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April 9, 2010

2:52 AM

Introduction to the theory of induction: about generalization and particularization

Eighth post in the ongoing series on important innovations in logic theory to be found in my works. First, let me apologize again to readers for not posting new blogs more often, but I am at this time very busy with new research and writing. In the present posting, I will present to you the major advance in induction theory published in my book Future Logic 20 years ago, called “factorial induction”. Readers are referred to my previous post on adduction for introductory comments.

The problem of formalizing generalization and particularization using actual/non-modal categorical propositions is simple enough (though some logicians and philosophers seem to have a lot of trouble with it still today). Before we can generalize, we need a particular proposition to generalize. Particulars can be known through direct observation, or less directly through adductive means or by deduction from other propositions so obtained. Note that we do not just assume a positive particular “some X are Y” to be true without reason: we need at least one case to convince us of it; for example, we would not accept that “some humans are blue-skinned” without empirical evidence. On the other hand, negative particulars – and indeed generalities – are often assumed when positive information was sought and not found; for example, “no humans are blue-skinned”.

Having in some way established that “some X are Y”, and not having found any reason to believe that “some X are not Y”, we can readily generalize and say that “all X are Y”. Why is that logically allowed and indeed recommended? Because whereas we do have evidence for the positive case, we have no evidence for the negative case. This is the basis for generalization that many logicians and philosophers have failed to understand. They think that generalization is an arbitrary act based only on particular evidence, not realizing the crucial role played by the absence of contrary evidence. Moreover, they forget that generalization is an inductive process – i.e. one subject to revision if further research uncovers contrary evidence. That is, having generalized from “some X are Y” to “all X are Y”, we are not stuck there for evermore; if we discover later on that “some X are not Y” – or even that assuming “all X are Y” leads us to some contradiction – we may and must particularize “all X are Y” to “only some X are Y” (i.e. some X are Y and some other X are not Y).

Needless to say, all this can in principle function the other way, starting from “some Y are not Y”, generalizing to “no X are Y”, then particularizing to “only some X are not Y”. Obviously, if we already know that “some X are Y” and “some X are not Y”, we would not bother generalizing either way, but would from the start adopt a contingent viewpoint. Generalization is not a must, but a rational option, to be exercised when appropriate; and it remains always tentative to some degree, with an open mind to retreat from it should new evidence or insight justify and demand particularization. Of course, in accord with the principles of adduction, the longer and more often the initial particular is confirmed by experience, and evidence is sought and not found for the subcontrary particular, the less tentative and uncertain does our generalization get. But it may still in principle be overturned at any time by means of just one contrary case, remember. All this is simple enough, as already said, when we are dealing with a ‘bestiary’ of only four actual forms, viz. “some X are Y”, “some X are not Y”, “all X are Y” and “no X are Y”.

However, when we start dealing with a larger bestiary of propositions, including notably de re modal propositions, first categorical and then more broadly conditional, we find the said simple approach no longer adequate as it stands. The question then arises: how far up the scale of generality (in the various modes of modality) can we rise, and how far back must we retreat if contrary evidence is found? For when dealing with more numerous propositions of various sorts, generalization and particularization depend not on one or two simple conditions, but on a variety of complex conditions; and we must be prepared to efficiently logically adapt to constant flux in our data base and reasoning processes.

This is where the processes of factorial induction come into play. This theory constitutes a sophisticated formal logic of induction, which foresees all possible permutations and combinations of categorical (and more broadly, conditional) propositions, in various modes of modality, singly and jointly, and by means of clear and persuasive principles foretells the valid inductive (and sometimes deductive) conclusion(s) to be drawn from them. No one has done this work before, or even thought that it ought to be done and tried to do it.

To start with, the ‘elementary’ propositions are identified and listed. Then all their ‘gross compounds’ (that is to say, their consistent combinations in twos, threes, or fours – including elements of either or both polarities) are identified and listed. The oppositions and eductions from these are considered, because these demonstrate that there are sometimes many possible paths of generalization or particularization from a given point of departure. In view of this, the need becomes evident to identify and list all possible ‘integers’ of propositions and their constituent ‘fractions’. The fractions refer to a subset of the subject-class concern, and the integers to the conjunction of such fractions. The advantage of integral formulae over gross compounds being that in the latter overlaps between different subsets cannot be handled, whereas the former leave no ambiguity in the distribution of cases.

Once this preparatory work is done, the ‘factorial analysis’ or ‘factorization’ of gross compounds can be pursued; i.e. we can identify and list the various alternative integers – now called ‘factors’ – that each gross compound can become in different contexts of knowledge. Some gross compounds have only one factor – one possible outcome in terms of integers (sometimes quite unexpectedly) – whereas most gross compounds have many factors. Induction – i.e. here understood in the sense of generalization and particularization processes – is now viewed as a pursuit of integers, consisting of ‘factor selection’ sometimes later corrected by means of ‘formula revision’.

These processes are guided and controlled by a single, universal law of generalization, which states: “in any factor selection, the strongest factor is the one to prefer”. The reason why this law is universal is that when new data appears, the resultant gross compound is changed, and therefore we refer to another row on the table of all possible factorizations to determine the appropriate strongest factor. When for some reason we know more directly that the previous strongest factor is inappropriate – for instance, if it leads to some contradiction – we can simply select the next available factor, since they are classed in order of their ‘strength’, that is how high they go on the scale of generalization. Precise rules of generalization, and thereafter of particularization, can thus be developed. The result is a precise list of valid moods of inductive argument.

This is work as original and revolutionary today as Aristotle’s formalization of the syllogism was in his day. No one claiming to be a logician can reasonably ignore this work and continue business as usual without reference to it. Once one has studied and understood it fully, one’s outlook on human knowledge changes entirely. Its essentially inductive nature is brought home forcefully and irreversibly. It is shocking to observe just how widespread still today is the ‘deductive’ approach to knowledge among logicians and epistemologists. The theory of factorial induction is designed to shatter this mind-set once and for all, and institute a truly ‘inductive’ approach to knowledge.

 

For more details on factorial induction, see FUTURE LOGIC, PART VI (CHAPTERS 50-59),

http://www.thelogician.net/2_future_logic/2_fl_part_6.htm

See also: FUTURE LOGIC, part VII, Chapter 67, about inductive logic.

And RUMINATIONS, part I, chapter 9, about negation.

 

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