www.TheLogician.net© Avi Sion All rights reserved

THE LOGIC OF CAUSATION

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

You can BUY online, Amazon.com (in paperback or kindle/.mobi form), at Lulu.com (in hardcover, paperback or e-book / .epub form ), and at many other online stores.

THE LOGIC OF CAUSATION

Phase One: Macroanalysis

Chapter 5 – Causative Syllogism.

1. Causal or Effectual Chains.

2. Some Instructive Examples.

3. Figures and Moods.

1. Causal or Effectual Chains.

The topic of concatenation of causations is an important field of research, though a tedious one. It is important not only to the natural sciences, which need to monitor or trace causal or effectual chains, but also to law and ethics.

To grasp its practical value in legal or ethical discourse, consider this example[1]: a motorist overruns a pedestrian, who in the hospital where he is rushed is additionally the victim of some medical mishap – can the motorist be blamed for the poor pedestrian’s subsequent misfortunes? Such questions can only be convincingly answered through a systematic and wide-ranging reflection on causal logic.

The concept of concatenation refers primarily to ‘chain reactions’: P causes Q, which causes R, and so on; or conversely, R is effected by Q, which is effected by P, and so forth.

Clearly, the concepts of cause and effect here are relative to each other. In the context of deterministic causality, nothing is absolutely a cause or absolutely an effect; it is always the cause or effect of something.

All we wish to point out here is the obvious: that a phenomenon Q which is a cause in relation to another phenomenon R may itself stand as effect in relation to yet a third phenomenon P. Similarly, a phenomenon Q which is an effect in relation to another phenomenon P may itself stand as cause in relation to yet a third phenomenon R.

When we speak in terms of chains like P-Q-R, we stand back from the underlying bipolar relations of cause and effect and focus on the wider picture. The items P, Q, R may then be referred to, more indifferently, as successive links in the chain.

Needless to say, concatenation of events implies but is not implied by the seriality of events (in whatever appropriate sense of the term ‘series’). Furthermore, even knowing that P causes Q and that Q causes R, we cannot presume concatenation. A series P-Q-R may be said to really form a chain, only if we can demonstrate that P, through the intermediary of Q, indeed causes R. This is not always feasible, for as we have seen the verb “causes” has a large variety of meaning.

You cannot just say “P causes Q and Q causes R, therefore P causes R” indiscriminately. This is one reason why a theoretical treatment of causal logic is essential to scientific thinking.

The search for concatenations varies in motive. Sometimes we are looking for the cause(s) of a cause, sometimes for the effect(s) of an effect, sometimes for some intermediary between a cause and an effect. We need not assume at the outset that all phenomena are bound to have causes and effects ad infinitum, nor that there has to be an infinity of intermediaries between any two given items.

A cause without apparent prior cause would be called a primary cause; an effect without apparent posterior effect would be called an ultimate effect. A cause and effect without apparent intermediary would be referred to as immediate or contiguous; if they have an intermediary, they would be referred to as mediated.

If we speculate that Existence as a whole has a Beginning and/or an End, then of course we may speak of that as a First Cause and/or a Last Effect. Likewise, we need not ab initio prejudice the issue concerning specific events within Existence, be it infinite or finite, and at least to start with make allowances for (in some sense) causeless or effectless phenomena.

We have so far mentioned what may be called orderly concatenation. We also search for chains in the context of parallelism of causes, or of effects. We may need to know whether parallel causes or parallel effects are themselves causally related, and thus order them in relation to the initial cause or terminal effect concerned. In such case, we are identifying one of the two causes or two effects (as the case may be) as an intermediary between the other two items.

It should be stressed, however, that the arguments about parallelism considered here cannot strictly-speaking tell us which one of the two causes (or two effects) causes the other; for as we have mentioned in the preceding chapters, sometimes there is a hidden issue of direction of causation to consider. This issue has to be resolved separately, with reference to spatial, temporal, or other conceptual or logical considerations[2]. We shall simply ignore this problem of ordering for now, and regard the tacit condition as always satisfied.

We should, additionally, in passing, mention the phenomenon of spiraling causation, which we commonly refer to as vicious circles. This phenomenon is a special case of concomitant variation[3]. It occurs when an increase or decrease in a cause C (C± x1) causes an increase or decrease in an effect E (E± y1), which in turn causes another increase or decrease in C (C± x1± x2), which in turn causes another increase or decrease in E (E± y1± y2), and so forth.

The spiral need not constitute an infinite chain, even if complete causation is involved at each step, because each of the causations involved is independent of (i.e. not formally implied by) its predecessors, note well. Even so, a spiral may come to a halt because it is in fact implicitly conditional, i.e. partial causation is involved at each step. But we can also conceive of infinite spirals, in the case of ongoing processes continuing as long as the universe lasts.

The problem of causal or effectual chains is, as we shall see, essentially syllogistic. We need to identify which syllogisms involving causative propositions as premises yield such propositions as conclusion. In this research, it is as important to expose the invalidity of certain syllogisms as to identify the valid syllogisms, for inappropriate reasoning is common[4].

2. Some Instructive Examples.

Before undertaking a systematic presentation and evaluation of causative syllogisms, I will propose some formal examples to acquaint the reader with some of the issues involved.

Consider, to begin with, the two causative syllogisms listed below (on the left):

Q is a complete cause of R;

=>

Given that if Q then R

P is a complete cause of Q;

=>

and that if P then Q,

so, P is a complete cause of R.

<=

it follows that if P then R.

Q is a necessary cause of R;

=>

Given that if notQ then notR

P is a necessary cause of Q;

=>

and that if notP then notQ,

so, P is a necessary cause of R.

<=

it follows that if notP then notR.

These typify orderly concatenation. Here, Q may be viewed as an intermediate cause of R after P, or as an intermediate effect of P before R. This arrangement of items is known to logicians as a ‘first figure’ syllogism. The first sentence in each case is called the ‘major premise’; the second one, the ‘minor premise’; the third, the ‘conclusion’.

In each case, notice, the premises and conclusion involve the same strong determination. We know that the conclusion may legitimately be drawn from the premises, because we can readily ‘reduce’ the argument to one previously known to logical science (shown on the left). That is, each premise given in the former implies a premise of the latter, whose conclusion in turn (granting certain provisions) implies that of the former.

The minimal provisions, as we have seen when defining these determinations, is as follows: in the case of complete causation, they are that P be possible and R be unnecessary; and in the case of necessary causation, they are that P be unnecessary and R be possible. We know these provisos are indeed met, in each case, being implied by the minor and major premises, respectively.

Ergo, these syllogisms are ‘valid’, they can be freely used, irrespective of what the items P, Q, R symbolize.

In contrast, consider the following two causative syllogisms:

If Q is a complete cause of R

=>

That if Q then R

and P is a necessary cause of Q,

=>

and if notP then notQ,

how are P and R then related?

yield only “if P, not-then notR”.

If Q is a necessary cause of R

=>

That if notQ then notR

and P is a complete cause of Q,

=>

and if P then Q,

how are P and R then related?

yield only “if notP, not-then R”.

These examples differ from the preceding two in that the premises are of different (though equally strong) determination, note. If we attempt to ‘reduce’ these arguments as before, we find no way to do so. We must thus admit that, in their case, we cannot demonstrably conclude either complete or necessary causation, and it would be misleading to think of the series P-Q-R as a chain. These combinations of premises are therefore ‘invalid’ arguments; we cannot reason with them without risking errors.

To be precise, these two arguments merely teach us that it would be wrong to deduce complete and/or necessary causation; but they do not exclude the possibility of such strong relations between P and R occurring independently of the intermediate item Q. The conclusion “if P, not-then notR” only precludes that “if P, then notR” (i.e. that P and R be incompatible), but not for instance that “if P, then R”. Similarly, the conclusion “if notP, not-then R” only precludes that “if notP, then R” (i.e. that P and R be exhaustive), but not for instance that “if notP, then notR”. Alternatively, for all we know, weaker forms of causation may apply or no causation at all.

Now, consider the following two causative syllogisms:

R is a complete cause of Q;

P is a necessary cause of Q;

therefore, P is a necessary cause of R.

And:

R is a necessary cause of Q;

P is a complete cause of Q;

therefore, P is a complete cause of R.

These typify parallelism of causes. Notice the positions of the items involved, here: Q is an effect in common to P and R (whereas in orderly concatenation it was an effect of P and a cause of R); this is known to logicians as ‘second figure’ argument. In this case, as we shall later show, the syllogisms are valid[5], i.e. logically acceptable, albeit their having premises of different (though equally strong) determinations. The conclusion, notice, has the same determination as the minor premise. On the other hand, as we will show later, if the premises (with P, Q, R in a similar arrangement) have the same determination, i.e. both concern complete causation or both necessary causation, we are not permitted to draw any causative conclusion.

Finally, consider the following two causative syllogisms:

Q is a complete cause of R;

Q is a necessary cause of P;

therefore, P is a complete cause of R.

And:

Q is a necessary cause of R;

Q is a complete cause of P;

therefore, P is a necessary cause of R.

These typify parallelism of effects. Notice the positions of the items involved, here: Q is a cause in common to P and R (whereas in orderly concatenation it was an effect of P and a cause of R, or in parallelism of causes it was an effect in common to P and R); this is known to logicians as ‘third figure’ argument. In this case, as we shall later show, the syllogisms are valid[6], i.e. logically acceptable, albeit their having premises of different (though equally strong) determinations. The conclusion, notice, has the same determination as the major premise. On the other hand, as we will show later, if the premises (with P, Q, R in a similar arrangement) have the same determination, i.e. both concern complete causation or both necessary causation, we are not permitted to draw any causative conclusion.

These examples reveal some of the complexities of causative argument.

We see from them that the ordering of the items involved in the premises affects the logical possibility of drawing a conclusion. In the first figure, two identical strong determinations yield a valid conclusion (of the same determination), whereas a mixture of such determinations is fruitless. In the second and third figures, the opposite is true; and furthermore, these differ from each other, in that a valid conclusion in the second figure follows the determination of the minor premise, whereas one in the third figure follows that of the major premise.

The problem becomes even more complicated when we investigate weak causations, which involve at least three items each (instead of two, as with strong causations). We discover, to give an extreme example, that whatever the figure considered, no conclusion can be drawn from two premises each of which concerns partial or contingent causation only. We then wonder what combinations of premises may be used to draw a conclusion about weak causation.

More broadly, considering that we have to deal with three figures, and eight possible determinations of causation for each premise, we have to examine 3*64=192 combinations, or ‘moods’ (as logicians say). What conclusion, if any, can be drawn from each one of those arguments; and how do we go about demonstrating it? Furthermore, we have so far mentioned syllogisms with only affirmative causative propositions; what of syllogisms involving propositions denying causation or a particular determination of causation?

Clearly, we cannot hope to reason correctly about causation without first dealing with causative syllogism in a thorough and systematic manner, so that we know precisely when an argument is valid and when it is not. If we limit our research to a few frequently used arguments, like those above shown, we will miss many opportunities for valid inference and risk making some invalid inferences. And in view of the volume of the problem, it has to be treated in as global a manner as possible.

This is our task in the next few chapters.

The research is tedious, because causative propositions are, as we have seen, very complex; they are each composed of two or more clauses, and most of these clauses are positive or negative conditional propositions, i.e. themselves complex.

In the simplest syllogisms, those involving strong determinations of causation only, and therefore the minimum number of (i.e. three) items, we can readily reduce causative reasoning to syllogism involving conditional propositions. The latter are reasonably well-known to logicians and to the public at large; a full treatment of them may be found in my work Future Logic.

But soon we find such simple methods inadequate. Syllogisms involving weak determinations or mixtures of strong and weak determinations are too complicated for us to feel secure with the results obtained by means of reduction. For certainty, I have had to develop a more complex method, called matricial analysis.

3. Figures and Moods.

A syllogism, we know thanks to Aristotle, consists of at least two premises and a conclusion. The premises together contain at least three items (terms or theses), at least one of which they have in common, and the conclusion contains at least two items, each of which was contained in a premise not containing the other.

Our job in syllogistic reasoning is to obtain from the premises, i.e. the given data, the information we need to construct the putative conclusion. If the premises, together and without reference to unstated assumptions, justify the conclusion, the syllogism proposed is valid deduction; otherwise it is invalid.

Validation (i.e. showing valid) justifies a form of reasoning; removing any uncertainty we may have about it or teaching us a new way of inference. Invalidation (i.e. showing invalid) is just as important, to contrast valid with invalid moods and thus set the limits of validity, and most of all to prevent us making mistakes in our thinking.

A putative conclusion may be invalid in the way of a non-sequitur, meaning that the conclusion does not conflict with anything in the premises, but just does not logically follow from them. Or, worse, it may be invalid in the way of antinomy, meaning that the conclusion is inconsistent (contradictory or contrary) with something in the premises[7].

In the case of a non-sequitur, we may be able to save the situation by stipulating some condition(s) under which the conclusion would follow; in that event, we may call the conclusion conditionally valid, or add the condition(s) to be satisfied to our premises as an additional premise to obtain an unconditionally valid conclusion, or again consider that we have a disjunctive conclusion whose alternatives include the satisfaction or non-satisfaction of the said condition(s).

In the case of an antinomy, we can redeem things by proposing the contradictory or a contrary of our invalid conclusion as a valid conclusion; if the invalid conclusion is a compound, we may be able to obtain a valid conclusion of the kind desired by negating some element(s) in it.

We are usually able to infer some information from the premises; but if this information does not add up to a causative proposition of some kind, we here consider the conjunction of the premises as a failure. For our task, in the present context, is not an investigation of deduction in the broadest sense, but specifically deduction of causative propositions from other causative propositions. Thus, do not be surprised if a syllogism is declared invalid even though some elements of a putative compound conclusion were inferable.

The evaluations of some moods may seem immediately or intuitively obvious; but some moods are too complicated for that and require careful examination. Some causative syllogisms, as already mentioned, can be validated by direct reduction to already established, non-causative syllogisms. Others are too complex for that, and can only be validated through matricial analysis, i.e. with painstaking reference to their corresponding matrix; this method will be described in detail later. Still others, though complex, can be validated by direct and/or indirect reduction (also known as reduction ad absurdum) to causative syllogisms already validated by other means (namely by matricial analysis).

Aristotle taught us that a syllogism may have one of three figures, according to the placement of the three items (terms or theses) in its premises and conclusion, as follows:

Table 5.1. The figures of (three-item) syllogism.

Figure 1

Figure 2

Figure 3

Major premise:

Q – R

R – Q

Q – R

Minor premise:

P – Q

P – Q

Q – P

Conclusion:

P – R

P – R

P – R

Notice, in each of the figures, the positions of the item found in both premises but not in the conclusion (namely Q; this is known as the middle item). Notice also the various positions of the other two items, one of which (R, the major item) is found only in the major premise (traditionally stated first) and conclusion (traditionally stated last), and the other of which (P, the minor item) is found only in the minor premise (traditionally stated second) and conclusion. The positions of the items tells us which ‘figure’ the reasoning is in.[8]

Each Aristotelian figure refers to three items (P, Q, R). But in the present context we are also dealing with some four-item (P, Q, R, S) arguments, which as we shall see can be combined in three different ways (and many more, which we shall deal with in a later chapter). Thus, we shall have to refer to subfigures. We can call Aristotle’s primary arrangement subfigure (a), and the three additional arrangements subfigures (b), (c), (d).

Table 5.2. Subfigures of each figure.

Subfigures

a

b

c

d

Definitions

both premises

strong only

major premise

strong only

minor premise

strong only

neither premise

strong only

Figure 1

QR

QR

Q(S)R

Q(P)R

PQ

P(S)Q

PQ

P(S)Q

PR

P(S)R

P(S)R

P(S)R

Figure 2

RQ

RQ

R(S)Q

R(P)Q

PQ

P(S)Q

PQ

P(S)Q

PR

P(S)R

P(S)R

P(S)R

Figure 3

QR

QR

Q(S)R

Q(P)R

QP

Q(S)P

QP

Q(S)P

PR

P(S)R

P(S)R

P(S)R

In (a), both premises involve strong determinations only; that is why there are only two items per premise (and in the conclusion). In (b) and (c), one premise (the major or minor, respectively) has only two items (implying the presence of only strong determination) and the other premise (and conclusion) has three items (implying the presence of joint strong and weak, or of only weak, determination). In (d), each premise (and the conclusion) involves three items (implying the absence of only-strong determination).

It is seen that the three-item symbolism (P, Q, R for the minor, middle and major items, respectively) is retained in four-item figures, except that we have an additional item (call it the subsidiary item, symbol S) appearing in a premise and the conclusion: S represents ‘outside interference’, as it were, in relation to the triad P-Q-R.

The important thing to note about this subsidiary item is that though it has to be mentioned in theoretical exposition and evaluation, as here, to place it and judge its impact, it need not be mentioned[9] in practice, because the conclusion follows the premises whatever its content happen to be. That is, the premise concerned and the conclusion need not specify “(complemented by S)”.

On the other hand, the clause “(complemented by P)” in the major premise of subfigures 1d, 2d and 3d, cannot be ignored in practice, since the middle item Q might well cause the major item R with some complement(s) other than the minor item P, rather than with P. Even though P is mentioned in association with Q in the minor premise, that in itself does not imply the causation in the major premise to be true with it: this knowledge must be obtained by other means to enable the inference of the conclusion.

If, in our present context, we specify as well as the figure the precise determination and polarity involved in each of the premises and in a putative conclusion, we have pinpointed the precise mood under discussion. This expression refers to the formal aspects of a syllogism, which distinguish it from all others. Thus, for each figure of syllogism, there are many conceivable moods.

The mood determinations (numbered 1-9 for reference) found in each subfigure are given in the table below. These tell us the determinations of the premises involved, which may be ‘strong only’ (abbreviation, so), a ‘mix of strong and weak’ (sw/ws), or ‘weak only’ (wo). Due to the numbers of items allowed for a premise in each subfigure, the number of determinations found in each subfigure vary.

Table 5.3. Determinations found in each subfigure.

Subfig.

a

b

c

d

Determ.

1

2

3

4

5

6

7

8

9

Major

so

so

so

sw/ws

wo

sw/ws

sw/ws

wo

wo

Minor

so

sw/ws

wo

so

so

sw/ws

wo

sw/ws

wo



There are 64 positive moods per figure, a total of 192 moods in all three of them. The system proposed now is to use three-digit identification numbers, or mood numbers. The hundreds will identify the figure 1, 2 or 3. The tens (#s 1-8, no 0 or 9) will tell us the major premise’s determination. The units (#s 1-8, no 0 or 9) will specify the minor premise’s determination. The subfigures (above labeled a-d) and modes (above labeled 1-9), shown in the preceding tables, are not explicitly mentioned in the mood number, but are tacitly implied by it.

Table 5.4. Subfigures, modes and moods.

Subfig.

Determ.

Premises

Moods Nos.

Qty

a

1

major

minor

strong only

strong only

tens 1, 4, 5

units 1, 4, 5

9

b

2

major

minor

strong only

sw, ws

tens 1, 4, 5

units 2, 3

6

3

major

minor

strong only

weak only

tens 1, 4, 5

units 6, 7, 8

9

c

4

major

minor

sw, ws

strong only

tens 2, 3

units 1, 4, 5

6

5

major

minor

weak only

strong only

tens 6, 7, 8

units 1, 4, 5

9

d

6

major

minor

sw, ws

sw, ws

tens 2, 3

units 2, 3

4

7

major

minor

sw, ws

weak only

tens 2, 3

units 6, 7, 8

6

8

major

minor

weak only

sw, ws

tens 6, 7, 8

units 2, 3

6

9

major

minor

weak only

weak only

tens 6, 7, 8

units 6, 7, 8

9

I could of course have used letters instead of numbers to symbolize the different moods, but fearing to confuse the reader with yet more letter-symbols (the science of logic abounds with them) I have preferred number-symbols. Note that there are no moods numbered 01-10, 19-20, 29-30, 39-40, 49-50, 59-60, 69-70, 79-80, or 89+. The table below clarifies the meaning of each mood number within any given figure.

Table 5.5. Mood numbers in each figure.

Minor

Major premise

premise

mn=1

mq=2

np=3

pq=4

m=5

n=6

p=7

q=8

mn=1

11

21

31

41

51

61

71

81

mq=2

12

22

32

42

52

62

72

82

np=3

13

23

33

43

53

63

73

83

pq=4

14

24

34

44

54

64

74

84

m=5

15

25

35

45

55

65

75

85

n=6

16

26

36

46

56

66

76

86

p=7

17

27

37

47

57

67

77

87

n=8

18

28

38

48

58

68

78

88



It is useful to expand the above table as done below, to show precisely what combination of determinations in the premises each mood number refers to.

Table 5.6. For each figure, mood numbers and determinations of major and minor premises.

Minor

Major premise

premise

mn=1

mq=2

np=3

pq=4

m=5

n=6

p=7

q=8

mn=1

11

21

31

41

51

61

71

81

major

mn

mq

np

pq

m

n

p

q

minor

mn

mn

mn

mn

mn

mn

mn

mn

mq=2

12

22

32

42

52

62

72

82

major

mn

mq

np

pq

m

n

p

q

minor

mq

mq

mq

mq

mq

mq

mq

mq

np=3

13

23

33

43

53

63

73

83

major

mn

mq

np

pq

m

n

p

q

minor

np

np

np

np

np

np

np

np

pq=4

14

24

34

44

54

64

74

84

major

mn

mq

np

pq

m

n

p

q

minor

pq

pq

pq

pq

pq

pq

pq

pq

m=5

15

25

35

45

55

65

75

85

major

mn

mq

np

pq

m

n

p

q

minor

m

m

m

m

m

m

m

m

n=6

16

26

36

46

56

66

76

86

major

mn

mq

np

pq

m

n

p

q

minor

n

n

n

n

n

n

n

n

p=7

17

27

37

47

57

67

77

87

major

mn

mq

np

pq

m

n

p

q

minor

p

p

p

p

p

p

p

p

q=8

18

28

38

48

58

68

78

88

major

mn

mq

np

pq

m

n

p

q

minor

q

q

q

q

q

q

q

q



Note that if you divide the above table in four equal squares, the top left square involves premises with only joint determinations, the bottom left one a joint major premise with a generic minor premise, the top right square involves a generic major premise with a joint minor premise, and finally the bottom right square premises with only generic determinations.

In my listing of moods in the next chapter, I do not follow their numerical order. Rather, I present the moods in a diagonal order with reference to the above table, starting with the strongest (top left hand corner) and ending with the weakest (bottom right hand corner).

Four moods, involving only strong determinations (namely, Nos. 11, 14, 41 and 44), have no ‘mirror images’; the remaining sixty moods may be treated in pairs, for each has a mirror image (thus, 12 and 13 are essentially the same, as are 21 and 31, and so forth). I present explicitly the more positive mood of each pair (e.g. 12), and only mention its mirror image (e.g. 13).

Moods with a stronger major premise are listed before moods with a stronger minor premise (e.g. 12, 13 before 21, 31). Moods with premises of uniform determination are listed before moods of mixed determination (e.g. 22, 33 before 23, 32). And so forth, the goal being to present all moods in a natural order.



[1] This example is based on one given by H. L. A. Hart and A. M. Honoré in Causation in the Law (Oxford: Oxford, 1959).

[2] Meaning, ultimately, with reference to our insights and hypotheses concerning the phenomena of nature in question, and more radically to our philosophical ordering of knowledge on a grand scale.

[3] See Appendix on J. S. Mill’s Methods.

[4] As a prescriptive science, Logic is ultimately only interested in valid argument. But as a descriptive one, it is very interested in knowing how (and how often) people tend to err in their reasoning processes. In this context we might apply the rule, if it can happen it will happen!

[5] Strictly speaking, the conclusion is permitted only if we can separately establish that the ordering of P and R as respectively cause and effect is acceptable. We shall deal with such details eventually.

[6] Strictly speaking, the conclusion is permitted only if we can separately establish that the ordering of P and R as respectively cause and effect is acceptable. We shall deal with such details eventually.

[7] Non-sequitur is a generic term, including both antinomy and non-antinomic non-sequitur (or ‘merely’ non-sequitur) as its species.

[8] There is, in fact a fourth figure, viz. Z-Y/Y-X/X-Z, in which the major and minor premises of the first figure are effectively transposed or whose conclusion is converse compared to a first figure conclusion. But as Aristotle argued, this is not a natural movement of thought, even though we can occasionally make some interesting inferences through it. Considering the matter insignificant, nothing more will be said about it, here. See Future Logic, p. 38.

[9] The content of S has to be ultimately known, otherwise the clauses involving it cannot be claimed to be known true. Nevertheless, if the premise involving S is a product of previous inductive and deductive arguments, and thus considered reasonably settled, the content of S can be ignored contextually. The same applies, of course, when there are a plurality of complements, i.e. when S itself stands for a composite of partial causes.

You can purchase a paper copy of this book Books by Avi Sion in The Logician Bookstore at The Logician’s secure online Bookshop.

2016-01-12T06:14:17+00:00