Pojam paradoksa najbolje opisuje rečenica „naizgled apsurdno, ali ipak istinito“. Paradoksi otkrivaju slabosti ljudskih misaonih sposobnosti. Čovečanstvo često nije znalo kako sa njima da izađe na kraj: scio me nihil scire. Paradoksi se koriste da njima budu opisane situacije koje su ironične. U njima moć rasuđivanja ne samo da postaje teška, već i uznemirujuća, kao kad se kaže „Toromana više nema.“ Dragan Toroman je bio rukovodilac Odeljenja za računarske tehnologije u Istraživačkoj stanici Petnica, bio je vrhunski IT stručnjak, dobitnik prestižne ICT nagrade „Diskobolos“ koju dodeljuje Jedinstveni informatički savez Srbije (JISA), stručni konsultant i saradnik na programu poboljšanja usluga zdravstvenih laboratorija u Srbiji. Bio je preduzetnik u oblasti informacionih tehnologija i član nekoliko međunarodnih razvojnih timova u IT oblasti. Rodio se u Pirotu 1969. godine, odrastao je u Nišu, a 13. aprila ovog proleća je preminuo od posledica bolesti izazvane koronavirusom. Ta nepodnošljiva vest posebno je pogodila petnički svet u kojem je Toroman bio prisutan najpre kao polaznik programa informatike od 1986. godine, a potom i kao rukovodilac programa računarstva od 1998. godine. Toroman je bio čovek vedrog duha, svirao je gitaru i voleo je pecanje. Znao je da bude i sitničav u prohtevima prema svojim polaznicima: „Koliko tačno eksera treba čoveku koji na pijaci traži da mu se spakuje 20-30 eksera?“ Imao je i neke čudne teorije: „Najbolja razlika u godinama između muškarca i žene je kada ona ima dvdesetdve“. Svaki put bi se čudio kada neko nije čuo za Miladina Šobića. Neke složene stvari je pokušavao da ilustruje kratkim rečenicama: „Što je ona duže čekala, telefon duže nije zvonio“. Tekst koji ovde donosimo Toroman verovatno nigde nije objavio. Miloš Savić se potrudio da ga pronađemo i predstavimo.

Resolving a paradox

with help of some chicken, eggs and time

Dragan Toroman

Petnica Science Center

November, 2009.

“There is a theory which states that if anybody ever discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.”

Douglas Adams, The Hitchhikers Guide to the Galaxy.

1. Paradox as Phenomena

Paradoxes are widely described as disambiguation, statements or arguments being self-contradictory or counter-intuitive.

Merriam-Webster’s Dictionary describes the word as follows:

Main Entry: par·a·dox

Pronunciation: \ˈper-ə-ˌdäks, ˈpa-rə-\

Function: noun

Etymology: Latin paradoxum, from Greek paradoxon, from neuter of paradoxos contrary to expectation, from para- + dokein to think, seem — more at decent

1 : a tenet contrary to received opinion
2 a : a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true b : a self-contradictory statement that at first seems true c : an argument that apparently derives self-contradictory conclusions by valid deduction from acceptable premises
3 : one (as a person, situation, or action) having seemingly contradictory qualities or phases

From the definition of the word itself and its roots coming from the Ancient Greek, we could assume that the paradoxes, as we know them, are the invention of the Greek philosophers. As a matter of fact, maybe the first known/recorded paradoxes in the western world are those of Zeno of Elea (Achilles and the Tortoise, Dichotomy, Rest Arrow, etc.). These paradoxes really show the essence of the idea of paradox.

Zeno’s paradoxes had no real use. Those were just a challenge for a philosopher’s mind, based on Plato’s Parmenides.

Being a challenge, paradoxes greatly influenced (and sometimes even became a kick-start) many theories and developments of ancient and modern Philosophy and Science. But, of course, it works the other way around, too. Many paradoxes emerged out of new theories.

Paradoxes are widely used in modern science and philosophy as a valid “Proof by Contradiction” method.

There is a large list of paradoxes known to men trough history. Many of them were resolved at some point, hence not being paradoxes any more. The later is no wonder, because we have to be aware of the time/era paradoxes were proposed. Some concepts that could make these paradoxes obsolete emerged much later, and could not even be imagined at the time. But the most interesting ones are the ones that “survived”, making them universal.

This article will mostly deal with logical natural language paradoxes, and later propose a “reasonable” resolution for one of the most popular ancient paradoxes.

There are several classifications of paradoxes. Classification could be done in regards of the nature of the paradox, but also the “field” or scope of the paradox.

So we have for instance W.V.Quine’s classification:

• A veridical paradoxproduces a result that appears absurd but is demonstrated to be true nevertheless.
• A falsidical paradox establishes a result that not only appears false but actually is false due to a fallacy in the supposed demonstration.
• A paradox which is in neither class may be an antinomy, which reaches a self-contradictory result by properly applying accepted ways of reasoning.

The fourth class has been described since Quine’s work:

• A paradox which is both true and false at the same time in the same sense is called a dialetheia.

Also, there could be another classification:

• Logical, non-mathematical
• Philosophical
• Scientific (mathematical, chemical, physical, economic, etc.)

Paradoxes were made sometimes just for fun, and sometimes for the sake of Philosophy, Science… But any kind of paradox not resolved was and is a great challenge for the free mind trough the History. And sometimes, these efforts really produce great results.

2. Paradoxes and Languages

First paradoxes (and majority of known paradoxes) are easily expressed in natural languages, and actually are proposed that way. Of course, there are some formal paradoxes, too. These are mostly the modern ones coming from scientific theories, trying to show internal inconsistencies of such theories or proving concepts which are very hard to describe and accept, like some Quantum paradoxes, or the Banach-Tarski paradox as an example.

Paradoxes are meant to be self-explanatory, meaning they contain all the necessary information to illustrate the paradox, usually using simple common logic. But also, the language itself could be the cause of the paradox, allowing a statement to be self-referencing and proposing its own truthfulness. The variation of the Liar’s paradox: “This sentence is false.” is a clear example of this.

Careful understanding of a proposed paradox is essential. Sometimes the sentence is not a paradox if properly understood. The example of this could be the sentence “I always lie.” It is just a tricky one which on first glance might look as a paradox. But this is only due to the false dichotomy – one can always lie or otherwise always tells the truth. Accepting that a person who does not always lie can tell a lie at any time, the sentence is proven not to be a paradox.

This example also shows some other interesting aspects of paradoxes. Paradoxes which we on first glance formalize as containing A and NOT A require careful investigation of what A and especially NOT A is (not in a logical sense or true and false).

Working with natural language paradoxes, one should take in consideration various aspects:

• The original language of the paradox. Is the paradox in question actually a paradox just because the language itself? Eastern culture paradoxes are sometimes hard to comprehend in western cultures, because of the difficulties of translation.
• Translating the paradox to its formal representation may not yield a truly compatible/equivalent paradox. Even if two paradoxes are taken as equivalent, the subtle difference may occur which would make the possible resolution found for one to be unacceptable for both, apart from their formal representation are considered to be the same.
• How self explanatory the paradox is? Does it refer to something “real”? Is it just a metaphorical paradox?
• Is it possible to resolve the paradox in its original state, not assuming anything outside the context?
• What system/set the paradox apply to? This is sometimes explicitly cited, and sometimes it is hidden as implicit information. There might be a system/set in which the paradox does not exist.
• Definition of words as meant in the paradox.

Probably the best and most pleasing resolution of a paradox is the one which is simple, and assumes the paradox in question literally. So this would be the first option to go.

This would involve:

• Try to interpret the paradox literally, with no hidden metaphorical meaning.
• Define the system for which the paradox apply
• Try to imagine a system in which the paradox does NOT apply.
• If the paradox has no restrictions on the system, this might be the resolution.
• If the system is banned from the paradox, this system might offer a way of resolving the paradox in other systems.
• Try to define used object/actions in a way that would be commonly acceptable (and not causing additional paradoxes by themselves) and can in combination make the paradox resolvable.
• Apply applicable logic reasoning to the paradox using the above.

Simple, isn’t it?

3. The “Paradox” of Time

As a member of the Eleatic School, Zeno was supporting Parmenides doctrine “all is one” which basically says that the world is an illusion. Plurality and change, and especially motion, is nothing but our false perception of the constant reality. Unfortunately his original paradoxes are known only trough work of Aristotle (Physics).

It is not in this article’s scope to deal with Zeno’s paradoxes, but it is worth emphasizing the importance of his ideas. Even if we disregard the idea of “the motion that is illusion”, the problem that actually was demonstrated trough his paradoxes deal with our perception and the nature of space and time. He also (without having any idea of relativism or how to explain it) made a point demonstrating differences of absolute and relative motion and time and also the principle of difference between position and momentum.

Mathematics and modern Physics offer “reasonable” resolutions for these paradoxes “as stated”, but even then, Zeno’s paradoxes are still in focus, 2,500 years after, if nothing more, then for educational purposes.

One of his points was the nature of time. Is it discrete or continuous? Is time made of small atomic “instants” or infinitely divisible?

These questions affect many subsequent theories in Physics and open many questions. What is a moment in time? Is it possible for something to happen to an object “at the same time”?

It is in our nature to perceive things as determined, finite, discrete and static. On the other hand, the way of describing the world to this stage is involving exploration, both physical and idealistic, leading so far to a constant change of perspective and perception, and sometimes even denying our own nature, or nature of existence.

Apart from the set of specific questions posted in the paradoxes, Zeno (and the Eleatic School) actually raised a very important issue – that of perception. The sole idea that the world really is different from what we sense/see is a great idea by itself. It does not even matter if their theory of what is the nature or the world was right or wrong.

All we know about the world/universe is based on our perception. But the perception is not something which is limited to the physical world. Our perception of ideas functions on the same principles. As perception is individual, and even an individual can have different perceptions of the same thing, it seems as a good idea to choose the “right” ones.

There is a good example in science, the one formerly known as a paradox, but nowadays known as the Parallax Mistake: “If from one angle of sight the liquid looks as if it is in the height I wish for in the tube, but from another it does not, which sight is actually correct?” Analyzing the proposed paradox, it was noticed that there is one “view” which was correct. So, the problem was solved and became a “mistake” instead of “paradox”.

Every theory (scientific, philosophic, religious…) is based on a set of chosen perceptions – physical or idealistic, accepted as being “right” by its followers. Quotes for “right” are used because it is actually very unlikely to have an absolute consensus on this statement according to the premises of perception and individuality.

One could say that Science has managed to explain and change our perception to the very high and even abstract levels. The “invisible”, which was at first part of the Philosophy corpus, became part of the hard evidence and proof enabled Science. Science now talks of energy, force, more than three dimensions, infinity, various abstract structures, relativity, duality, etc. And not just contemplating, but offering experimental and formal proofs.

Knowing all that, is it then common for the reader of this text to accept or perceive the existence of a four dimensional cube? Or perhaps infinity (not to mention different kinds of infinities)?

The key word is accept. One is actually accepting the idea of such concepts, without the actual perception in place. The idea is perceived, not the underlying object, fact or action. So, if we accept infinity, for instance, is it possible to imagine it? Or perceive any instance of an infinite object? Or “nothing”?

Any kind of theory that embraces existance in any form, must (even if it claims that it’s not acceptable) accept the lack of existance or non-existance as its counter-part, if nothing more but as the idea itself. It is not much different from infinity.

Let’s talk about geometry and numbers.

In Euclidean geometry, there are lines and points. Geometric lines and points. Infinitely long lines, infinitely thin. And infinitely small points. Yet Euclidean geometry does not talk about infinity. Is is just accepted. It is abstract by definition. Is it possible to have geometric point in nature? As far as we know, not. Can we measure a point of where something is in some referent coordinate system. Closely, yes, with limited precision, but absolutely, no.

It is similar with real numbers, that we use for “accurate” calculations and measurement. They are called real for a reason, aren’t they?

Remember this?

1/9 = 0.1111111111*

2/9 = 0.2222222222*

3/9 = 0.3333333333*

4/9 = 0.4444444444*

5/9 = 0.5555555555*

6/9 = 0.6666666666*

7/9 = 0.7777777777*

8/9 = 0.8888888888*

9/9 = 0.9999999999*

Well, isn’t 9/9 = 1? Yes. But it is also 0.9999999999*

It is not in scope of this document to discuss the proof in detail, but take a look:

9/9 = 9*(1/9)

1 = 9*(0.1111111111*)

1 = 0.9999999999*

Another paradox? No. Mathematicians say it’s just a matter of representation and our limited perception of working with infinities.

So, is time like that? We don’t know for sure. But at the same time, it would only matter if we would be able to produce a measuring device with zero-time measuring cycle. Until then, Planck made his point on this subject.

Quantum theory and theory of relativism, apart from being a breakthrough in our (for now) understanding of the world, is much less consistent than presented to the “average scholar”. Lots of paradoxes emerged from these theories, which is self-explanatory. But these theories are affected by our limited physical perception. But our mind is much less restricted, allowing us to perform abstract calculation, abstract experiments and abstract models. So, we are satisfied with our theories, as far as we can confirm or prove them using our perception. When we encounter inconsistencies, we try better. So, is our perception a limiting factor? Yes, and no. If we see it as our present lack of understanding the physical nature of existence, then yes. But if we see it as a challenge and tool, then no. Even if one day, we find a model that would explain everything around us with no inconsistencies, that would not necessarily mean that existence really functions this way. Either we haven’t yet encountered something inconsistent or it doesn’t really matter.

Let’s escape for a moment from the philosophic or scientific meaning of what does it mean to be not-A as opposed to A, as irrelevant for the sake of this example:

A <and> not-A = True – “Can something be and not be?”

Rewrite this as:

A <and_at_the_same_time> not-A = True (this is commonly taken as implicit in the above expression)

or

A <and_at_different_time> not-A = True (very often a “real-world”, dynamic, or duality situation)

Is there a difference between the two? Obviously the first one would be wrong, and second might be right if we expand the logic to that of and beyond Temporal logic.

So, using the U operator of Temporal logic, this could be written as:

A U not-A = T (the U operator denotes “Until”)

Science adopted dynamic nature of the world, and actually any philosophy that does not deny Time implicitly adopted the dynamic paradigm.

Does being dynamic automatically exclude the possibility of something being “static” in such a system? Something that does not change its value/existence (state) trough time could be static in some sense. We might even propose existence of an object which state does not change infinitely trough time, an infinitely lasting, permanent object. Is such object possible? It seems to be that it is not. While adopting the temporal dimension, the position of this object would change at least in time. So, in a n-dimensional system of which one dimension is time, there will always be at least one dimension that would make the object “mutable”. The object could be considered constant (or static) only in dimensional systems which does not include time hence having number of dimensions <= n-1 or if we, for instance, freeze the value of the problematic dimension(s) (at least Time in this case)

Back to the position and momentum problem, and following the above, it is easily seen that position refers to a “static” property fixed in an indivisible moment of time, while the momentum refers to a dynamic property dependent of a time interval and change of position, making them not compatible. Figuratively speaking, determining a position of a moving object would require taking ultimately fast photograph of it, lasting zero time, providing us with no chance for motion blur at all. But then, this photograph could not provide information about the motion itself or any dynamic properties (including the momentum). This was in some way, Zeno’s point.

4. Chicken and (its?) Egg

“Which came first, the chicken or the egg?”

This paradox is one of the widely known paradoxes, and maybe the one that is mostly used as a figure of speech denoting meaningless questions. But, it seems to be that this is the only consequence that is left out of this paradox, as it is shown that actually it is not one. Modern interpretation classifies it just as a metaphorical paradox.

So why is this paradox important then? It is a very good example and illustration of the concepts stated in the Chapter 2 of this article.

Let’s again spend some time analyzing this paradox. First, we would have to decide which version of the paradox we are going to analyze. According to Aristotle, we are talking about a bird, not a chicken. Later, Plutarch is talking of a hen (chicken). Let’s assume it is possible to generalize these two versions to the one stated in the beginning of this chapter and make the less general statement a representative of the more general question involving a bird. It is possible to show that this is true, by applying proposed resolution(s) for the “chicken paradox” applicable to the “bird paradox”.

If we tend to make this paradox formal we might say “Which came first, X that can’t come without Y, or Y that can’t come without X?” But this actually does not represent equivalent to the original question. It is just explaining its metaphorical nature, which shows the important concept of circular reference.

At the time this paradox was raised, there was no difference between the real and the metaphorical meaning of the question. Ancient philosophers, trying to resolve this paradox, came to question the creation of the world (which would provide the ultimate – not metaphorical resolution, thus making the paradox real, not metaphorical).

So it is more desirable to provide non-metaphorical answers to a paradox, as it makes the resolution more convincing. The resolution of a metaphorical paradox would have to be metaphorical too, which then makes it impossible to be questioned or proved. So, purely metaphorical paradoxes can be ignored.

Many paradoxes have somewhat dual nature. It is possible to interpret them metaphorically or non-metaphorically. The resolution of these paradoxes should deal with the “real’ part as much as possible (because sometimes it is very hard to make a clear line between those parts – especially for the philosophical paradoxes).

We can demonstrate the interpretation/perception problem very easy. Imagine someone put a live grown chicken and a fresh egg in front of you and asked you the question: “Which came first, the chicken or the egg?” The answer would be trivial, and the question would not be a paradox at all. One might say that this interpretation is destroying the idea of the paradox itself, which is true, but it still remains a valid proof of context related issues.

But, as it was shown by various authors, this paradox is solvable nowadays even in its more general form. This brings us to the next level. Are we talking about a chicken and a chicken egg, end more specifically – are we talking about a chicken and that chicken egg? Clearly, for the paradox not to be considered trivial, even at the time it was created, the later is not the case. So, we have come to a conclusion that we could be talking of a chicken and that species egg or of a chicken (bird) and any species egg. The solutions might be different according to different accepted theories of life creation, but resolvable in any of proposed theories.

So, any “realistic” interpretation for this “paradox” leads to a resolved state, and the only one unresolved is the metaphorical one, which is not resolvable anyway. To emphasize the non-resolvable issue of the metaphorical interpretation it is necessary to point out that in the metaphorical interpretation there is actually no chicken and no egg whatsoever.

The purpose of this was not to re-resolve the chicken-egg paradox, but to demonstrate some interesting aspects applicable to some paradoxes.

The Omnipotent being (God, The-all-mighty, or just Him) Paradox

“Could an omnipotent being create a stone so heavy that even he could not lift it?”

This is one of the paradoxes that is quoted so often, and used so many times as a “Scientific proof” for atheists that omnipotent being (read “God”) could not exist. Well, it is applicable to religion, for sure, but the philosophical and fun aspects of it remain, too.

The origin of this paradox uncertain, leaving us with no context of beliefs, culture, time or mind creating it. Nevertheless, it was a reason for endless discussions and arguments, ever since. And the outcome is number of interpretations ranging from denying omnipotence to denying the paradox.

Some people see this paradox just as a “trick-question” made for fun, and thus has no deeper meaning. So, this means no resolution is needed, or even expected. The rest of the world, seeing it as a paradox, tried to resolve or deny it.

The most popular modern resolutions claim He can, however with some limitations. Let’s make a brief tour trough the most known answers.

Here comes a list of common explanations:

Let’s first see the straight-forward, “non-believers” point of view, or the so-called “atheist proof”:

1. Either God can create a stone which he cannot lift, or he cannot create a stone which he cannot lift.
2. If God can create a stone which he cannot lift, then he is not omnipotent (since he cannot lift the stone in question).
3. If God cannot create a stone which he cannot lift, then he is not omnipotent (since he cannot create the stone in question).
4. Therefore God is not omnipotent.

J.L.Cowan, “The Paradox of Omnipotence Revisited”

As the premise is that God is omnipotent, then there is no God. Simple as that. But, some disagree.

The next approach denies the possibility of omnipotence (being divine or not), making the question obsolete.

1) If a being exists, then it must have some active tendency.

2) If a being has some active tendency, then it has some power to resist its creator.

3) If a being has the power to resist its creator, then the creator does not have absolute power.

The “mild” version embraces omnipotence, but recognizes different levels of omnipotence, leaving a possibility of resolving the paradox in favor of the omnipotent being (under condition of accepting the proposed omnipotence definition).

Different levels, according to different religions/faiths:

1. A deity is able to do anything that is logically possible for it to do (Aquinas).
2. A deity is able to do anything that it chooses to do (St.Augustine).
3. A deity is able to do anything that is in accord with its own nature (thus, for instance, if it is a logical consequence of a deity’s nature that what it speaks is truth, then it is not able to lie).
4. Hold that it is part of a deity’s nature to be consistent and that it would be inconsistent for said deity to go against its own laws unless there was a reason to do so (Polkinghorne).
5. A deity is able to do anything that corresponds with its omniscience and therefore with its world-plan
6. A deity is able to do absolutely anything, even the logically impossible.

The level No.2 in this list corresponds best with the theological concept of Willpower. This concept is one of the most popular modern theological answers to the paradox. Having Willpower, God can create a stone that He cannot lift if He wishes to, and does not wish to lift the stone. Afterwards, if He wishes, He might do anything to the stone, including lifting it, or even turn it into a smaller stone and lift it.

There are some modern debates on omnipotence, claiming that language (and philosophy) cannot explain the meaning of omnipotence, making it a semantic problem.

These approaches could resolve the paradox with a properly chosen level of omnipotence, either by making it obsolete or trivial. But not all are satisfied with this kind of answers. The real challenge is not to deny the paradox with some answer which basically evaluates to “Because it is convenient.” If this is shown to be the only way of resolving the paradox, then it might be accepted. On the other hand, there are other ways, not including denying of logic, or making things depend on “will”. Willpower could open a debate raising the question “If someone does not want to do something, what does it say about his ability to perform it?”

So back to the beginning, the “hard” version denies the question itself. Basically, according to this theory, the question is not a paradox, but just plain “nonsense”, a meaningless trick-question not to be considered. The question asked in the paradox is of the same level as asking “What is the name of the present King of United States of America?”, or asking a non-smoker “Does your mother know you are a smoker?” expecting yes or no for an answer.

Also, for some, it is inappropriate to even question the omnipotence of God because:

1. It is not applicable
2. It is not comprehendible

…

All of these proposals are subject to argument, and none has managed to clearly satisfy many.

Having in mind all the great thinkers and different approaches involved in resolving this paradox, now it’s time to try some slightly different approaches. Let’s begin by analyzing the paradox once again.

Well, it is clear that the paradox questions the possibility of omnipotence. But also it represents a question about some “ordinary” object from the physical world, involving a stone and lifting. This paradox belongs to a group of similar paradoxes questioning the same “greater” question “Can there be omnipotence?” Why not just ask? The beauty of paradoxes is actually in illustrating the idea, giving us an example to work with. So, resolving the particular paradox in question (the stone paradox) in either way does not prove that there is or there is not an omnipotent being. It can just prove that the paradox is not really a paradox, hence not a good representative of the “great question” or the “great idea” of omnipotence.

We should make a difference between this kind of paradox and for instance the “This sentence is false” paradox. In the later, the point is to demonstrate flaws of (some) logic systems and languages allowing us to make such a statement, which makes it rather linguistic. So, the outcome is rather “unusable” and demonstrates something real, something that is, and just recognizes it. But, on the other hand, the first is dealing with the unknown, trying to prove non-existence of something rather abstract by its definition.

Paradoxes of this kind are created to make a statement. There was a motive for this. Someone creating it for sure doubted the idea of omnipotence and illustrated it trough this question, possibly in an argument, giving the opposite side “an unsolvable task” in defending its stand. This is very much the principle of philosophical argumentation. Zeno’s paradoxes are exactly that. Why would he talk about motion otherwise knowing he believed that motion is just an illusion?

Let’s analyze the sentence in the first place. It might be rather important. Remember, we are talking of the paradox in form of the sentence:

“Could an omnipotent being create a stone so heavy that even he could not lift it?”

What could be an answer to that question? It is, obviously, yes or no. But, also, we should explain why.

So, what we have in it?

• Omnipotent being (this we are trying to question actually)
• Creation of stone (not any stone, but one that is “too heavy”)
• The stone (the one created)
• Lifting the stone (the one created)

We will for now skip the first one.

The question itself is somewhat realistic. It is talking about some ordinary objects like stones, and some ordinary actions, like lifting. If the question was about men (or other non-omnipotent) being, it would be a normal question, with the answer “yes”. The change from a non-omnipotent to the omnipotent being is what makes it a paradox. So, let’s try to stay as close to the “real” question in this analysis.

First thing that hits the eye of a scientific puritan, would be the stone itself. Some would say that there is a limit for the stone before it self-collapses. But let’s not consider this limit as an option, partly for the sake of deeper analysis, and partly because of the omnipotence which could prevent this from happening.

Similar to that, some would say that absolute (infinite) power should be challenged with infinite tasks. So the stone should have infinite mass/size (well, even more than infinite) in order to challenge the omnipotent. This, if proven wrong, would very much help resolving.

Now for the lifting part. This action is so common here on Earth that we usually don’t think of it much. It’s got to do with gravity, so in order to lift an object could be the action of applying a force opposite of the one of gravity. This would be very realistic, physical approach. As a matter of fact, it would be the weight, not the mass that is involved in the “lifting” process. Lifting same masses in different gravities would not be the same.

We can also rewrite this paradox to a formal representation:

C(x) function that creates a stone of weight x

L(x) = lifting function of stone x

w = the weight

(for every w), C(w) <and> L(C(w)) <=> Omnipotent

This meaning of this would be:

If one can create stone of all sizes (including these impossible to lift) and can lift that stone, then he is omnipotent. (This is the error in the above expression. It is because if a being can perform specified actions, we don’t know if it is able to do anything else, so we cannot call it omnipotent based on two actions it could perform only.)

If one is omnipotent than he must be able to create stones of any size, including (including ones impossible to lift) and be able to lift them.

So, the better one would be:

(for every w) Omnipotent => C(w) <and> L(C(w))

This expression could be taken in a more general sense by making the creation and lifting functions more general as “creation of anything” and “action on that”. But, let us stay within the scope of the original sentence.

This formal representation would automatically lead to the straight-forward solution mentioned above:

F => F = T

not_omnipotent, can_create, can’t_lift

not_omnipotent, can’t_create, can_lift

not_omnipotent, can’t_create, can’t_lift

F => T = T

not_omnipotent, can_create, can_lift

T => F = F

omnipotent, can_create, can’t_lift

omnipotent, can’t_create, can_lift

omnipotent, can’t_create, can’t_lift

T => T = T

omnipotent, can_create, can_lift

The first one tells us that not being omnipotent means you can’t create or can’t lift or both.

The second and fourth mean that if you can create and can lift, it does not necessarily mean that you are omnipotent, but you might be, as well.

The third evaluates to false, so it is not considered.

At some point in this chapter we loosened the definitions of a stone and lifting, having in mind the omnipotence of the being performing the actions. It is a premise that omnipotence should not be limited by some physical laws. So we come again to the definition of omnipotence. This, as shown before, could be one of the key factors in resolving this paradox. Thomas Aquinas was aware of the omnipotence problem, and the lack of its definition. As we could see, lots of proposed resolutions rely on some form of protected or restricted omnipotence.

Absolute omnipotence, as proposed by Descartes, would make all the paradoxes including it obsolete and impossible. This definition of omnipotence is above any law, including those of logic. One would think that this kind of omnipotence would be immediately embraced by theologies, but it is not. It seems to be too “nihilistic”.

There was also an effort to respond to the paradox using the Gödel’s incompleteness theorems, using the set “all actions” containing all the things an omnipotent being could do. Its conclusion is similar to some other, claiming that the paradox does not deserve evaluation, because it is logically inconsistent. It is not clear if the Gödel’s theorems are applicable to this paradox, but there is something interesting in the approach, namely, the “all actions” set.

It is time to try a somewhat “realistic” definition of omnipotence, which could help us in resolving the paradox which is also somewhat “realistic”. There must be a System in which the omnipotence is demonstrated. So, the omnipotence might be a set of all actions that are “somehow possible” to perform in such a System. That would be enough to demonstrate omnipotence to an observer from within the System.

Here comes an example that would be appropriate. For a being to demonstrate its omnipotence in our System (the Universe) it would be enough for Him to be able to lift (manipulate) all the mass it contains and not a bit more. That explains the context of omnipotence for a System. There is no need for Him to be more omnipotent then necessary. This does not make him limited. There is just no need for more. So we could define his “lifting” ability as an ability to manipulate all available mass in the System. As the mass in the system is not (and does not need to be) infinite, we do not fall in the trap of infinities.

Let’s explain this a little deeper. “Somehow possible” does not limit the omnipotence in any real way (having in mind that the System is real).

1. For instance, creating anything new in the System is not a problem. The omnipotent being could for sure create something that is possible to reside in the System.
2. It the omnipotent being creates something entirely new to the System, something that shouldn’t exist by the laws known to the System, this would just mean that something is wrong with the System’s knowledge of the System’s nature.

The same goes for the actions.

In short, it is not necessary to have infinite powers at all times, in order to be omnipotent. The point is that one should have enough power to be omnipotent at any time. This is due to the fact that omnipotence does not mean anything without a System to be demonstrated by, including System’s creation. The amount of this power should be considered a dynamic property.

We could even close our discussion here and say “As there is no stone, there is nothing really to lift, so the question is meaningless at that time, as lifting imaginary objects is obsolete. Even I can think of creating a stone which is bigger than anything I can lift. I might even make it with some use of TNT. And then, I just might make a machine that would lift it – even Archimedes said he would need a lever and a place to stand in order to move the Earth. This way I would clearly create a stone that I cannot lift, but I would be able to lift it with some machinery. People do it all the time.” But, no, let us make it more evident.

The paradox itself implicitly is “dynamic”. Apart from the “contradictory” part, creating and lifting are two actions which are in our System, sequential in time. When the question is asked, there is no stone of such kind. It is to be created. Being able to create anything it would be possible for the Him to create a stone bigger than anything known to the System. Talking about mass and Universe, this would be a Stone bigger then the mass of Universe. If we denote the present mass of the Universe as X, the stone of mass, say, X+1 would be considered impossible to manipulate. As the present omnipotence allows manipulation of mass X.

Not limited by this, He creates such a stone, entering it into the System. The key point is that of creation. It doesn’t even matter if nature of time is discrete or not, there is a time the stone does not exist, and after some point, time the stone exists. Even in the process of creation, the stone still does not exist, as the stone is a “finished product” something assuming creation as a finite process. Creation may take some time or zero time. But after the process finishes, the stone exists, and can or can not be lifted.

Before that particular time the stone begins its existence, it does not pose a “threat” to the omnipotent. After creation of the stone of mass X+1, the mass of the System is 2X+1.

But, having in mind that we defined omnipotence as a dynamic property, and said that He can manipulate all of the mass in the System, from that moment He’s omnipotence is (translated to “static”) manipulating mass 2X+1.

So, yes, He can create such a stone, as before its creation, it is an idea of a stone that would be too hard to lift, and then lift it (with no help of “Willpower”) because his omnipotence would be applicable to anything existing. Problem solved, by meaning that here was no problem at all. So, the paradox is resolved, and as such, not a paradox any more.

Conclusions

Well, he paradox is resolved. What does it prove? Almost nothing. Just that THIS (in this form) is not a paradox, if it is literally taken to some point, and if things are analyzed having time in mind. This does not imply that Omnipotent being exists. Modifications of this paradox might make it more “bullet-proof”.

“Could an omnipotent being create a stone so heavy that even he could never lift?”

This might be a harder one, but then we would have to define “never” and also we would have to think of the ability of an omnipotent to manipulate the time maybe.

This text tends to give a “realistic” view of paradoxes, and proposes a dynamic approach in cases where applicable. In this case it may prove sufficient.

Even if one personally does not believe in omnipotence, just having the ability to think about it is a great achievement of human mind. We should be guided, but not limited by the physical world. Many times in history, our “view” of the physical world has changed, driven by Science. If we can do it, omnipotent would for sure be able to.

“There is a theory which states that if anybody ever discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.”

Douglas Adams, The Hitchhikers Guide to the Galaxy.