Causality. And a bit miscellaneous.

0. I’ve been too busy in my day-job to write anything at any one of my blogs, but recently, a couple of things happened.


1. I wrote what I think is a “to read” (if not a “must read”) comment, concerning the important issue of causality, at Roger Schlafly’s blog; see here [^]. Here’s the copy-paste of the same:

1. There is a very widespread view among laymen, and unfortunately among philosophers too, that causality requires a passage of time. As just one example: In the domino effect, the fall of one domino leads to the fall of another domino only after an elapse of time.

In fact, all their examples wherever causality is operative, are of the following kind:

“If something happens then something else happens (necessarily).”

Now, they interpret the word `then’ to involve a passage of time. (Then, they also go on to worry about physics equations, time symmetry, etc., but in my view all these are too advanced considerations; they are not fundamental or even very germane at the deepest philosophical level.)

2. However, it is possible to show other examples involving causality, too. These are of the following kind:

“When something happens, something else (necessarily) happens.”

Here is an example of this latter kind, one from classical mechanics. When a bat strikes a ball, two things happen at the same time: the ball deforms (undergoes a change of shape and size) and it “experiences” (i.e. undergoes) an impulse. The deformation of the ball and the impulse it experiences are causally related.

Sure, the causality here is blatantly operative in a symmetric way: you can think of the deformation as causing the impulse, or of the impulse as causing the deformation. Yet, just because the causality is symmetric here does not mean that there is no causality in such cases. And, here, the causality operates entirely without the dimension of time in any way entering into the basic analysis.

Here is another example, now from QM: When a quantum particle is measured at a point of space, its wavefunction collapses. Here, you can say that the measurement operation causes the wavefunction collapse, and you can also say that the wavefunction collapse causes (a definite) measurement. Treatments on QM are full of causal statements of both kinds.

3. There is another view, concerning causality, which is very common among laymen and philosophers, viz. that causality necessarily requires at least two separate objects. It is an erroneous view, and I have dealt with it recently in a miniseries of posts on my blog; see https://ajitjadhav.wordpress.com/2017/05/12/relating-the-one-with-the-many/.

4. Notice, the statement “when(ever) something happens, something else (always and/or necessarily) happens” is a very broad statement. It requires no special knowledge of physics. Statements of this kind fall in the province of philosophy.

If a layman is unable to think of a statement like this by way of an example of causality, it’s OK. But when professional philosophers share this ignorance too, it’s a shame.

5. Just in passing, noteworthy is Ayn Rand’s view of causality: http://aynrandlexicon.com/lexicon/causality.html. This view was basic to my development of the points in the miniseries of posts mentioned above. … May be I should convert the miniseries into a paper and send it to a foundations/philosophy journal. … What do you think? (My question is serious.)

Thanks for highlighting the issue though; it’s very deeply interesting.

Best,

–Ajit


3. The other thing is that the other day (the late evening of the day before yesterday, to be precise), while entering a shop, I tripped over its ill-conceived steps, and suffered a fall. Got a hairline crack in one of my toes, and also a somewhat injured knee. So, had to take off from “everything” not only on Sunday but also today. Spent today mostly sleeping relaxing, trying to recover from those couple of injuries.

This late evening, I naturally found myself recalling this song—and that’s where this post ends.


4. OK. I must add a bit. I’ve been lagging on the paper-writing front, but, don’t worry; I’ve already begun re-writing (in my pocket notebook, as usual, while awaiting my turn in the hospital’s waiting lounge) my forth-coming paper on stress and strain, right today.

OK, see you folks, bye for now, and take care of yourselves…


A Song I Like:

(Hindi) “zameen se hamen aasmaan par…”
Singer: Asha Bhosale and Mohammad Rafi
Music: Madan Mohan
Lyrics: Rajinder Krishan

 

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Relating the One with the Many

0. Review and Context: This post is the last one in this mini-series on the subject of the one vs. many (as understood in the context of physics). The earlier posts in this series have been, in the chronological and logical order, these:

  1. Introducing a very foundational issue of physics (and of maths) [^]
  2. The One vs. the Many [^]
  3. Some of the implications of the “Many Objects” idea… [^]
  4. Some of the implications of the “One Object” idea… [^]

In the second post in this series, we had seen how a single object can be split up into many objects (or the many objects seen as parts of a single object). Now, in this post, we note some more observations about relating the One with the Many.

The description below begins with a discussion of how the One Object may be separated into Many Objects. However, note that the maths involved here is perfectly symmetrical, and therefore, the ensuing discussion for the separation of the one object into many objects also just as well applies for putting many objects together into one object, i.e., integration.


In the second and third posts, we handled the perceived multiplicity of objects via a spatial separation according to the varying measures of the same property. A few remarks on the process of separation (or, symmetrically, on the process of integration) are now in order.

1. The extents of spatial separation depends on what property you choose on the basis of which to effect the separation:

To begin with, note that the exact extents of any spatial separations would vary depending on what property you choose for measuring them.

To take a very “layman-like” example, suppose you take a cotton-seed, i.e. the one with a soft ball of fine cotton fibres emanating from a hard center, as shown here [^]. Suppose if you use the property of reflectivity (or, the ability to be seen in a bright light against a darker background), then for the cotton-seed, the width of the overall seed might come out to be, say, 5 cm. That is to say, the spatial extent ascribable to this object would be 5 cm. However, if you choose some other physical property, then the same object may end up registering quite a different size. For instance, if you use the property: “ability to be lifted using prongs” as the true measure for the width for the seed, then its size may very well come out as just about 1–2 cm, because the soft ball of the fibres would have got crushed to a smaller volume in the act of lifting.

In short: Different properties can easily imply different extensions for the same distinguished (or separated)“object,” i.e., for the same distinguished part of the physical universe.

2. The One Object may be separated into Many Objects on a basis other than that of the spatial separation:

Spatial attributes are fundamental, but they don’t always provide the best principle to organize a theory of physics.

The separation of the single universe-object into many of its parts need not proceed on the basis of only the “physical” space.

It would be possible to separate the universe on the basis of certain basis-functions which are defined over every spatial part of the universe. For instance, the Fourier analysis gives rise to a separation of a property-function into many complex-valued frequencies (viz. pairs of spatial undulations).

If the separation is done on the basis of such abstract functions, and not on the basis of the spatial extents, then the problem of the empty regions vaporizes away immediately. There always is some or the other “frequency”, with some or the other amplitude and phase, present at literally every point in the physical universe—including in the regions of the so-called “empty” space.

However, do note that the Fourier separation is a mathematical principle. Its correspondence to the physical universe must pass through the usual, required, epistemological hoops. … Here is one instance:

Question: If infinity cannot metaphysically exist (simply because it is a mathematical concept and no mathematical concept physically exists), then how is it that an infinite series may potentially be required for splitting up the given function (viz. the one which specifies the variations the given property of the physical universe)?

Answer: An infinite Fourier series cannot indeed be used by way of a direct physical description; however, a truncated (finite) Fourier series may be.

Here, we are basically relying on the same trick as we saw earlier in this mini-series of posts: We can claim that what the truncated Fourier series represents is the actual reality, and that that function which requires an infinite series is merely a depiction, an idealization, an abstraction.

3. When to use which description—the One Object or the Many Objects:

Despite the enormous advantages of the second approach (of the One Object idea) in the fundamental theoretical physics, in classical physics as well as in our “day-to-day” life, we often speak of the physical reality using the cruder first approach (the one involving the Many Objects idea). This we do—and it’s perfectly OK to do so—mainly because of the involved context.

The Many Objects description of physics is closer to the perceptual level. Hence, its more direct, even simpler, in a way. Now, note a very important consideration:

The precision to used in a description (or a theory) is determined by its purpose.

The purpose for a description may be lofty, such as achieving fullest possible consistency of conceptual interrelations. Or it may be mundane, referring to what needs to be understood in order to get the practical things done in the day-to-day life. The range of integrations to be performed for the day-to-day usage is limited, very limited in fact. A cruder description could do for this purpose. The Many Objects idea is conceptually more economical to use here. [As a polemical remark on the side, observe that while Ayn Rand highlighted the value of purpose, neither Occam nor the later philosophers/physicists following him ever even thought of that idea: purpose.]

However, as the scope of the physical knowledge increases, the requirements of the long-range consistency mandate that it is the second approach (the one involving the One Object idea) which we must adopt as being a better representative of the actual reality, as being more fundamental.

Where does the switch-over occur?

I think that it occurs at a level of those physics problems in which the energetics program (initiated by Leibnitz), i.e., the Lagrangian approach, makes it easier to solve them, compared to the earlier, Newtonian approach. This answer basically says that any time you use the ideas such as fields, and energy, you must make the switch-over, because in the very act of using such ideas, implicitly, you are using the One Object idea anyway. Which means, EM theory, and yes, also thermodynamics.

And of course, by the time you begin tackling QM, the second approach becomes simply indispensable.

A personal side remark: I should have known better. I should have adopted the second approach earlier in my life. It would have spared me a lot of agonizing over the riddles of quantum physics, a lot of running in loops over the same territory (like a dog chasing his own tail). … But it’s OK. I am glad that at least by now, I know better. (And, engineers anyway don’t get taught the Lagrangian mechanics to the extent physicists do.)

A few days ago, Roger Schlafly had written a nice and brief post at his blog saying that there is a place for non-locality in physics. He had touched on that issue more from a common-sense and “practical” viewpoint of covering these two physics approaches [^].

Now, given the above write-up, you know that a stronger statement, in fact, can be made:

As soon as you enter the realm of the EM fields and the further development, the non-local (or the global or the One Object) theories are the only way to go.


A Song I Like:

[When I was a school-boy, I used to very much like this song. I would hum [no, can’t call it singing] with my friends. I don’t know why. OK. At least, don’t ask me why. Not any more, anyway 😉 .]

(Hindi) “thokar main hai meri saaraa zamaanaa”
Singer: Kishore Kumar
Music: R. D. Burman
Lyrics: Rajinder Krishan


OK. I am glad I have brought to a completion a series of posts that I initiated. Happened for the first time!

I have not been able to find time to actually write anything on my promised position paper on QM. … Have been thinking about how to present certain ideas better, but not making much progress… If you must ask: these involve entangled vs. product states—and why both must be possible, etc.

So, I don’t think I am going to be able to hold the mid-2017 deadline that I myself had set for me. It will take longer.

For the same reasons, may be I will be blogging less… Or, who knows, may be I will write very short general notings here and there…

Bye for now and take care…

 

Some of the implications of the “Many Objects” idea…

0. Context and Review:

This post continues from the last one. In the last post, we saw that the same perceptual evidence (involving two moving grey regions) can be conceptually captured using two entirely different, fundamental, physics ideas.

In the first description, the perceived grey regions are treated as physical objects in their own right.

In the second description, the perceived grey regions are treated not as physical objects in their own right, but merely as distinguishable (and therefore different) parts of the singleton object that is the universe (the latter being taken in its entirety).

We will now try to look at some of the implications that the two descriptions naturally lead to.

1. The “Many Objects” Viewpoint Always Implies an In-Principle Empty Background Object:

To repeat, in the first description, the grey regions are treated as objects in their own right. This is the “Many Objects” viewpoint. The universe is fundamentally presumed to contain many objects.

But what if there is one and only one grey block in the perceptual field? Wouldn’t such a universe then contain only that one grey object?

Not quite.

The fact of the matter is, even in this case, there implicitly are two objects in the universe: (i) the grey object and (ii) the background or the white object.

As an aside: Do see here Ayn Rand’s example (in ITOE, 2nd Edition) of how a uniform blue expanse of the sky by itself would not even be perceived as an object, but how, once you introduce a single speck of dust, the perceptual contrast that it introduces would allow perceptions of both the speck and the blue sky to proceed. Of course, this point is of only technical importance. Looking at the real world while following the first description, there are zillions of objects evidently present anyway.

Leaving aside the theoretically extreme case of a single grey region, and thus focusing on the main general ideas: the main trouble following this “Many Objects” description is twofold:

(i) It is hard to come to realize that something exists even in the regions that are “empty space.”

(ii) Methodologically, it is not clear as to precisely how one proceeds from the zillions of concrete objects to the singleton object that is the universe.  Observe that the concrete objects here are physical objects. Hence, one would look for a conceptual common denominator (CCD) that is narrower than just the fact that all these concrete objects do exist. One would look for something more physical by way of the CCD, but it is not clear what it could possibly be.

2. Implications of the “Many Objects” Viewpoint for Causality:

In the first description, there are two blocks and they collide. Let’s try to trace the consequences of such a description for causality:

With the supposition that there are two blocks, one is drawn into a temptation of thinking along the following lines:

the first block pushes on the second block—and the second block pushes on the first.

Following this line of thought, the first block can be taken as being responsible for altering the motion of the second block (and the second, of the first). Therefore, a certain conclusion seems inevitable:

the motion of the first block may be regarded as the cause, and the (change in) the motion of the second block may be regarded as the effect.

In other words, in this line of thought, one entity/object (the first block) supplies, produces or enacts the cause, and another entity/object (the second block) suffers the consequences, the effects. following the considerations of symmetry and thereby abstracting a more general truth (e.g. as captured in Newton’s third law), you could also argue that that it is the second object that is the real cause, and the first object shows only effects. Then, abstracting the truth following the consideration of symmetry, you could say that

the motion (or, broadly, the nature) of each of the two blocks is a cause, and the action it produces on the other block is an effect.

But regardless of the symmetry considerations or the abstractness of the argument that it leads to, note that this entire train of thought still manages to retain a certain basic idea as it is, viz.:

the entity/actions that is the cause is necessarily different from the entity/actions that is the effect.

Such an idea, of ascribing the cause and the effect parts of a single causal event (here, the collision event) to two different object not only can arise in the many objects description, it is the most common and natural way in which the very idea of causality has come to be understood. Examples abound: the swinging bat is a cause; the ball flying away is the effect; the entities to which we ascribe the cause and the effect are entirely different objects. The same paradigm runs throughout much of physics. Also in the humanities. Consider this: “he makes me feel good.”

Every time such a separation of cause and effect occurs, logically speaking, it must first be supposed that many objects exist in the universe.

It is only on the basis of a many objects viewpoint that the objects that act as causes can be metaphysically separated, at least in an event-by-event concrete description, from the objects that suffer the corresponding effects.

3. Implications of the “Many Objects” Viewpoint, and the Idea of the “Empty” Space:

Notice that in the “many objects” description, no causal role is at all played by those parts of the universe that are “empty space.” Consider the description again:

The grey blocks move, come closer together, collide, and fly away in the opposite directions after the collision.

Notice how this entire description is anchored only to the grey blocks. Whatever action happens in this universe, it is taken by the grey blocks. The empty white space gets no metaphysical role whatsoever to play.

It is as if any metaphysical locus standi that the empty space region should otherwise have, somehow got completely sucked out of itself, and this locus standi then got transferred, in a way overly concentrated, into the grey regions.

Once this distortion is allowed to be introduced into the overall theoretical scheme, then, logically speaking, it would be simple to propagate the error throughout the theory and its implication. Just apply an epistemologically minor principle like Occam’s Razor, and the metaphysical suggestion that this entire exercise leads to is tantamount to this idea:

why not simply drop the empty space out of any physical consideration? out of all physics theory?

A Side Remark on Occam’s Principle: The first thing to say about Occam’s Principle is that it is not a very fundamental principle. The second thing to say is that it is impossible to state it using any rigorous terms. People have tried doing that for centuries, and yet, not a single soul of them feels very proud when it comes to showing results for his efforts. Just because today’s leading theoretical physics love it, vouch by it, and vigorously promote it, it still does not make Occam’s principle play a greater epistemological role than it actually does. Qua an epistemological principle, it is a very minor principle. The best contribution that it can at all aspire to is: serving as a vague, merely suggestive, guideline. Going by its actual usage in classical physics, it did not even make for a frequently used epistemological norm let alone acted as a principle that would necessarily have to be invoked for achieving logical consistency. And, as a mere guideline, it is also very easily susceptible to misuse. Compare this principle to, e.g., the requirement that the process of concept formation must always show both the essentials: differentiation and integration. Or compare it to the idea that concept-formation involves measurement-omission. Physicists promote Occam’s Principle to the high pedestal, simply because they want to slip in their own bad ideas into physics theory. No, Occam’s Razor does not directly help them. What it actually lets them do, through misapplication, is to push a wedge to dislodge some valid theoretical block from the well-integrated wall that is physics. For instance, if the empty space has no role to play in the physical description of the universe [preparation of misapplication], then, by Occam’s Principle [the wedge], why not take the idea of aether [a valid block] out of  physics theory? [which helps make physics crumble into pieces].

It is in this way that the first description—viz. the “many objects” description—indirectly but inevitably leads to a denial of any physical meaning to the idea of space.

If a fundamental physical concept like space itself is denied any physical roots, then what possibly could one still say about this concept—about its fundamental character or nature? The only plausible answers would be the following:

That space must be (a) a mathematical concept (based on the idea that fundamental ideas had better be physical, mathematical or both), and (b) an arbitrary concept (based on the idea that if there is no hard basis of the physical reality underlying this concept, then it can always be made as soft as desired, i.e. infinitely soft, i.e., arbitrary).

If the second idea (viz., the idea that space is an arbitrary human invention) is accorded the legitimacy of a rigorously established truth, then, in logic, anyone would be free to bend space any which way he liked. So, there would have to be, in logic, a proliferation in spaces. The history of the 19th and 20th centuries is nothing but a practically evident proof of precisely this logic.

Notice further that in following this approach (of the “many objects”), metaphysically speaking, the first casualty is that golden principle taught by Aristotle, viz. the idea that a literal void cannot exist, that the nothing cannot be a part of the existence. (It is known that Aristotle did teach this principle. However, it is not known if he had predecessors, esp. in the more synthetic, Indic, traditions. In any case, the principle itself is truly golden—it saves one from so many epistemological errors.)

Physics is an extraordinarily well-integrated a science. However, this does not mean that it is (or ever has been) perfectly integrated. There are (and always have been) inconsistencies in it.

The way physics got formulated—the classical physics in particular—there always was a streak of thought in it which had always carried the supposition that there existed a literal void in the region of the “gap” between objects. Thus, as far as the working physicist was concerned, a literal void could not exist, it actually did. “Just look at the emptiness of that valley out there,” (said while standing at a mountain top). Or, “look at the bleakness, at the dark emptiness out there between those two shining bright stars!” That was their “evidence.” For many physicists—and philosophers—such could be enough of an evidence to accept the premise of a physically existing emptiness, the literal naught of the philosophers.

Of course, people didn’t always think in such terms—in terms of a literal naught existing as a part of existence.

Until the end of the 19th century, at least some people also thought in terms of “aether.”

The aether was supposed to be a massless object. It was supposed that “aether” existed everywhere, including in the regions of space where there were no massive objects. Thus, the presence of aether ensured that there was no void left anywhere in the universe.

However, as soon as you think of an idea like “aether,” two questions immediately arise: (i) how can aether exist even in those places where a massive object is already present? and (ii) as to the places where there is no massive object, if all that aether does is to sit pretty and do nothing, then how is it really different from those imaginary angels pushing on the planets in the solar system?

Hard questions, these two. None could have satisfactorily answered these two questions. … In fact, as far as I know, none in the history of physics has ever even raised the first question! And therefore, the issue of whether, in the history of thought, there has been any satisfactory answer provided to it or not, cannot even arise in the first place.

It is the absence of satisfactory answers to these two questions that has really allowed Occam’s Razor to be misapplied.

By the time Einstein arrived, the scene was already ripe to throw the baby out with the water, and thus he could happily declare that thinking in terms of the aether concept was entirely uncalled for, that it was best to toss it into in the junkyard of bad ideas discarded in the march of human progress.

The “empty” space, in effect, progressively got “emptier” and “emptier” still. First, it got replaced by the classical electromagnetic “field,” and then, as space got progressively more mathematical and arbitrary, the fields themselves got replaced by just an abstract mathematical function—whether the spacetime of the relativity theory or the \Psi function of QM.

4. Implications of the “Many Objects” Viewpoint and the Supposed Mysteriousness of the Quantum Entanglement:

In the “many objects” viewpoint, the actual causal objects are many. Further, this viewpoint very naturally suggests the idea of some one object being a cause and some other object being the effect. There is one very serious implication of this separation of cause and effect into many, metaphysically separate, objects.

With that supposition, now, if two distant objects (and metaphysically separate objects always are distant) happen to show some synchronized sort of a behavior, then, a question arises: how do we connect the cause with the effect, if the effect is observed not to lag in time from the cause.

Historically, there had been some discussion on the question of “[instantaneous] action at a distance,” or IAD for short. However, it was subdued. It was only in the context of QM riddles that IAD acquired the status of a deeply troubling/unsettling issue.

5. Miscellaneous:

5.1

Let me take a bit of a digression into philosophy proper here, by introducing Ayn Rand’s ideas of causality at this point [^]. In OPAR, Dr. Peikoff has clarified the issue amply well: The identity or nature of an entity is the cause, and its actions is the effect.

Following Ayn Rand, if two grey blocks (as given in our example perceptual field) reverse their directions of motions after collision, each of the two blocks is a cause, and the reversals in the directions of the same block is the effect.

Make sure to understand the difference in what is meant by causality. In the common-sense thinking, as spelt out in section 2. of this post, if the block `A’ is the cause, then the block `B’ is the effect (and vice versa). However, according to Ayn Rand, if the block `A’ is the cause, then the actions of this same block `A’ are the effect. It is an important difference, and make sure you know it.

Thus, notice, for the time being, that in Ayn Rand’s sense of the terms, the principle of causality actually does not need a multiplicity of objects.

However, notice that the causal role of the “empty” space continues to remain curiously unanswered even after you bring Ayn Rand’s above-mentioned insights to bear on the issue.

5.2:

The only causal role that can at all be ascribed to the “empty” space, it would seem, is for it to continuously go on “monitoring” if a truly causal body—a massive object—was impinging on itself or not, and if such a body actually did that, to allow it to do so.

In other words, the causal identity of the empty space becomes entirely other-located: it summarily depends on the identity of the massive objects. But the identity of a given object pertains to what that object itself is—not to what other objects are like. Clearly, something is wrong here.


In the next post, we shall try to trace the implications that the second description (i.e. The One Object) leads to.


A Song I Like:

(Hindi) “man mera tujh ko maange, door door too bhaage…”
Singer: Suman Kalyanpur
Music: Kalyanji Anandji
Lyrics: Indivar


[PS: May be an editing pass is due…. Let me see if I can find the time to come back and do it…. Considerable revision done on 28 April 2017 12:20 PM IST though no new ideas were added; I will leave the remaining grammatical errors/awkward construction as they are. The next post should get posted within a few days’ time.]

On whether A is not non-A

This post has its origin in a neat comment I received on my last post [^]; see the exchange starting here: [^].


The question is whether I accept that A is not non-A.

My answer is: No, I do not accept that, logically speaking, A is not non-A—not unless the context to accept this statement is understood clearly and unambiguously (and the best way to do that is to spell it out explicitly).

Another way to say the same thing is that I can accept that “A is not non-A,” but only after applying proper qualifications; I won’t accept it in an unqualified way.

Let me explain by considering various cases arising, using a simple example.


The Venn diagram:

Let’s begin by drawing a Venn diagram.

Draw a rectangle and call it the set R. Draw a circle completely contained in it, and call it the set A. You can’t put a round peg to fill a rectangular hole, so, the remaining area of the rectangle is not zero. Call the remaining area B. See the diagram below.

The Venn Diagram

Case 1: All sets are non-empty:

Assume that neither A nor B is empty. Using symbolic terms, we can say that:
A \neq \emptyset,
B \neq \emptyset, and
R \equiv A \cup B
where the symbol \emptyset denotes an empty set, and \equiv means “is defined as.”

We take R as the universal set—of this context. For example, R may represent, say the set of all the computers you own, with A denoting your laptops and B denoting your desktops.

I take the term “proper set” to mean a set that has at least one element or member in it, i.e., a set which is not empty.

Now, focus on A. Since the set A is a proper set, then it is meaningful to apply the negation- or complement-operator to it. [May be, I have given away my complete answer right here…] Denote the resulting set, the non-A, as A^{\complement }. Then, in symbolic terms:
A^{\complement } \equiv R \setminus A.
where the symbol \setminus denotes taking the complement of the second operand, in the context of the first operand (i.e., “subtracting” A from R). In our example,
A^{\complement } = B,
and so:
A^{\complement } \neq \emptyset.
Thus, here, A^{\complement } also is a proper (i.e. non-empty) set.

To conclude this part, the words “non-A”, when translated into symbolic terms, means A^{\complement }, and this set here is exactly the same as B.

To find the meaning of the phrase “not non-A,” I presume that it means applying the negation i.e. the complement operator to the set A^{\complement }.

It is possible to apply the complement operator because A ^{\complement } \neq \emptyset. Let us define the result of this operation as A^{\complement \complement}; note the two ^{\complement}s appearing in its name. The operation, in symbols becomes:
A^{\complement \complement} \equiv R \setminus A^{\complement} = R \setminus B = A.
Note that we could apply the complement operator to A and later on to A^{\complement} only because each was non-empty.

As the simple algebra of the above simple-minded example shows,
A = A^{\complement\complement},
which means, we have to accept, in this example, that A is not non-A.

Remarks on the Case 1:

However, note that we can accept the proposition only under the given assumptions.

In  particular, in arriving at it, we have applied the complement-operator twice. (i) First, we applied it to the “innermost” operand i.e. A, which gave us A^{\complement}. (ii) Then, we took this result, and applied the complement-operator to it once again, yielding A^{\complement\complement}. Thus, the operand for the second complement-operator was A^{\complement}.

Now, here is the rule:

Rule 1: We cannot meaningfully apply the complement-operator unless the operand set is proper (i.e. non-empty).

People probably make mistakes in deciding whether A is not non-A, because, probably, they informally (and properly) do take the “innermost” operand, viz. A, to be non-empty. But then, further down the line, they do not check whether the second operand, viz. A^{\complement} turns out to be empty or not.

Case 2: When the set A^{\complement} is empty:

The set A^{\complement} will be empty if B = \emptyset, which will happen if and only if A = R. Recall, R is defined to be the union of A and B.

So, every time there are two mutually exclusive and collectively exhaustive sets, if any one of them is made empty, you cannot doubly apply the negation or the complement operator to the other (nonempty) set.

Such a situation always occurs whenever the remaining set coincides with the universal set of a given context.

In attempting a double negation, if your first (or innermost) operand itself is a universal set, then you cannot apply the negation operator for the second time, because by Rule 1, the result of the first operator comes out as an empty set.


The nature of an empty set:

But why this rule that you can’t negate (or take the complement of) an empty set?

An empty set contains no element (or member). Since it is the elements which together impart identity to a set, an empty set has no identity of its own.

As an aside, some people think that all the usages of the phrase “empty set” refers to the one and the only set (in the entire universe, for all possible logical propositions involving sets). For instance, the empty set obtained by taking an intersection of dogs and cats, they say, is exactly the same empty set as the one obtained by taking an intersection of cars and bikes.

I reject this position. It seems to me to be Platonic in nature, and there is no reason to give Plato even an inch of the wedge-space in this Aristotlean universe of logic and reality.

As a clarification, notice, we are talking of the basic and universal logic here, not the implementation details of a programming language. A programming language may choose to point all the occurrences of the NULL string to the same memory location. This is merely an implementation choice to save on the limited computer memory. But it still makes no sense to say that all empty C-strings exist at the same memory location—but that’s what you end up having if you call an empty set the empty set. Which brings us to the next issue.

If an empty set has no identity of its own, if it has no elements, and hence no referents, then how come it can at all be defined? After all, a definition requires identity.

The answer is: Structurally speaking, an empty set acquires its meaning—its identity—“externally;” it has no “internally” generated identity.

The only identity applicable to an empty set is an abstract one which gets imparted to it externally; the purpose of this identity is to bring a logical closure (or logical completeness) to the primitive operations defined on sets.

For instance, intersection is an operator. To formally bring closure to the intersection operation, we have to acknowledge that it may operate over any combination of any operand sets, regardless of their natures. This range includes having to define the intersection operator for two sets that have no element in common. We abstractly define the result of such a case as an empty set. In this case, the meaning of the empty set refers not to a result set of a specific internal identity, but only to the operation and the disjoint nature the operands which together generated it, i.e., via a logical relation whose meaning is external to the contents of the empty set.

Inasmuch as an empty set necessarily includes a reference to an operation, it is a concept of method. Inasmuch as many combinations of various operations and operands can together give rise to numerous particular instances of an empty set, there cannot be a unique instance of it which is applicable in all contexts. In other words, an empty set is not a singleton; it is wrong to call it the empty set.

Since an empty set has no identity of its own, the notion cannot be applied in an existence-related (or ontic or metaphysical) sense. The only sense it has is in the methodological (or epistemic) sense.


Extending the meaning of operations on an empty set:

In a derivative sense, we may redefine (i.e. extend) our terms.

First, we observe that since an empty set lacks an identity of its own, the result of any operator applied to it cannot have any (internal) identity of its own. Then, equating these two lacks of existence-related identities (which is where the extension of the meaning occurs), we may say, even if only in a derivative or secondary sense, that

Rule 2: The result of an operator applied to an empty set again is another empty set.

Thus, if we now allow the complement-operator to operate also on an empty set (which, earlier, we did not allow), then the result would have to be another empty set.

Again, the meaning of this second empty set depends on the entirety of its generating context.

Case 3: When the non-empty set is the universal set:

For our particular example, assuming B = \emptyset and hence A = R, if we allow complement operator to be applied (in the extended sense) to A^{\complement}, then

A^{\complement\complement} \equiv R \setminus A^{\complement} = R \setminus (R \setminus A) = R \setminus B = R \setminus (\emptyset) = R = A.

Carefully note, in the above sequence, the place where the extended theory kicks in is at the expression: R \setminus (\emptyset).

We can apply the \setminus operator here only in an extended sense, not primary.

We could here perform this operation only because the left hand-side operand for the complement operator, viz., the set R here was a universal set. Any time you have a universal set on the left hand-side of a complement operator, there is no more any scope left for ambiguity. This state is irrespective of whether the operand on the right hand-side is a proper set or an empty set.

So, in this extended sense, feel free to say that A is not non-A, provided A is the universal set for a given context.


To recap:

The idea of an empty set acquires meaning only externally, i.e., only in reference to some other non-empty set(s). An empty set is thus only an abstract place-holder for the result of an operation applied to proper set(s), the operation being such that it yields no elements. It is a place-holder because it refers to the result of an operation; it is abstract, because this result has no element, hence no internally generated identity, hence no concrete meaning except in an abstract relation to that specific operation (including those specific operands). There is no “the” empty set; each empty set, despite being abstract, refers to a combination of an instance of proper set(s) and an instance of an operation giving rise to it.


Exercises:

E1: Draw a rectangle and put three non-overlapping circles completely contained in it. The circles respectively represent the three sets A, B, C, and the remaining portion of the rectangle represents the fourth set D. Assuming this Venn diagram, determine the meaning of the following expressions:

(i) R \setminus (B \cup C) (ii) R \setminus (B \cap C) (iii) R \setminus (A \cup B \cup C) (iv) R \setminus (A \cap B \cap C).

(v)–(viii) Repeat (i)–(iv) by substituting D in place of R.

(ix)–(xvi) Repeat (i)–(viii) if A and B partly overlap.

E2: Identify the nature of set theoretical relations implied by that simple rule of algebra which states that two negatives make a positive.


A bit philosophical, and a form better than “A is not non-A”:

When Aristotle said that “A is A,” and when Ayn Rand taught its proper meaning: “Existence is identity,” they referred to the concepts of “existence” and “identity.” Thus, they referred to the universals. Here, the word “universals” is to be taken in the sense of a conceptual abstraction.

If concepts—any concepts, not necessarily only the philosophical axioms—are to be represented in terms of the set theory, how can we proceed doing that?

(BTW, I reject the position that the set theory, even the so-called axiomatic set theory, is more fundamental than the philosophic abstractions.)

Before we address this issue of representation, understand that there are two ways in which we can specify a set: (i) by enumeration, i.e. by listing out all its (relatively concrete) members, and (ii) by rule, i.e. by specifying a definition (which may denote an infinity of concretes of a certain kind, within a certain range of measurements).

The virtue of the set theory is that it can be applied equally well to both finite sets and infinite sets.

The finite sets can always be completely specified via enumeration, at least in principle. On the other hand, infinite sets can never be completely specified via enumeration. (An infinite set is one that has an infinity of members or elements.)

A concept (any concept, whether of maths, or art, or engineering, or philosophy…) by definition stands for an infinity of concretes. Now, in the set theory, an infinity of concretes can be specified only using a rule.

Therefore, the only set-theoretic means capable of representing concepts in that theory is to specify their meaning via “rule” i.e. definition of the concept.

Now, consider for a moment a philosophical axiom such as the concept of “existence.” Since the only possible set-theoretic representation of a concept is as an infinite set, and since philosophical axiomatic concepts have no antecedents, no priors, the set-theoretic representation of the axiom of “existence” would necessarily be as a universal set.

We saw that the complement of a universal set is an empty set. This is a set-theoretic conclusion. Its broader-based, philosophic analog is: there are no contraries to axiomatic concepts.

For the reasons explained above, you may thus conclude, in the derivative sense, that:

“existence is not void”,

where “void” is taken as exactly synonymous to “non-existence”.

The proposition quoted in the last sentence is true.

However, as the set theory makes it clear and easy to understand, it does not mean that you can take this formulation for a definition of the concept of existence. The term “void” here has no independent existence; it can be defined only by a negation of existence itself.

You cannot locate the meaning of existence in reference to void, even if it is true that “existence is not void”.

Even if you use the terms in an extended sense and thereby do apply the “not” qualfier (in the set-theoretic representation, it would be an operator) to the void (to the empty set), for the above-mentioned reasons, you still cannot then read the term “is” to mean “is defined as,” or “is completely synonymous with.” Not just our philosophical knowledge but even its narrower set-theoretical representation is powerful enough that it doesn’t allow us doing so.

That’s why a better way to connect “existence” with “void” is to instead say:

“Existence is not just the absence of the void.”

The same principle applies to any concept, not just to the most fundamental philosophic axioms, so long as you are careful to delineate and delimit the context—and as we saw, the most crucial element here is the universal set. You can take a complement of an empty set only when the left hand-side operator is a universal set.

Let us consider a few concepts, and compare putting them in the two forms:

  • from “A is not non-A”
  • to “A is not the [just] absence [or negation] of non-A,” or, “A is much more than just a negation of the non-A”.

Consider the concept: focus. Following the first form, a statement we can formulate is:

“focus is not evasion.”

However, it does make much more sense to say that

“focus is not just an absence of evasion,” or that “focus is not limited to an anti-evasion process.”

Both these statements follow the second form. The first form, even if it is logically true, is not as illuminating as is the second.

Exercises:

Here are a few sentences formulated in the first form—i.e. in the form “A is not non-A” or something similar. Reformulate them into the second form—i.e. in the form such as: “A is not just an absence or negation of non-A” or “A is much better than or much more than just a complement or negation of non-A”. (Note: SPPU means the Savitribai Phule Pune University):

  • Engineers are not mathematicians
  • C++ programmers are not kids
  • IISc Bangalore is not SPPU
  • IIT Madras is not SPPU
  • IIT Kanpur is not SPPU
  • IIT Bombay is not SPPU
  • The University of Mumbai is not SPPU
  • The Shivaji University is not SPPU

[Lest someone from SPPU choose for his examples the statements “Mechanical Engg. is not Metallurgy” and “Metallurgy is not Mechanical Engg.,” we would suggest him another exercise, one which would be better suited to the universal set of all his intellectual means. The exercise involves operations mostly on the finite sets alone. We would ask him to verify (and not to find out in the first place) whether the finite set (specified with an indicative enumeration) consisting of {CFD, Fluid Mechanics, Heat Transfer, Thermodynamics, Strength of Materials, FEM, Stress Analysis, NDT, Failure Analysis,…} represents an intersection of Mechanical Engg and Metallurgy or not.]

 


A Song I Like:

[I had run this song way back in 2011, but now want to run it again.]

(Hindi) “are nahin nahin nahin nahin, nahin nahin, koee tumasaa hanseen…”
Singers: Kishore Kumar, Asha Bhosale
Music: Rajesh Roshan
Lyrics: Anand Bakshi

[But I won’t disappoint you. Here is another song I like and one I haven’t run so far.]

(Hindi) “baaghon mein bahaar hain…”
Music: S. D. Burman [but it sounds so much like R.D., too!]
Singers: Mohamad Rafi, Lata Mangeshkar
Lyrics: Anand Bakshi

[Exercise, again!: For each song, whenever a no’s-containing line comes up, count the number of no’s in it. Then figure out whether the rule that double negatives cancel out applies or not. Why or why not?]


 

[Mostly done. Done editing now (right on 2016.10.22). Drop me a line if something isn’t clear—logic is a difficult topic to write on.]

[E&OE]