THE LOGIC OF CAUSATION
Phase Three: Software Assisted Analysis
Chapter 17 – Resuming the Research
1. History of the Research.
I have been dreaming of systematizing causal logic since my teens, I think, when I first studied works on logic and philosophy.
My first book, Future Logic (1990), mentions the manifest modal foundations of causality and indeed the tacit causal foundations of modality, stressing that different types (or modes) of causality exist reflecting the different types of modality (see chapters 11-12 there). And of course, knowing approximately the basic definitions of causation in terms of conditional propositions, the work done on the latter in Future Logic was incidentally work on the logic of causation (see parts III and IV there).
Moreover, having understood the formal continuity between categorical and de re conditional propositions, and indeed between the different modes of modality (including the logical), the work done with regard to factorial induction of categoricals was also significant in the long run to induction of conditionals – and thence to that of causative propositions (which are, after all, just conjunctions of selected conditional and categorical ones).
I made some general remarks relating to causal logic in my book Judaic Logic, first published in 1995 (see there chapter 10.2, binyan av), showing my continuing interest.
My research efforts into the logic of causation per se started in earnest in the late 1990s, with a macroanalytic approach to the problem. My purpose then was, simply put, to clearly define all the varieties of causation (its determinations, indicative of degrees of causation), then correlate them all (oppositions, eductions) and work out all syllogistic reasoning possible between them (which necessitated the development of matricial analysis). It was, I believe the first time anyone had ever tried so ambitious a project in this field of logic. This first phase of the research was published in October 1999 as The Logic of Causation. However, I soon realized that there were some problems in these initial results, and tried to improve on them in a second edition published in July 2000.
But it was by then clear that I needed to develop a much deeper and more systematic approach to obtain reliable results. It was, I think, not until the later half of 2002 that I found the time to proceed with microanalysis of causation, the second phase of my research. The massive amount of work involved was completed rather quickly, because I devoted all my time and concentration to it. By about March 2003, I was able to publish the results. This work involved very many painstaking manual ‘calculations’, and produced a very profound understanding of causation, which allowed me to formally settle some age-old difficult issues concerning it.
Thus, various “laws of causation” traditionally proposed were examined and evaluated. Criticisms of causation as such, such as those of Nagarjuna or David Hume, were rebutted. The notion of natural spontaneity used in modern quantum physics, as well as the Buddhist notion of interdependence were scrutinized and judged. And a critical analysis of J. S. Mill’s proposed methodology for identifying causation was made possible. (See chapter 16 and appendix 1, here.) In same period, I wrote two other works which had some bearing on the understanding of causality, namely Buddhist Illogic (2002 – see chapters 7 & 8) and Phenomenology (2003 – see chapter 2.5).
However, even as I was completing this new phase of the research, it was clear to me that some uncertainties remained, due to the manual method of calculation used (subject in principle to human error, though all results were double-checked) and because some problems could not be solved by considering only three items. It was clear to me that a third phase of research, involving a more mechanical approach (using spreadsheets, database software or ad hoc programming) to increase reliability, and a larger scope (i.e. at least four items) to increase reliability, were needed. There and then, I started doing some work in that direction; but ran out of time, having to deal with many mundane matters.
In 2004, I devoted my time to writing Volition and Allied Causal Concepts, a study relevant to causation by implication. I continued thinking about causation in 2005, writing down my insights in Ruminations (see part I chapter 8 there), and even made some effort to advance phase III causation research. In 2006, my time was taken up writing Meditations, and in 2007-8 writing Logical and Spiritual Reflections. The latter work including some insights relating to causality (notably in book 1, chapters 3 & 6, and book 3, chapter 11). I also made some more effort in 2008 to advance phase III research, but was soon stopped by other concerns. The year 2009 was devoted to improving my website and to creating an online bookshop to sell my books.
I first posted some phase III results in my website, TheLogician.net, in October 2009, partly to encourage myself to pursue the matter further. In January 2010, I decided to try and complete phase III – and the work done is described in the following pages.
My initial idea with regard to phase III research was to develop a computer program capable of ‘calculating’ the value of causative propositions and syllogisms directly from the matrix relating the items concerned[1]. Realizing that in the absence of professional help such programming was beyond my immediate capabilities, I thought instead of using database software, such as Access. I began indeed doing so, but soon realized I had difficulty visualizing the interrelationships involved, not having made use of such software for many years. I therefore decided that the best way for me to proceed was through the use of spreadsheets, namely Excel software; and this is what I did.
2. Matrices of the Frameworks.
As explained in phase II, a ‘matrix’ is a condensed statement or catalogue of all logically conceivable ontological situations relating any number of items; note that ignorance or uncertainty is not counted as a situation, being merely epistemological. Each such completely defined situation has called a ‘modus’.
The first major task of our resumed research was to develop a matrix for four items similar to the matrices for two and three items developed in chapter 12. It was soon clear that to achieve that, I had to transpose the earlier tables so that the modus numbers henceforth appeared as a column instead of as a row as heretofore. It is much easier to view and work with 65,536 rows than so many columns. This simple change of perspective makes all processes so much easier, I wish I had thought of doing it from the start – it would certainly have saved me much time and trouble! Sometimes by hurrying blindly we slow ourselves down.
Anyway, in this manner Table 12.1, cataloguing the 16 moduses for 2 items (PR), became Table 17.1 shown below. As can be seen, the modus numbers 1 to 16 are in the first column, labeled ID. The next four column headings signify the different possible combinations of the two items concerned, P and R (0 and 1 here meaning present and absent, note well) – ‘11’ meaning (P + R), ‘10’ meaning (P + notR), ‘01’ meaning (notP + R), and ‘00’ meaning (notP + notR). Note that 11>10>01>00.
The ‘summary’ column merely summarizes the information in the preceding four (ignoring leading zeros); notice that the numbers in it range from 0 (i.e. ‘0000’) to 1111 in an orderly manner (i.e. each number is greater than the one above it). It is evident from the summary that the moduses are not numbered randomly, but in increasing magnitude. The last column counts the zeros in each of the moduses[2]; this number is an indication of the degree of freedom or lack of it in the situation signified by the modus concerned (since here, 0 means ‘impossible’ while 1 means ‘possible’ combination). Using a spreadsheet program, it is easy to generate the modus numbers, the summaries and count of zeros; and to verify that the summaries are indeed in order.
Table 17.1 List of 16 Moduses for 2 Items PR
ID | 11 | 10 | 01 | 00 | summary | number of zeros |
1 | 0 | 0 | 0 | 0 | 0000 | 4 |
2 | 0 | 0 | 0 | 1 | 0001 | 3 |
3 | 0 | 0 | 1 | 0 | 0010 | 3 |
4 | 0 | 0 | 1 | 1 | 0011 | 2 |
5 | 0 | 1 | 0 | 0 | 0100 | 3 |
6 | 0 | 1 | 0 | 1 | 0101 | 2 |
7 | 0 | 1 | 1 | 0 | 0110 | 2 |
8 | 0 | 1 | 1 | 1 | 0111 | 1 |
9 | 1 | 0 | 0 | 0 | 1000 | 3 |
10 | 1 | 0 | 0 | 1 | 1001 | 2 |
11 | 1 | 0 | 1 | 0 | 1010 | 2 |
12 | 1 | 0 | 1 | 1 | 1011 | 1 |
13 | 1 | 1 | 0 | 0 | 1100 | 2 |
14 | 1 | 1 | 0 | 1 | 1101 | 1 |
15 | 1 | 1 | 1 | 0 | 1110 | 1 |
16 | 1 | 1 | 1 | 1 | 1111 | 0 |
Similarly, Table 12.3, cataloguing the 256 moduses for 3 items (PR), became Table 17.2, consisting of rows labeled 1 to 256 and a matrix of 8 columns labeled 111, 110, 101, 100, 011, 010, 001, 000, followed by a summary column, with numbers ranging from 0 (i.e. ‘00000000’) to 11111111, and a count of zeros.
Finally, a new table cataloguing the 65,536 moduses for 4 items (PQRS), Table 17.3, was generated. Note that the latter table, having a matrix of 16 columns, implied some high summary numbers to be 16 digits long; this was a technical problem in that Excel software cannot handle more than 15 digits, so that the last digit is made 0 instead 1 in certain cases (e.g. 1111111111111111). Here two a final column was added on, showing the number of zeros in each modus.
For the record: the matrix of a single item P comprises 2*4=8 cells (see Table 13.9). The matrix for two items, say P and R, comprises 4*16=48 cells (Table 17.1). That for three items, PQR has 8*256=2,048 cells (Table 17.2). The corresponding table for four items has 16*65,536=1,048,576 cells (Table 17.3). Each cell represents a bit of information (about a possible or impossible combination of the items concerned) that needs to be induced or deduced to determine the modus applicable; only when all the cells adding up to a modus are so determined can we claim to know that there is or is not a relation of causation or whatever between the items. Note this well, because it shows how far true causal logic is from the simplistic claims of the likes of David Hume.
Thus, our first three tables display the matrices at the root of the three frameworks, which we will in the present phase of our study: These matrices contain the basic data from which all other tables will be constructed – i.e. they house the information needed to develop the definitions of all conjunctive, conditional and causative forms, their oppositions, eductions from them, and syllogisms involving them. These three tables are:
Table 17.1 – 2-Item Matrix: 16 Moduses. (1 page in pdf file).
Table 17.2 – 3-Item Matrix: 256 Moduses. (4 pages in pdf file).
Table 17.3 – 4-Item Matrix: 65,536 Moduses. (565 pages in pdf file).
The first of these tables, being brief, is reproduced above, though most of its content is already given in Table 12.1, in order to show how the original was transposed. The second, could have been reprinted here, but most of its information is already given in Table 12.3, so there is no point. The third table is of course far too long to include in the present printed report. However, this table and the two preceding it, and indeed all subsequent tables relating to phase III work, are made available as .pdf files in my website[3], at the following address (there click on ‘Phase III’):
http://www.thelogician.net/LOGIC-OF-CAUSATION/Tables-and-Diagrams.htm
Please do carefully examine the phase III tables there, for they are the purpose of the whole research! Note that the first three tables were developed successively, in order of complexity, starting with two items, then three, then four. The advantage of this method is that the formulae used in each table to generate data or calculate results can be passed on to the next table, i.e. to the larger framework, with appropriate modifications to adjust to the increased complexity at hand. Though I may not point it out repeatedly, keep in mind that this pattern of development is used throughout the present research. It has made things rather easy for me!
3. Comparing Frameworks.
The next challenge was to compare moduses in the different frameworks. This is not only done out of curiosity, but to understand in detail just how each larger framework passes on information previously found and provides new information.
We had already answered this question in part in Table 12.6, listing the correspondences between 2-item moduses and 3-item moduses – but the work was then performed manually, and now it needed to be done mechanically, i.e. using formulae in a spreadsheet. This new work resulted in the following two tables, displayed in the website: Table 17.4 (equivalent to Table 12.6), comparing the 2- and 3- item frameworks, and Tables 17.5 and 17.6 (two new ones), comparing the 3- and 4- item frameworks.
These tables were produced in stages, briefly put, as follows. I started with a 3-item matrix with 8 columns, and used it to produce another table with only 4 columns. Note well – rather than move from 2 to 3 items, I worked backwards from 3 to 2 items. Each of the cells in the latter had an appropriate formula deriving it from relevant cells in the same row of the former. A summary column was then added to this derivative table, from which – using the vertical lookup function of Excel – each 3-item row was given a 2-item modus number (ranging from 1 to 16), and the job was done.
Thereafter, it was easy to derive a further table listing and then counting the 3-item moduses corresponding to each 2-item modus. See the additional notes at the bottom of these tables. All information obtained was checked with reference to the relevant tables produced in phase II and found consistent.
The same method was used to identify the 3-item modus number corresponding to each 4-item modus number. However, whereas for the ‘2 to 3 items’ comparison the results are lumped together in one pdf (Table 17.4), for the ‘3 to 4 items’ comparison the results are split into two pdf files (Tables 17.5 & 17-6), in view of the mass of data involved. The first part lists all the moduses for 4 items and next to each of them the corresponding the 3-item modus, and it also lists the count of 4-item moduses corresponding to each 3-item modus. The second part specifies the 4-item modus numbers corresponding to each 3-item modus number, and counts them.
Needless to say, the latter three tables all provide us with some valuable new information not previously generated. Thus, regarding comparison of frameworks, the following three tables are made available for your scrutiny on The Logician website:
Table 17.4 – From 2 to 3 Items Moduses. (6 pages in pdf file).
Table 17.5 – From 3 to 4 Items Moduses – 1st part. (1192 pages in pdf file).
Table 17.6 – From 3 to 4 Items Moduses – 2nd part. (2792 pages in pdf file).
We can now look into the moduses applicable to each of the forms of causation and various other forms.
[1] This is why I have called this phase that of Software Assisted Analysis, although of course its ultimate motive is to investigate the 4-item framework.
[2] This information was previously given in the second column of Table 12.6.
[3] Note that some tables were produced in ‘landscape’ (instead of ‘portrait’) orientation. Adobe pdf Reader shows such tables sideways – you have to click on View then on Rotate View Clockwise to redress them. Needless to say, if the image seems too small, you can increase its size as much as you want using the appropriate button on the Reader tool bar.