PART VI.FACTORIAL INDUCTION.
2.Induction of Particulars.
1.Elements and Compounds.
4.More Factorial Formulas.
5.Open System Analysis.
APPENDIX 1– Factorial Analysis of Elements and Gross Formulas (Open System).
2.Equality of Status.
3.Stages of Induction.
5.The Paradigm of Induction.
6.The Pursuit of Integers.
2.The Uniformity Principle.
3.The Law of Generalization.
1.Closed Systems Results.
2.Some Overall Comments.
3.Rules of Generalization.
4.Review of Valid Moods.
5.Open System Results.
2.Kinds of Revision.
3.Unequal Gross Formulas.
4.Equal Gross Formulas.
1.Adding Fractions to Integers.
2.Reconciliation of Integers.
3.Indefinite Denial of Integers.
4.Other Formula Revisions.
5.Revision of Deficient Formulas.
Summary of findingsin the chapters of this part:
Part VI.The Logic of Factorial Induction.This is a completely new field of logic — the first genuinely formal theory of generalization and particularization in history. Again, this sets entirely new, extremely precise standards for all future science, and answers some of the most fundamental questions of epistemology (if I may be forgiven for sounding such a loud trumpet).
50. The problem of induction was to begin with posed with regard to actual categorical propositions — how are they known, whether particular or general? The solution is so simple, with relation to actuals, that it seems puerile; but as we later see, when modal propositions are considered, the solution appears much more interesting.
51. In order to deal with modal induction, we first had to develop a precise theory of all the logically possible combinations of modal propositions. The forms considered thus far were mere elements, that may be compounded in a certain number of ways, according to their oppositions. We noted that compounds give rise to special arguments; also, that directional issues may be raised.
52. Next, we introduced the concepts of fractions and integers, which describe states of being more definitely than elements or compounds can do. The former are parts of the latter. These concepts and the resulting formulas depend on the logics ofde-reconditionals, and different systems evolve according to the mixtures ofde-remodality we choose to consider, and whether we ignore or include directional issues.
53. These preliminaries led us to a formal theory of factorial analysis of elements and compounds, and indeed of all states of knowledge concerning anything. A factorial formula consists of all the alternative integers which logically may come out of any given item of knowledge. In some cases, only one alternative is formally acceptable, so that an unexpected deductive situation occurs.
54. Thereafter, we proceeded to formally demonstrate the knowability of all types of necessity, whether extensional (generality), natural or temporal, or any combination of these. We described the stages of induction, and defined the central issue of induction as a pursuit of solitary integers.
55. After discussing the philosophical aspects of induction, we proposed an exact ‘Law of Generalization’ in formal, factorial terms — one which precisely determines the factor selection from any given datum whatsoever. As later shown, this same Law also controls the formula revisions we call particularization.
56. We applied the Law of Generalization to all possible elementary and compound forms, and showed the predicted valid inductive conclusion in each and every case to be rationally credible, thus also demonstrating the correctness, value and validity of the Law as a whole.
57. We then considered context changes, which require us to amplify previous conclusions or harmonize them with new data. This was called formula revision, and the difficulties it involves were clearly described, in order to show the power of their formal resolution.
58. We applied the Law of Generalization to all inconsistent conjunctions of elements or compounds, and obtained precise inductive conclusions from all of them.
59. Lastly, we applied the Law of Generalization to other situations requiring formula revision, like adding fractions to integers, reconciliation of integers, indefinite denial of integers, among others.
In this way, we demonstrated that we have evolved a single, uniform, consistent tool for dealing with all knowledge contexts. We used the specific example of categorical propositions involving different mixtures ofde-remodalities; but we also indicated how expanded application to still more complex situations is to be effected.