I thought about you before I knew you!

Having a somewhat romantic disposition, I used to indulge in the following fantasy, in my former, bachelor years, when my passion for philosophy and science left little room for a long term relationship: 

"Suppose I fall in love sometime in the future, and meet the girl of my dreams. If that indeed happens, then she is alive today somewhere, going about her everyday life, oblivious to the fact that we're going to be together, and the fact that I'm thinking about her at this very moment." I indulged in the fact that I could think of her before we even met. 

But was I really thinking of her, or merely about a set of people who merely satisfied the predicate 'my next girlfriend'? This is analogous to thinking about the winner of a race before it has started. But it was difficult for me to accept that I'm merely thinking about a potential person, given that she already existed. Surely it wasn't determined who she is (going to be), before we actually met, and therefore at the time of my romantic deliberations. But such speculations aside, surely the predicate picks out a single person, although it can be said to apply to a number of potential candidates, as it were. At least the romantic mind tends to folly in this manner (I am thinking about her) where cold reason suggests otherwise (I am thinking about a set of potential the ones).

However, presently when I say to her "I thought about you before we met" it seems to be true, because now we know that it would actually be her. This is truly romantic! If so, then we have logically validated, or at least salvaged some of the reality underlying my spells of romantic imagination! And at the same time, it is also true that in some sense I was thinking about her before I met her.

Worlds either exist eternally (without a beginning or end), or don't exist at all.

I reject the notion of the world (i.e. everything that exists) coming into being. However, I do permit, that nothing may have existed - or to put this less contentiously - that nothing, rather than something may have turned out. That is, worlds either exist eternally - without a beginning or end in time - or don't exist at all. Since our world exists, it is eternal. In this sense, phenomena like The Big Bang are temporally local.

Note that this doesn't rule out the possibilty of there existing more than one world. Of course, the typical spatiotemporal notions, applicable to any one world, don't apply accross worlds. That is, notions like "x happening in this world at the same time (or before/after) as y happening in another world" or "this world being x distance away from some other world" are meaningless. This doesn't however rule out an imposition of meaningful relations on the possibility space itself.

Eksplikacja reżyserska do filmu "Bracia"

Film „Bracia” to przypowiastka, która zestawia elementy natury ludzkiej takie jak obłuda i próżność aby alegorycznie przedstawić pewien archetyp ludzkich interakcji.

Film wstrzymuje się od stwierdzenia morału, czyli od jednoznacznego ocenienia wynikającej manipulacji, a zaledwie zwraca uwagę na nieunikniony brak kompatybilności pewnych postaw, a raczej ich kompatybilność, która jest skazaną na groteskę. Wielowarstwowość przekazu symbolicznego udostępnia szeroki wachlarz interpretacji – na przykład, mimo dość uniwersalnego przekazu, istnieje też możliwa interpretacja przypowiastki jako komentarza historycznego odnośnie systemów politycznych, opartych na utopijnej wizji świata, czy jako nawiązania do motywu rywalizacji braterskiej – w szerszym sensie rozumianego jako rywalizacji rodzeństwa – często występującego w biblijnych przypowieściach oraz w mitologii, czy w końcu osobistej interpretacji jako metafory podobnych sytuacji z naszego codziennego życia. Innymi słowy, film bez narzucania właściwej interpretacji, skłania widza do ogólnej refleksji nad nieodłącznymi elementami natury ludzkiej. 

Motyw obłudy, który w sposób kluczowy podszywa fabułę całego filmu, uwidacznia się wyraziście w fałszywej wizji lepszego świata przedstawionej przez Wielkiego Brata. Ten fałszywy obraz nawiązuje nie tylko do symptomów systemów politycznych, opartych na utopijnych mrzonkach, ale też bardziej powszechnego zjawiska manipulacji społecznej – cynicznej propagandy służącej wypaczeniu obrazu rzeczywistości dla celów osobistych lub wybranej grupy ludzi. Tym samym filmowi można przypisać bezpośrednie odniesienie do kanonu literackiego krytycznych esejów i powieści takich jak Moore’a (Utopia), Huxley’a (Nowy wspaniały świat), Orwella (Rok 1984), czy domyślne odniesienie do jeszcze szerszego, proto-filozoficznego oraz filozoficznego kanonu traktującego kwestie moralności – od Starego Testamentu i światowej mitologii począwszy a na współczesnych rozważaniach etycznych skończywszy.

Wybór formy fabuły jako bajki narracyjnej został umotywowany uniwersalnością tej formy literackiej – która stosownie odpowiada uniwersalności poruszonej tematyki – oraz łatwością adaptacji do medium filmu animowanego. A więc ten film wpisuje się tradycję, tak starą jak ludzkość – sięgając tradycji oralnej – rozpowszechnianej przez starożytnych i nowożytnych bajkopisarzy takich jak Ezop, Fedrus, czy La Fontaine, i nadal kultywowaną w nowożytnej oraz współczesnej twórczości polskich poetów, pisarzy, i filozofów takich jak Krasicki, Mickiewicz, Lem, czy Kołakowski.

Symboliczny motyw dzbana w sposób absurdalny karykaturyzuje obłudę podszywającą interakcję bohaterów oraz właściwą próżność jego roli jako identyfikatora statusu. Poza tym symbolika dzbana jest dość obszerna – chociażby jego rola w przypowieściach biblijnych czy sądzie skorupkowym starożytnej Grecji. W języku polskim dzban ma też bardzo ciekawe, współczesne znaczenie, które oferuje nam właściwą dwuznaczność – jako pejoratywny epitet, dzban znaczy tyle co głupiec. W internetowym plebiscycie, organizowanym przez Wydawnictwo Naukowe PWN, dzban wygrał tytuł młodzieżowego słowa roku 2018. Dzban to jednocześnie próżność i głupota Dużego Brata, który daje się uwieść wizji „wielkości urojonej”, ale zarazem próżnia ontologiczna realności tej wizji. Czyli symbolika jest rzeczywiście bogata i uniwersalna, więc powinna przemawiać do wszystkich – poza dzbanami, naturalnie.

Phenomenology of Non-existence

Whenever asked to explain the key idea underlying his latest masterpiece 'Phenomenology of Non-existence', Joseph would often make use of an enigmatic Gedankenübung (thought-exercise). He'd invite the typical inquisitive offtologist, or the occasional ontologist, as the case may be, to consider how seriously they take themselves. That it was always too much, Joseph explained, was the first step to understanding the main themes covered in his work.

Consciousness is not an illusion.

A dream you'd rather not have.

Imagine the following situation. A Dignitas patient-client ingests the lethal dose of the pentobarbital, and soon after falling asleep (the sleep which gradually deepens into a coma, and eventually leads to respiratory arrest and death) they have a lucid dream whereby they're aware of what is happening, but for some reason in that dream they change their mind regarding ending it all. Naturally, they won't be able to wake up, and they know that in the dream.

THE ILLUSION OF DECISION MAKING (an arument for epiphenomenalism)

There's a high correlation between what we like to call our decisions---actions to which we'd like to attribute intention---and their alleged causal consequences. For example I decide to take a sip of coffee, and it happens. I decided to write this essay, and I'm writing it. The correlation value appears to diminish with decisions and plans regarding actions/events that are more distant in the future. In effect, this argument would be to support the ephiphenomenalist view, that the feeling of self agency (illusion of free will) is a real but ineffectual phenomenon.

Now, I'd like to propose an interpretation of what we would normally call decisions, in the absence of free will. That is, I propose an interpretation of what we would call and identify as decisions in the absence of free will----an interpretation which explains the correlation between those intensional events we identify as decisions and the alleged outcomes of those decisions. So to clarify. There persists a strong feeling that we have an autonomous, causal influence over our actions. It seems obvious that we make decisions and then carry them out! Is it possible that we're not actively and freely initiating our actions?  I propose an explanation of this strong feeling/conviction of decision-making and its apparent efficacy on the world, even in the absence of free will.

MAIN PART
Here it goes: assuming absence of free will, what we identify as decisions are in fact predictions of what we (our bodies) will do. We're quite good at making those predictions since we know our bodies quite well, and have a well developed intuition that helps in the accuracy of the prediction. This would explain the correlation between the illusory decisions---which are in fact predictions---and their outcomes. It should be stressed that predictions are not intended acts. We don't choose to make those predictions---they just happen and we merely witness them as if they were revealed to us, and we only interpret them as our decisions. To put it another way, we receive the predictions as 'reports' from our system, so we remain decisively passive in this regard. That's about it for the core of the idea. 

To be clear the phenomenological aspect of what we would call a decision is still the same. That is, I'm saying that decisions are in fact no more than predictions, and the belief of their causal efficacy is illusory: I'm going to watch this movie still appears in our minds but it's merely a passively received prediction and it has no causal effect on the world, unlike a real decision employed by free will. It's like shifting from a geocentric to a heliocentric explanation of the Sun's path across the sky. It still looks the same (it is the same image on our retinas), but we're not at the center of the Solar system. No longer the protagonists we thought we were.

It should be noted that instances where the predictions fail, correspond exactly to circumstances that would get in the way of acting on a decision, i.e. external circumstances that are unexpected, and as such difficult to account for from the reading of our inner state alone. For example, if I decide to go to the local cinema tomorrow, but it burns down, then I won't go. But this is also something that would be difficult to predict---the likelihood of an accident happening at the cinema is precisely reflected by the degree of my confidence that I will carry out the plan of seeing the movie. Changing one's mind is to be interpreted as merely passively updating/refining/revising a prediction, i.e. being presented with such an update.

It just seems to me that each instance of what we identify as a decision could be equally well explained in terms of a revealed prediction. And it seems that failures to realize previously decided upon acts are equally amenable to being plausibly explained by unforeseen circumstances that are not being accounted for in the prediction.

The picture that emerges here is one where we're mere passive witnesses to our own existence who create (or are presented with) a meaningful narrative which places us at its center as the protagonists, i.e. we live out a geocentric illusion within a heliocentric reality.

LETHAL SPELL ABILITY thought experiment.

I had thought of this interesting hypothetical scenario recently, which could easily be extended to a thought experiment. Imagine that each person, over the age of, say, 16 was given the magical ability of having another person drop dead by thought alone, e.g. 'Adolf!', and Adolf drops dead. The ability would entail perfect concealment of he act, i.e. it would be impossible to know who had cast the deadly spell. This part is important, so those exercising this magical power would do so with full knowledge of its effectiveness, and impunity. Moreover, I believe that to many, only the presence of such anonymity would be a necessary condition for casting the spell at all.

This is interesting for a number of reasons. I wonder how many people would go ahead and actually use the spell? Also, suppose the version of the magic ability such that the total number of spells was unlimited, with perhaps some daily limit. And that each spell would have to be cast at an individual, e.g. general spells like "may all people above 6 feet drop dead" would be disallowed, which doesn't mean that one couldn't find out who the people like this are and cast the spell on them. At what rate and in what sorts of patterns would people start dropping dead? 

I have been thinking about this, and wondering what kind of people would jump at the opportunity of exercising such power.

For example, suppose that in the original scenario (of having a single spell only), we let the time to cast the spell last for a month. The point being that existing grudges, prejudices, and feelings of hate would be sorted out right away, so to speak, instead of having the scenario allow to wait for a time when an otherwise unwilling to use the spell person would be forced to do so in self defense. 
So suppose a month goes by and all the spells had been cast. It follows that we're left with a population that is at most smaller than before by the whole number of people (magicians) who were endowed with the ability to cast the spell (the case when no magicians are killed before casting their own spell), and roughly (±1) at least smaller than before by half the number of magicians (the case when half the magicians are killed before casting their spell).

Now imagine 2 alternative outcome-worlds:
--world 1 consisting of all the spell casters and the remaining youth under 16. So this world is left with those that though it to be better to cleanse it one way or another. And those who got eliminated no longer exist.
--world 2 consisting of all the spell victims, and all of the under 16 that were present before any spell was cast, i.e. the original, under 16 population. So basically this world consists of everyone minus the spell-casters.

I wonder how those worlds would evolve in time. World 1 has less people remaining, but let's suppose the populations to be sufficiently large, as not to be in danger of extinction after the cleansing process. I wonder how those worlds would differ, given what kind of people occupy it.

I also wonder if people would just annihilate themselves into extinction, on the version of the scenario where there would be no limit on the number of lethal spells. It would be like giving anyone access today to the current nuclear arsenal deployment.

I'm sure there are other aspects of this thought experiment that I haven't raised, and which could be quite interesting. I'm thinking of writing a short story based on this scenario as the underlying premise. No prizes for guessing that Borges and Lem are among my inspirations! 

ALOSZA I DRZWI

Mój szczur po prostu nie radzi sobie z pojęciem drzwi. Jak są zamknięte to je obwąchuje z ewidentną ciekawością. Gdy drzwi są otwarte to jego zdziwienie wcale nie ustaje---obwąchuje próg i framugę z zaciekawieniem; rozgląda się z widocznym osłupieniem: jak to może być, że perspektywa wcześniej nie widoczna, nagle się ujawnia? Nagła zmiana w topologii podłogi musi robić wrażenie. Ale, mimo trudności z ogarnięciem swoim szczurzym umysłem pojęcia drzwi, trzeba temu ciekawskiemu gryzoniowi przyznać godną pochwały wytrwałość.

Asortyment Nieskończoności

Jest to prezentacja Powerpoint, prelekcji której udzieliłem w Otwartej Kolonii (filii Wolskiego Centrum Kultury) 20 kwietnia. Ten wykład to krótka historia pojęcia niskończoności, oraz wstęp do arytmetyki liczb kardynalnych poprzez prezentację dowodów Cantora na równoliczność zbiorów nieskończonych. Poniższa prezentacja zawiera trochę definicji oraz pomocy ilustracyjnych, ale oczywiście brakuje jej niezbędnej narracji która wyjaśnia wszystkie pojęcia oraz ich relacje.

Asortyment Nieskończoności PDF by Mariusz Popieluch on Scribd

Ad hoc theory of one of the reasons why time appears to flow faster in dreams.

A perplexing yet strangely familiar experience to anyone who has ever dreamed, would undoubtedly have been making the occasional observation that much more seems to occur in our dreams than the duration of our sleep alone can account for. After a night of intense dreaming we often remember the equivalent of much a longer time period then we had actually spent sleeping (which is generally much longer than the time spent dreaming). Or when we take a short nap during the day, we're often surprised to learn upon waking that only a few minutes have past after what had seemed like an epic adventure. 
So apparently there's a clear subjective disconnect between dream state and waking state with regard how we perceive the flow of time. I make the assumption that time doesn't actually undergo some substantial warping in our heads in a way that would account for this phenomenon.

Since we're under the impression that more has happened during our dream than the time being asleep can account for, there seem to be a few possibilities as to why that may be. I should mention that I'm writing this blog post without having consulted any particular, specialist literature on the matter, hence the post title. 
One possibility is that our brain processes inner-conjured dream-films with a higher temporal resolution. That is, the dream-films are presented to us with increased speed, but we perceive them as normal, since within our dream state there's no other reference frame other than the dream itself. This hypothesis appears to be amiable to empirical exploration.

The other possibility, which occurred to me recently, is closely related with how our memory works. So the following explanation requires both some assumptions about the mechanics of our memory, but also some assumptions about the actual content of dream-films. The vast majority of us, whenever recollecting some series of events---say our last birthday party, or the recent trip to the countryside---tends to recall vivid snapshots of the more intense impressions, which are more or less arranged in chronological order. The point is that most of us don't remember the entire experiencial continuum, but rater a collection of short vivid fragments. Now, if we think about it, this is how dreams often seem to manifest---in a bunch of , more or less, disconnected situations combined into one weird stream.

Of course when I talk about this, I have only my memory to rely on, so if our memory indeed works as I just have outlined, then I cannot justifiably rely on it to speak of the character of the dream-films. But the point is that it is possible that dreams may be arranged and have the actual content analogous to the way our memory works. That is, dreams may consist of vivid fragments connected haphazardly with a vague chronological cause-effect narrative sense, like a movie trailer. But because our memory also works, as assumed earlier, such that we tend to only recall movie-trailer-like imagery, and only infer all the in-between events that must have occurred, when presented with the dream-film trailer, we inflate its content by inferring that there must have been intervals in-between.

For example, in real life, when recalling our plane journey from one city to another, we may only vividly recall say, the departure and arrival, plus a few more interesting cloud formations viewed through the window on the way, but that doesn't mean that the journey only consisted of those few intervals (the sum of which may consist of less than a minute of vivid mnemonic imagery), because as a matter of fact the journey is much longer (say, a few hours) than the sum of our recollected and vivid fragments---we infer that there must have been in-between intervals, other than those that we can vividly access.

But it's possible that my dream of a similar flight would be a kind of film trailer actually consisting of only the vivid fragments of the departure, a view of the more interesting cloud formations, and the arrival, arranged in a sequence of one vivid fragment occurring immediately after the previous one, without any intervals in-between at all. However upon recalling this 3-scene act when we wake up, it seems much longer because our memory mechanism inflates this vividly dreamed sequence by filling it with inferential content, i.e. a journey like that must have lasted a few hours (there must have been many more scenes in this act).

So the ad hoc explanation of why time appears to flow faster in dreams is that dreams are much like film trailers, but because of how our memory works, we tend to have the impression that we've watched the entire film.

I've also written about about how our memory works in a recent post, and a short story written years ago.

Group memory quality as a function of group size and interaction---an explanation.

I read somewhere, years ago, that species of birds that live in flocks tend to display better group-memory. I don't remember (pun intended) how that was measured, but have observed this to be also true of human interactions. Aside from mere cognitive stimulation---in a group we also tend to interact more, so there's a greater chance of certain facts being continually refreshed---I believe that there's more to what enhances group memory. That is, I think that there's another factor that influences group-memory, not by virtue of cognitive stimulation alone, but the kind of stimulation that has greater likelihood of occurring as the group size increases.



It has to do with both the abilities and limitations of each individual, and the fact (conjecture) that those abilities/limitations vary significantly among individuals. The hypothesis underlining this idea is simple---it's very plausible that given that we've been a social species for most of our evolutionary history, it would appear to be an adaptive trait to distribute cognitive abilities related to memory and recollection over the entire group. In particular, there may be a significant variation of the temporal ranges that individuals are capable of recollecting proficiently. That is some people would be better are remembering recent events, with high fidelity, whereas other's mid-term or long-term memory may be much better. After all it's the sum of all memories, fed into the social collective that counts, so it needn't be preserved by each individual---that would be an unnecessary over investment of neural-power.



From careful observation of my family and friends I have noticed that some of us display great proficiency in recollecting great amount of detailed information of current affairs, i.e. information gathered in the recent days or weeks, but may struggle holding that detail for a long period of time, and their memory of current affairs progressively fades over time. Others, on the other hand, display much better long term memory than others.

Perhaps such traits can be shown to be more generally diversified with respect to temporal ranges of optimal recollection. Moreover, there may be variance regarding the kind of of information that is remembered---some people may be better at remembering how to perform certain actions, others at phenomena that occurs in the surrounding environment, and others still may be really good at keeping a moral tally of group members, i.e. those displaying altruistic actions and those that exploit the benefits offered by the group. Combining those individual abilities in a social group would obviously enhance the group memory, simply by virtue of mutual compensation of individual limitations. So, such a social mnemonic mechanism would result in the group super-organism as a whole displaying much better memory than each of its members alone, on the condition that information flow is unhindered.

Those ruminations on the nature of memory are an unintended interest of mine. I just can't help but make conjectures about it. I've explored this subject matter before, via a short story---in particular how perception influences our memory and recollection.

Auto Capgras delusion.

Capgras delusion is a disorder whereby a person holds a delusion that a friend, spouse, parent, or other close family member (or pet) has been replaced by an identical-looking impostor. 

I was wondering this morning what it would be like, if possible at all (I think it may be) to suffer from the peculiar case of Capgras delusion, whereby one believes themselves to be an impostor.

Cognitive dissonance and belief revision in perilous political discourse.

This is only a free rumination on one aspect of the role that cognitive dissonance plays in political belief revision, with particular attention to the role of perceived self-image.

We need look no further than the comments sections beneath a random sample of online articles or videos that contain overt political content, to realize that public political discourse is—more often than not—far from polite and respectful. Tendentious rhetoric and insults are common there. And no one is spared—even those who have (naively enough) found themselves attempting the conversation with a genuinely open mind about having their own views revised if a good argument that challenges them presented itself. This phenomenon is most pronounced in cases where clearly identifiable, dichotomous for/against positions are present. Now I'll sketch out how cognitive dissonance regarding perceived self-image can hinder belief revision, in particular how it can lead to a tendency of entrenching one's views—even those that have been previously held only tentatively. Central to this dissonance is the contribution of unpleasant experiences acquired from engaging in careless political discourse with anonymous others.

Here I focus on that entrenchment being precisely the result of cognitive dissonance rather than a rational process. One needs to look no further than the aforementioned discourse space, to find people engaging in disingenuous rhetoric or contemptuous criticism, which often escalate to outright insults. The insults in most cases center around the following personal attributes: knowledge, cognitive faculties, and moral integrity. It's safe to assume that most people don't ascribe to a self-image as described by such disingenuous criticism or outright pejorative combinations of epithets aimed at the aforementioned personal attributes. That is, people prefer to think of themselves as not being completely ignorant, especially when speaking out on issues that matter to them; also, unless a medical diagnosis states otherwise, they prefer to think of themselves as not being in any way cognitively impaired; and finally they prefer to think of their beliefs as not being morally questionable. So by making the occasional error of engaging in discussions on unmoderated forums, where trolls abound and where other interlocutors often display no more conversational etiquette than trolls, one puts one's self-image at peril, and as a result gathers painful resentment toward "the other side" over time.

As a consequence one will tend to become more entrenched in one's beliefs. Why? Because changing one's view (in cases of issues with clear-cut for/against positions) would almost amount to agreeing with those who had previously expressed contradictory views to one's own regarding their self-image. But this is unacceptable (to most), as it would be treading very closely to tacitly condoning views that contradict one's self-image. After all, conceding that someone who holds such contradictory views can be right about some things—in particular those things that prompted them to make a contradictory judgement regarding others' self-image—is not too far from saying that they may also be right about that very judgement. Or to put it another way, if someone is right about some things, then their judgement about those who fail to see those things as right may very well also be right. Mental discomfort resulting from such doubt is an instance of cognitive dissonance.

Generally, conceding to the correctness of a view previously held as incorrect may indeed prompt many to consider revising their self-image—there's an implication of certain unfitness of some of the aforementioned personal attributes—after all, how could one have been wrong for so long? In this case however the inclination to revise one's belief is additionally hindered because it entails agreeing with people who explicitly hold contradictory views regarding one's self-image. Hence the strong inclination to escape this cognitive dissonance, resulting in a tendency to (irrationally) stick with one's original position/view. Indeed, seeing the degree of emotional engagement in political discourse that people display, would make one think that it's not the broader issues that are being defended but something more personal.

There is no distinction without similarity.

Moreover, it seems that there even can't be a distinction without similarity. For it's impossible to distinguish any two objects without both of them being at least similar in the sense of being objects of consideration in the first place (or just objects, existent or not, for that matter)---that being their necessary condition (similarity) for comparability and hence distinction. This gives similarity a foundational character. The converse need not be true however, since in order to consider a single object as identical (maximally similar) to itself it doesn't seem obvious that one needs to resort to the consideration of it being distinct from anything else. The immediate corollary of this, if not a mere reformulation, is that there are no absolutely distinct (absolutely independent) things (there, the proof is in the formulation itself---things!). If this is true, then absolutely everything is similar in one way or another and as such admits to a similarity analysis. And there don't seem to be any paradoxes at the limits either, since even any totality of things (even the Russel set) is similar to itself, as is nothing, for that matter. Even everything and nothing are similar in the sense of belonging to the same ontological category. So the only questions to which much ontological inquiry reduces are "in what manner are things similar?" and "to what extent are they similar?"

What causes the heuristic 'if you're not with us, then you're against us' so persistent? Rumination on some discourse leaden reasons that lead to polarization of political views.

One of the reasons that (general) political view declarations are important, and persist, is because they play a key role in aiding the interpretation of what is being said. Of course what is said, often strongly implies such a declaration, but not always; that's why it becomes useful to declare "what side one is on", even when one wishes to remain neutral (until one's understanding of the issue at hand has matured), unless one is prepared to have their utterances treated with suspicion. This is a more general phenomenon, not only restricted to political views, but I believe that politics makes the phenomenon more pronounced. Much more could be said about this, but here I'll just give an example, which illustrates the idea. I will use an abstract example (to avoid the risk of creating a distraction).

Let the context be one where there is a stark divide in views regarding some issue in it. Now, say there's a proposition 'A' that expresses an attitude regarding that issue in the mentioned context. That is, some would think, and feel very strongly that A, whereas others would think, and feel very strongly that not A.

EXAMPLE 1
Person 1: "A"
Undeclared: "Of course"

EXAMPLE 2
Person 1: "Not A"
Undeclared: "Of course"

In both cases, it's very likely that Person 1 (or person 2) will inquire "what do you mean?" And why are they doing this? Was the Undeclared's statement not clear enough? It was, but coming from an Undeclared it may be taken as meaning the opposite of what it says, e.g. being a sarcastic remark; the existence of such remarks, being common where political views clash, adds to the ambiguity of the actual intended content of undeclared's statements. The degree of this phenomenon, I suspect, would be a function of the context's scope, i.e. given some context, what is the extent of the Undeclared's lack of declaration.

Needless to say, this leaves those who wish to maintain neutrality (which is more often than not the wise attitude, I believe) in an uncomfortable position of their opinions being either notoriously misunderstood or treated with suspicion, so there exists a pressure to declare oneself politically, even if one is neutral. In a "war of words", which often is the form of political discourse outside of echo chambers, being declared facilitates unpacking the non-explicit, intended content of the actual statements that are made. So enhancing one's clarity is a tempting reward for the mere cost of an instrumental declaration (even when it is inconsistent with one's actual view). Consequently, this mechanism inadvertently facilitates the proliferation and sustainment of extreme views, at the cost of the more balanced ones.

A valuable lesson.

Mother to child---'If you learn how to tell the time from an analogue clock by the end of the week, you shall be rewarded'. Within a week the child returns, demonstrating mastery in the ability of telling the time from a clock, and reminds the mother about the reward that was promised. 'You have already received your reward'---answeres the mother.

Best of imperfect worlds.

Lev somewhat dissatisfied with his previous cosmic project, decided to embark on a more careful enterprise of world creation, and turn down the perfection parameter from maximum. This time he decided to play with the parameters of fundamental values, and observe how they'd influence the hedonistic dynamics. The idea was to run a simulation where hugs are set as having principal value, and consequently become the sought after currency -- in other words, hugs in that world were to be the sole wealth determinant. But in what sense 'hugs'? -- one may ask. Receiving them of course! And naturally they'd have to be of genuine sincerity; neither bought nor forced in any way. That is, hugs have value only if they're sincere and welcomed. But then how does one accumulate such wealth, given that hugs are such ephemeral phenomena? Surely, one can't be in possession of a great number of hugs. The only way one can accumulate wealth of this kind, that is, become a prosperous hugee, is to guarantee and maintain the existence of those willing to give those hugs, i.e. huggers (aka ready-to-hug beings). In other words, one can attain a high flux of hugs, and wealth would be interpreted as maintaining a high hug-flux. This can be done in many ways of course, and Lev calibrated the simulation with no limit on the degrees of freedom concerning valid hug-acquisition. Naturally those who attain the ability to reach and maintain a high hug-flux steady state, aka hugagogues  are sought after as raw models for guidance, whose wisdom would guarantee the attainment of wealth in that world. Lev has hypothesized that such a world would be among those that are the closest to being perfect without giving rise to any absurd consequences, which inevitably accompany perfection.

On the (nontrivial) non-uniqueness of the empty collection.

Consider a property P : "is green all over, and is not green all over".

Alice and Bob are friends. Bob reasons about the world, and accordingly conditions his understanding of what is or isn't logically possible to classical logic. In particular, not only there are no objects with property P, but there cannot be any such objects since P is a logically impossible property. Alice on the other hand, having travelled far and wide, and having seen curiosities such that no classically (as per classical logic) minded philosopher has even dreamed of, accepts paraconsistent logic as the correct way to reason about the world and its many wonders.

Alice and Bob agree that 'empty' (or 'the empty collection', denoted (e)) is an absence of any-thing, i.e. it is an absence of any object (or simply 'a set with no elements'). They also adopt a covention whereby saying that 'some-thing is contained by the empty collection' means the same as saying that 'that some-thing does not exist', e.g. 'four sided triangles are contained by the empty collection' is just another way of saying that 'four sided triangles don't exist'. This is the convention both Alice and Bob adopt, and agree on. Nota bene, impossibilia (impossible objects) are also contained by the empty collection, for since they can't exist, in particular they don't actually exist.

The duo however disagree about what counts as an object. Whereas Bob considers any object with property P as impossible, i.e. an object that cannot possibly exist, Alice doesn't. As a consequence, within their discussion about the world and its many wonders, the term 'empty collection' despite being correctly understood by both friends as having the same extension, is incorrectly assumed to have the same intension. The extension is the same since it is an absence of any object, but its intension is distinct, due to our protagonists' differences in reasoning about the world, which in turn bear on what counts as possible and what doesn't.

In particular, for Bob the empty collection has the property of containing all "objects" with property P, since that amounts to saying that such objects don't exist (since they can't exist!). Alice would disagree with Bob with regards to such a property --- she would negate it having such a property outright, by saying that it's not the case that the empty collection contains all objects with property P, since (after all) the existence of such objects is consistent with her view of the world. So if she were to accept that the empty collection had such a property, it would amount to saying that the empty collection is not empty after all. Instead, Alice asserts the negation, i.e. that there are some objects with property P that are not contained by the empty collection.

What does this mean? That Alice and Bob aren't talking about a unique empty collection, but rather two distinct ones. To eliminate confusion they decide to introduce yet another convention whereby they index those distinct empty collections by what makes them distinct --- in this case the distinction is conditioned on who is considering the empty collection, i.e. (e)-A, and (e)-B. But since the differences in the intension of those two terms are in virtue of the reasoning system (logic) adopted by either friend, we can conclude that the intension of "empty collection" is logic relative.

Unlike the act of conditioning the meaning of 'the empty collection' to contexts which deliver contingent distinctions, e.g. what a caveman, a poet, a chemist or a physicist would consider as 'empty', the difference in the meaning of "the empty collection" in the case of Alice and Bob arises out of how such concepts can be reasoned about in principle. As such it is a distinction in the meaning of 'the empty collection' in principle, i.e. in principle that concept has no unique meaning.

Also this is a nontrivial claim, since it's possible (conceivable) that Alice and Bob never met, and so never discovered the discrepancy in the meaning of 'the empty collection'. In other words, it's a nontrivial result since it's conceivable that there could be rational agents confined to a single reasoning system only, thereby not being capable (lacking the necessary epistemic condition, which is the act of abstracting away from the preferred reasoning system) of seeing the fundamental non-uniqueness of the notion of 'the empty collection'.

To sum up: Alice's notion of 'the empty collection' is not the same as Bob's notion of it is. Concisely speaking (e)-A is not identical to (e)-B, which yields the truth of the claim '(e)-LP is not identical to (e)-classical'.

But let's assume per impossible that (e)-LP is identical to (e)-classical; hence x is in (e)-LP iff x is in (e)-classical. Also pick the object a (such that Pa) as an LP-possible object (a is an LP possibilia). This is a fair assumption since not all contradictory properties need be impossible in LP, thus allowing objects with such properties to be legitimate possibilia. Hence a is not in (e)-LP, according to the convention adopted by Alice and Bob. But according to the same convention, all x such that Px are elements of (e)-classical, i.e. all x such that Px are classically impossible (are classical impossibilia). In particular a is in (e)-classical, but that means that a is in (e)-LP given the hypothesis (identity assumption), but we assumed that a is not in (e)-LP, which yields a contradiction.

Therefore, adopting classical logic as the one governing this proof (the metalogic here), and reductio ad absurdum as a valid proof method, it follows that (e)-LP is not identical to (e)-classical, as required.

The key idea of the above discussion can be compressed to saying that although the extension of classically impossible objects can be expressed as the extension of the set (CI) of objects that satisfy some inconsistent property, since those two are necessarily coextensive in classical logic, but the extension of paraconsistently impossible objects cannot be expressed as the extension of CI. Hence, the intension of 'no possible objects' is logic relative (obviously?).

Note: 'element of' and 'belongs to' and 'is in' are terms expressing binary relations that I tentatively use to denote some kind of association of an impossibilia with the empty collection, or the empty set (or empty sets, since I'm arguing that they're needn't be unique across distinct theories). Note also that for each single theory the empty set still remains a unique object/element, as theories are usually based on a single logic. (This assumpton needn't be correct.)

One may rightfully object, and some have, that there's a fundamental problem with asserting that anything is an element of the empty collection. But suppose I claimed that the objects it 'contains' are non-existent, but by doing so were I to wish to retain the meaningfulness of such 'objects' I would be committing to some kind of non-eistic ontology, which posits non-existent objects. But those non-existent objects are objects of some kind after all, and thus cannot be said to be elements of the empty collection, which contains no objects whatsoever. So suppose I reject non-eism, and say that when I say 'a is an element of the empty collection', in no way do I wish to commit to a's existence --- not even as a non-existent object --- i.e. there is no a. But then what is it that I am talking about when I refer to a? I'm not refering to anything apparently, by definition.

I choose the second route, i.e. the one that doesn't commit to an non-eistic ontology. 

(To be continued.)

How empty is { }? Different orders of the Empty Set. Coursera Forum discussion.

Here is a raw cut-and-paste of the exchange on this topic from a Coursera forum.

I'd like to thank those who provided constructive criticism, or just picked at the idea with an unbiased razor of skepticism. Apparently most of the interlocutors were inspired, and some have even admitted to having been led to think in a new way --- to a philosopher, there's no greater reward :)

It's this sort of exchange that allows the idea to mature, and hopefully catch on :)
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Mariusz Popieluch· 2 months ago
Let me introduce the discussion with a little humor.

The Poet, the Chemist, and the Physicist.
The trio met one day over some wine and discussed matters of rhyme, thyme and time. Soon the conversation turned to the notions of nothingness and emptiness - what is empty? The poet finished his glass, and pointed to it -- as far as I am concerned this glass is empty, as it is devoid of wine - the sparkling grape, the drink of gods. Hold on a minute! -- exclaimed the chemist -- surely it's not empty as it contains air - we'd have to pump all the air out of the glass, creating a vacuum in it, and then and only then would it be empty. It wouldn't make any difference to me -- replied the poet, shrugging his shoulders. Please let me interject at this juncture dear fellows -- interjected the physicist -- and let me put an end to your obvious confusion. Vacuum, as you describe it doesn't cut it at all, since we know that even empty space is a breeding ground for virtual particles whose immediate annihilation results in what we observe and call vacuum energy. The poet looked at the physicist with a frown of suspicion -- I know nothing when I see it good man, and I won't let anyone tell me otherwise - let the bartender settle this matter - haloo, good fellow! Another round please!

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The above humorous scenario pertains to natural language and some theories of varying degree of conceptual refinement, but the idea extends nicely (I believe) to formal languages, theories and logics too, thus generalizing the idea of emptiness. In particular it intends to do to the notion of emptiness and it's various formalizations such as the set theoretic ∅, what Cantor has done with the notion of infinity - show that there is structure within it, and varying orders of it.
I know this is a radical idea, but rest assured that the idea of varying orders of infinity wasn't initially taken lightly either - even today many still think it's preposterous. But interestingly enough, there seems to be a demand for such a hierarchy - hinted at in the literature - but I'm not aware of a formal theory of various orders of the empty set being developed.
The general claim is that what counts as empty is dependent on the strength of a logic underlying the theory in which it is defined and the expressibility (richness) of the language of the theory. That is, in some theories there are more objects that fail to be members of the empty set (e.g. to the physicist there is more to the contents of the glass than wine and air), thus rendering the empty set somehow emptier (to put it simply).
Or to put it another way - we could just define the order of the empty set by the cardinality of all the impossible objects in a given language/logic/theory. Those impossible objects are not actually members of the empty set, but rather are assigned to it in some sense - by saying that some object (this includes linguistic objects) is an element of the empty set (in a given theory), is just another way of saying that this object is impossible (in that theory).
Example.
Consider classical logic CL and some paraconsistent logic, say LP. In LP contradictions are not considered as impossible. Now define a property G: of being green all over, and not being green all over. Now, in CL any object with that property is considered as an impossible object, take the object a, so in CL we obviously have Ga ∈∅, but in LP we have Ga ∉∅. Hence the suggestion to index ∅ by logics - this may become crucial when talking of various theories in some metalanguage/meta theory.

Philosophically speaking "{ }" is an object with a fixed extension across theories (it has no elements), yet varying intension. What the above aims to show is that the varying intension can be accompanied with a varying quasi-extension, i.e. rarity as opposed to cardinality.

Mariusz Popieluch· 2 months ago

Consider two urns in a marble shop. The shopkeeper labelled them in the following way - urn 1 holds only (and can only hold) white marbles, whereas urn 2 can hold both white and black marbles.

Now consider a new assistant that has just been offered an apprenticeship in the shop of his dreams (he loves playing marbles), and is not as yet familiar with the arcana of the labeling system, i.e. he hasn't as yet been told by the shopkeeper what the restrictions on the contents of the urns are.

Now consider the statements 'urn 1 contains no white marbles' and 'urn 2 contains no white marbles' - they entail different things, depending who is exposed to that information - the shopkeeper or the assistant. In particular, to the shopkeeper it means that 'urn 1 is empty'.

On a more technical side, we can substitute the terms 'has no white marbles' and 'is empty' interchangeably salva veritate in the context of urn 1 (charity permitted, for naturally one could generate oddly sounding sentences). This cannot however be done in the context of urn 2.

David M. Kaziska· 2 months ago
In your first post the set, {wine in the glass}, is empty after the wine is consumed (neglecting any residual wine).  For the poet's own purposes it is empty but we may his or her statement that it is empty as colloquial in light of the chemist's and physicist's later statements.  I'm not seeing mathematical consequences which would require redefining the notion empty sets.  {wine in the glass} may be an empty set but {air in the glass} may be nonempty.  

In your later post, the shopkeeper reasons as follows.  (P1)  All marbles in Urn 1 are white, (P2)  There are no white marbles in Urn 1, therefore (C)  Urn 1 is empty.  The assistant has (P2) but not (P1) and cannot draw the conclusion.  Urn 1 is still empty, though, it's not a matter which is relative to the observer.

Mariusz Popieluch· 2 months ago
Thank you for your reply David. I'll do my best to address your observations, and further clarify the idea. :)

The later post intends to illustrate the idea that in some theory T based on some logic L1 some formula F may express an impossibility (which we can express as that formula being "an element" of, or assigned to, the empty set), whereas the exactly same formula in an analogue to T based on a logic L2 which is stronger than L1 may not come out as impossible, and as such cannot be expressed as "belonging"/being assigned to the empty set.

Note: "element of" and "belonging" are terms I tentatively use to denote some kind of assignment of F to the empty set (or empty sets, since I'm arguing that they're needn't be unique across analogous theories based on distinct logics). Note also that for each single theory the empty set still remains a unique object/element, as theories are usually based on a single logic.

The epistemic states of the protagonists of the marble shop illustration (informal context) correspond to the assumptions/axioms/conditions/semantics of theories based on distinct logics in the formal context.

Some logics distinguish semantically the propositions "p & ~p" and "p & q". In particular they consider the first proposition to take a fixed value for all valuations (contradictions are always false, and only false in classical logic). Some other logics do not make that distinction.

Likewise the shopkeeper distinguishes the two propositions concerning the urn contents, whereas the assistant doesn't. Try to think of the urn labelling system and the protagonists' distinct epistemic states as an analogy for the distinct meta-linguistic propoerties of analogous theories based on distinct logics, and the propositions concerning the urn content as object language entities, i.e. formulae.

But if the informal analogies muddle, rather than clarify the idea, I'll be happy to talk more formally about it. :)


Louise Craven· 2 months ago
Every set is a subset of the relevant domain of discourse. Thus if we are only talking of natural numbers, the empty set is the set containing no natural numbers. The poet was thinking of drinks when he said his glass was empty, so it truly was. The chemist and physicist tried to confuse him by exploiting the fact that he had not explicitly stated the domain of discourse, and they were also right about the nature of the empty set within their individual domains of discourse.  In the marble example, the difference between the shopkeeper and the assistant is not in the 'size' of the empty set, but in the inferences they can draw from the fact of emptiness, which depends on other information.

I am, however, intrigued by your suggestion and it may be that in some applications of mathematics to real-world problems, the size of the empty set in the model may need to vary according to the significance of 'nothing' in the area modelled. For instance, complete extinction of a species is of much greater significance than the remaining existence of a number of breeding pairs much smaller than the population at a previous time.

I shall go now and give it more thought....

Mariusz Popieluch· 2 months ago
Hi Louise - thank you for taking the time to read the OP.

The informal anecdotes are mere illustrations of the general idea.

In the first scenario I intended to show, via the relevant domains of discourse illustration,that given the same language (in this case natural language), but distinct theories, identical propositions (formulae, i.e. linguistic objects) entail different things - in particular, where in one theory "no wine in glass" does entail "glass is empty", in other theories it doesn't. I stress again, that this is merely an illustration of meta theoretic properties, rather than a discussion concerning restricted quantification to relevant domains of discourse per se.

Think of the Poet, the Chemist and the Physicist as analogues of theories of "The stuff on the table, at the local restaurant.", albeit based on dictinct logics. In the poet's logic "there's no wine in the glass" can be associated with the empty set - this however isn't true in the chemist's or physicist's logics. Likewise in the chemist's logic "there is no liquid and no gas in the glass" can be associated with the empty set - but again, this isn't true on the physicist's logic.

As for the second scenario, please see my above reply to David. And please do not shy away from addressing the formal content of the OP.

Andrew Kelley· 2 months ago
First let me say that I enjoyed reading this post. It made me think in a way I hadn't thought before.

I think you may be on to something with recognizing different orders of the empty set. However I am as of yet unable to think of any examples where the knowledge of such different orders is relevant to solving a problem. Are you?

Mariusz Popieluch· 2 months ago
It may serve as the ontological foundation for a theory of comparative impossibility.

Francisco Vasconcelos· 2 months ago
I think that the analogy between empty set and infinite set is not the most correct one. The evident oppositions would be between empty and complete set, and infinite and infinitesimal amounts.

So on the empty/complete side, I'm not aware that there are different orders of completeness. Both these terms are usually used as binary categories applied to other sets, both finite and infinite. Yes, their meaning varies with language context, but so do everything else, such as the number "2", the operator "+", or the word "number".

On the other hand, I think it's easier to make the claim that there are different orders of infinitesimal quantities. For example, think about the probability of picking the number "5" at random from the set of all natural numbers. Since the set is infinite, the probability is 0. However, it seems that this "0" is still bigger than the "0" probability of random sampling the number "5" from the set of all real numbers, since the pooling set is infinite to a higher degree than the previous one.

Andrew Kelley· 2 months ago
I'm not buying that you can say that one 0 is bigger than another 0. Zero is zero. Can you prove it?

Mariusz Popieluch· 2 months ago
Francisco, thank you for your reply -- you're right that it's not the most correct analogy, as it's not intended to be a directanalogy, i.e. a dual of sorts. It's a weak kind of analogy, in the sense that in both cases of infinity and nothingness/emptiness we're dealing with concept which apparently don't admit to degrees. Whereas Cantor developed the idea of bijection as the criterion for equinumerosity, I'm using it in developing the notion of the rarity of some empty set, which is currently tentatively defined to be the cardinality of the set of all formulae which express an impossibility (aka impossibilia) in a given logic L, thus yielding an indexed empty set ∅_L.

As I said before, in the above response to Louise, the contexts of relevant discourse analogy is again merely an illustrationof the more precise idea. See the above reply.

The observation you expressed in your last paragraph puts you in good company. In the context of probability theory the idea of different orders of “zero” has been hinted at by Andrey Kolmogorov and Bruno de Finetti as a possible candidate to solving some probability theory paradoxes – “Like Kolmogorov, de Finetti is occupied mostly with probabilities defined directly on arbitrary uncountable sets; but he views additivity differently, and is led to such anomalies as an unlimited sequence of layers, like an onion, or different orders of zero probabilities that add up to one, etc. ” (E.T. James: Probability Theory, the Logic of Science, 2011, p.656).

Francisco Vasconcelos· 2 months ago
Andrew, try not to read my "0" as the integer number 0, but as an infinitesimal quantity that is infinitely close to zero, something like the result of limx→∞1x. As infinity can have different sizes, the result of this limit quotient should be able to have different sizes too.

Mariusz, thanks for the info

" different orders of zero probabilities that add up to one"

this is definitely interesting and worth checking out.

Regarding your emptiness orders:

"I'm using it in developing the notion of the rarity of some empty set, which is currently tentatively defined to be the cardinality of the set of all formulae which express an impossibility (aka impossibilia) in a given logic L, thus yielding an indexed empty set ∅_L."

I get it now. So I guess you're trying to say that in different theories the empty set can be connected to a set of propositions that can have a different finite or infinite size.
Another question: hypothetically speaking, do you think that within a single theory L (maybe with number references), it is possible to represent empty sets with different sizes in this sense?

Andrew Kelley· 2 months ago
We can mathematically prove that infinity can have different sizes by using bijections. But we cannot mathematically prove that limit of 1/x as x approaches infinity is not equal to the integer zero. In fact, we *can* prove that, for example, 0.999999 repeating is exactly equal to 1. Not a number which has an infinitesimally small difference than 1. 1.

Francisco Vasconcelos· 2 months ago
Andrew I'm still trying to figure out what different kind of "0" can mean, it seems that this concept raises some issues in probability theory, but lets forget that for a moment and define the following:

For any functions f(x) and g(y), assume that
limx→∞f(x)=0
limy→∞g(y)=0

Now suppose the following hypothesis:
For any arbitrary ϵ, there is NOT a bijection between all possible values of g(y) and f(x), for x,y≥ϵ

If this proposition is true, then the limit of g(y) and f(x) being "0" might have different meanings, since we know that for any given ϵ, one of the sets will always be larger than the other.


Mariusz Popieluch· 2 months ago
@Francisco: "Another question: hypothetically speaking, do you think that within a single theory L (maybe with number references), it is possible to represent empty sets with different sizes in this sense?"

Well, personally I'm not entirely on-board with that idea, for the reasons I stated in my reply to David above: "Note also that for each single theory, the empty set still remains a unique object/element, as theories are usually based on a single logic."

My position is such, due to what I mean by "different orders of emptiness". But this is not to say that I'm not open to the variation of this idea, which you and the company of famous thinkers suggest. :)

Andrew Kelley· 2 months ago
Francisco, thank you for this example. I think you may be right but I am finding it extremely hard to wrap my brain around it.

Hayden VanIderstine· 2 months ago
Francisco, in order for the hypothesis that there exists no bijection between all possible values of g(y) and f(x) when x,y>ϵ, then it is necessary that the cardinality of the set of all possible g(y) with y>ϵ be different from the set of all possible f(x) with x>ϵ. For this to be the case, then both sets cannot both be of any of the following cardinalities:

Finite
Countably infinite (set of naturals)
Uncountably infinite 1 (set of reals)
Uncountable infinite 2 (set of all functions defined from the set of reals to the set of reals)

What I am wondering, is what is the domain and range of f and g, because if they both have a range being a subset of the real numbers (or the entire set of real numbers), then there necessarily exists a bijection between them.

Francisco Vasconcelos· 2 months ago
Hayden,

I'm not sure I follow you. The way I see it, for there to be no bijection it is only required that the domain of f(x),x>ϵ has a different cardinality of g(y),y>ϵ. So, for example if the domain of f(x) is a subset of the rational numbers, and the domain of g(y) is an interval of the real numbers, then there is no bijection.


Hayden VanIderstine· 2 months ago
In my opinion, excellent post Mariusz. I am thinking in a new way now thanks to you.

What we consider as empty is dependent on what we acknowledge exists.

{n∈N | 1 < n < 2} =  ∅
Exactly  |N| elements fail to be in the set {n∈N | 1 < n < 2}

But exactly |R| elements fail to be in the set {x∈R | 1 < x < 2}.

According to your definition of the rarity of some empty set, would the rarity of the empty set in a logic which isn't strong enough to construct the reals, but is strong enough to construct the naturals, be |N|, because there are only a countably infinite number of sentences like 1 < n < 2 which could be made which are impossible?

(btw, I'm not well versed in formal logic, nor am I sure that there exists a logic which is strong enough for the naturals, but not enough for the reals).

Mariusz Popieluch· 2 months ago
Hayden - thank you for your insightful reply and interpretation of the ideas in this thread. Your observations, and question also made me think more carefully about what I'm proposing.

The natural numbers, or Peano arithmetic, is a first order theory, whereas the it is not possible to characterize the reals with first-order logic alone since the supremum axiom of the reals quantifies over subsets of the naturals, and is therefore a second-order logical statement. See the list of axioms here: https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach

The stronger logic in which the axioms of R are expressed is second order logic, which is stronger than first order. So to answer your question directly - first order logic is the logic "which is strong enough for the naturals, but not enough for the reals", for the reasons given above.

(Note on the terminology: Logic B is stronger than logic A iff all theorems of A are B therorems, and there exists some B theorem that's not an A theorem. Or equivalently B is a stronger logic than A iff the set of A theorems is a proper subset of B theorems.)

Mariusz Popieluch· 2 months ago
Also, to clarify - "What we consider as empty is dependent on what we acknowledge exists." is not entirely correct.

Rather "What we consider as empty is dependent on what we acknowledge can exist.".

Pedro Forquesato· 2 months ago
Hello Mariusz,

Nice post! It brought a good discussion, and that is the purpose of this forum. I think your stories are good examples of the importance of clear and rigorous (for example mathematical) sentences in philosophy (and other studies), and the danger of "spoke communication".

Let's try to solve it by translating to "mathematical language" what they are saying. In the case of the urns, for example, when the apprentice says "the urn is empty", he means (lets define U as urn and W as the set of white marbles):
U = { }
While when the master says "the urn is empty", rigorously he means:
U intersection W = { }

So while in English what they say is the same, actually they are saying different logical propositions, and thus it is not paradoxical that the second might be true while the first is not. It is not the empty sets that are different, it is the translation from English to logic that differs. (Naturally the same argument is valid for the poet's glass).

Mariusz Popieluch· 2 months ago
Thank you for your reply Pedro - if you're interested in a mathematical treatment of the idea, don't look at the intuitive illustrations of the marble shop, and the trio drinking wine. Instead focus on the content of the OP that follows the phrase "The general claim is...". :)

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To be continued...