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RUMINATIONS

© Avi Sion, 1990-2005 All rights reserved.

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RUMINATIONS

Part I – Chapter 6

About “Modern Logic”

1. A School of Logicians

2. Alleged New Methods

3. Non-Aristotelian “Logic”

4. Postmodern “Logic”

5. Mere Manipulations

6. Thinking Reflexively

7. Conventional Logic

8. Absolute Truths

9. Untouched by Consciousness

10. Logical Atomism

11. Exclusive Judgments

12. Empty Terms

1. A School of Logicians

“Modern logic” is the name of a school (or set of schools) of logicians. The term refers specifically to logicians with certain anti-traditional tendencies; it is not intended to include all logicians of modern times.

For example, though Jean Piaget is a 20th Century logician, I would not class him as a “modern” logician in this sense. Moreover, most logicians are only in part “modern” in this pejorative sense; they still adhere to some traditional premises and conclusions. An example of this half-half class in my view is Bertrand Russell.

2. Alleged New Methods

Some “modern logicians” claim to have developed “new methods” of validation of syllogism. This claim seems pretentious to me, just a way for these people to give themselves a place in the history of logic. For the question is: do these new methods arise in response to actual problems – i.e. errors – in the old methods, or were the latter only a bit wishy-washy? Why is Aristotle’s exposition of ‘Barbara’, say, considered insufficient? It causes no error, as far as I know; at worst, it is perhaps a bit vague. Also, are these methods really new, or just applications of Aristotle’s teachings? If we look closely, we notice the latter.

Any improvements in clarity, rigor and credibility, the moderns have made are of course welcome. But this achievement remains relatively modest in comparison to Aristotle’s original work in that field, unless they have identified errors in the latter’s approach. If their only claim to fame is that Aristotle was ‘too intuitive’, we can reply that their allegedly ‘more scientific’ insights are also ultimately just intuitions. That is, all logical science is ultimately based on conceptual insight.

As for the specific techniques used by the moderns, they are all mere derivatives of the Aristotelian schema of syllogistic reasoning; they do not stand over and above it, or prior to it. They are just further ways of better digesting the already known – which is all well and good, but does not justify blowing any trumpets.

The modern revolutions that occurred in mathematics – such as non-Euclidean geometry – were (so I have been taught) due to the perception of errors in the old methods, which made it necessary to develop new foundations. I do not see such necessity involved in the development of modern logic; the motive seems rather to have been an intense desire of self-assertion by certain academics. Logic was already adequately “validated” – legitimatized.

Aristotle’s work has not been displaced by modern logic, in the way that Ptolemaic astronomy was replaced by Copernicus. The relationship between Aristotelian and modern logic is not even one of inclusion in a larger theory, akin to that between Newton and Einstein, because whereas Einstein found limits to Newton, the moderns did not fundamentally circumscribe the applications of Aristotle. The syllogistic he developed remains valid.

This does not mean that new discoveries have not been made. Some have indeed been very enlightening and fruitful. For example, the studies of classification, hypothetical propositions, of paradoxes, of modalities, of induction, have greatly evolved.

For my part, I think the most important rule for logicians to follow is this: any theory of knowledge proposed must fully account for its own genesis within the theory. A logician must always consider his own thought processes, and whether he has verified their consistency, explained their role and demonstrated their validity within his theorizing about logic. And with regard to this crucial criterion, I must say that so-called modern logicians have all too often fallen short.

3. Non-Aristotelian “Logic”

As already stated, many “modern logicians” – since the late 19th Century – have yearned to do for (or to) Logic, what Copernicus did in Astronomy, or later what Einstein did in Physics. Each one of them was, it seems, fired by the grandiose desire to be the equivalent great modern revolutionary in the field of logic.

They thus inaugurated a persistent assault on Reason, a veritable carnival of Unreason, which has lasted for over a hundred years, with disastrous consequences for many a poor mind and for social peace and wellbeing.

Their conceptual model was non-Euclidean geometry. Just as modern mathematicians came to consider certain Euclidean axioms to be debatable, if not arbitrary, so these modern logicians sought to put in doubt or discard the Aristotelian “laws of thought”, and found some new system – a “non-Aristotelian logic”.

But this is an impossible exercise, because[1] the laws of thought are more fundamental to reason than Euclid’s axioms (in particular, that regarding parallels). The geometrical model of axioms and theorems is only superficially applicable to logic, because it is itself an aspect or teaching of (Aristotelian) logic.

When mathematicians decided to review the traditional axioms of geometry, they were using reasoning by means of the laws of thought. They argued: “we see no self-contradiction, or doctrinal inconsistency, or even (eventually) contradiction to experience in proposing some alternative axioms and systems; therefore, Euclid’s assumptions are not exclusive and irreplaceable.”

The same cannot be argued in the case of logic itself, without self-contradiction. We cannot, say, point to the particle-wave duality and say “it seems that contradictions do exist in the world, therefore we shall review the logical axiom of non-contradiction” – we cannot do so, for the reason that such review is motivated and rendered credible precisely by the law of non-contradiction, in the way of an attempt to restore an apparently lost consistency.

The very method used of reviewing one’s premises in the face of contradiction and abandoning or at least modifying one or more of them to recover consistency – this very methodology is a teaching of Aristotelian logic! We cannot say: “I understand that if I advocate contradiction, I open myself to being contradicted; but that does not bother me, because it is a consistency of sorts – I accept self-contradiction.”

In the very act of making such a superficially reasonable proposal, we are reasserting the universality of the laws of thought, their being at the very root of reason, inherent in the very act of reasoning. The only way we could conceivably abandon these laws would be to give up all thought, all attempt at rational knowledge. Logic cannot be used against itself: it is the very paradigm and paragon of consistency.

We can suggest: “A can be non-A”, or some such “new axiom” for logic, but the resulting discourse will still be nonsense – however nicely wrapped up and ordered, however well “systematized” stealing the methods of Aristotelian logic. Such proposals are an imposture.

Those who propose such ideas are swindlers, profiting from the gullibility and intimidation of many people. It is like in the story of the emperor’s new clothes, in which con men sold the emperor invisible clothes, which no one dared to deny were clothes – till a child pointed out he was naked.

There simply is no such thing as “non-Aristotelian logic” (i.e. a logical system that denies one, two or all three laws of thought). To come forward with such a system is merely to pronounce words. These words have no collective content, no meaning; there is nothing behind them other than the imagination that there might be something behind them because the phrase is composed of individually meaningful words.

No “Copernican revolution” is conceivable in the field of logic: it would not merely be anti-Aristotelian but anti-rational. Logicians must abandon such vain ambitions, and more modestly continue to expand the scope of logical analysis and the depth of understanding of logic. The role of logicians is to do logic, not undo it. Reason is a precious value for mankind, and logicians ought to be its guardian.

Would you entrust your life to, say, an airplane built by engineers practicing “non-Aristotelian logic”, people who feel cozy in the midst of contradictions and in between truth and falsehood? Similarly, in all fields of human endeavor and interaction: logic is a guarantee of sanity and safety.

4. Postmodern “Logic”

As if such irrational currents were not enough, there is (I gather) a new generation of “postmodern” logicians and philosophers who eschew even the pretense of accountability, considering that any discourse that seems to be about “logic” is acceptable. These are of course part of a wider trend, not limited to our field.

Being relativists, these people are not directly attacking anything or anyone. They are not mere anti-rationalists: they are so indifferent to the niceties of reason that they feel no need to justify themselves. They are of course the natural offspring of the moderns, taking their teachings to their ‘logical’ conclusion. They are more consistently illogical than their predecessors, no longer owing a semblance of allegiance to reason, not needing even to pay lip service to it. Absurdity does not bother them, so they need no logical window dressing for their doctrines.

Indeed, these people take pride in their fashionable madness. They strive to be as confusing and incomprehensible as possible, considering that what others cannot possibly understand must be very deep indeed. They have only a very vague notion of what logic is about, but seek to impress other people with meaningless symbolic constructs and use of fancy pseudo-scientific terminology. They prattle away, eruditely formulating fake theories immune to any empirical or rational review. They function as (con) artists rather than scientists.

Yes, such people do exist; some even have teaching positions in prestigious universities. Because most people – including some in high academic positions, including some who are hired to teach logic – know or understand little about logic, they are easily intimidated by such intellectual posturing and imposture. They fear to reveal their own poverty in the course of questioning or debate.

Besides, it is no use denouncing the swindle; no one apparently cares, because few people realize the importance of logic (apart from some simple formulas needed in computer programming). Reason is out of fashion, has been for generations. Logic is too abstract; you cannot show artistic footage of it on TV. It cannot be very entertaining: it requires an effort of thought.

5. Mere Manipulations

Most “modern logicians” base their approach to logic on the manipulation of pre-existing knowledge[2]. They do not properly ask: “how are concepts and propositions in the first place produced?” but are content to look into how they think these ready-made products should be ordered relative to each other. Another example: relations between the modalities are discussed conventionally, without having clarified how they are apprehended and how they may be comprehended.

What logicians develop in such manner cannot even rightly be called (as they call it) a “deductive system”; it is just a set of invented schemas for ordering given units. Some place the chicken before the egg; others prefer placing the egg before the chicken. They do not ask where both chicken and egg came from. They place their systems in orbit, but do not ground them anywhere. But the proverbial buck has got to stop somewhere!

They do not consider the possibility that their proposed epistemology is bound to skew the results, i.e. give a misleading image of the nature of knowledge.

They have not understood that deduction is only fully comprehensible within an “inductive system” of logic (such as the one proposed in Future Logic and my other works). These people fail to grasp the essentially epistemological task of logical science, which is to find out how humans tend to and should organize knowledge, i.e. how knowledge actually develops and how such development can be optimized.

A true system of logic is one that treats the issue of knowledge as a whole – and in that perspective, knowledge is essentially an inductive enterprise, in which deduction is one of the tools used. Knowledge cannot be likened to a construction using “building blocks” (or atoms of knowledge). It is something much more fluid, a process; yet it has apprehensible behavioral patterns and rules.

Knowledge starts with experience of appearances (phenomena, intuitions, and logical insights), out of which cognitive entities (concepts, propositions) are gradually formed (through more or less logical arguments) by humans, in an effort to comprehend and sort out the experiences. Appearances are the ground of all knowledge.

Symbols invented by logicians can never be effective “placeholders” for such basic data. Logicians must never forget that their theories are abstractions without meaning if not firmly anchored to their empirical sources. Logic is not only about final, static relations; the ongoing process of induction must always be kept in mind.

6. Thinking Reflexively

Logicians and philosophers must learn to think reflexively – and always ask themselves how they arrived at and can justify their own beliefs and proposals. Even concepts and propositions that seem obvious and reasonable enough must be subjected to reflective scrutiny.

For example, when Wittgenstein II claims that ‘understanding’ consists in knowing the conditions of truth, i.e. the rules of verification – he sounds credible. But upon reflection, one might ask how such knowledge (of correct procedures) is itself to be discovered and established. Surely, the basis of it cannot be previously known procedures, and so on ad infinitum. If we only refer to the said thesis, ‘understanding’ remains ultimately unexplained. Therefore, it is inadequate to the theoretical task at hand.

That is, some ‘understanding’ must be accepted as primary – i.e. some knowledge content and logical insights must be irreducible, capable of informing and convincing us directly and fully. Broad principles like the laws of thought must be among these first understandings. Only after they are apprehended and comprehended is it possible to develop specific deductive, and indeed inductive, verification procedures.

Again, Frege insists that thought is not possible without language[3] – relying for his credibility on a very limited sense of the word ‘thought’ and totally ignoring the issue of how language itself is to be grasped without prior thought. He demands defined terms throughout – but such a starting premise for ‘language theory’ is unjustifiable, since it generates infinite regression. These are just the hang-ups of a narrow-minded formalist.

Very few terms are predefined in the way Frege expects and demands. With careful observation of our mental behavior, it becomes evident that most terms have inductive definitions that develop gradually by trial and error, going through adaptive changes as relevant data and thoughts emerge); and indeed, some terms are never defined (very basic ones like ‘existence’ are irreducible primaries).

It is ironic that such people, who claim to be logicians, have not understood the basic teaching of logic – that cogency depends on complete consistency.

7. Conventional Logic

Logic is not a convention, an arbitrary setup agreed between self-styled logicians.

What do we mean by “conventional logic”? Here is an example: “If the green traffic light goes on, it is permitted and safe to move on; whereas if the red one goes on, it is not.” This is a social convention, useful for living in the world of people.

Many of our propositions are of this sort: they signify an agreement among all participants (which may be imposed by authorities, but must be made known to all others) as to what certain symbols are intended to mean. There may be (indeed, must be) some underlying factual (i.e. non-conventional) truth; for example, whether the light is green or red, and whether accidents are less likely if the rules convened are obeyed. But some aspect is arbitrary, i.e. it could have been otherwise if we had so willed it; for example, we could have used the red color for “pass” and the green for “wait”.

Buddhist philosophers, by the way, use the word “conventional” very freely, with reference to any view they want to discredit. They regard all ordinary – i.e. non-enlightened – knowledge as conventional. That is clearly incorrect usage – at least for those of us who have not personally encountered the enlightened view. For the term may only be used in contrast to something non-conventional; it cannot be literally universal without self-contradiction.

In my view, logic in general is very definitely not “conventional”. Sorting out conventions is one of the tasks of logic, a very minor task. Logic is much broader than that, concerned with the ways to arrive at “knowledge of reality”, whatever that be.

Note also, in passing, that opinions people label as “conventional wisdom” are often neither granted by everyone nor wise. The expression is often just false advertising, to make believe.

8. Absolute Truths

We must admit some truths to be absolute.

Even if reality is relative to consciousness in some way, as some philosophers advocate, then that observation becomes the framework for “realism” – i.e. that is the fundamental truth independent of the observer. Realism does not have to be equated to extreme materialism, but some sort of fixed “fact” must be admitted.

In such case, if consciousness somewhat affects reality – as the idea of relativity here seems to suggest – what sort of impact does the subject and his consciousness have on the object it relates to? Is consciousness (so conceived) a veil, a distortion, or a modifying or creative force? Whatever its effect, the important issue would be whether we can somehow become aware of such effect and correct our reading for it.

The techniques of induction are such that they are in principle capable of discovering such eventual effects and correcting our knowledge accordingly. Inductive knowledge is a result of an ongoing process of hypothesizing and confronting our hypotheses with experience. It is a holistic enterprise, which does not statically depend on specific beliefs. It is the cunning way we are able to transcend our actual, or even just conceivable, limitations or faults.

In this context, I hasten to add, the proposed hypothesis of an unknowable (and not merely unknown) “thing in itself” is the inductively weakest speculation, being by definition unverifiable with regard to any experience whatsoever. Consciousness must be admitted to get some part of its object right, if only its realization that it is getting some part of its object wrong. If it were completely wrong, it would not even be able to conceive of an object beyond its ken.

9. Untouched by Consciousness

Beneath all Bolzano’s deviant logical terminology, and theoretical misconceptions, one discerns the shadow of Kant. This is part of the ravage caused by the latter’s pretentious “thing-in-itself”, his notion of a “noumenon”, of something beyond the phenomenal unknowable to anyone (but Kant himself, of course) and yet open to discussion (somehow, in spite of the inherent contradiction – indeed because of it, because of the perverse twist in it).

In the last analysis, Bolzano is not interested in studying ordinary abstraction from experience, the ways we come to know the unknown; he is instead pursuing a Kant-like “transcendental logic”, a means to somehow get to know the unknowable. His sought after object is not real, but “surreal”. He wants to do the impossible and inconceivable: to cognize the “in-itself” – i.e. something untouched by consciousness – ignoring that the moment he did cognize it, it would not longer fit his requirement.

Note that I am not taking the position that nothing is untouched by consciousness. I believe some things exist beyond consciousness (at least, human consciousness), based on the observation that my own knowledge is variable and different from that of others. I am merely pointing out that there is no need to look for some pristine object unspoiled by cognition; everything is pure and virginal until cognized by someone, and consciousness does not necessarily pollute its object.

10. Logical Atomism

Modern logicians are inclined to “atomism”, cutting statements or texts into parts and then considering the interrelations between these parts and their relations to the whole. The study of the relations of whole and parts has been dubbed “mereology”. The parts are viewed as atoms, and together they build up the whole; the relations between them are the structure that keeps them together, their cohesion.

But my question would be: are the relations between the parts not themselves parts? The answer would surely be: yes – if our analysis of the whole into parts is to be fully explicit. In that case, one might go on, and ask if the relations do not have relations among themselves and with the remaining parts? The answer again has to be: yes, they do.

From which it follows that there are an infinity of relations and parts – and the proposed atomistic method of analysis is in fact impracticable. Note that it is not ‘infinity’ per se that is the problem, here – since presumably the world is a whole made up of an infinity of parts and relations. The problem is the need to verbalize all that, i.e. to repeat an infinite world in words.

Clearly, the error of such atomism is to regard all units of thought as concrete items; specifically, they are words. Thought is confused with its outward symbols, the words of our discourse. In this view, even abstract items are concrete, since they have no real existence till they are put into words. Clearly, the proponents of this view have not thought their proposal through; had they done so, they would have realized its absurdity.

This is in contrast to the classical, Aristotelian, approach, which makes a distinction between form and content. The words, the symbols, are only forms – distinct from their contents, the underlying meanings, the realities (or at least, appearances) that they are intended to refer to. The relations exist abstractly, even when not verbalized; and verbalizing them does not make them concrete, it merely tags on a concrete label to them.

For this reason, there is no infinity of relations over and above the first or second relations. There are (abstract) relations between non-relations; then there are (more abstractly) relations between relations; and then nothing more. You cannot propose ‘relations between relations and non-relations’, because these are identical to the first category, i.e. ‘relations between non-relations’. You cannot propose ‘relations between relations between relations’ (and so on, ad infinitum) because all these are already covered by the second category, i.e. ‘relations between relations’.

In the latter cases, the words may differ, but the underlying referent is still the same. As soon as you have a ‘relation’ between two or more (concrete) things, you have not only the (abstract) glue between the things, but also the glue between that glue and each of these things. There is no new glue to stick the glue; it is that very same glue all through. On the other hand, comparison between this glue and the glue between other sets of things requires a new, more abstract ‘relation’ – another kind of ‘glue’. But that additional ‘relation’ is singular – it is simply ‘glueness’; that is, no further levels of abstraction are possible beyond it.

Moreover, this concrete image of ‘glue’ to explain ‘relations’ should not be taken too literally. The abstract has a much less ‘real’ existence than the concrete. It refers to common measures or degrees between things in some respect(s). These are in a sense ‘out there’, because we can directly or indirectly compare things; for instance, we can take a measuring tape and observe the proportion between the widths of two bodies. But in another sense, abstracts are not quite ‘out there’, but depend for their actual existence on there being an observer able to compare. Till then, abstracts have only potential existence.

The results of comparisons (if carefully made) are ‘objective’ in the sense that they reflect ultimately concrete events beyond the observer; but they still depend on the presence of an observer – a ‘Subject’ engaged in measurement. The latter proposition about subjectivity, too, if true, is an objective truth of sorts; note well, it claims to be as factual as any other fact (concerning concretes).

We might thus say that the abstract is a more potential being, compared to the actuality of the concrete, insofar as its existence is observable less directly, i.e. it requires additional cognitive processes (of measurement by someone). Note that results of measurement are in principle repeatable, although in practice the opportunity to do so may pass us by too quickly.

Note lastly: the distinction of ‘form and content’ may be used not only for ‘words and meanings’ (as done above); in some contexts, it is intended to refer to ‘abstract and concrete’ or ‘concept and percept’ and other such pairs. The underlying image is that of container and contained.

11. Exclusive Judgments

A lot of ‘modern’ logic and philosophy seems to have arisen because of exclusive judgments of the form “Q, but not P”, instead of the inclusive “Q, as well as P”. Instead of amplifying past ideas with new insights (for example, adding to Aristotle’s subject-predicate logic, by investigating comparatives like “A > B”[4]) – the tendency was to provocatively belittle, or try to reject and replace the old, so as to ascribe more importance to the new. I can’t help seeing such behavior as pretentious and arrogant.

To discover that some thesis “P” does not cover all the ground of some area of knowledge does not justify saying “not P”, but only “not only P” or “P is not the whole story”. Because it is only the assumption that P was all, the excessive generalization of it, that can be faulted, and not the item P as such. Particularization is only partial denial; to equate it to thorough denial is wrong inference; it is extremism.

Conversely, we might say that such people themselves over-generalize. Thus, for example, as I explained in Future Logic[5], Godel builds his theory of logic with reference to a too-limited pool of propositional forms. Or again, the underlying fallacy committed by Frege in his linguistic analysis (literally: cutting up statements into constituent parts) – is to take one example, one kind of case, and to generalize his treatment from there, without attention to the possibility of other cases.

Frege assumes that all statements can be split up (at will, by the imagination) into two parts: an ‘argument’ and a ‘function’. Thus, in “Caesar conquered Gaul”, “Caesar” is the argument and “conquered Gaul” is the function; the latter is like a container (‘unsaturated’) and the former fills it with a definite content, completing it (‘saturating’ it)[6]. But, as I have shown in Future Logic[7], in my treatment of the Russell paradox, such cutting up of a sentence is not always logically permissible: for instance, statements about membership cannot be permuted without producing contradiction.

12. Empty Terms

With regard to empty terms – i.e. terms devoid of referents. Human knowledge is built in part through the imagination. A term may be imaginary, meaning that its referents are knowingly fictional (i.e. we know there is no such animal in fact), or tentatively assumed for inductive purposes (until actual cases are observed).

We often conceive of things we have not yet actually experienced, e.g. in constructing a theory, and then try and find out whether our construct can be confirmed. This is a standard practice of inductive logic. Sometimes, we eventually come to the conclusion that our assumption was unjustified, and the imagined term is in fact empty. Sometimes, we arrive at such a negative conclusion, after for a long time believing the term not empty, and then after further investigation discovering to our surprise that it is empty.

A proposition involving a term known to be empty is, strictly speaking (i.e. factually), “false”. A proposition with a fictional term may be considered conventionally true – for example, “unicorns are horses with a horn and wings”. This is conventionally true, in the sense that the definition rightly describes our mental image of a unicorn; but it is factually false, in that there are no unicorns in the material world.

A proposition involving a term of uncertain status in this respect, i.e. we think but do not know for sure that the term has referents, is “either true or false”. Frege’s claim that such statements are “neither true nor false” is not correct, and sows confusion[8].

Some statements are indeed neither true nor false – for example, “this is false” or “this is true”[9]. But, though composed of words that are meaningful in other contexts and are here put together in a grammatically valid way, such statements are on closer scrutiny found to be meaningless verbal constructs; they have neither referents nor sense. But statements with empty terms, or possibly empty terms, are either true or false.

There are also of course propositions that are false, though all of their terms have referents – because the conjunction of their terms is inappropriate; i.e. the terms do not belong together in the way proposed.

In conclusion, empty terms can only be properly understood through consideration of inductive logic. If they are analyzed with a narrowly deductive logic outlook, like Frege’s, they will be misunderstood.



[1] As I have explained repeatedly in Future Logic.

[2] I would classify this approach as Neo-Cartesian, save for my respect for Descartes. Worse still, they end up manipulating mere symbols (becoming Nominalists). Among the “logicians” intended here, I count even Bolzano, although in his case the manipulation involved is not one of symbols, but of artificial concepts.

[3] Incidentally, if it were true that thought without language is impossible, one would have to continuously speak to oneself, whether in one’s head or out loud. Yet, when we become conscious of doing that, we commonly reprove ourselves for being excessively talkative, i.e. for verbalizing things much more than necessary. This shows that we commonly consider words not always needed for thought. The same is true in interpersonal communication – we are annoyed by people who speak too much, preferring those who can control their tongues.

[4] I give this as an example of a proposition not yet permuted into the form “S is P”. I could equally give as an example a sentence like “A loves B”. See why further on.

[5] Chapter 66.2.

[6] See Jones, p. 147 – “Never ask for the meaning of a word in isolation,” etc. The funny thing is that this is precisely Frege’s own error here!

[7] Chapter 45.

[8] As does his claim that the only referents of any statement are its truth or falsehood! If this were so, surely all statements would have one of two meanings: true ones the meaning true and false ones the meaning false.

[9] See my analysis of the liar paradox in Future Logic, chapter 32.2.

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2016-06-13T11:27:44+00:00