

We shall, to repeat, in the present chapter only list the moods and their valid conclusion(s) if any, and state succinctly the basis of these results. In the next two chapters, we will show how these results were obtained, systematically and in detail; i.e. we will justify our claims.

Note that, in accord with the tradition in logic, if a mood is valid, only the correct conclusion(s) is/are mentioned in the listing; other conclusions, not mentioned, are tacitly implied to be incorrect. But it is well to keep both the valid and invalid conclusions in mind; for the purpose of the whole exercise is not only to instruct us in proper reasoning, but also to save us from improper reasoning!

As will be seen, some conclusions have to be validated or invalidated by matricial analysis; moods with at least one conclusion treated by matricial analysis may be called primary. The remaining conclusions may be validated or invalidated by reduction to the primary moods; moods all of whose conceivable conclusions have been treated by reduction may be called secondary or derived.

As for moods invalid due to inconsistency between the premises, they need not of course be subjected to matricial analysis or reduction. Note that it may be possible to affirm or deny some conclusion(s) from some of their clauses, if the inherent contradiction is disregarded; but that would be nonsensical, for if all the clauses are taken into consideration, we have to admit that the premises in question cannot in fact come together to yield such conclusion(s).

All evaluations could be performed by matricial analysis; but this process is long-winded, so we try and avoid it as much as possible. Such avoidance is anyway not sheer laziness on our part, for it is instructive to be aware of the interrelations between moods which reduction reveals. We learn, in this way, that causative syllogisms together constitute a close-knit totality, a system.

It should be stressed that the issue of direction of causation is ignored throughout the present formal treatment. In figure 1, this is no problem; i.e. given the directions of causation implied in the premises (namely, from P to Q and from Q to R), the direction of causation implied in an eventual valid conclusion (viz. from P to R) follows necessarily. But in figures 2 and 3, any eventual valid conclusions must be regarded as conditionally valid, i.e. on the proviso that the implied direction of causation (viz. from P to R) is established by other means.

However, if it turns out that a figure 2 or 3 conclusion is found not to satisfy this condition, the underlying implications between the items concerned (P and R) may still in certain cases result in a causative conclusion in the reverse direction. Such cases are formally predictable, simply by transposition of the premises concerned. If such transposition has some causative conclusion, then the direction of causation implied by that conclusion (i.e. from R to P) will be unconditionally valid. For if there is causation between P and R, it is bound to be in one direction or the other.

a. Strong determinations. If two premises yield the conclusion ‘P is a complete cause of R’, then their transposition will yield the converse conclusion ‘R is a necessary cause of P’. If we do not know the direction of causation, we cannot know which of these conclusions is the correct one, but we do know that at least one of them must be. If we know that it is not this one, then we know it must be that one. Similarly, with the eventual conclusions ‘P is a necessary cause of R’ and ‘R is a complete cause of P’.[1]

b. Weak determinations. If two premises yield the conclusion ‘P (complemented by S) is a partial cause of R’, and this conclusion is found unjustified with regard to the issue of direction of causation, then its converse has to be admitted as valid, viz. ‘R (complemented by notS) is a contingent cause of P’ (note well the change of polarity of the complement). Similarly, if we know that an eventual conclusion of the form ‘P (complemented by S) is a contingent cause of R’ is inapplicable with respect to the issue of direction of causation, then we may affirm ‘R (complemented by notS) is a partial cause of P’ instead.

The following statistics, based on the listings below, are of interest:

· In figure 1, out of 64 conceivable positive moods, 30 are valid and 34 are invalid (of which 10, due to inconsistency in the premises).

· In figure 2, out of 64 conceivable positive moods, 18 are valid and 46 are invalid (of which 6, due to inconsistency in the premises).

· In figure 1, out of 64 conceivable positive moods, 18 are valid and 46 are invalid (of which 10, due to inconsistency in the premises).

Thus, out of the 192 positive moods considered, 66 (34%) are valid and 126 (66%) are invalid. Obviously, in view of this validity rate, such reasoning cannot be left to chance!

2. Moods in Figure 1.

§1. Mood No. 111 = mn/mn/mn.

VALID

Q is a complete and necessary cause of R;

P is a complete and necessary cause of Q;

so, P is a complete and necessary cause of R.

by reduction to moods 155, 166.

No mirror mood.

§2. Mood No. 112 = mn/mq/mq.

VALID

Q is a complete and necessary cause of R;

P (complemented by S) is a complete and contingent cause of Q;

so, P (complemented by S) is a complete and contingent cause of R.

by reduction to moods 118, 155.

No. 113 = mn/np/np (similarly, through 117, 166).

§3. Mood No. 121 = mq/mn/mq.

VALID

Q (complemented by S) is a complete and contingent cause of R;

P is a complete and necessary cause of Q;

so, P (complemented by S) is a complete and contingent cause of R.

by reduction to moods 155, 181.

No. 131 = np/mn/np (similarly, through 166, 171).

§4. Mood No. 122 = mq/mq.

INVALID

Q (complemented by P) is a complete and contingent cause of R;

P (complemented by S) is a complete and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 133 = np/np (similarly).

§5. Mood No. 123 = mq/np.

INVALID

Q (complemented by P) is a complete and contingent cause of R;

P (complemented by S) is a partial and necessary cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 132 = np/mq (similarly).

§6. Mood No. 114 = mn/pq/pq.

VALID

Q is a complete and necessary cause of R;

P (complemented by S) is a partial and contingent cause of Q;

so, P (complemented by S) is a partial and contingent cause of R.

by reduction to moods 117, 118.

No mirror mood.

§7. Mood No. 141 = pq/mn/pq.

VALID

Q (complemented by S) is a partial and contingent cause of R;

P is a complete and necessary cause of Q;

so, P (complemented by S) is a partial and contingent cause of R.

by reduction to moods 171, 181.

No mirror mood.

§8. Mood No. 124 = mq/pq/q.

VALID

Q (complemented by P) is a complete and contingent cause of R;

P (complemented by S) is a partial and contingent cause of Q;

so, P (complemented by S) is a contingent cause of R.

by reduction to 128 or 184 and by matricial analysis.

No. 134 = np/pq/p (similarly, through 137 or 174 and MA).

§9. Mood No. 142. pq/mq.

INVALID

Q (complemented by P) is a partial and contingent cause of R;

P (complemented by S) is a complete and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 143 = pq/np (similarly).

§10. Mood No. 144 = pq/pq/pq.

VALID

Q (complemented by P) is a partial and contingent cause of R;

P (complemented by S) is a partial and contingent cause of Q;

so, P (complemented by S) is a partial and contingent cause of R.

by reduction to moods 147+148, or 174+184.

No mirror mood.

§11. Mood No. 115 = mn/m/m.

VALID

Q is a complete and necessary cause of R;

P is a complete cause of Q;

so, P is a complete cause of R.

by reduction to moods 111, 112, 155.

No. 116 = mn/n/n (similarly, through 111, 113, 166).

§12. Mood No. 151 = m/mn/m.

VALID

Q is a complete cause of R;

P is a complete and necessary cause of Q;

so, P is a complete cause of R.

by reduction to moods 111, 121, 155.

No. 161 = n/mn/n (similarly, through 111, 131, 166).

§13. Mood No. 125 = mq/m/m.

VALID

Q (complemented by S) is a complete and contingent cause of R;

P is a complete cause of Q;

so, P is a complete cause of R.

by reduction to 121, 155 and by matricial analysis.

No. 136 = np/n/n (similarly, through 131, 166 and MA).

§14. Mood No. 126 = mq/n.

INVALID

Q (complemented by S) is a complete and contingent cause of R;

P is a necessary cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 121 and by matricial analysis.

No. 135 = np/m (similarly, through 131 and MA).

§15. Mood No. 152 = m/mq/m.

VALID

Q is a complete cause of R;

P (complemented by S) is a complete and contingent cause of Q;

so, P is a complete cause of R.

by reduction to 112, 155 and by matricial analysis.

No. 163 = n/np/n (similarly, through 113, 166 and MA).

§16. Mood No. 153 = m/np.

INVALID

Q is a complete cause of R;

P (complemented by S) is a partial and necessary cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 113 and by matricial analysis.

No. 162 = n/mq (similarly, through 112 and MA).

§17. Mood No. 117 = mn/p/p.

VALID

Q is a complete and necessary cause of R;

P (complemented by S) is a partial cause of Q;

so, P (complemented by S) is a partial cause of R.

by reduction to 113, 114 and by matricial analysis.

No. 118 = mn/q/q (similarly, through 112, 114 and MA).

§18. Mood No. 171 = p/mn/p.

VALID

Q (complemented by S) is a partial cause of R;

P is a complete and necessary cause of Q;

so, P (complemented by S) is a partial cause of R.

by reduction to 131, 141 and by matricial analysis.

No. 181 = q/mn/q (similarly, through 121, 141 and MA).

§19. Mood No. 127 = mq/p.

INVALID

Q (complemented by P) is a complete and contingent cause of R;

P (complemented by S) is a partial cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 124 and by matricial analysis.

No. 138 = np/q (similarly, through 134 and MA).

§20. Mood No. 128 = mq/q/q.

VALID

Q (complemented by P) is a complete and contingent cause of R;

P (complemented by S) is a contingent cause of Q;

so, P (complemented by S) is a contingent cause of R.

by reduction to 122, 124 and by matricial analysis.

No. 137 = np/p/p (similarly, through 133, 134 and MA).

§21. Mood No. 172 = p/mq.

INVALID

Q (complemented by P) is a partial cause of R;

P (complemented by S) is a complete and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 183 = q/np (similarly).

§22. Mood No. 173 = p/np.

INVALID

Q (complemented by P) is a partial cause of R;

P (complemented by S) is a partial and necessary cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 182 = q/mq (similarly).

§23. Mood No. 145 = pq/m.

INVALID

Q (complemented by S) is a partial and contingent cause of R;

P is a complete cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 141 and by matricial analysis.

No. 146 = pq/n (similarly, through 141 and MA).

§24. Mood No. 154 = m/pq.

INVALID

Q is a complete cause of R;

P (complemented by S) is a partial and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to 114, 124 and by matricial analysis.

No. 164 = n/pq (similarly, through 114, 134 and MA).

§25. Mood No. 147 = pq/p/p.

VALID

Q (complemented by P) is a partial and contingent cause of R;

P (complemented by S) is a partial cause of Q;

so, P (complemented by S) is a partial cause of R.

by reduction to mood 144 and by matricial analysis.

No. 148 = pq/q/q (similarly, through 144 and MA).

§26. Mood No. 174 = p/pq/p.

VALID

Q (complemented by P) is a partial cause of R;

P (complemented by S) is a partial and contingent cause of Q;

so, P (complemented by S) is a partial cause of R.

by reduction to mood 134 and by matricial analysis.

No. 184 = q/pq/q (similarly, through 124 and MA).

§27. Mood No. 155 = m/m/m.

VALID

Q is a complete cause of R;

P is a complete cause of Q;

so, P is a complete cause of R.

by reduction to 111, 112 and by matricial analysis.

No. 166 = n/n/n (similarly, through 111, 113 and MA).

§28. Mood No. 156 = m/n.

INVALID

Q is a complete cause of R;

P is a necessary cause of Q;

does it follow that P is a cause of R? No!

by reduction to moods 111, 113, 121.

No. 165 = n/m (similarly, through 111, 112, 131).

§29. Mood No. 157 = m/p.

INVALID

Q is a complete cause of R;

P (complemented by S) is a partial cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 113, 114, 124.

No. 168 = n/q (similarly, through 112, 114, 134).

§30. Mood No. 158 = m/q.

INVALID

Q is a complete cause of R;

P (complemented by S) is a contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 112, 114, 152.

No. 167 = n/p (similarly, through 113, 114, 163).

§31. Mood No. 175 = p/m.

INVALID

Q (complemented by S) is a partial cause of R;

P is a complete cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 131, 135.

No. 186 = q/n (similarly, through 121, 126).

§32. Mood No. 176 = p/n.

INVALID

Q (complemented by S) is a partial cause of R;

P is a necessary cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 131, 136, 141.

No. 185 = q/m (similarly, through 121, 125, 141).

§33. Mood No. 177 = p/p.

INVALID

Q (complemented by P) is a partial cause of R;

P (complemented by S) is a partial cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to 133, 134 and by matricial analysis.

No. 188 = q/q (similarly, through 122, 124 and MA).

§34. Mood No. 178 = p/q.

INVALID

Q (complemented by P) is a partial cause of R;

P (complemented by S) is a contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 134, 138.

No. 187 = q/p (similarly, through 124, 127).

3. Moods in Figure 2.

§1. Mood No. 211 = mn/mn/mn.

VALID

R is a complete and necessary cause of Q;

P is a complete and necessary cause of Q;

so, P is a complete and necessary cause of R.

by reduction to mood 111.

No mirror mood.

§2. Mood No. 212 = mn/mq/mq.

VALID

R is a complete and necessary cause of Q;

P (complemented by S) is a complete and contingent cause of Q;

so, P (complemented by S) is a complete and contingent cause of R.

by reduction to mood 112.

No. 213 = mn/np/np (similarly, through 113).

§3. Mood No. 221 = mq/mn/n.

VALID

R (complemented by S) is a complete and contingent cause of Q;

P is a complete and necessary cause of Q;

so, P is a necessary cause of R.

by reduction to mood 256 and by matricial analysis.

No. 231 = np/mn/m (similarly, through 265 and MA).

§4. Mood No. 222 = mq/mq.

INVALID

R (complemented by P) is a complete and contingent cause of Q;

P (complemented by S) is a complete and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by matricial analysis.

No. 233 = np/np (similarly, through MA).

§5. Mood No. 223 = mq/np.

INVALID

R (complemented by P) is a complete and contingent cause of Q;

P (complemented by S) is a partial and necessary cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 232 = np/mq (similarly).

§6. Mood No. 214 = mn/pq/pq.

VALID

R is a complete and necessary cause of Q;

P (complemented by S) is a partial and contingent cause of Q;

so, P (complemented by S) is a partial and contingent cause of R.

by reduction to mood 114.

No mirror mood.

§7. Mood No. 241 = pq/mn.

INVALID

R (complemented by S) is a partial and contingent cause of Q;

P is a complete and necessary cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by matricial analysis.

No mirror mood.

§8. Mood No. 224 = mq/pq.

INVALID

R (complemented by P) is a complete and contingent cause of Q;

P (complemented by S) is a partial and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by matricial analysis.

No. 234 = np/pq (similarly, through MA).

§9. Mood No. 242. pq/mq.

INVALID

R (complemented by P) is a partial and contingent cause of Q;

P (complemented by S) is a complete and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 243 = pq/np (similarly).

§10. Mood No. 244 = pq/pq.

INVALID

R (complemented by P) is a partial and contingent cause of Q;

P (complemented by S) is a partial and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by matricial analysis.

No mirror mood.

§11. Mood No. 215 = mn/m/m.

VALID

R is a complete and necessary cause of Q;

P is a complete cause of Q;

so, P is a complete cause of R.

by reduction to mood 115.

No. 216 = mn/n/n (similarly, through 116).

§12. Mood No. 251 = m/mn/n.

VALID

R is a complete cause of Q;

P is a complete and necessary cause of Q;

so, P is a necessary cause of R.

by reduction to mood 161.

No. 261 = n/mn/m (similarly, through 151).

§13. Mood No. 225 = mq/m.

INVALID

R (complemented by S) is a complete and contingent cause of Q;

P is a complete cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 221, 222.

No. 236 = np/n (similarly, through 231, 233).

§14. Mood No. 226 = mq/n/n.

VALID

R (complemented by S) is a complete and contingent cause of Q;

P is a necessary cause of Q;

so, P is a necessary cause of R.

by reduction to moods 221, 256.

No. 235 = np/m/m (similarly, through 231, 265).

§15. Mood No. 252 = m/mq.

INVALID

R is a complete cause of Q;

P (complemented by S) is a complete and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 162.

No. 263 = n/np (similarly, through 153).

§16. Mood No. 253 = m/np/n.

VALID

R is a complete cause of Q;

P (complemented by S) is a partial and necessary cause of Q;

so, P is a necessary cause of R.

by reduction to mood 163.

No. 262 = n/mq/m (similarly, through 152).

§17. Mood No. 217 = mn/p/p.

VALID

R is a complete and necessary cause of Q;

P (complemented by S) is a partial cause of Q;

so, P (complemented by S) is a partial cause of R.

by reduction to mood 117.

No. 218 = mn/q/q (similarly, through 118).

§18. Mood No. 271 = p/mn.

INVALID

R (complemented by S) is a partial cause of Q;

P is a complete and necessary cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 231, 241.

No. 281 = q/mn (similarly, through 221, 241).

§19. Mood No. 227 = mq/p.

INVALID

R (complemented by P) is a complete and contingent cause of Q;

P (complemented by S) is a partial cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 224.

No. 238 = np/q (similarly, through 234).

§20. Mood No. 228 = mq/q.

INVALID

R (complemented by P) is a complete and contingent cause of Q;

P (complemented by S) is a contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 222.

No. 237 = np/p (similarly, through 233).

§21. Mood No. 272 = p/mq.

INVALID

R (complemented by P) is a partial cause of Q;

P (complemented by S) is a complete and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 283 = q/np (similarly).

§22. Mood No. 273 = p/np.

INVALID

R (complemented by P) is a partial cause of Q;

P (complemented by S) is a partial and necessary cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 233.

No. 282 = q/mq (similarly, through 222).

§23. Mood No. 245 = pq/m.

INVALID

R (complemented by S) is a partial and contingent cause of Q;

P is a complete cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 241.

No. 246 = pq/n (similarly, through 241).

§24. Mood No. 254 = m/pq.

INVALID

R is a complete cause of Q;

P (complemented by S) is a partial and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 164.

No. 264 = n/pq (similarly, through 154).

§25. Mood No. 247 = pq/p.

INVALID

R (complemented by P) is a partial and contingent cause of Q;

P (complemented by S) is a partial cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 241.

No. 248 = pq/q (similarly, through 244).

§26. Mood No. 274 = p/pq.

INVALID

R (complemented by P) is a partial cause of Q;

P (complemented by S) is a partial and contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 234.

No. 284 = q/pq (similarly, through 224).

§27. Mood No. 255 = m/m.

INVALID

R is a complete cause of Q;

P is a complete cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 165.

No. 266 = n/n (similarly, through 156).

§28. Mood No. 256 = m/n/n.

VALID

R is a complete cause of Q;

P is a necessary cause of Q;

so, P is a necessary cause of R.

by reduction to mood 166.

No. 265 = n/m/m (similarly, through 155).

§29. Mood No. 257 = m/p.

INVALID

R is a complete cause of Q;

P (complemented by S) is a partial cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 167.

No. 268 = n/q (similarly, through 158).

§30. Mood No. 258 = m/q.

INVALID

R is a complete cause of Q;

P (complemented by S) is a contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 168.

No. 267 = n/p (similarly, through 157).

§31. Mood No. 275 = p/m.

INVALID

R (complemented by S) is a partial cause of Q;

P is a complete cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 231, 241.

No. 286 = q/n (similarly, through 221, 241).

§32. Mood No. 276 = p/n.

INVALID

R (complemented by S) is a partial cause of Q;

P is a necessary cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 231, 233.

No. 285 = q/m (similarly, through 221, 222).

§33. Mood No. 277 = p/p.

INVALID

R (complemented by P) is a partial cause of Q;

P (complemented by S) is a partial cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 233.

No. 288 = q/q (similarly, through 222).

§34. Mood No. 278 = p/q.

INVALID

R (complemented by P) is a partial cause of Q;

P (complemented by S) is a contingent cause of Q;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 234.

No. 287 = q/p (similarly, through 224).

4. Moods in Figure 3.

§1. Mood No. 311 = mn/mn/mn.

VALID

Q is a complete and necessary cause of R;

Q is a complete and necessary cause of P;

so, P is a complete and necessary cause of R.

by reduction to mood 111.

No mirror mood.

§2. Mood No. 312 = mn/mq/n.

VALID

Q is a complete and necessary cause of R;

Q (complemented by S) is a complete and contingent cause of P;

so, P is a necessary cause of R.

by reduction to mood 365 and matricial analysis.

No. 313 = mn/np/m (similarly, through 356 and MA).

§3. Mood No. 321 = mq/mn/mq.

VALID

Q (complemented by S) is a complete and contingent cause of R;

Q is a complete and necessary cause of P;

so, P (complemented by S) is a complete and contingent cause of R.

by reduction to mood 121.

No. 331 = np/mn/np (similarly, through 131).

§4. Mood No. 322 = mq/mq.

INVALID

Q (complemented by P) is a complete and contingent cause of R;

Q (complemented by S) is a complete and contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 333 = np/np (similarly).

§5. Mood No. 323 = mq/np.

INVALID

Q (complemented by P) is a complete and contingent cause of R;

Q (complemented by S) is a partial and necessary cause of P;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 332 = np/mq (similarly).

§6. Mood No. 314 = mn/pq.

INVALID

Q is a complete and necessary cause of R;

Q (complemented by S) is a partial and contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by matricial analysis.

No mirror mood.

§7. Mood No. 341 = pq/mn/pq.

VALID

Q (complemented by S) is a partial and contingent cause of R;

Q is a complete and necessary cause of P;

so, P (complemented by S) is a partial and contingent cause of R.

by reduction to mood 141.

No mirror mood.

§8. Mood No. 324 = mq/pq.

INVALID

Q (complemented by P) is a complete and contingent cause of R;

Q (complemented by S) is a partial and contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by matricial analysis.

No. 334 = np/pq (similarly, through MA).

§9. Mood No. 342. pq/mq.

INVALID

Q (complemented by P) is a partial and contingent cause of R;

Q (complemented by S) is a complete and contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 343 = pq/np (similarly).

§10. Mood No. 344 = pq/pq.

INVALID

Q (complemented by P) is a partial and contingent cause of R;

Q (complemented by S) is a partial and contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by matricial analysis.

No mirror mood.

§11. Mood No. 315 = mn/m/n.

VALID

Q is a complete and necessary cause of R;

Q is a complete cause of P;

so, P is a necessary cause of R.

by reduction to mood 116.

No. 316 = mn/n/m (similarly, through 115).

§12. Mood No. 351 = m/mn/m.

VALID

Q is a complete cause of R;

Q is a complete and necessary cause of P;

so, P is a complete cause of R.

by reduction to moods 151.

No. 361 = n/mn/n (similarly, through 161).

§13. Mood No. 325 = mq/m.

INVALID

Q (complemented by S) is a complete and contingent cause of R;

Q is a complete cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 126.

No. 336 = np/n (similarly, through 135).

§14. Mood No. 326 = mq/n/m.

VALID

Q (complemented by S) is a complete and contingent cause of R;

Q is a necessary cause of P;

so, P is a complete cause of R.

by reduction to mood 125.

No. 335 = np/m/n (similarly, through 136).

§15. Mood No. 352 = m/mq.

INVALID

Q is a complete cause of R;

Q (complemented by S) is a complete and contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 312 and by matricial analysis.

No. 363 = n/np (similarly, through 313 and MA).

§16. Mood No. 353 = m/np/m.

VALID

Q is a complete cause of R;

Q (complemented by S) is a partial and necessary cause of P;

so, P is a complete cause of R.

by reduction to moods 313, 356.

No. 362 = n/mq/n (similarly, through 312, 365).

§17. Mood No. 317 = mn/p.

INVALID

Q is a complete and necessary cause of R;

Q (complemented by S) is a partial cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 313, 314.

No. 318 = mn/q (similarly, through 312, 314).

§18. Mood No. 371 = p/mn/p.

VALID

Q (complemented by S) is a partial cause of R;

Q is a complete and necessary cause of P;

so, P (complemented by S) is a partial cause of R.

by reduction to mood 171.

No. 381 = q/mn/q (similarly, through 181).

§19. Mood No. 327 = mq/p.

INVALID

Q (complemented by P) is a complete and contingent cause of R;

Q (complemented by S) is a partial cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 324.

No. 338 = np/q (similarly, through 334).

§20. Mood No. 328 = mq/q.

INVALID

Q (complemented by P) is a complete and contingent cause of R;

Q (complemented by S) is a contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 324.

No. 337 = np/p (similarly, through 334).

§21. Mood No. 372 = p/mq.

INVALID

Q (complemented by P) is a partial cause of R;

Q (complemented by S) is a complete and contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 383 = q/np (similarly).

§22. Mood No. 373 = p/np.

INVALID

Q (complemented by P) is a partial cause of R;

Q (complemented by S) is a partial and necessary cause of P;

does it follow that P is (complemented by S) a cause of R? No!

due to inconsistency of premises.

No. 382 = q/mq (similarly).

§23. Mood No. 345 = pq/m.

INVALID

Q (complemented by S) is a partial and contingent cause of R;

Q is a complete cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 146.

No. 346 = pq/n (similarly, through 145).

§24. Mood No. 354 = m/pq.

INVALID

Q is a complete cause of R;

Q (complemented by S) is a partial and contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 314.

No. 364 = n/pq (similarly, through 314).

§25. Mood No. 347 = pq/p.

INVALID

Q (complemented by P) is a partial and contingent cause of R;

Q (complemented by S) is a partial cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 344.

No. 348 = pq/q (similarly, through 344).

§26. Mood No. 374 = p/pq.

INVALID

Q (complemented by P) is a partial cause of R;

Q (complemented by S) is a partial and contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 334.

No. 384 = q/pq (similarly, through 324).

§27. Mood No. 355 = m/m.

INVALID

Q is a complete cause of R;

Q is a complete cause of P;

does it follow that P is a cause of R? No!

by reduction to mood 156.

No. 366 = n/n (similarly, through 165).

§28. Mood No. 356 = m/n/m.

VALID

Q is a complete cause of R;

Q is a necessary cause of P;

so, P is a complete cause of R.

by reduction to mood 155.

No. 365 = n/m/n (similarly, through 166).

§29. Mood No. 357 = m/p.

INVALID

Q is a complete cause of R;

Q (complemented by S) is a partial cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 313, 314.

No. 368 = n/q (similarly, through 312, 314).

§30. Mood No. 358 = m/q.

INVALID

Q is a complete cause of R;

Q (complemented by S) is a contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to moods 312, 314.

No. 367 = n/p (similarly, through 313, 314).

§31. Mood No. 375 = p/m.

INVALID

Q (complemented by S) is a partial cause of R;

Q is a complete cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 176.

No. 386 = q/n (similarly, through 185).

§32. Mood No. 376 = p/n.

INVALID

Q (complemented by S) is a partial cause of R;

Q is a necessary cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 175.

No. 385 = q/m (similarly, through 186).

§33. Mood No. 377 = p/p.

INVALID

Q (complemented by P) is a partial cause of R;

Q (complemented by S) is a partial cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 334.

No. 388 = q/q (similarly, through 324).

§34. Mood No. 378 = p/q.

INVALID

Q (complemented by P) is a partial cause of R;

Q (complemented by S) is a contingent cause of P;

does it follow that P is (complemented by S) a cause of R? No!

by reduction to mood 334.

No. 387 = q/p (similarly, through 324).



[1] We can on this basis anticipate, in figures 2 and 3, the validity or invalidity of some moods on the basis of others. For the order of the premises in these figures is arbitrary.

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2016-01-12T06:15:01+00:00