Inductive Logic

A Thematic Compilation by Avi Sion

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16. The Logic of Induction


1.    Degrees of Being

Before determining where the philosophy of science stands today, I would like to highlight and review some of the crucial findings of our own research in this volume.

The first thing to note are the implications of certain of our findings in modal logic. We saw earlier that, contrary to what has been assumed throughout the history of logic, the premises:


All M can be P and

This S can be P (or: This S is P, or: must be P)


…do not yield the conclusion ‘therefore, This S can be P’, but a more disjunctive result, namely:


therefore, This S can (get to) be or become P.


Thus, the mode ppp is valid, but only provided we take transitive propositions into consideration. Past logicians, including moderns, failed to take the existence of change into account, in their analysis of modal logic, and for this reason did not spot this important alternative conclusion from a merely potential first-figure major premise. It is true that Aristotle analyzed change with great perspicacity in his ontological works — and indeed, my own formalization of change is based on his insights — but even he did not integrate this relation into his formal logic.

The immediate formal significance of this finding is that natural modality is not permutable. Although in common discourse we rephrase ‘S can be P’ as ‘S is {capable of being P}’ or as ‘S is {potentially P}’, in strict terms, we may not do so — we may not enclose the modality within the predicate, and consider these ‘is’ copulae as having the same meaning as that in an actual ‘S is P’. If this is true of potentiality, it has to be equally true of natural necessity, since the oppositional relations between modal forms have to be maintained. By similar argument, we can show that temporal modality is impermutable.

These formal findings force upon us certain ontological inferences of the highest import. I was myself surprised by the conclusions; I had not intentionally ‘built them into’ my system. The implication is, that we may not regard a potential relation as signifying the presence of an actual ‘mark’ in the subject; the subject contains, within its ‘identity’, the potentiality as such, and not by virtue of some actuality. Thus, there really are ‘degrees of being’. We may not reduce all being to the actual; there are lesser degrees of being, called potentialities, and (by extension) higher degrees of being called natural necessities.

In between these extremes, therefore, the degrees of natural probability are also different degrees of being. And likewise, temporal modalities have to be so interpreted. Note well, none of this is speculative: these positions are imposed upon us by formal logic, by the requirement of impermutability (which, incidentally, was also useful in understanding the Russell Paradox). Thus, we are not making a vague metaphysical statement, but referring to precise technical properties which reveal and demonstrate the ‘self-evidence’ (in the formal sense, of logical necessity) of the concept of degrees of being.

Thus, although the concepts of modality are at first presented as purely statistical characterizations of relations, we come to the final conclusion (on formal grounds) that this numerical aspect is merely a symptom of a real ontological variation in the meaning of ‘is’. Aristotle left us with a limited vision of the scope of the copula ‘is’, because of the restrictions of his nonmodal logic; but now we see that there are real nuances in the sense of that copula, which only a modal logic can bring out into the open for our consideration.

We see, in this way, the impact modal deductive logic may have on ontology. But, as we shall see, the ramifications in modal inductive logic are even more significant, for epistemology. However, beforehand, I would like to make some incidental remarks.

Until now, the formal theory of classification, or class logic, has been notoriously simplistic. No one can deny how valuable it has been to science: for instance, Aristotle, and in modern times the Swedish biologist Carolus Linnaeus, have used it extensively in constructing their taxonomies of plant and animal life, and indeed every systematization involves reference to genus-species-individual relations. However, this approach has always seemed somewhat rigid and static.

Our world is conspicuously a world of change. Things come and go, there is generation and corruption, alteration, development, and evolution. What was yesterday a member of one class, may tomorrow be a member of another instead. Something may belong to a class only conditionally. And so forth. Only a modal class logic can assimilate such dynamic relations. Science needs this methodological tool, to fully depict the world of flux it faces.

Instant ‘state of affairs’ pictures are not enough; there is need to specify the avenues and modalities of transition (or absence of transition) from one state to another, as well as the causal relations involved. It is not enough to say vaguely what things ‘are’: we have to specify what they ‘must be’, what they ‘can be’, and from what to what and via what, and in which circumstances, they go: only thus can science fulfill its responsibilities.

For this reason, formal logic is obligated to study transitive categoricals and de-re conditioning of all types, in great detail. Without such a prolegomenon, many philosophical and scientific controversies will remain alive indefinitely. Right now, there is no formal logic (other than the one here proposed) which provides a language and neutral standards of judgment for, say, Darwin’s evolutionary theory or Hegel’s dialectic of history.

It is just so obvious that someone who is aware of the complexities of dynamic relations, is more likely to construct interesting and coherent theories on whatever subject-matter.

Returning now to modality. You will recall that we distinguished between types of modality and categories of modality, and we said that a modality is ‘fully’ specified only when both its type and its category are specified. Upon reflection, now, we can say that even then, the modality is not quite fully specified: to do so, we would still need to pinpoint the exact compound of modality it is an expression of, and indeed, we must do this in both directions of the categorical relations. Furthermore, to complete our description of the relation, we would need to specify the precise de-re conditions of its actualization.

Now, just as natural necessity, actuality, and potentiality form a continuum of ‘degrees of being’, and likewise for temporal modalities — so all the subdivisions of these modalities implied in the previous paragraph clarify the various degrees of being. That is, once we grasp the ontological significance of modality, as we did, then by extrapolation all the other formal distinctions, which occur within the types of modality in question, acquire a real dimension (of which we were originally unaware).

Moreover, the very concept of ‘degrees of being’ can be carried over into the field of extensional modality, in view of the powerful analogies which exist between it and the natural and temporal fields. This is not a mere generalization, because we from the start understood extensional modality as more than mere statistics; it relates to the possibilities inherent in ‘universals’ as units. Thus, ‘Some S are P’ and ‘All S are P’ are different degrees in which S-ness as such may ‘be’ related to P-ness as such. Thus, the quantifier is not essentially something standing outside the relation, but is ultimately a modification of the copula of being.

Going yet further, the valid modes of the syllogism, and indeed all argument, like nnn or npp for instances — they too may be viewed as informing us of the inherent complexities of modal relations. That ‘All S must be P’ implies only ‘some P can be S’ tells us something about being ‘in rotation’, as it were. That premises np yield conclusion p (rather than n or a) tells us something about the causal interactions of these different degrees of being. Likewise, for all types and mixtures of modality. All these so-called processes, therefore, serve to define for us the properties of different types and measures of being, giving us a fuller sense of their connotations.

Which brings us, at last, to the most radical extrapolation of all, and the most relevant to induction theory. Since, as we saw, in principle, logical necessity implies (though it is not implied by) natural necessity, and logical possibility is implied by (though it does not imply) potentiality — we may interpret these logical modalities as, in turn, themselves stronger or weaker degrees of being.  The inference is not as far-fetched as it may at first seem. That something is such that its negation is ‘inconceivable’ or such as to be itself ‘conceivable’ is a measure of its belonging in the world as a whole (including the ‘mental’ aspects thereof).

Between minimal logical possibility (which simply means, you will recall, having at all appeared in the way of a phenomenon, with any degree of credibility) and logical necessity (which means that the negation has not even a fictional, imaginary place in the world), are any number of different degrees of logical probability. If our extrapolation is accepted, then high and low logical probability are measures of ‘being’, not merely in a loose epistemological sense, but in a frankly ontological one. This continuum overlaps with but is not limited to the continua of being in a natural, temporal or extensional sense.

‘Truth’, the de-dicto sense of ‘realization’, and ‘singular actuality’ in the natural/temporal and extensional sense, become one and the same in (concrete or abstract) phenomena. The really here and now is the level of experience of phenomenal appearances (in the most open senses of those terms); we might even say of concrete and abstracts that they are also different degrees of presence, in their own way. Beyond that level of the present in every respect, ‘existence’ fans out into various ways of stronger and weaker being. Thus, logical probability may be viewed as in itself informative concerning the object, and not merely a somehow ‘external’ characterization of the object.

This suggestion is ultimately made to us by formal logic itself, remember; it is rooted in the concept of impermutability. Thus, the contention by some that Werner Heisenberg’s Principle of Uncertainty signifies an objective indeterminism, rather than merely an impossibility to measure — may well have significance. I am myself surprised by this possible conclusion, but suddenly find it no longer unthinkable and shocking: once one accepts that there are ‘degrees of being’ in a real sense, then anything goes.

Thus, we may also view the mental and the physical, the conceptual and the perceptual, the ‘universal’ and the individual, the ideal and the real, knowledge and fact, and why not even the absolute and the relative — as different types and degrees of being. Being extends into a large variety of intersecting continua. In this way, all the distinct, and seemingly dichotomous, domains of our world-view are reconcilable.


2.    Induction from Logical Possibility

Let us now return to the main topic, that of induction, and consider the impact of what has been so far said. We acquainted ourselves with two major processes of induction: adduction and factorial induction.

Adduction concerns theory formation and selection. The logical relation between postulates and predictions, consists of a probabilistic implication of some degree, conditioned by the whole context of available information. The postulates logically imply, with more or less probability (hopefully, lots of it) the predictions; and the latter in turn logically imply with more or less probability (anything from minimal possibility, even to logical necessity) the postulates. The logical relations note well are mutual, though to different degrees, and in flux, since they depend on a mass of surrounding data.

Thus, the adduced probability, in any given context, of any single proposition, be it frankly theoretical or seemingly empirical, is the present result of a large syndrome of forces, which impact on each other too. Theories are formed (appear to us), and are selected (by comparison of their overall-considered probabilities, to those of any modifications or alternative theories), with reference to the totality of our experiences.

Concrete experience, note, is by itself informing, even when it is not understood; abstract theories are also in a sense experiences, to be taken into account. Empirical phenomena determine our theories, and they in turn may affect our particular interpretations of empirical phenomena. There is a symbiotic give and take between them, which follows from the holistic, organic, nature of their logical relation.

Thus, adduction may be viewed as the way we generally identify the degree of being of any object, relative to the database present to our consciousness. Within the domain delimited by our attention, each object has a certain degree of being; and this degree is objective, in the sense that from the present perspective the object indeed appears thus and thus. The appearance may not be the central ‘essence’ of that object, but it is in a real sense a facet of it, a projection of it at level concerned. In that way, we see that logical probabilities, and logical modality in general, ultimately have a de-re status too: their way of ‘being’ may be more remote, but it is still a measure of existence.

Deduction is merely one tool, within the larger arsenal of adductive techniques. Deductive processes are, apart from very rare exceptions of self-evidence (in the formal sense), always contextual, always subject to adductive control in a wider perspective. Modern logicians, so-called Rationalists, who attempt to reduce knowledge to deductive processes, fail to grasp the aspect of holistic probability. Our knowledge is not, and can never be made to be, a static finality; the empirical reality of process must be taken into account for a truly broad-based logic. Likewise, the opposite extreme of Empiricism is untenable, because fails to explain how it allowed itself to be formulated in a way that was clearly far from purely empirical terms.

Now, factorial induction is another major tool at our disposal in the overall process of induction. In fact, we may view all induction as essentially adductive, and say that deduction and factorial induction are specific forms or methods of adduction. Essentially, factorial induction is built on the adductive method of listing all the alternative ‘explanations’ about a ‘given datum’ — in our case, the given datum is the gross element or compound, and the list of eventual explanations is the factorial formula; that is, the formally exhaustive series of integers compatible with the gross formula, and therefore constituting logically possible outcomes of it.

In the general adductive relation, the hypothetical proposition ‘these predictions probably imply those postulates (and thus the theory as a whole)’, the terms of the antecedent categorical need not be the same as the terms of the consequent categorical. Thus, the terms of the hypothesis may be mere constructs, of broader meaning and application than the more singular, actual and real terms of the allegedly empirical ground. That there are degrees of being, implies not only that there are degrees of truth (as explained, logical modality has a de-re status too), but also that there are degrees of meaning (again, in the objective sense that something has at least appeared).

The terms of a theory may be at first vague, almost meaningless concepts, but gradually solidify, gaining more and more definition, as well as credence. This evolution of meaning and credibility, as we look at the apparent object every which way, may be viewed a change in the degree of ‘being’; as long as the apparent object does not dissolve under scrutiny, it carries some weight, some ‘reality’, however weak. It remains true that any alternative with apparently more weight of credibility and meaning, has a ‘fuller’ reality, more ‘being’. Thus, even though ‘truth’ is a comparative status, it may still be regarded as an objective rendering of the ‘world’ of our context.

In contrast, factorial induction deals with generalization and particularization of information. What distinguishes it from adduction (in a generic sense) is the uniformity of the terms in its processes. Factorial induction concerns the selection and revision of ‘laws’. We generalize ‘this S is P’ to ‘all S must be P’ or some less powerful compound (some other integer), with reference to precise rules. Here, note well, the terms are the same. This sameness is at least nominal; for it is true that by generalizing the singular actual to a general natural necessity (or whatever), we modify the degree of being and meaning of the terms somewhat. This modification is not arbitrary, but is determined by the whole context, including the rules followed.

But anyway, factorial induction is obviously a case of adduction (though a special case because of the continuity of terms). That means that the terms themselves may well be more or less theoretical, in the sense of having lower degrees of meaning. Also, the seeming empiricism of their singular actual relation may or may not be true; that is, it too has degrees of credibility and truth, determined by the overall context. At all levels, from the seemingly empirical, through factorial induction, to the adduction of overt constructs — there is some interactive reference to overall context.

Thus, the rules of factorial induction remain the same, however meaningful or true the terms appear at a given stage: they are formal rules, which continue to apply all along the development of knowledge. At each stage, they determine a certain answer, or a range of answers, depending on how definite and credible the terms and relations involved appear to us at the time, taking into consideration all available information. The factorial approach to induction is distinguished by its utter formalism, and independence from specific contents.

I want to stress here the profound importance of such an integrated theory of modal induction. Through it we see graphically that there is no essential discontinuity between logical (de-dicto) modality and the de-re modalities. The modality of a thing’s being, is the meeting point of all these aspects: on the outer edge, its logical meaning and truth, ranging from logical necessity to extremely dilute conceivability; closer to the center, the de-re modalities at play; at the very center, the empirical realization of the essence, towards which we try to tend.

Truth and full definition are approached in a spiral motion, as it were. We can tell that we are closer, but there is always some amount of extrapolation toward some presumed center. Our position at any stage, however composed of theoretical constructs and generalizations, always has some reality, some credibility, some meaning — it just may not be as advanced as that which someone else has encountered or which we will ourselves encounter later. But it is still a product of the Object, the whole world of appearances, and as such may well be acknowledged to have some degree of objective being in any case.

Another way to view inductive processes is as follows. Since logical possibility is a subaltern of natural possibility (potential), we can generalize (subject to appropriate rules of corrective particularization) from logical possibility to natural possibility, just exactly as we generalize (under particularizing restrictions) from, say, natural possibility (potentiality) to temporal possibility (temporariness). This means that adduction in general (that is, even with imaginary terms) is a species of factorial induction.

We have already developed a definitive inductive logic for the de-re modalities (with the example of categoricals — de-re conditionals can similarly be dealt with, almost entirely by a computer: we know the way). This de-re inductive logic can now be extended further to de-dicto aspects, simply by introducing more factors into our formulas. We saw that the combinations of the natural and extensional types of modality gave rise to 12 plural elements, and thence to 15 factors. When temporal modality is additionally taken into consideration, the result is 20 plural elements and 63 factors. It is easy (though a big job) to extend the analysis further, with reference to the fourth type of modality, namely the logical.

Roughly speaking (I have not worked out all the details), we proceed as follows. Each previously considered element becomes three elements: a logically necessary version (say, prefixed by an N), a just-true version (without prefix), and a logically possible version (say, with prefix P). These more complex elements are then combined into fractions, and thence into integers; the resulting number of integers is the new maximum number of factors a formula may consist of.

Every gross formula is then given a factorial interpretation, comprising a disjunction of one to all the available factors. The factors must of course be ordered by modal ‘strength’, to allow for easy application of the law of generalization. Logical necessity or impossibility are ‘stronger’ than logical contingency coupled with truth or falsehood. The overall factorial formula for any event is accordingly much longer, but with the factors ordered by ‘strength’, factor selection or formula revision proceeds in accordance with exactly the same unique law of generalization.

Thus, our manifesto for modal induction is not limited to the special field of de-re categoricals (and eventually de-re conditionals), but is capable of coherently and cohesively encompassing even logical modalities (applied categorically, or eventually hypothetically). We have therefore discovered the precise mechanics of all adduction. At any stage in knowledge, it should henceforth therefore be possible to characterize any apparent proposition with reference to a precise integer, the strongest allowed by the context.

This refers, not only to simple generalization of ‘laws’ (observed regularities), but to determining the status as well as scope of any complex ‘theory’ whatever (however abstract or even constructed be its terms, even if their definitions are still notional and their truths still hypothetical). Of course, the terms still have to be at least minimally intuitively meaningful and credible. But the selection (subject to revision) of the strongest available factor precisely determines a proposition (or its negation) as true. There is no appeal to some rough extrapolation on vague grounds, toward the central ‘truth’; we now have a formal depiction of the process of pin-pointing the truth at any time.


3.    The Novelty of My Work

[With regard to the history of inductive logic,] one thing is clear at the outset: no one has to date formulated any theory remotely resembling factorial induction. Adduction is well known — it is the hypothetical-deductive method, attributed to Bacon and Newton; actual induction may, I believe, be attributed to Aristotle (I certainly learned it from his work); but factorization, factor selection and formula revision (not to mention the prior logics of transition and of de-re modal conditioning) are completely without precedent.

These constitute, I am happy to report, a quantum leap in formal logic. I stress this not to boast, but to draw attention to it. It was the most difficult piece of intellectual problem-solving (it took 2 or 3 months) this logician has been faced with, and the most rewarding. The problem was finding a systematic way to predict and interpret all consistent compounds of (categorical) modal propositions; many solutions were unsuccessfully attempted, until the ideas of fractions and integers, and of factorial analysis, presented themselves, thanks God.

The historical absence of a formal approach to induction, or even the idea of searching for such an approach, is the source of many enduring controversies, as we shall see. Once a formal logic of induction exists, as it now does, many doubts and differences become passé. Just as formal deductive logic set standards which precluded certain views from the realm of the seriously debatable, so precisely the formal inductive logic made possible by factorial analysis of modal propositions simply changes the whole ball game.


Drawn from Future Logic (1990), Chapter 67:1-2, 3(part).


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