>The present king of France is bald
What did he mean by this?
>So we do not have a failure of the Law of Excluded Middle: "the present King of France is bald" (i.e. ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]) is false, because there is no present King of France. The negation of this statement is the one in which 'not' takes wide scope: ~∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]. This statement is true because there does not exist anything which is currently King of France.
When did logic become so obtuse?
>>645193
>The present king of France is bald
I know fuck all about the history of france or philosophy but here's my guess: the real leader of france doesn't wear a crown
>>645202
How is that obtuse?
>>645209
Looking at non-numerical equations makes my eyes gloss over, dulls my interest in learning, and chips away at my intellectual security.
Discussing logic with numbers just rubs me the wrong way. Yes, I might be an idiot.
>>645240
>numbers
boolean algebra. it was on the tip of my tongue. remembered the term after posting.
>>645240
>on-numerical equations
Those are identity statements.
>dulls my interest in learning,
That's too bad because this stuff is actually pretty interesting once you stop being actively offended by the fact that someone else understands it and you don't.
>and chips away at my intellectual security.
Isn't that a good thing? I don't know what intellectual security is supposed to be but I'm pretty sure Socrates BTFO anyone pretending it exists.
>>645202
>∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]
>∀y[PKoF(y) → y=x]
What does this mean?
>>645193
>There is, in fact, an element of sour grapes in Stoicism. We can't be happy, but we can be good; let us therefore pretend that, so long as we are good, it doesn't matter being unhappy. This doctrine is heroic, and, in a bad world, useful; but it is neither quite true nor, in a fundamental sense, quite sincere.
The natural rebuttal to Russell here is that he has misunderstood the Stoic understanding of Happiness. In choosing their actions and goods in a principle of "Rational decision in accordance with nature", Stoics do not deny what would make them happy. Happiness for the Stoics just is making that choice willfully.
Perhaps Russell might be right were he to say that rational decisions in accordance with nature may not give as much scope for pleasure, in a sense in which pleasure extends to things like delight and lust. But for the Stoic, that's not a real kind of happiness. You can still as a Stoic enjoy, for example, food, substances and pursuits in accordance with your own personal nature, without being dominated by the passions that might drive us to pursue these things at the expense of other aspects of our natural selves. Giving in to temptation, while maybe fun, doesn't ultimately yield true happiness, which the Stoics would say is reached through rational choice rather than impulse.
>>645276
>There is some X such that X is the present King of France, and for every Y if Y is the present King of France then Y = X, and X is bald
>For every Y, if Y is the present King of France, then Y = X
>>645257
>Isn't that a good thing?
It is, but I don't feel drawn to mending my aversions and at no point do I point fingers at anything but myself for this.
>offended
I'm not offended. I'm genuinely glad that there are people to whom this sort of thinking is compelling.
>intellectual security
I'm studying in a university I'm only attending through a stroke of luck and currently succeeding purely due to regular memorization, meticulous note taking, and creating connections with better students.
I'm afraid that if I put aside my usual methods and attempt to truly learn, not memorize, I'll find that I'm a lucky fool who simply pretends for everyone around me. An impostor. Can I accept this? Sure. How others would react frightens me.
The basis for Russel, and most of the analytic philosophy is that language works like computer programs: statements about things are all boolian and they have fixed unchanging definations.
Wittgenstein showed us that this isn't is the case with his PI. So essentially Russels entire career was a waste.
>>645300
Oh I get it. Usually x and y are used to describe elements from different sets X and Y but here they are not.
>>645329
Yes, it's an identity statement. Just as the (PKoF)x is a predication statement.
>>645207
god you're dumb
>>645360
He's not wrong, there are no Jewish kings
>>645307
Then why is Wittgenstein a central figure in analytic philosophy and why do so many analytic philosophers contest not only his views but also the countless interpretations of them that exist?
>>645202
That logic notation is fucking disgusting
>>645301
Sounds like you a straight BITCH lmao
But no really imposter syndrome is pretty common especially in academia.
Infertile discussions.
>>645992
What do you have against predicate logic?
>>645202
ITT continentals get confused by rigorous symbolic logic and vacuous truths
>tfw Socrates runs
>tfw the empty set is both open and closed
>tfw they can't grasp this
>>646032
Propositional logic is superior
>>646055
That's a pretty odd opinion to bandy about. Why do you think that?
>>645301
>regular memorization, meticulous note taking, and creating connections with better students.
Sounds like academia is for you
That isn't meant as a compliment
>>646061
It's easier to read
>>646075
That's a pretty bad reason to hold that opinion.
>>646050
Empty set doesn't exist in the real world. There's nothing to grasp.
>>646071
If that's how philosophy academia is like then no wonder no one takes them seriously
>>646077
Easier to read = easier to learn
>>646078
>doesn't know the natural numbers that we count with is built up of successive nestings of empty sets
>tfw they think counting doesn't exist
>tfw they think the existence of things somehow had any bearing on how truths are deduced
>>646094
I'm not even a philosopher or an expert in the area, but it seems obvious that natural numbers are direct abstractions from reality, there's nothing empty about them.
>>646084
I'm finishing up a philosophy major at an analytic university, the levels of autism some of the professors here have are impressive.
>>646090
Again, pretty bad reason to hold that opinion. Easy =/= good. Easier =/= better.
Have you actually tried to learn predicate logic or do you just get confused by the notation every time?
>>645295
You seem to be under the misapprehension that this is not a troll thread.
>>645202
Makes perfect sense if you've taken higher level math
>>646107
>but it seems obvious that natural numbers are direct abstractions from reality
If that's so obvious why is the other guy contesting it?
>>646107
I suggest you search up the construction of the naturals from ZF before you post anything else and embarrass yourself.
>>646109
I'm saying that if they both accomplish the same thing, and propositional logic is easier to learn, then propositional is superior
I'm not a PHIL major, so I've only had the opportunity to learn logic through an elective, which only covered propositional logic
>>646124
>not actually willing to learn by himself
I'll spoonfeed you:
{} is the empty set
1 = |{{}}|,
2 = |{{}, {{}}}|,
3 = |{{}, {{}}, {{{}}}}|,
Etc.
Existence of an empty set and the fact that we can do this indefinitely is guaranteed by axioms of ZF. If you find the natural to be so natural you must also find axioms of ZF natural, since it is the so far the least set of axioms that are able to develop Peano arithmetic.
>>646140
>Not quite true.
Please explain. I thought the main difference was the symbols
>>646134
Different person to who you talking to, but propositional logic most certainly does NOT accomplish all the same things that predicate logic does. Predicate logic is much more powerful than propositional logic, and everything that can be done with propositional logic can also be done with predicate logic. The only reason PL is even still taught is for pedagogical reasons: it makes learning other logics easier.
>>646140
> don't think you should post in philosophy threads anymore.
Socrates would agree with me, desu.
>>646141
>not actually willing to learn by himself
You are contesting what I said, so you must be able to show me you understand exactly what you are saying and not be a messenger boy.
You still didn't explain to me how emptyness can exist in the real world.
>>646156
No, I was teyong to guide you to seek out knowledge for yourself instead of staying in the echo chamber that is your own head.
So you're saying something (the naturals) can come from nothing (the empty set)?
Christians btfo I guess
>>646144
They're not contradictory, to begin with. Propositional logic can incorporate predicate logic and predicate logic can incoproprate propositional logic.
Propositional logic deals with propositions and the relations between them (P, P->Q, : . Q)
Predicate logic deals with logical objects and the properties they have (∃x[PKoF(x), there is some X such that X is the present king of France)
>>646156
>Socrates would agree with me, desu.
You'll have to prove this, now.
>>646166
>teyong
Kek. Trying**
>>646165
No problem. Given that you've already learnt propositional logic, you should really look into quantificational predicate logic. You could learn it in a few days effort and it's much more interesting and insightful than PL. If you're interested, I recommend the introductory textbook Tomassi, Paul, "Logic". It's exceptionally easy to follow on your own.
>>646166
If you don't know what knowledge is, how can you teach me the way? You're the only one running away from your own conclusions.
I said the opposite, I said natural numbers are symbols of something blatantly real, which is quantity.
>>646171
Socrates usually agrees when someone says the truth without any shady interest.
>>646195
>Socrates usually agrees when someone says the truth without any shady interest.
Socrates is best known for making anybody who claims to know the truth look like a fool. Whether or not your interests are shady, and whether or not Socrates' opinion matters, philosophy isn't really a place to just say "Oh, it's obvious that X is true" and expect everyone to go along with you.
>>646195
Holy fuck you can't be this stupid.
I've shown that the naturals can be constructed from sets of empty sets. You posit that the natural numbers exist while the empty set doesn't, implying that something real can come from (be built up of) something not real.
Do you understand now? Is this how much I can expect from /his/? And people still insists that this isn't just another shitposting board.
>>646289
>hurr whatever I said is teue because I assume that it's true
It doesn't matter, I already constructed them in the previous post regardless of whether they're real or not.
>can't grasp abstract concepts
I'm done. You're actually this stupid. I'm sorry I insulted you.
>>646302
If you don't believe in truth and can't recognize it when you see, then it's not worth it to talk with you. Sophist.
>>646323
>intuitionist
>whatever is obviousis true
This is why no one takes you seriously. Absolutely payhetic.
>>646344
Jesus Christ fucking phone
>obvious is*
>pathetic*
>>646344
Kek, you will never achieve truth with you negative power adn sarcasm. Why can't you be courageous at least once in your life and begin to believe that you can perceive at least small truths instead of "yuo cant know anythng".
>>646374
That's literally irrelevant to my point. The fact that you still think the view "empty sets don't real while naturals do" isn't absolutely absurd is what makes you pathetic.
>he thinks he somehow has say over what is true or not even after he's shown to be embarrassingly wrong
Literally anti-intellectualism.
>>646406
Your point is irrelevant, and you're not an intellectual.
I said that emptyness doesn't exist in reality, so empty set is only an abstraction that expresses a relation between two concrete things, not a thing in itself. These "concrete" things can be expressed by natural numbers. So empty set and real sets are two completely different things.
>>646443
Except that the naturals are made of empty sets you fucktard holy shit.
Are you selectively blind? For fuck's sake stop pisting anytime and kill yourself.
>>646457
Natural numbers are either a language or a concrete thing, it's not "made of empty sets".
>>646464
>literally ignoring >>646141
You can just admit that you don't understand it you know.
The point is that they can be, implying that real things can be constructed from unreal (according to you) things.
This is your stance. There's no argument about it. Now the argument is about how stupid the stance is, which apparently you don't know why.
>>646477
I ignore it because saying "this is an ampty set= {}" doesn't mean anything, he literally said:
This is 0=0
1 = 0
2 = 00
3 = 000
You're not explaining anything, only wraping your mind in a tautological universe.
>>646488
So you're literally retarded. Must be tough to be as stupid as the dog turd on the ground.
Hope you have an easy life.
>>646506
>you're ugly
Now I see that the only thing that matters to you is pretend to be smart, maybe that's why you love all this pseudo-philosophy stuff.
>>646520
>doesn't underatand what sets are
>interprets notation in his own absolutely moronic way even though they're rigorous mathematical objects not open to interpretation
>dismisses what he doesn't underatand
Literally anti-intellectualism. I don't need to prove anything to you; you've already done that plenty.
>>646536
I'm pretty sure the one who doesn't understands nothing is you, animufag, you showed to be a fraud who can't even walk with your own legs without your false authority appeals and "you don't understand my language, haha, little ugly kid".
You don't even know what intellectualism means, stop embarassing intellectuals.
I think he was referring to Charles the Bald, grandson of Charlemagne
>>646551
So you're still not going to look up what a set or an empty set is? Should I spoonfeed you again?
How pathetic is it that you're not only ignorant but also arrogant
>>646289
>
You're completely unprepared. You are claiming the truth in the sense Socrates would be against, I wasn't.
I don't understand what you mean. Are you actually claiming that Socrates would adore you, and that because Socrates would adoore you you're correct?
>Analytical "Philosophy"
This thread hurts my brain.
>>645307
>boolian
>definations
>this severe misunderstanding of both Russell and Wittgenstein's works
Classic bait.
>>645193
There is no consensus among philosophers about many of the central problems concerned with the philosophy of science, including whether science can reveal the truth about unobservable things and whether scientific reasoning can be justified at all. In addition to these general questions about science as a whole, philosophers of science consider problems that apply to particular sciences (such as biology or physics). Some philosophers of science also use contemporary results in science to reach conclusions about philosophy itself.
While philosophical thought pertaining to science dates back at least to the time of Aristotle, philosophy of science emerged as a distinct discipline only in the middle of the 20th century in the wake of the logical positivism movement, which aimed to formulate criteria for ensuring all philosophical statements' meaningfulness and objectively assessing them. Thomas Kuhn's landmark 1962 book The Structure of Scientific Revolutions was also formative, challenging the view of scientific progress as steady, cumulative acquisition of knowledge based on a fixed method of systematic experimentation and instead arguing that any progress is relative to a "paradigm," the set of questions, concepts, and practices that define a scientific discipline in a particular historial period.[1]
>>648512
Subsequently, the coherentist approach to science, in which a theory is validated if it makes sense of observations as part of a coherent whole, became prominent due to W. V. Quine and others. Some thinkers such as Stephen Jay Gould seek to ground science in axiomatic assumptions, such as the uniformity of nature. A vocal minority of philosophers, and Paul Feyerabend (1924–1994) in particular, argue that there is no such thing as the "scientific method", so all approaches to science should be allowed, including explicitly supernatural ones. Another approach to thinking about science involves studying how knowledge is created from a sociological perspective, an approach represented by scholars like David Bloor and Barry Barnes. Finally, a tradition in continental philosophy approaches science from the perspective of a rigorous analysis of human experience.
Philosophies of the particular sciences range from questions about the nature of time raised by Einstein's general relativity, to the implications of economics for public policy. A central theme is whether one scientific discipline can be reduced to the terms of another. That is, can chemistry be reduced to physics, or can sociology be reduced to individual psychology? The general questions of philosophy of science also arise with greater specificity in some particular sciences. For instance, the question of the validity of scientific reasoning is seen in a different guise in the foundations of statistics. The question of what counts as science and what should be excluded arises as a life-or-death matter in the philosophy of medicine. Additionally, the philosophies of biology, of psychology, and of the social sciences explore whether the scientific studies of human nature can achieve objectivity or are inevitably shaped by values and by social relations.
>>648514
Good overviews of the more recent history of analytic philosophy are Burge's Philosophy of Language and Mind: 1950-2000 and Philosophy of Mind: 1950-1990 (ch.20), the philosophy of science side in a very lively and polemical form is described in Zammito's Nice Derangement of Epistemes, "the best history of post-positivist philosophy and sociology of science we are likely ever to get", according to Giere's review.
That the linguistic turn has exhausted itself is generally shared. Burge writes: "The philosophy of language became a vibrant, semi-autonomous discipline in the 1960s and early 1970s. In fact, it was considered by many to be the new "first philosophy... But by the late 1970s or early 1980s philosophy of language no longer seemed the obvious propaedeutic for dealing with central philosophical problems. As I have intimated, one ground for this shift was that many philosophers felt that philosophy of language had done its job that the natural development of philosophical reasoning led into the philosophy of mind, or other adjacent areas". From publisher's abstract:"Zammito shows how problems that Quine and Kuhn saw in the philosophy of the natural sciences inspired a turn to the philosophy of language for resolution. This linguistic turn led to claims that science needs to be situated in both historical and social contexts, but the claims of recent "science studies" only deepened the philosophical quandary".
Philosophies of mind and science however remain vibrant. Sure, we do not live in the time of Kant and Hegel, but Unger's charge of "emptiness" targets Lewis, Putnam and Kripke. What about Quine, Kuhn, Davidson and Dummett?
The present king of France cannot be bald because there is no present king of France. You can't say he's bald or has hair because he doesn't exist.
Why do we need the ∃x[PKoF(x) & ∀y[PKoF(y) → y=x] & B(x)]) horseshit to say that?
>>648520
Russell claims yet another clueless victim.
>>646745
>So you're still not going to look up what a set or an empty set is? Should I spoonfeed you again?
he thinks the empty set exists
>>648520
>>645193
The last king of France died a long time ago. He's just a skellington now. Skulls have no hair.
Philosophers in the continental philosophical tradition are not traditionally categorized as philosophers of science. However, they have much to say about science, some of which has anticipated themes in the analytical tradition. For example, Nietzsche advanced the thesis in his "The Genealogy of Morals" that the motive for search of truth in sciences is a kind of ascetic ideal.[54]
Hegel with his Berlin students
Sketch by Franz Kugler
In general, science in continental philosophy is viewed from a world-historical perspective. One of the first philosophers who supported this view was Georg Wilhelm Friedrich Hegel. Philosophers such as Pierre Duhem and Gaston Bachelard also wrote their works with this world-historical approach to science, predating Kuhn by a generation or more. All of these approaches involve a historical and sociological turn to science, with a priority on lived experience (a kind of Husserlian "life-world"), rather than a progress-based or anti-historical approach as done in the analytic tradition. This emphasis can be traced through Edmund Husserl's phenomenology, the late works of Merleau-Ponty (Nature: Course Notes from the Collège de France, 1956–1960), and Martin Heidegger's hermeneutics.[55]
The largest effect on the continental tradition with respect to science was Martin Heidegger's critique of the theoretical attitude in general which of course includes the scientific attitude.[56] For this reason the continental tradition has remained much more skeptical of the importance of science in human life and philosophical inquiry. Nonetheless, there have been a number of important works: especially a Kuhnian precursor, Alexandre Koyré. Another important development was that of Foucault's analysis of the historical and scientific thought in The Order of Things and his study of power and corruption within the "science" of madness. Post-Heideggerian authors contributing to the continental philosophy of science in the second half of the 20th century include Jürgen Habermas (e.g., "Truth and Justification", 1998), Carl Friedrich von Weizsäcker ("The Unity of Nature", 1980), and Wolfgang Stegmüller ("Probleme und Resultate der Wissenschafttheorie und Analytischen Philosophie", 1973–1986).
>empty sets don't exist
This is the level of some people here
>>648540
>>648540
>Englishman
stopped reading right there tbqh
>2016
>falling for the logics and mathematics troll
There is no empirical evidence for the existence of any number, there is no empirical evidence for the existence of any word so fuck off with your le logical philosopher crap.
>>649383
LOL
>>649413
>the present king of France is bald
Anything can be a king
Anything can be bald
That sentence, and any sentence you can possibly come up with that is semantically correct, can and is both true and not true at the same time, because it really doesn't matter, it's just a sentence.
>the present king of France is bald
Can be untrue since there is no king in France. Can be true if the "king of france" metaphorically represents some famous bald dude. You can argue literally anything. The present king of France could arguably be Steve Jobs
>Steve Jobs is the dude from apple
>Apple is for faggots
>Frenchmen are faggots
Also, words don't have meaning in themselves, and don't objectively exist, get over it
>>649444
>and don't objectively exist
Are you literally telling me that there aren't symbols qua intentional objects of consciousness?
>>649451
There are symbols but every symbol needs to point at something that actually exists in order to be useful. Yeah, sure, somewhere inside your brain there are a couple of neurons in charge of the word "tree" and another neurons whose job is to make you a picture of a nice evergreen so when you hear the word you know what it means.
But "tree" as in a platonically perfect thing, an actual objective thing, it doesn't exist. You have a representation of a tree in your brain but everyone has and everyone has a different one. And real "tree", the thing the symbol points at, does exist, but it is not the same as your representation. "Tree" is a category where you put your evergreens, but also your apple trees and oaks and everything that is minimally similar to the trunk+leaves configuration. In order for you to actually talk of real things you'd need to narrow down your categories. Eventually you'd get to the point of being absolutely specific and saying that, of all apple trees of this height and this thickness and this many leaves, you are talking of the one that is located in this and this coordinates. And even then you would not have an objective word, you'd have a fuckload of symbols that point to something objective and real.
Words are categories to explain shit that exists, but words don't exist, they are just symbols and they can literally mean whatever you like them to
>>649490
>But "tree" as in a platonically perfect thing, an actual objective thing, it doesn't exist.
Where in this thread has someone said otherwise, though? It seems like you're just making pseudo-Wittgensteinian assertions directed at strawmen. What you're saying doesn't even seem like an objection to any of Russell's actual (more nuanced than you understand) positions on the nature of symbols. This is such basic stuff that you should be embarrassed to put it out there as if it proves anyone wrong or as if this is a particularly insightful formulation of what has been called the sense/reference distinction.
>>649505
No, I'm using that to say that words can pretty much mean whatever you want them to mean, so whilst logic can be used as a model, it's a fake. Anything can and is true and lies at the same time. Argumentation at the end of the day is pretty much about who can convince the best, who is more likeable and who is more contageous personally or through text.
Mathematics work somewhat similar. It's all a fake. Taking it as anything other than a tool is stupid
>>649528
>It seems like you're just making pseudo-Wittgensteinian assertions directed at strawmen.
>Anything can and is true and lies at the same time
That's basically what Russell is trying to say in >>645202, though. I still don't understand what your actual issue is. Nobody ITT has made the claims you're arguing against.
>>648520
The problem such sentences pose isn't that they can't be formulated in logic. That wasn't the problem that Russell was attempting to solve. The problem is that such sentences appear to violate the law of the excluded middle, which seems deeply counterintuitive. To say that such a sentence is FALSE results in its negation being true. But the natural interpretation of the negated sentence is that the present King of France has hair, which is also false (which results in its negation being true, and leading to an infinite regress).
Frege, who Russell is responding to in On Denoting where this discussion takes place, proposed an even weirder solution: that sentences which refer to things which don't exist, such as the present King of France, are in fact NEITHER true nor false, and therefore don't threaten the law of the excluded middle.
Russell believed this to be a faulty solution to the problem, as such sentences appear identical in form to other true conducive sentences. Instead, Russell's solution is to reformulate the logical representation of proper names in quantified predicate logic as one which is not strictly referential, and instead merely conceives of proper names are disguised definite descriptions, I.e descriptions which uniquely pick out one object. Then, via scope negation, we are able to merely negate the wide scope quantification of the new logical sequent rather than negate it in a manner which results in such sentences creating an infinite regress.
This has massive ramifications throughout philosophy in ways which most people are deeply surprised, and is far from trivial word play.
>>646977
what should I study in order to understand this thread?
>>649543
>This has massive ramifications throughout philosophy in ways which most people are deeply surprised, and is far from trivial word play.
This, I find it amusing that the same Continental loyalists who talk about the great advances in Continental semiotics are so quick to dismiss this kind of thing just because it makes their heads hurt or seems trivial.
>>649538
Wow sorry didn't know that. But why would he even waste time doing all the logic sentences and the whatnot if he knows that is just wordplay and literally play
>>649568
>if he knows that is just wordplay and literally play
Because he values wordplay more than you, as someone who doesn't yet understand that philosophy is pure verbiage, do.
>>646050
Nonempty sets can be open and closed in topological spaces. It's equivalent to the space being disconnected.
>>646745
Arrogance and preciousness of Analytic Philosophy on display here. If you truly understand this then you should have no problem explaining this to a lay person without prior knowledge.
>>649608
>I'll just keep repeating the same things instead of reading something new so that I won't find something that conflicts with my stupid world view
>babu babu feed me momma
>what is Google
https://en.m.wikipedia.org/wiki/Set_(mathematics)
https://en.m.wikipedia.org/wiki/Empty_set
>>649591
Half open neighborhoods are neither closed nor open, champ.
>not a union of open sets
>does not contain all of its limit points
The only sets in a topology that can be both open and close are the empty set and the whole space, and the latter doesn't need vacuous truths to prove that it's clopen.
Good try, little guy. The best attempt in this thread yet.
>>649608
You know, you can explain even the most abstract parts of Hegel, Nietzche, and Plato without self-referncing.
Analytics can't even communicate basic sentences about the hair of the king of france without circle-jerking themself.
Proving once and for all that continental is where all the serious discussion happens and analytic is for obscurist sophists.
>>650621
>Analytics can't even communicate basic sentences about the hair of the king of france without circle-jerking themself.
I think you missed the point
>>645193
It's really not that hard guys. He's talking about Obama, i.e. France is no longer a sovereign state and likes to suck off the US.
>>650641
Kek but Obama isn't bald either
>>645193
>russel
>made frege JUST
unforgivable
>>650680
Universal quantifies are literally replacements for words. You could understand predicate logic if you're not a complete retard.
>>650763
huh?
>>645193
This is Henri VII the current count of Paris and head of the House of France (note that House of France =/= he's still king of France, he just doesn't wield any power). Defining Henri as bald is quite correct.
>>648520
and this is....?
>>650772
Different anon.
Universal quantifies are literally replacements for words. You could understand predicate logic if you're not a complete retard.
>>650569
I never said anything about half open spaces.
Let me break down what I was saying, because I guess you misunderstood. You said "empty set is both open and closed" and that implies to me that you believe that this is a property that it inherits because of vacuity, but it isn't. Let me explain:
The standard definition of a topology requires the empty set to be open. Since the complement of the empty set is the universe (and thus open), the empty set is also closed. This isn't an example of a vacuous truth, because it isn't reliant on properties of "elements" of the empty set, but instead properties of the empty set itself.
>The only sets in a topology that can be both open and close are the empty set and the whole space
This is incredibly wrong. Here a few examples of proper clopen subsets:
-In the topological space [0,1]U[2,3] equipped with its subspace topology, the set [0,1] is clopen.
-Similarly, in (0,1]U[2,3] equipped with its subspace topology, (0,1] is clopen (note that (0,1] is neither open nor closed in the ambient space).
-Let M be a smooth, orientable manifold with 2^n orientations. Then M has n connected components, each of which are clopen.
-Let G be a Lie group with idenitity e. The subgroup G' generated by a connected neighborhood of e is clopen. If G/G' is nontrivial, it is easy to see that G' is properly contained in G.
>>653153
I'm pretty sure it's not by definition that the complements of opens sets are closed. I'm saying that the empty set can be proven to be closed with a vacuous truth. Also most of what was said was in the context of R.
The last two examples are interesting, care to give proof sketches? I'm assuming you're using the atlas on the manifold as the basis for the topology right?
Also I don't see how a subgroup G' generated by an (open) neighborhood of e in G is closed.
>>653184
>I'm pretty sure it's not by definition that the complements of opens sets are closed
Depends on if you're doing topology or analysis. For most cases, the complement definition is the best, though. To a person who mainly works with topological definitions, "empty set being clopen is a vacuous truth" just looks inherently wrong. I can see why you would think that if you think a closed set is "a set containing all its limit points"- which is just part of a TFAE for me.
The last two examples are interesting, care to give proof sketches? I'm assuming you're using the atlas on the manifold as the basis for the topology right?
So, for the orientable manifold one, just let M be a connected manifold, remember that the volume form is unique, and you're pretty much done (yes, manifolds always have as their basis smooth balls they inherit from the coordinate maps).
For the Lie group one, check out Lee, Smooth Manifolds Prop 7.14.
>>653241
>smooth balls
I meant just "balls."
Sorry, in my world everything is smooth, so that adjective usually never affects anything.
>>646094
>things don't have to exist to bear truth value in the real world
kek, get the fuck out with your mathematical idealism Plato.
>>653241
Right, since if the volume form is closed in one connected component, it must be closed in any component and by Stokes the components themselves are closed sets? Where does the (I'm assuming distinct) orientations come in?
>>646094
>he literally think that we can form a set which is not finite.
I bet you work in FOL, like the pleb you are.
>>653272
Wild Burger pls
>>650569
>>The only sets in a topology that can be both open and close are the empty set and the whole space,
lel, found the 20 yo undergrad.
anime is cancer indeed.
So, is there anything of interest in analytic philosophy if you don't care about math or linguistics?
>>653268
For some reason, I gave the case where M is a (compact) Lie group.
Here's the real proof sketch: M a connected manifold. It's easy to prove with ON frames that M has exactly 2 orientations. Thus, disjoint unions of n connected manifolds (of equal dimension) must have 2^n orientations (it's just a basic calculation). It's clear that the converse is also true: a manifold with 2^n orientations has n connected components, n! clopen sets.
Connected components are ALWAYS clopen. I think you might want to read the first few chapters of a text like Munkres' Topology.
>>653321
If you don't care about language why would you care about philosophy?
