THE LOGIC OF CAUSATION

Phase One: Macroanalysis

Chapter 8 –Matricial Analyses.

1. Matricial Analysis.

2. Crucial Matricial Analyses in Figure 1.

Evaluation of mood # 221.(Similarly,mutadis mutandis, formood # 231.)

Major premise: R is a complete and (complemented by S) a contingent cause of Q:

(i)If R, then Q;

(ii)if notR, not-then Q;

(iii)where: R is possible.

(iv)If (notR + notS), then notQ;

(v)if (R + notS), not-then notQ;

(vi)if (notR + S), not-then notQ;

(vii)where: (notR + notS) is possible.

Minor premise: P is a complete and necessary cause of Q:

(viii)If P, then Q;

(ix)if notP, not-then Q;

(x)where: P is possible.

(xi)If notP, then notQ;

(xii)if P, not-then notQ;

(xiii)where: notP is possible.

Putative conclusionis P a complete or (complemented by S) a partial cause of R?

NO! P is not implied to be a complete cause of R:

 If P, then R? denied by (vi) + (xi); if notP, not-then R implied by (iv) + (vii) + (viii); where: P is possible same as (x).

Nor (complemented by S) a partial cause of R:

 If (P + S), then R? denied by (vi) + (xi); if (notP + S), not-then R? open; if (P + notS), not-then R? denied by (iv) + (viii); where: (P + S) is possible implied by (vi) + (xi).

 Table 8.16.Evaluation of mood 221. P Q R S see (i) + (iii), or (viii) + (x), or (xii) P Q R notS implied possible by (v) + (xi) P Q notR S implied possible by (vi) + (xi) P Q notR notS implied impossible by (iv) P notQ R S implied impossible by (i) or (viii) P notQ R notS implied impossible by (i) or (viii) P notQ notR S implied impossible by (viii) P notQ notR notS implied impossible by (viii) notP Q R S implied impossible by (xi) notP Q R notS implied impossible by (xi) notP Q notR S implied impossible by (xi) notP Q notR notS implied impossible by (iv) or (xi) notP notQ R S implied impossible by (i) notP notQ R notS implied impossible by (i) notP notQ notR S see (ii), or (ix), or (xi) + (xiii) notP notQ notR notS implied possible by (iv) + (vii) + (viii)

Evaluation of mood # 222.(Similarly,mutadis mutandis, formood # 233.)

Major premise: R is a complete and (complemented by P) a contingent cause of Q:

(i)If R, then Q;

(ii)if notR, not-then Q;

(iii)where: R is possible.

(iv)If (notR + notP), then notQ;

(v)if (R + notP), not-then notQ;

(vi)if (notR + P), not-then notQ;

(vii)where: (notR + notP) is possible.

Minor premise: P is a complete and (complemented by S) a contingent cause of Q:

(viii)If P, then Q;

(ix)if notP, not-then Q;

(x)where: P is possible.

(xi)If (notP + notS), then notQ;

(xii)if (P + notS), not-then notQ;

(xiii)if (notP + S), not-then notQ;

(xiv)where: (notP + notS) is possible.

Putative conclusionis P (complemented by S) a cause of R?

NO! P is not implied to be a complete cause of R:

 If P, then R? open; if notP, not-then R implied by (i) + (xi) + (xiv); where: P is possible same as (x).

Nor (complemented by S) a partial cause of R:

 If (P + S), then R? open; if (notP + S), not-then R? open; if (P + notS), not-then R? open; where: (P + S) is possible? open.

Nor a necessary cause of R:

 If notP, then notR? denied by (v) + (xi), or (iv) + (xiii); if P, not-then notR? open; where: notP is possible implied by (v) or (vii) or (ix) or (xiii) or (xiv).

Nor (complemented by S) a contingent cause of R:

 If (notP + notS), then notR implied by (i) + (xi); if (P + notS), not-then notR? open; if (notP + S), not-then notR implied by (v) + (xi), or (iv) + (xiii); where: (notP + notS) is possible same as (xiv).

 Table 8.17.Evaluation of mood 222. P Q R S see (i) + (iii), or (viii) + (x) P Q R notS see (i) + (iii), or (viii) + (x), or (xii) P Q notR S see (vi), or (viii) + (x) P Q notR notS see (vi), or (viii) + (x), or (xii) P notQ R S implied impossible by (i) or (viii) P notQ R notS implied impossible by (i) or (viii) P notQ notR S implied impossible by (viii) P notQ notR notS implied impossible by (viii) notP Q R S implied possible by (v) + (xi), or (iv) + (xiii) notP Q R notS implied impossible by (xi) notP Q notR S implied impossible by (iv) notP Q notR notS implied impossible by (iv) or (xi) notP notQ R S implied impossible by (i) notP notQ R notS implied impossible by (i) notP notQ notR S see (ii), or (iv) + (vii), or (ix) notP notQ notR notS implied possible by (i) + (xi) + (xiv)

Evaluation of mood # 224.(Similarly,mutadis mutandis, formood # 234.)

Major premise: R is a complete and (complemented by P) a contingent cause of Q:

(i)If R, then Q;

(ii)if notR, not-then Q;

(iii)where: R is possible.

(iv)If (notR + notP), then notQ;

(v)if (R + notP), not-then notQ;

(vi)if (notR + P), not-then notQ;

(vii)where: (notR + notP) is possible.

Minor premise: P (complemented by S) is a partial and contingent cause of Q:

(viii)If (P + S), then Q;

(ix)if (notP + S), not-then Q;

(x)if (P + notS), not-then Q;

(xi)where: (P + S) is possible.

(xii)If (notP + notS), then notQ;

(xiii)if (P + notS), not-then notQ;

(xiv)if (notP + S), not-then notQ;

(xv)where: (notP + notS) is possible.

Putative conclusionis P (complemented by S) a cause of R?

NO! P is not implied to be a complete cause of R:

 If P, then R? denied by (i) + (x); if notP, not-then R implied by (i)+(ix), or (i)+(xii)+(xv); where: P is possible implied by (vi) or (x) or (xi) or (xiii).

Nor (complemented by S) a partial cause of R:

 If (P + S), then R? open; if (notP + S), not-then R implied by (i) + (ix); if (P + notS), not-then R implied by (i) + (x); where: (P + S) is possible same as (xi).

Nor a necessary cause of R:

 If notP, then notR? denied by (iv) + (xiv), or (v) + (xii); if P, not-then notR? open; where: notP is possible implied by (v) or (vii) or (ix) or (xiv) or (xv).

Nor (complemented by S) a contingent cause of R:

 If (notP + notS), then notR implied by (i) + (xii); if (P + notS), not-then notR? open; if (notP + S), not-then notR implied by (iv) + (xiv), or (v) + (xii); where: (notP + notS) is possible same as (xv).

 Table 8.18.Evaluation of mood 224. P Q R S see (i) + (iii), or (viii) + (xi) P Q R notS see (i) + (iii), or (xiii) P Q notR S see (vi), or (viii) + (xi) P Q notR notS see (vi), or (xiii) P notQ R S implied impossible by (i) or (viii) P notQ R notS implied impossible by (i) P notQ notR S implied impossible by (viii) P notQ notR notS implied possible by (i) + (x) notP Q R S implied possible by (iv) + (xiv), or (v) + (xii) notP Q R notS implied impossible by (xii) notP Q notR S implied impossible by (iv) notP Q notR notS implied impossible by (iv) or (xii) notP notQ R S implied impossible by (i) notP notQ R notS implied impossible by (i) notP notQ notR S implied possible by (i) + (ix) notP notQ notR notS implied possible by (i) + (xii) + (xv)

Evaluation of mood # 241.

Major premise: R (complemented by S) is a partial and contingent cause of Q:

(i)If (R + S), then Q;

(ii)if (notR + S), not-then Q;

(iii)if (R + notS), not-then Q;

(iv)where: (R + S) is possible.

(v)If (notR + notS), then notQ;

(vi)if (R + notS), not-then notQ;

(vii)if (notR + S), not-then notQ;

(viii)where: (notR + notS) is possible.

Minor premise: P is a complete and necessary cause of Q:

(ix)If P, then Q;

(x)if notP, not-then Q;

(xi)where: P is possible.

(xii)If notP, then notQ;

(xiii)if P, not-then notQ;

(xiv)where: notP is possible.

Putative conclusionis P (complemented by S) a cause of R?

NO! P is not implied to be a complete cause of R:

 If P, then R? denied by (vii) + (xii); if notP, not-then R implied by (ii)+(ix), or (v)+(viii)+(ix); where: P is possible same as (xi).

Nor (complemented by S) a partial cause of R:

 If (P + S), then R? denied by (vii) + (xii); if (notP + S), not-then R implied by (ii) + (ix); if (P + notS), not-then R? denied by (v) + (ix); where: (P + S) is possible impl. by (i)+(iv)+(xii), or (vii)+(xii).

Nor a necessary cause of R:

 If notP, then notR? denied by (iii) + (ix); if P, not-then notR impl. by (i)+(iv)+(xii), or (vi)+(xii); where: notP is possible same as (xiv).

Nor (complemented by S) a contingent cause of R:

 If (notP + notS), then notR? denied by (iii) + (ix); if (P + notS), not-then notR implied by (vi) + (xii); if (notP + S), not-then notR? denied by (i) + (xii); where: (notP + notS) is possible impl. by (iii)+(ix), or (v)+(viii)+(ix).

 Table 8.19.Evaluation of mood 241. P Q R S implied possible by (i) + (iv) + (xii) P Q R notS implied possible by (vi) + (xii) P Q notR S implied possible by (vii) + (xii) P Q notR notS implied impossible by (v) P notQ R S implied impossible by (i) or (ix) P notQ R notS implied impossible by (ix) P notQ notR S implied impossible by (ix) P notQ notR notS implied impossible by (ix) notP Q R S implied impossible by (xii) notP Q R notS implied impossible by (xii) notP Q notR S implied impossible by (xii) notP Q notR notS implied impossible by (v) or (xii) notP notQ R S implied impossible by (i) notP notQ R notS implied possible by (iii) + (ix) notP notQ notR S implied possible by (ii) + (ix) notP notQ notR notS implied possible by (v) + (viii) + (ix)

Evaluation of mood # 244.

Major premise: R (complemented by P) is a partial and contingent cause of Q:

(i)If (R + P), then Q;

(ii)if (notR + P), not-then Q;

(iii)if (R + notP), not-then Q;

(iv)where: (R + P) is possible.

(v)If (notR + notP), then notQ;

(vi)if (R + notP), not-then notQ;

(vii)if (notR + P), not-then notQ;

(viii)where: (notR + notP) is possible.

Minor premise: P (complemented by S) is a partial and contingent cause of Q:

(ix)If (P + S), then Q;

(x)if (notP + S), not-then Q;

(xi)if (P + notS), not-then Q;

(xii)where: (P + S) is possible.

(xiii)If (notP + notS), then notQ;

(xiv)if (P + notS), not-then notQ;

(xv)if (notP + S), not-then notQ;

(xvi)where: (notP + notS) is possible.

Putative conclusionis P (complemented by S) a cause of R?

NO! P is not implied to be a complete cause of R:

 If P, then R? denied by (i) + (xi), or (ii) + (ix); if notP, not-then R? open; where: P is possible implied by (ii) or (iv) or (vii) or (xi) or (xii) or (xiv).

Nor (complemented by S) a partial cause of R:

 If (P + S), then R? open; if (notP + S), not-then R? open; if (P + notS), not-then R implied by (i) + (xi), or (ii) + (ix); where: (P + S) is possible same as (xii).

Nor a necessary cause of R:

 If notP, then notR? denied by (v) + (xv), or (vi) + (xiii); if P, not-then notR? open; where: notP is possible implied by (iii) or (vi) or (viii) or (x) or (xv) or (xvi).

Nor (complemented by S) a contingent cause of R:

 If (notP + notS), then notR? open; if (P + notS), not-then notR? open; if (notP + S), not-then notR implied by (v) + (xv), or (vi) + (xiii); where: (notP + notS) is possible same as (xvi).

 Table 8.20.Evaluation of mood 244. P Q R S see (i) + (iv), or (ix) + (xii) P Q R notS see (i) + (iv), or (xiv) P Q notR S see (vii), or (ix) + (xii) P Q notR notS see (vii), or (xiv) P notQ R S implied impossible by (i) or (ix) P notQ R notS implied impossible by (i) P notQ notR S implied impossible by (ix) P notQ notR notS implied possible by (i) + (xi), or (ii) + (ix) notP Q R S implied possible by (v) + (xv), or (vi) + (xiii) notP Q R notS implied impossible by (xiii) notP Q notR S implied impossible by (v) notP Q notR notS implied impossible by (v) or (xiii) notP notQ R S see (iii) or (x) notP notQ R notS see (iii), or (xiii) + (xvi) notP notQ notR S see (v) + (viii), or (x) notP notQ notR notS see (v) + (viii), or (xiii) + (xvi)

Evaluation of mood # 312.(Similarly,mutadis mutandis, formood # 313.)

Major premise: Q is a complete and necessary cause of R:

(i)If Q, then R;

(ii)if notQ, not-then R;

(iii)where: Q is possible.

(iv)If notQ, then notR;

(v)if Q, not-then notR;

(vi)where: notQ is possible.

Minor premise: Q is a complete and (complemented by S) a contingent cause of P:

(vii)If Q, then P;

(viii)if notQ, not-then P;

(ix)where: Q is possible.

(x)If (notQ + notS), then notP;

(xi)if (Q + notS), not-then notP;

(xii)if (notQ + S), not-then notP;

(xiii)where: (notQ + notS) is possible.

Putative conclusion: is P a complete or (complemented by S) a partial cause of R?

NO! P is not implied to be a complete cause of R:

 If P, then R? denied by (iv) + (xii); if notP, not-then R implied by (iv) + (x) + (xiii); where: P is possible implied by (vii) + (ix), or (xi), or (xii).

Nor (complemented by S) a partial cause of R:

 If (P + S), then R? denied by (iv) + (xii); if (notP + S), not-then R? open; if (P + notS), not-then R? denied by (i) + (x); where: (P + S) is possible implied by (xii).

 Table 8.21.Evaluation of mood # 312. P Q R S see (i) + (iii), or (v), or (vii) + (ix) P Q R notS implied possible by (i) + (xi) P Q notR S implied impossible by (i) P Q notR notS implied impossible by (i) P notQ R S implied impossible by (iv) P notQ R notS implied impossible by (iv) or (x) P notQ notR S implied possible by (iv) + (xii) P notQ notR notS implied impossible by (x) notP Q R S implied impossible by (vii) notP Q R notS implied impossible by (vii) notP Q notR S implied impossible by (i) or (vii) notP Q notR notS implied impossible by (i) or (vii) notP notQ R S implied impossible by (iv) notP notQ R notS implied impossible by (iv) notP notQ notR S see (ii), or (iv) + (vi), or (viii) notP notQ notR notS implied possible by (iv) + (x) + (xiii)

Evaluation of mood # 314.

Major premise: Q is a complete and necessary cause of R:

(i)If Q, then R;

(ii)if notQ, not-then R;

(iii)where: Q is possible.

(iv)If notQ, then notR;

(v)if Q, not-then notR;

(vi)where: notQ is possible.

Minor premise: Q (complemented by S) is a partial and contingent cause of P:

(vii)If (Q + S), then P;

(viii)if (notQ + S), not-then P;

(ix)if (Q + notS), not-then P;

(x)where: (Q + S) is possible.

(xi)If (notQ + notS), then notP;

(xii)if (Q + notS), not-then notP;

(xiii)if (notQ + S), not-then notP;

(xiv)where: (notQ + notS) is possible.

Putative conclusion: is P (complemented by S) a cause of R?

NO! P is not implied to be a complete cause of R:

 If P, then R? denied by (iv) + (xiii); if notP, not-then R impl. by (iv)+(viii), or (iv)+(xi)+(xiv); where: P is possible impl. by (vii) + (x), or (xii), or (xiii).

Nor (complemented by S) a partial cause of R:

 If (P + S), then R? denied by (iv) + (xiii); if (notP + S), not-then R implied by (iv) + (viii); if (P + notS), not-then R? denied by (i) + (xi); where: (P + S) is possible implied by (vii) + (x), or (xiii).

Nor a necessary cause of R:

 If notP, then notR? denied by (i) + (ix); if P, not-then notR implied by (i)+(vii)+(x), or (i)+(xii); where: notP is possible impl. by (viii), or (ix), or (xi) + (xiv).

Nor (complemented by S) a contingent cause of R:

 If (notP + notS), then notR? denied by (i) + (ix); if (P + notS), not-then notR implied by (i) + (xii); if (notP + S), not-then notR? denied by (iv) + (vii); where: (notP + notS) is possible implied by (ix), or (xi) + (xiv).

 Table 8.22.Evaluation of mood # 314. P Q R S implied possible by (i) + (vii) + (x) P Q R notS implied possible by (i) + (xii) P Q notR S implied impossible by (i) P Q notR notS implied impossible by (i) P notQ R S implied impossible by (iv) P notQ R notS implied impossible by (iv) or (xi) P notQ notR S implied possible by (iv) + (xiii) P notQ notR notS implied impossible by (xi) notP Q R S implied impossible by (vii) notP Q R notS implied possible by (i) + (ix) notP Q notR S implied impossible by (i) or (vii) notP Q notR notS implied impossible by (i) notP notQ R S implied impossible by (iv) notP notQ R notS implied impossible by (iv) notP notQ notR S implied possible by (iv) + (viii) notP notQ notR notS implied possible by (iv) + (xi) + (xiv)

Evaluation of mood # 324.(Similarly,mutadis mutandis, formood # 334.)

Major premise: Q is a complete and (complemented by P) a contingent cause of R:

(i)If Q, then R;

(ii)if notQ, not-then R;

(iii)where: Q is possible.

(iv)If (notQ + notP), then notR;

(v)if (Q + notP), not-then notR;

(vi)if (notQ + P), not-then notR;

(vii)where: (notQ + notP) is possible.

Minor premise: Q (complemented by S) is a partial and contingent cause of P:

(viii)If (Q + S), then P;

(ix)if (notQ + S), not-then P;

(x)if (Q + notS), not-then P;

(xi)where: (Q + S) is possible.

(xii)If (notQ + notS), then notP;

(xiii)if (Q + notS), not-then notP;

(xiv)if (notQ + S), not-then notP;

(xv)where: (notQ + notS) is possible.

Putative conclusion: is P (complemented by S) a cause of R?

NO! P is not implied to be a complete cause of R:

 If P, then R? open; if notP, not-then R impl. by (iv)+(ix), or (iv)+(xii)+(xv); where: P is possible implied by (vi), or (viii) + (xi), or (xiii), or (xiv).

Nor (complemented by S) a partial cause of R:

 If (P + S), then R? open; if (notP + S), not-then R implied by (iv) + (ix); if (P + notS), not-then R? denied by (i) + (xii); where: (P + S) is possible implied by (vi) + (xii), or (viii) + (xi), or (xiv).

Nor a necessary cause of R:

 If notP, then notR? denied by (v) + (viii), or (i) + (x); if P, not-then notR implied by (i)+(viii)+(xi), or (i)+(xiii), or (vi)+(xii); where: notP is possible implied by (v), or (ix), or (x), or (xii) + (xv).

Nor (complemented by S) a contingent cause of R:

 If (notP + notS), then notR? denied by (v) + (viii), or (i) + (x); if (P + notS), not-then notR implied by (i) + (xiii); if (notP + S), not-then notR? denied by (iv) + (viii); where: (notP + notS) is possible implied by (v) + (viii), or (x), or (xii) + (xv).

 Table 8.23.Evaluation of mood # 324. P Q R S implied possible by (i) + (viii) + (xi) P Q R notS implied possible by (i) + (xiii) P Q notR S implied impossible by (i) P Q notR notS implied impossible by (i) P notQ R S implied possible by (vi) + (xii) P notQ R notS implied impossible by (xii) P notQ notR S see (ii) or (xiv) P notQ notR notS implied impossible by (xii) notP Q R S implied impossible by (viii) notP Q R notS implied possible by (v) + (viii), or (i) + (x) notP Q notR S implied impossible by (i) or (viii) notP Q notR notS implied impossible by (i) notP notQ R S implied impossible by (iv) notP notQ R notS implied impossible by (iv) notP notQ notR S implied possible by (iv) + (ix) notP notQ notR notS implied possible by (iv) + (xii) + (xv)

Evaluation of mood # 344.

Major premise: Q (complemented by P) is a partial and contingent cause of R:

(i)If (Q + P), then R;

(ii)if (notQ + P), not-then R;

(iii)if (Q + notP), not-then R;

(iv)where: (Q + P) is possible.

(v)If (notQ + notP), then notR;

(vi)if (Q + notP), not-then notR;

(vii)if (notQ + P), not-then notR;

(viii)where: (notQ + notP) is possible.

Minor premise: Q (complemented by S) is a partial and contingent cause of P:

(ix)If (Q + S), then P;

(x)if (notQ + S), not-then P;

(xi)if (Q + notS), not-then P;

(xii)where: (Q + S) is possible.

(xiii)If (notQ + notS), then notP;

(xiv)if (Q + notS), not-then notP;

(xv)if (notQ + S), not-then notP;

(xvi)where: (notQ + notS) is possible.

Putative conclusion: is P (complemented by S) a cause of R?

NO! P is not implied to be a complete cause of R:

 If P, then R? denied by (ii) + (xiii); if notP, not-then R implied by (iii)+(ix), or (v)+(x), or (v)+(xiii)+(xvi); where: P is possible implied by (ii), or (iv), or (vii) + (xiii), or (ix) + (xii), or (xiv) or (xv).

Nor (complemented by S) a partial cause of R:

 If (P + S), then R? denied by (ii) + (xiii); if (notP + S), not-then R implied by (v) + (x); if (P + notS), not-then R? denied by (i) + (xiii); where: (P + S) is possible implied by (ii) + (xiii), or (vii) + (xiii), or (ix) + (xii), or (xv).

Nor a necessary cause of R:

 If notP, then notR? denied by (vi) + (ix); if P, not-then notR implied by (i)+(ix)+(xii), or (i)+(xiv), or (vii)+(xiii); where: notP is possible implied by (iii), or (vi), or (viii), or (x), or (xi), or (xiii) + (xvi).

Nor (complemented by S) a contingent cause of R:

 If (notP + notS), then notR? denied by (vi) + (ix); if (P + notS), not-then notR implied by (i) + (xiv); if (notP + S), not-then notR? denied by (v) + (ix); where: (notP + notS) is possible implied by (iii) + (ix), or (vi) + (ix), or (xi), or (xiii) + (xvi).

 Table 8.24.Evaluation of mood # 344. P Q R S implied possible by (i) + (ix) + (xii) P Q R notS implied possible by (i) + (xiv) P Q notR S implied impossible by (i) P Q notR notS implied impossible by (i) P notQ R S implied possible by (vii) + (xiii) P notQ R notS implied impossible by (xiii) P notQ notR S implied possible by (ii) + (xiii) P notQ notR notS implied impossible by (xiii) notP Q R S implied impossible by (ix) notP Q R notS implied possible by (vi) + (ix) notP Q notR S implied impossible by (ix) notP Q notR notS implied possible by (iii) + (ix) notP notQ R S implied impossible by (v) notP notQ R notS implied impossible by (v) notP notQ notR S implied possible by (v) + (x) notP notQ notR notS implied possible by (v) + (xiii) + (xvi)

Evaluation of mood # 352.(Similarly,mutadis mutandis, formood # 363.)

Major premise: Q is a complete cause of R:

(i)If Q, then R;

(ii)if notQ, not-then R;

(iii)where: Q is possible.

Minor premise: Q is a complete and (complemented by S) a contingent cause of P:

(iv)If Q, then P;

(v)if notQ, not-then P;

(vi)where: Q is possible.

(vii)If (notQ + notS), then notP;

(viii)if (Q + notS), not-then notP;

(ix)if (notQ + S), not-then notP;

(x)where: (notQ + notS) is possible.

Putative conclusion: is P a necessary cause of R?

NO! P is not implied to be a necessary cause of R:

 If notP, then notR? open; if P, not-then notR implied by (i) + (viii); where: notP is possible? implied by (v), or (vii) + (x).

 Table 8.25.Evaluation of mood # 352. P Q R S see (i) + (iii), or (iv) + (vi) P Q R notS implied possible by (i) + (viii) P Q notR S implied impossible by (i) P Q notR notS implied impossible by (i) P notQ R S see (ix) P notQ R notS implied impossible by (vii) P notQ notR S see (ii) or (ix) P notQ notR notS implied impossible by (vii) notP Q R S implied impossible by (iv) notP Q R notS implied impossible by (iv) notP Q notR S implied impossible by (i) or (iv) notP Q notR notS implied impossible by (i) or (iv) notP notQ R S see (v) notP notQ R notS see (v), or (vii) + (x) notP notQ notR S see (ii) or (v) notP notQ notR notS see (ii) or (v), or (vii) + (x)

You can purchase a paper copy of this bookat The Logician’s secure online Bookshop.