I think, it is not easy to correctly define paradox. Hence, I propose that all of us give e.g. a proposal of a paradox definition and, first of all, examples of paradoxes. Then, after studying all proposals and examples we would try to formulate definition of paradox. The subject is very interesting, because many scientific theories give sometimes paradoxical results in solution of single problems. My first, ad hoc,  proposal of paradox: "Paradox is solution of a problem which is not consistent with common sense or is inconsistent with existing theory (knowledge)". Example of "paradox" taken from interval arithmetic. We add two uncertain intervals A=[1,5] plus B=[2,7]. It is possible that sum of the components is a precise number with  infinitely small uncertainty, for example: A+B=[1,5]+[2,7]=[9,9]=9? If such result would be possible than it could be called "paradox".

Hi Greame,

I agree with Andrzej's definitions.

Underlying both of them however, I see the following possibly more general definition:

A paradox is a conclusion rigorously arrived at from a set of premises, one or more of which is(are) false, typically due to such premise(s) having been insufficiently ascertained as being valid.

To Andre text: there exists shallow thinking and deep thinking. False premises are results of too shalow thinking about  (analysis of) a problem. Greetings!

HI Andrzej,

I agree also with this. And would add that even deep thinking can take into account only what we know already, or think we know.

If our knowledge about one of the premises is incomplete even if we think it is complete, the premise may be false also unbeknownst to us.

Hi Peter,

I always thought of sets as requiring at least one coherence criterion.

I see a coherence criterion as one that appears common to some elements in a set (a relative superset, in context) and that allows extracting these elements to form a subset that will be devoid, by definition, of any exception to this criterion.

I also see sets as possibly being determined by more than one criterion.

Here enters the notion of Reference frame.

A Reference frame would then be the set of all criteria that are common to all elements in a set.

I wonder how this fits (if it does) with Russell and von Newman's conclusions.

Would you have a personal opinion on this?

I am sorry I am not familiar with Russells and von Newman's conclusions. However, the topic is a condition for paradox, how does that relate?

The relation is the following.

If a paradox is a logical conclusion drawn from a faulty set of premises, in the sense that at least one of the initial premises being considered to draw the conclusion would be false for whatever reason, then the manner in which premises sets are defined is what relates to defining the concept of paradox

Hi Peter,

I have been thinking about your statement:

"Classically speaking, a paradox is a statement that does not have a unique valuation in the interpretation defined for the grammar. Or if you prefer, it is an unsound interpretation (aka "valuation")…"

Implying, if I understand correctly a possible "unsound interpretation" of the set of premises that led to the conclusion, by opposition to a possible "sound interpretation" leading to a correct conclusion drawn from the same set of premises.

I make the parallel with Andrzej's idea of "shallow thinking" leading to a possible "unsound interpretation" versus "deep thinking" leading to a possible "sound interpretation".

As I was digging into these concepts, I ended up doing away with more than one term to describe sets hierarchies (class, set) because it blocked me from exploring upwards into sets hierarchies, and fell back rather to the Russell's sets of sets, that is, hierarchies of sets, which allows considering each set as being located within a hierarchy between a superset (the set that encloses it) and one or more subsets (sets that can be abstracted from it).

This made me very aware that we can't know in advance whether all the elements that we assign to a set righty belong to it before specific verification and confirmation of each element. and that with any set not completely verified, even sound interpretation can only lead to wrong conclusion if even one element in the set considered is wrong or doesn't belong for whatever reason.

It seemed to me then that to avoid dealing with a possibly faulty set of premises when verification is difficult or even impossible for lack of sufficient information, that if sets are chosen by means of criterion coherence, then this could be avoided. Sort of reverse engineering.

Here is an example, simplified in the extreme. From the set of all particles defined in the Standard Model of fundamental physics (set A), you can extract the set of all particles that are known to be scatterable (Set B) scatterable meaning that they can be made to rebound after having collided with each other. Right there you have a more limited set that excludes virtual particles (conceived of to mathematize some descriptions). From set B, you extract all particles that are considered massive and stable (set C), you tentatively end up with only 3 particles (electron, proton, neutron).

Now with the elements of set C you can build all elements in the periodic table (superset B1), from set B1, you can build all matter in the universe (Superset A1). In this case not grounded on axioms, but on observed events.

I see recursivity in the process and hierarchy in the structure of sets. I observed that it seems possible to explore any issue in this manner without leading to paradoxes.

I agree with you that recursions must stop at one of the axiomatically defined "base" sets,

Do you know if this avenue has been explored before?

I am not a professional logician but someone with a deep interest in paradoxes. From a layman's point of view, a set of valid premises leading to an unsound conclusion would constitute a paradox. On Scrutiny, one of the valid premises would be revealed to be unsound, perhaps leading to a resolution of the paradox. Clearly it has more to do with semantics and the limits of unambiguous expression in the language.

Prof. Breuer's statement that "a paradox is a statement that does not have a unique valuation in the interpretation defined for the grammar" brought to my mind visual illusions that defy a single interpretatation, but depend rather on the state of the observer/perspective (for e.g., bistable illusions). In contrast it appears to me that most paradoxes might resolve to a singular interpretation. Interestingly, there is a scale to rate paradoxes, depending on their resistance to resolution.

Ashok,

I think you hit it right on the button with your statement:

"From a layman's point of view, a set of valid premises leading to an unsound conclusion would constitute a paradox. On Scrutiny, one of the valid premises would be revealed to be unsound, perhaps leading to a resolution of the paradox. :

You say "from a layman's point of view". I can tell you that this is also true from a formal logic point of view. In fact, this was established more than 2000 years ago by the Greek, and formal training to verify premises is provided by resolving the complete set of Euclid's theorems.

I have come to the conclusion that your statement can be formulated in absolute terms: If an unsound conclusion is drawn from a set of "assumed" valid premises, this can only mean that one of the premises is unsound and does not belong or that a required premise is missing in the set. If all required premises are confirmed as being sound and present then the conclusion can only be true.

This, in fact, has to do with the way neural networks operate. Our neocortex is a 6-layer neural network and operates by coherence perception in sets like all neural networks. This is what Donald Hebb explored in the 1940's.

If you feed it sound data sets, it will provide sound coherences (conclusions). From an unsound data set, it will provide what coherence is possible from this unsound data set.

Andre,

I think I agree. The question is, would an unsound dataset cohere? I don't know.

Ashok,

Yes. It happens all the time. Whenever we find out that we previously drew a wrong conclusion after finding the right answer, about any issue whatsoever, this is because we found coherence in a less sound data set.

Whatever coherence is possible in any data set, sound our not, the network will provide it. It is up to us to verify the premises of any conclusion after having reached it and make certain that they were sound.

Dear Andre,

Thanks for your reply. Let me interpret your premise:

"Whenever we find out that we previously drew a wrong conclusion after finding the right answer, about any issue whatsoever, this is because we found coherence in a less sound data set."

That says more about the nature of the conclusion than that the premises "cohered". The agent in question presumed the conclusion -- which might be different from saying that the the dataset "cohered" to the conclusion. If we could agree on some objective definition of coherence, then it would be the case that the conclusion was presumed absent coherence. Could an agent presume a conclusion absent coherence?  It's an open question.

Dear Ashok,

When the agent reached his first conclusion, there was no notion of presumption, because he did not know yet the new element that caused him later to reach a better conclusion.

Indeed, the definition of terms is of utmost importance.

When I use the phrase "perceiving a coherence" in a data set, I mean exactly the same thing as "reaching a conclusion" from a set of premises.

A coherence is a connection perceived between the elements of a set, a conclusion drawn from a set of premises.

Example: we know that a long time ago, with the little information that our forebears had about the Earth,  it took quite some time before sufficient knowledge was accumulated for them to be able to correctly conclude that the Earth was not flat but was rather spherical. It took time also before it could be correctly concluded that the earth was orbiting the Sun, and not the opposite, etc.

But before these facts that are now known and verified were gathered, people could not presume of these conclusions. They concluded from what knowledge they had at each stage, and the conclusions refined progressively as the amount of knowledge expanded, because the set of data that conclusions could be drawn from expanded to become more precise as time passed.

Hi Peter,

The fact that we come from entirely different backgrounds and even different mother languages, and use different terms to describe the various aspects of the concept makes it a little difficult to clearly exchange ideas about the concept. But I think this can be circumvented.

I will try to analyze what you explain:

I:G->M

I = function (conclusion)

G= series of sentences describing M

M= what is being described

(1) I is well defined if G correctly describes M

(2) I totally describes M if every sentence of G has meaning

Now if I understand correctly what I just summarized, then I understand what you mean here and I agree:

"I suggested that a paradox is a partial interpretation (1 holds, 2 fails), which cannot be extended to any total interpretation (1 & 2 hold). If one tries, contradiction follows. "

But this pre-supposes that all that can be known about M is already known and confirmed to be true, and that we need only to carefully examine it to be able to obtain the case of (1 & 2 hold).

This would be ok with me theoretically, of course, if all elements of M were confirmed axioms.

But in physical reality, we usually are confronted with situations that we do not have access to complete information about, or that some info about may be wrongly assumed true, unbeknownst to us, and even if we are very careful in interpreting M, we may not be able to reach the state (1 & 2 hold) for lack of some important information.

So it seems to me that just carefully interpreting M is not sufficient in and of itself for us to always reach the state (1 & 2 hold) .

All of the data required about M to even have access to the state (1 & 2 hold) needs to first have been gathered and correctly verified to be true and to apply.

Please correct me if I misunderstood the meaning of I:G->M

Dear Andre,

Somehow I remain unconvinced. For one thing, we could draw a distinction between tentative conclusions (based on incomplete information) and absolute conclusions, if in hindsight (for instance, there was never conclusive evidence that the earth was flat).

I feel your discussion involves ideal agents, whereas agents in our world make "hasty" conclusions all the time ( without much substance or coherence -- examples abound). Perhaps this is a commentary on the agents then.

PS: edited.

Dear Ashok

Hasty conclusions is precisely what is done when all premises (or data or facts) on which conclusions are based are not all completely verified.

hindsight, is already reconsideration of a previous conclusion after having obtained more information. When first conclusion is drawn, we may have thought that we had all the fact even if this was not so.

Objective conclusions can be had only when all premises or data or facts on which conclusions are based are all completely verified.

In real life, we are not always able to obtain all information to completely verify some premises, data or facts, even if we think we have at some point, and this is why it is difficult, even with deep thinking on potentially incompetely verified premises, data or facts, to obtain objective final conclusions based on completely verified premises, data or facts.

This is why we may be confronted with "apparent" paradoxes even with deep thinking about "what we think that we know" of premises, data or facts at the moment of first conclusion. If we become certain that the paradox is true, and do not reconsider and try gathering more info, then we keep thinking the paradox is objectively true.

The neocortex can work only with the set of information, data or fact that it has at the moment of conclusion. We always tend to be certain of our conclusions, but we must not be totally certain, particularly in cases of apparent paradoxes, but also for conclusions that seem to be ok, in case more info gathered at some point in the future eventually shows us that the first conclusion was based on partial information.

If we become certain that the first conclusion is final and true, we may not pay sufficient attention if new information becomes available to us and we may ignore the new info that would show that the first conclusion is not ok and we may not reconsider even when more complete information becomes available.

This is what Socrates, Platon, Descartes and many others explained, each in his own way.

I try to explain as clearly as possible, but I may not be able to do so. Sorry.

Dear Andre,

From my reading of our discussions, it seems that the issue is about the art of concluding from incomplete information.

I would argue that there is a qualitative difference to the coherence achieved with tentative conclusions relative to that with final conclusions.

Best, Ashok

Dear Ashok

You understood me correctly:

You wrote "From my reading of our discussions, it seems that the issue is about the art of concluding from incomplete information."

This is right. Because objectively occurring events in physical reality can have an indefinite number of characteristics, of which we generally can become aware of only a part.

If we are not careful (if we do not completely verify), we end up concluding with insufficient information to form an objective opinion (conclusion).

We can analyze only what we allow ourselves to become aware of.

We can also chose to be thoroughly methodical about a posteriori verification and confirmation of premises.

In summary, there is

1- Objective physical reality with its indefinite number of characteristics.

2- What we perceive of this objective physical reality

3- What we conclude from what we perceived of this objective physical reality, which can be possibly correct even without verifying premises (if we are lucky) of incorrect (partial, paradoxes, etc) or confirmed (if premises are completely verified and found to be true)

The important fact is that we are physiologically unable to directly analyze 1. We are physiologically able to analyze only 2.