Inductive Logic

A Thematic Compilation by Avi Sion

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31. The Significance of Negation


1.    Formal Consequences

Returning to logic – our insight here into the nature of negation can be construed to have formal consequences. The negative term is now seen to be a radically different kind of term, even though in common discourse it is made to behave like any other term.

We cannot point to something as ‘negative’ except insofar as it is the negation of something positive. This remark is essentially logical, not experiential. The term ‘not’ has no substance per se – it is a purely relative term. The positive must be experienced or thought of before the negative can at all be conceived, let alone be specifically sought for empirically. This is as true for intuitive as for material or mental objects; and as true for abstracts as for concretes.

One inference to draw from this realization of the distinction of negation is: “non-existence” is not some kind of “existence,” Non-existent things cannot be classed under existence; they are not existent things. The term “non-existence” involves no content of consciousness whatsoever – it occurs in discourse only as the verbal repository of any and all denials of “existence,” Existentialist philosophers have written volumes allegedly about “non-being,” but as Parmenides reportedly stated:


“You cannot know not-being, nor even say it.”


This could be formally expressed and solidified by saying that obversion (at least that of a negative – i.e. inferring “This is nonX” from “This is not X”) is essentially an artificial process. If so, the negative predicate (nonX) is not always inferable from the negative copula (is not). In other words, the form “There is no X” does not imply “There is non-X;” or conversely, “X does not exist” does not imply “nonX exists,”

We can grant heuristically that such eductive processes work in most cases (i.e. lead to no illogical result), but they may be declared invalid in certain extreme situations (as with the term “non-existence”)! In such cases, “nonX” is ‘just a word’; it has no conscionable meaning – we have no specific thing in mind as we utter it.

Logicians who have not yet grasped the important difference of negation are hard put to explain such formal distinctions. I know, because it is perhaps only in the last three years or so that this insight about negation has begun to dawn on me; and even now, I am still in the process of digesting it.

Note that a philosophical critic of this view of negation cannot consider himself an objective onlooker, who can hypothesize ‘a situation where absence exists but has not or not yet been identified’. For that critic is himself a Subject like any other, who must explain the whence and wherefore of his knowledge like anyone else – including the negatives he appeals to. No special privileges are granted.

That is, if you wish to deny all the above, ask yourself and tell me how you consider you go about denying without having something to deny! Claiming to have knowledge of a negative without first thinking of the corresponding positive is comparable to laying personal claim to an absolute framework in space-time – it is an impossible exercise for us ordinary folk.

It should also be emphasized that the above narrative describes only the simplest kind of negation: negation of a perceptual item. But most of the time, in practice, we deal with far more complex situations. Even the mere act of ‘pointing’ at some concrete thing involves not only a positive act (“follow my finger to this”), but also the act of negation (“I do not however mean my finger to point at that”).

Again, a lot of our conceptual arsenal is based on imaginary recombinations of empirical data. E.g. I have seen “pink” things and I have seen “elephants,” and I wonder whether “pink elephants” perhaps exist. Such hypothetical entities are then tested empirically, and might be rejected (or confirmed). However, note, abstraction does not depend only on negation, but on quantitative judgments (comparing, and experiencing what is more or less than the other).

Abstraction starts with experiences. These are variously grouped through comparisons and contrasts. Negation here plays a crucial role, since to group two things together, we must find them not only similar to each other but also different from other things. This work involves much trial and error.

But at this level, not only denial but also affirmation is a rational act. For, ‘similarity’ means seemingly having some quality in common in some measure, although there are bound to be other qualities not in common or differences of measure of the common quality. The essence of affirmation here is thus ‘measurement’.

But Nature doesn’t measure anything. Every item in it just is, whatever it happens to be (at any given time and place). It is only a Subject with consciousness that measures: this against that, or this and that versus some norm.

This weighing work of the cognizing Subject is not, however, arbitrary (or ought not to be, if the Subject has the right attitudes). As in the above case of mere negation, the conclusion of it does proceed from certain existing findings. Yet, it is also true that this work only occurs in the framework of cognition.


2.    Negation and the Laws of Thought

Logic cannot be properly understood without first understanding negation. This should be obvious from the fact that two of the laws of thought concern the relation between positive and negative terms. Similarly, the basic principle of adduction, that hypotheses we put forward should be empirically tested and rejected if they make wrong predictions – this principle depends on an elucidation of negation.

a.         The so-called laws of thought are, in a sense, laws of the universe or ontological laws – in that the universe is what it is (identity), is not something other than what it is (non-contradiction) and is something specific (excluded middle).

They have phenomenological aspects: appearances appear (identity); some are in apparent contradiction to others (a contradiction situation); in some cases, it is not clear just what has appeared (an excluded middle situation).

They may also be presented as epistemological laws or laws of logic, in that they guide us in the pursuit of knowledge. However, they are aptly named laws of thought, because they really arise as propositions only in the context of cognitive acts.

To understand this, one has to consider the peculiar status of negation, as well as other (partly derivative) major processes used in human reasoning, including abstraction, conceiving alternative possibilities and making hypotheses.

b.         The impact of this insight on the laws of thought should be obvious. The law of identity enjoins us primarily to take note of the positive particulars being perceived. But the laws of non-contradiction and of the excluded middle, note well, both involve negation. Indeed, that’s what they are all about – their role is precisely to regulate our use of negation – to keep us in harmony with the more positive law of identity!

Their instructions concerning the subjective act of negation, at the most perceptual level, are as follows. The law of non-contradiction forbids negating in the perceptible presence of the thing negated. The law of the excluded middle forbids accepting as final an uncertainty as to whether a thing thought of is currently present or absent.

We are unable to cognize a negative (not-X) except by negation of the positive (X) we have in mind; it is therefore absurd to imagine a situation in which both X and not-X are true (law of non-contradiction). Similarly, if we carefully trace how our thoughts of X and not-X arise in our minds, it is absurd to think that there might be some third alternative between or beyond them (law of the excluded middle.)

Thus, these two laws are not arbitrary conventions or happenstances that might be different in other universes, as some logicians contend (because they have unfortunately remained stuck at the level of mere symbols, “X” and “non-X,” failing to go deeper into the cognitive issues involved). Nor are they wholly subjective or wholly objective.

These laws of thought concern the interface of Subject and Object, of consciousness and existence – for any Subject graced with rational powers, i.e. cognitive faculties that go beyond the perceptual thanks in part to the possibility of negation.

They are for this reason applicable universally, whatever the content of the material and mental universe faced. They establish for us the relations between affirmation and denial, for any and every content of consciousness.

c.         On this basis, we can better comprehend the ontological status of the laws of thought. They have no actual existence, since the concrete world has no use for or need of them, but exists self-sufficiently in positive particulars.

But the laws are a potential of the world, which is actualized when certain inhabitants of the world, who have the gifts of consciousness and freewill, resort to negation, abstraction and other cognitive-volitional activities, in order to summarize and understand the world.

In a world devoid of humans (or similar Subject/Agents), there are no negations and no ‘universals’. Things just are (i.e. appear) – positively and particularly. Negation only appears in the world in relation to beings like us who can search for something positive and not find it. Likewise for ‘universals’ – they proceed from acts of comparison and contrast.

Consciousness and volition are together what gives rise to concepts and alternative possibilities, to hypotheses requiring testing. It is only in their context that logical issues arise, such as existence or not, reality or illusion, as well as consistency and exhaustiveness.

It is important to keep in mind that the laws of thought are themselves complex abstractions implying negations – viz. the negative terms they discuss, as well as the negation of logical utility and value in contradictory or ‘middle’ thinking. Indeed, all the ‘laws’ in our sciences are such complex abstractions involving negations.

d.         The insight that negation is essentially a volitional act allied to cognition explains why the laws of thought are prescriptive as well as descriptive epistemological principles.

The laws of thought are prescriptive inasmuch as human thought is fallible and humans have volition, and can behave erratically or maliciously. If humans were infallible, there would be no need for us to study and voluntarily use such laws. There is an ethic to cognition, as to all actions of freewill, and the laws of thought are its top principles.

The laws of thought are descriptive, insofar as we commonly explicitly or implicitly use them in our thinking. But this does not mean we all always use them, or always do so correctly. They are not ‘laws’ in the sense of reports of universal behavior. Some people are unaware of them, increasing probabilities of erroneous thinking. Some people would prefer to do without them, and eventually suffer the existential consequences. Some people would like to abide by these prescriptions, but do not always succeed.

These prescriptions, as explicit principles to consciously seek to abide by, have a history. They were to our knowledge first formulated by a man called Aristotle in Ancient Greece. He considered them to best describe the cognitive behavior patterns that lead to successful cognition. He did not invent them, but realized their absolute importance to human thought.

Their justification is self-evident to anyone who goes through the inductive and deductive logical demonstrations certain logicians have developed in this regard. Ultimately it is based on a holistic consideration of knowledge development.

Our insights here about the relativity of negation and abstraction, and the realization of their role in the laws of thought serve to further clarify the necessity and universality of the latter.


3.    Consistency is Natural

It is important to here reiterate the principle that consistency is natural; whereas inconsistency is exceptional.

Some modern logicians have come up with the notion of “proving consistency” – but this notion is misconceived. Consistency is the natural state of affairs in knowledge; it requires no (deductive) proof and we are incapable of providing such proof, since it would be ‘placing the cart before the horse’. The only possible ‘proof’ of consistency is that no inconsistency has been encountered. Consistency is an inductive given, which is very rarely overturned. All our knowledge may be and must be assumed consistent, unless and until there is reason to believe otherwise.

In short: harmony generally reigns unnoticed, while conflicts erupt occasionally to our surprise. One might well wonder now if this principle is itself consistent with the principle herein defended that negatives are never per se objects of cognition, but only exist by denial of the corresponding positives. Our principle that consistency is taken for granted seems to imply that we on occasion have logical insights of inconsistency, something negative!

To resolve this issue, we must again emphasize the distinction between pure experience and the interpretations of experience that we, wordlessly (by mere intention) or explicitly, habitually infuse into our experiences. Generally, almost as soon as we experience something, we immediately start interpreting it, dynamically relating it to the rest of our knowledge thus far. Every experience almost unavoidably generates in us strings of associations, explanations, etc.

The contradictions we sometimes come across in our knowledge do not concern our pure experiences (which are necessarily harmonious, since they in fact exist side by side – we might add, quite ‘happily’). Our contradictions are necessarily contradictions between an interpretation and a pure experience, or between two interpretations. Contradictions do not, strictly-speaking, reveal difficulties in the raw data of knowledge, but merely in the hypotheses that we conceived concerning such data.

Contradictions are thus to be blamed on reason, not on experience. This does not mean that reason is necessarily faulty, but only that it is fallible. Contradictions ought not be viewed as tragic proofs of our ignorance and stupidity – but as helpful indicators that we have misinterpreted something somewhere, and that this needs reinterpretation. These indicators are precisely one of the main tools used by the faculty of reason to control the quality of beliefs. The resolution of a contradiction is just new interpretation.

How we know that two theories, or a theory and some raw data, are ‘in contradiction’ with each other is a moot question. We dismiss this query rather facilely by referring to “logical insight,” Such insight is partly ‘experiential’, since it is based on scrutiny of the evidence and doctrines at hand. But it is clearly not entirely empirical and involves abstract factors. ‘Contradiction’ is, after all, an abstraction. I believe the answer to this question is largely given in the psychological analysis of negation.

There is an introspective sense that conflicting intentions are involved. Thus, the ‘logical insight’ that there is inconsistency is not essentially insight into a negative (a non-consistency), but into a positive (the intuitive experience of conflict of intentions). Although the word inconsistency involves a negative prefix, it brings to mind something empirically positive – a felt tension between two theses or a thesis and some data.

For this reason, to say that ‘consistency is assumable, until if ever inconsistency be found’ is consistent with our claim that ‘negations are not purely empirical’. (Notice incidentally that we did not here “prove” consistency, but merely recovered it by clarifying the theses involved.)

The above analysis also further clarifies how the law of non-contradiction is expressed in practice. It does not sort out experiences as such, but concerns more abstract items of knowledge. To understand it fully, we must be aware of the underlying intentions. A similar analysis may be proposed to explain the law of the excluded middle.

In the latter case, we would insist that (by the law of identity) ‘things are something, what they are, whatever that happen to be’. Things cannot be said to be neither this nor the negation of this, because such characterizations are negative (and, respectively, doubly negative) – and therefore cannot constitute or be claimed as positive experience. Such situations refer to uncertainties in the knower, which he is called upon to eventually fill-in. They cannot be proclaimed final knowledge (as some modern sophists have tried to do), but must be considered temporary postures in the pursuit of knowledge.


4.    Status of the Logic of Causation

It should be pointed out that the theory of negation here defended has an impact on our theory of causation. If causation relates to the conjunctions and non-conjunctions of presences and absences of two or more items – then our knowledge of causes (i.e. causatives) is subsidiary to judgments of negation. It follows that the logic of causation is not “purely empirical,” but necessarily involves acts of reason (namely the acts of negation needed to declare something absent or two or more things not conjoined).

Incidentally, we can also argue that causative judgments are not purely empirical with reference to the fact that it always concerns kinds of things rather than individual phenomena. Truly individual phenomena are by definition unrepeated and so cannot strictly be said to be present more than once, let alone said to be absent. Causation has to do with abstractions – it is conceptual, it concerns classes of things. In this regard, too, causation depends on rational acts.

These features of causation do not make it something non-existent, unreal or invalid, however. The skeptic who tries to make such a claim is also engaged in negation and abstraction – and is therefore implicitly suggesting his own claim to be non-existent, unreal or invalid! One cannot use rational means to deny reason. It is obviously absurd to attempt such intellectual convolutions, yet many have tried and keep trying.

The polemics of Nagarjuna and David Hume are examples of such sophism. As I have shown in previous writings, they try to deny causation without even defining it properly (and likewise for other rational constructs). This is a case of the fallacy I have identified more generally in the present reflections – namely, the attempt to deny something before one even has something to deny. What are they disputing if indeed there is nothing to discuss?

As we have seen, awareness of the distinctiveness of negative terms can have consequences on logical practice. Generally speaking, a negative term (i.e. one contradicting a positive term) is more naturally a predicate rather than a subject of (categorical) propositions. Similarly, the negation of a proposition is more naturally a consequent than an antecedent.

Using a negative term as a propositional subject is sometimes a bit artificial, especially if the proposition is general. When we so use a negative term, we tacitly understand that a set of alternative contrary positive terms underlie it. That is to say, given “All non-A are B,” we should (and often do) look for disjuncts (say C, D, E, etc.) capable of replacing non-A.

In the case of a causative proposition, the positive side of the relation may be more effective than the negative side, even when the latter is the stronger. That is, when the causative seems on the surface to be a negation, we should (and often do) look deeper for some positive term(s) as the causative.

This recommendation can only, however, be considered heuristic. Formal rules remain generally valid.


5.  Zero, One and More

Another consequence of the theory of negation has to do with the foundations of mathematics. What is the number ‘zero’ (0)? It refers to the ‘absence’ of units of some class in some domain. And of course, we can here reiterate that there is no possibility of concretely identifying such absence, without having first sought out the presence of the units concerned. Therefore, here too we can say that there is a sort of relativity to a Subject/Agent (who has to seek out and not find a certain kind of unit).

But of course, not only zero is ‘relative’ in this sense. We could say that the only purely empirical number is the unit, one (1). It is the only number of things that can be perceived directly, without processing information. As we said earlier, there are only positive particulars. We may here add: each of them is ‘only a unit’, never ‘one of many’.

Such units may be mentally (verbally or even just intentionally) grouped together, by means of some defining rule (which may just be a circle drawn in the dust around physical units, or a more abstract common and exclusive characteristic). We thus form natural numbers larger than one (such as 2, 3, etc.) by abstraction. It follows that any number larger than one (as in the case of zero) can be actualized only if there is someone there to do the counting.

Thus, zero and the natural numbers larger than one are less directly empirical than the unit; they are conceptual constructs. It still remains true that ‘2+2 = 4’ or false that ‘2+2 = 5’ – but we do not get to know such truth or falsehood just by ‘looking’ out at the world: a rational process (partly inductive, partly deductive) is required of us. If no one with the needed cognitive powers was alive, only units would actually exist – other numbers would not appear.

And if this dependence on someone counting is true of whole numbers, it is all the more true of fractions, decimals and even more abstract numerical constructs (e.g. imaginary numbers). As for ‘infinity’, it is obviously the most abstract of numerical constructs – considering, too, the negativity it involves by definition.

But we can go one step further in this analysis, and re-examine our above notion of a purely empirical unit! Implicit in this notion is that what appears before us (in the various sensory media, and their mental equivalents) is a multiplicity of distinct units. This already implies plurality – the existence of many bits and pieces in a given moment of appearance (different shapes, colors, sounds, etc.), and/or the existence of many moments of appearance (across ‘time’, as suggested by ‘memory’).

But multiplicity/plurality does not appear before us through mere observation. It is we (those who are conscious of appearances) who ‘sort out’ the totality of appearance into distinct bits and pieces (e.g. physical or mental, or sights and sounds, or blue and white), or into present phenomena and memories of phenomena. We do this by means of intentions and mental projections (acts of will, sometimes involving imagination), in an effort to summarize and ‘make sense of’ the world we face.

Thus, to speak of ‘positive particulars’ as pure percepts (or in some cases, as objects of intuition) is not quite accurate as phenomenology. The starting data of all knowledge is a single undifferentiated mass of all our experience. This is split up and ordered in successive stages.

Consider my field of experience at a given moment – say, for simplicity, I look up and see a solitary bird floating in the blue sky, i.e. two visual objects (ignoring auditory and other phenomenal features), call them x and y.

Initially (I postulate), they are one experience. Almost immediately, however, they are distinguished from each other (I postulate this true even for a static moment[1], but it is all the more easy to do as time passes and the bird flies through different parts of the sky, and other birds and clouds come into the picture).

This basic distinction is based on the fact that the bird has a shape and color that visually ‘stand out’ from the surrounding blue of the sky, i.e. by virtue of contrast. This may be called ‘imagined separation’, and involves a mental projection (or at least, an intention) of imaginary boundaries between the things considered.

It need not (I again postulate) involve negations. That is, I make a distinction because x is x and y is y, not because x is not-y and y is not-x. The latter negations can only logically occur as an afterthought, once the former contrasts give me separate units I can negate.

The acknowledgment of ‘many’ things within the totality of experience (a sort of epistemological initial ‘big bang’) is already a stage of ratiocination. Negation is yet another of those stages, occurring perhaps just a little after that. Numbers are yet a later stage, dependent on negation (since to explicitly distinguish things from each other we need negation).

By the way, the arising of multiplicity does not only concern external objects; we must also take into consideration the Cartesian cogito ergo sum. This refers to the development of successive pluralities relating to the psyche, notably:

Cognized and cognizing, and also cognition; thus, Subject – consciousness – Object.

Self and other; or further, soul/spirit, mind, body and the rest of the world (the latter also spiritual, mental and material/physical).[2]

Everything beyond the totality of experience depends on judgment, the cognitive activity we characterize as rational. Such judgment exists in varying amounts in humans. It also seems to exist to a lesser degree in higher animals (since they search for food or look out for predators, for instances), and even perhaps a little in the lowest forms of sentient life (though the latter seem to function almost entirely by reflex).[3]


Drawn from Ruminations (2005), Chapter 9:5-6,8-10.


[1]             Of course, the observer of the static moment takes time to make a distinction between items within it. But there is no inconsistency in our statement, since we are not claiming our world as a whole to be static but merely mentally considering a static moment within it.

[2]             The distinction between internal and external objects varies with context, of course. ‘Internal’ may refer to spiritual intuitions (own cognitions, own volitions, own appraisals, and self), mental phenomena (memories, mental projections, emotions), or bodily phenomena (sensations and visceral sentiments). ‘External’ then means, respectively, phenomena in one’s own mind-body and beyond, or only those in one’s body and beyond it, or again only the world outside one’s body.

[3]             A good argument in favor of this thesis, that mental separation and negation are distinct stages of distinction, is the possibility it gives us (i.e. biology) of supposing that lower animals are aware of multiplicity but unable to negate (because the latter requires a more pronounced level of imagination).

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