The Full Import of Paradoxes
I've been a paradox hunter/buff for as long as I can remember. Even as pimply, awkard teen, I was fascinated by them and my love for them, I realized only now, hasn't faded/diminshed even a bit.
What are paradoxes?
Loosely defined, paradoxes are counterintuitive, surprising, ironical, etc. but the philosophical meaning is contradiction.
Why are paradoxes/contradictions (so) important?
Their significance to all (real) thinkers is that renders trivial the logical systems in which they arise.
What do we mean a logical system is trivial? Simply this: every proposition is true (in that logical system) via ex falso quodlibet (the principle of explosion).
All of philosophy, math especially, uses classical logic made up of categorical logic (Aristotle), sentential logic (Chrysippus), and predicate logic (Frege).
Most if not all thinkers are under the impression that they're using classical logic - they don't take too kindly to contradictions. This is ok of course except for the small matter of genuine paradoxes (vide Wikipedia for a list of alleged paradoxes).
The existence of paradoxes (contradictions) means
1. Classical logic has to use Occam's broom (sweep paradoxes under the rug) otherwise, via ex falso quodlibet, concede that classical logic is trivial.
2. We're using some version of paraconsistent logic and we're not aware of it.
A penny for your thoughts.
What are paradoxes?
Loosely defined, paradoxes are counterintuitive, surprising, ironical, etc. but the philosophical meaning is contradiction.
Why are paradoxes/contradictions (so) important?
Their significance to all (real) thinkers is that renders trivial the logical systems in which they arise.
What do we mean a logical system is trivial? Simply this: every proposition is true (in that logical system) via ex falso quodlibet (the principle of explosion).
All of philosophy, math especially, uses classical logic made up of categorical logic (Aristotle), sentential logic (Chrysippus), and predicate logic (Frege).
Most if not all thinkers are under the impression that they're using classical logic - they don't take too kindly to contradictions. This is ok of course except for the small matter of genuine paradoxes (vide Wikipedia for a list of alleged paradoxes).
The existence of paradoxes (contradictions) means
1. Classical logic has to use Occam's broom (sweep paradoxes under the rug) otherwise, via ex falso quodlibet, concede that classical logic is trivial.
2. We're using some version of paraconsistent logic and we're not aware of it.
A penny for your thoughts.
Comments (284)
Not sure this is relevant but I generally accept that humans are clever animals who use language to help manage their environment. As a consequence, meanings and worldviews are riddled with inconstancies and subversions, some of them more striking than others. When I encounter a paradox it tends to remind me of the poetic, imprecise nature of language and the manufactured character of human understanding.
My main concern is the existence/nonexistence of (true) paradoxes. If they exist then, classical logic is trivial unless it excludes some rule of natural deduction that prevents ex falso quodlibet. The rule that most logicians choose to exclude from natural deduction in order to prevent explosion is disjunction introduction/addition. Should we do that? It seems the right course of action assuming there are (true/real) paradoxes.
The bottom line is this: either resolve all paradoxes OR accept paraconsistent logic.
What about Occam's shaving gel? Makes the razor run smooth... Shave those needless hairs away! Smoooooth.....
Just ask yourself, how many paradoxes involve this. Starting from Russell's paradox.
Perhaps I am not a (real) thinker, but all the excitement about paradoxes goes over my head. I just can't see how they have any practical meaning.
Quoting Tom Storm
It strikes me that many (most? all?) so-called paradoxes are really just playing with language. There was a discussion a month or so ago on the forum about whether the Liar's sentence/Russell's paradox undermine the validity of mathematics. Apparently Alan Turing actually believed that, because of those paradoxes, bridges designed with mathematics might fall down. I find that hard to grasp. [irony]Turing was somewhat smarter than I am.[/irony] I don't understand how he could believe that.
Quoting Agent Smith
I wasn't familiar with the idea of logical explosion, so I opened my trusty Wikipedia. Here's the example used in that article:
As a demonstration of the principle, consider two contradictory statements—"All lemons are yellow" and "Not all lemons are yellow"—and suppose that both are true. If that is the case, anything can be proven, e.g., the assertion that "unicorns exist", by using the following argument:
To me, that example is based on a case of philosophical bait and switch. The two propositions in question "Not all lemons are yellow" and "All lemons are yellow," are concrete examples from the real world. Please show me a "paradox" like that. I can't think of any. The contradictory examples that tangle philosopher's and mathematician's shorts are all language paradoxes, e.g. "This sentence is false."
On an unrelated, or at least only semi-related subject, does the fact that light has both a wave and particle nature constitute a valid example of a real-life, concrete paradox, which I just denied the existence of?
:up:
Escher's paradoxical ever up or down going stairs is about the angle of vision (that resolved the seeming contradiction). The twin paradox is about everyday experience and gravity, resolved by general relativity. "Contra-diction" is not always about diction.
You're right. After I wrote that about language, I thought of Zeno's paradox. Now I'm trying to figure out how Zeno's and Russell's paradoxes are different from the liar's paradox, if they are. I think the twin paradox is only a paradox if you don't understand general relativity, which, of course, I don't. As for Escher, I would call the things he drew optical illusions. Is that the same thing as a paradox, just in a visual rather than a verbal medium? I'm not sure.
I don't think that changes my impression that the strain paradoxes supposedly put on philosophy is illusory. It seems pretty unlikely that I've got it right while some of the smartest people in history have it wrong, so I'm hoping to be enlightened.
It's not that complicated. Given the natural deduction rules (there are 18 of 'em) of predicate logic, the existence of a true paradox means predicate logic is what logicians call trivial - it proves every statement conceivable is true via ex falso quodlibet (explosion).
The only way out is to adopt paraconsistent logic which accepts the existence of true contradictions, but prevents explosion by tweaking the rules of natural deduction e.g. it does away with the disjunction introduction/addition rule.
The yellow lemons, unicorn example argument of an ex falso quodlibet uses the disjunction introduction/addition rule in line [3].
I've seen at least two negative self-referential paradoxes: the liar sentence and Curry's paradox.
Your point?
I didn't say that the idea of paradoxes goes over my head, I said the excitement about them does. I just don't see why it's a big deal. They're not that hard to recognize. It's not like they can sneak up on you.
Oh! Sorry, my bad. You didn't read my post thoroughly. I explain why paradoxes are a big deal.
Paradoxes break (classical) logic.
Question: I'm told that any system of logic in which every sentence can be proven true is trivial. I know what "trivial" means, but in this case, it probably has a deeper meaning. Anyone knows what that is? Any help will be deeply appreciated. Thanks
I did read your post thoroughly, Mr. Snooty. Agent Snooty. The explanation doesn't make sense to me. What difference does it make other than providing a bit of agita to some philosophers and mathematicians?
:lol: I consider your actions an act of war!!
Read my last post, the post before you lost your mind. :smile:
I went back and reread your posts. I don't think there is any misunderstanding between us about the issue on the table. We just disagree on the implications. I have four answers to the question "What difference does it make that language paradoxes seem to undermine the value of logic?" Those answers are, in no particular order, none, zero, zilch, and nada.
On first hearing of this, I wondered if we were meant to experience what is it like to be a [s]bat[/s] stone? but then no mushin no shin isn't no mind, it's mind without mind. Centrism/madhyamaka/the middle path.
:ok: Great!
Three types of paradox
Falsidical – Logic based on a falsehood.
Veridical – Truthful.
Antinomy – A contradiction, real or apparent, between two principles or conclusions, both of which seem equally justified.
Maybe the optical illusion paradox should be included.
Only when you use different types. The principle of explosion, based on two contradicting statements, was removed by the introduction of paraconsistent logìc, as its name suggests. It was used by Zermelo and Frankel to save set theory from disaster, Hilbert's paradox, with those strange sets including themselves, no longer paradoxed and the village barber could shave his own beard without repercussions...
Is an optical illusion a paradox? Can things move while standìng still? Can stairs go up while keeping you leveled? Can clocks run dìfferently for different observers? The last paradox is not an optical paradox. It's a veridical one.
What does it mean when you show a word you crossed out? Thought process?
A crashed computer can't be compared with a mind (is that why you crossed it out?). Zen means excluding all thoughts. Only experiencing the moment. But some structural organization remains. A mourning dove, a ticking clock, gas flowing in a pipe, an ache in the chest, a fly zooming in a glass, a door shutting, a fucking fridge, the whistle of a neighbor. It blends together in one... oh oooh...
Usually the mathematical paradoxes/logical paradoxes are structured this way.
Do notice, with the same structure is also made theorems like Gödel's Incompleteness Theorem or Turings answer to the Halting Problem.
Self reference + Denial/Negation = Paradox.
I don't speak English! Oh, I just did! The only possibility, post elimination of the impossible, this (all I've written) is not English. :chin:
Give a reply to my comment that you won't never give in this forum.
Are there those comments that @Agent Smith doesn't give in this forum? Of course. Can you give them or utter them as @Agent Smith? Of course not! You are who you are.
The power of negative self reference.
Yaay! :smile:
I want to pick your brains on something. Why did you bring up negative self-reference? Do you have a specific reason for doing so? Are you, if I may ask, trying to say that all paradoxes can be reduced to a negative self-referential paradox?
I'd like to see you do that with Zeno's paradoxes if you don't mind that is. Can you?
Thanks.
We're using, we have to, paraconsistent logic if we're not to end up as confused masses of protoplasm (Zen koans, aporia, mushin no shin). In other words, any philosophical argument that depends on disjunction elimination/addition has to be taken as invalid. Know any?
Mind giving an example of a paradox that exists, and logical people still hold as true?
List of paradoxes
Knock yourself out.
I don't think I was clear. It wasn't a list of paradoxes I was looking for. I was looking for paradoxes that logical people hold as true, or thoughts that logical people still hold despite it falling into a paradox. Most paradoxes are fun accidents that I know of, and no serious logical thinker that I am aware of, entertains a thread of logic that necessarily leads into a paradox.
The Wikipedia link I provided is a list of paradoxes (which logical people hold as true).
You might know they hold them as true, but how do you know they are logical?
I'll try one more time in case you aren't understanding my request. I know those are paradoxes that people have come up with. What logical thinker holds onto something that leads into a paradox, agrees that the paradox is sound, but still insists on holding onto logic that leads to that specific paradox?
You're noting that people sweep paradoxes under the bridge to hold certain logical arguments. Which arguments? Which logical argument are people holding onto despite it leading directly into a paradox?
There are arguments whose conclusions are paradoxes. Check out Wikipedia or the Stanford Encyclopedia of Philosophy, etc.
Quoting Philosophim
That's all of us Philosophim. The Wikipedia list (of paradoxes) I linked to applies to every person, assuming they're genuine paradoxes (no one, as of yet, has attempted to resolve all them; from the multiple disciplines involved, it'll require a team).
Since everyone knows there are paradoxes in classical
logic, we have to stop explosion (ex falso quodlibet) and one way of doing that is doing away with the disjunction introduction/addition rule in natural deduction. That's paraconsitent logic! We're supposed to use it, not the old classical logic systems of Aristotle, Chrysippus, and Frege.
1. P
2. P v Q [disjunction introduction/addition]
Line 2 should not be allowed.
That post and most of what is said in the poster's following posts is terribly ill-premised.
Classical logic itself does not result in contradictions. Rather, adding certain non-logical axioms results in contradictions. For example, classical first order logic is consistent, but if we add an axiom schema of unrestricted comprehension (which is a set of non-logical axioms) then we get the contradiction known as 'Russell's paradox'.
A logical principle is one that is true in all models. A non-logical principle is one that is not true in at least one model.
It is fatal mistake not to understand that difference between logical axioms and rules (the bare bones of deduction) and non-logical axioms or rules.
Classical logic, which includes the explosion principle, is consistent. It is nonsense to claim that classical logic can be consistent only if it eschews the explosion principle.
There are many other terrible misconceptions stated by the poster, but at least it's a start to point out the lack of distinction between the logic itself and what can be derived using the logic from additional non-logical axioms.
You need to first learn basic symbolic logic. I recommend 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague and Mar. Then basic set theory in which the systems of logic (including paraconsistent logic) are formalized. I recommend 'Elements Of Set Theory' by Enderton. Then mathematical logic, which is a deeper study of logic. I recommend 'A Mathematical Introduction To Logic' by Enderton, supplemented with 'Introduction To Logic' by Suppes for his discussion of mathematical definition, which is the very best discussion of that subject I've seen.
That will give you a clear and rigorous understanding as opposed to flitting about among various bits from improperly edited Wikipedia pages, taken out of context or even grossly misunderstood and then irresponsibly misrepresented by you on this Internet forum.
This is ill-phrased. (P & ~P) does not "break" logic because it is not a theorem. That is, it cannot be deduced in classical logic.
A situation in which (P & ~P) would be one in which "classical" logic was inapplicable. There are, as you note, other logics.
Again, the appropriate way to think of this is as choosing what language - which logic - is applicable to a given circumstance.
Hence it is Quoting T Clark but that is not to trivialise logic; playing with language is what we do.
Ah, I see @TonesInDeepFreeze has made much the same point.
Well, paradoxes break math (Bertrand Russell and his set of all sets that don't contain itself) for the reason, I was told, that ex falso quodlibet renders such mathematical systems trivial (all well-formed formulae are true). :grin:
Why I said what I said was because the LNC (the law of noncontradiction) is an axiom (vide the 3 laws of logic); paradoxes, if real, imply that the LNC can no longer be an axiom, it has to go. That's what I meant.
On the larger point...
If true paradoxes exist (re heterological paradox), we're either
1. Using paraconsistent logic, a version of it
or
2. We're ignoring them but not in a systematic manner (picture a bomb disposal unit at work)
Bear with me...this is mainly guesswork.
No, it doesn't. 2+2 still makes 4, regardless of Russell.
Quoting Agent Smith
Indeed. Take @TonesInDeepFreeze' advice and study up on logic some more. There is a lot to be said for doing a formal course, since only in that close interaction will someone be able to pick apart the multiple, small errors in your comments.
:snicker: 2 + 2 = 5 or chimpanzee too if Russell's set is allowed in math based on set theory. I recall reading how Russell attempted damage control by, in a sense, making his set of all sets that don't contain itself illegal in a manner of speaking.
What sayest thou, kind sir?
Nothing Russell might say, and certainly no paradox, can lead us to conclude that 2+2=5. You have the tail wagging the dog. If one's argument concludes that 2+2=5, one has made an error.
(p & ~p) is false in both propositional logic and first order predicate calculus. Hence, it is difficult to see what you mean by Quoting Agent Smith
We are animals? Sometimes, I say this to my wife. Sometimes she agrees...
Yes they do.
The paradox is that approximations in math are the exact solution. An apparent contradiction. A oaradix, eeehhh... paradox.
I don't see why you should object to what is the official position on Russell's paradox. Russell's set (of all sets that don't contain itself), in colloquial language, breaks math; to be precise, it, via ex falso quodlibet, means every mathematical statement is true. In my world math is broken when that (trivialization of math) happens; it happens because the set theory based axioms of mathematics allows Russell's set to, well, exist; what happens next is common knowledge.
That's not a thing.
Quoting SEP, Russell’s Paradox
I didn't notice this earlier, so I'll try to give an answer to this. I'm no mathematician, so the answer can be quite difficult to understand. Hopefully I make sense to you.
As we know, Zeno's paradoxes are about infinity. Modern math has "no problem" with Zeno's paradoxes of Dichotomy, the Arrow or the Tortoise as it uses limits (or the infinitesimal). And modern math just takes infinity as an axiom, and axioms don't have to be explained. Hence we still have a lot of questions about infinity, because we don't have an understanding about it. (The people who say we do, then should answer the Continuum Hypothesis for us)
Now when you think about infinity, the self reference should be obvious. It's pretty hard going from the finite to the infinite: you cannot just add finite numbers to other finite numbers and get infinite. Or, you add them an infinite times (hence the self reference). Cantor himself did understand the paradoxical nature of infinity, but could make something about with the proof of there being more reals as natural numbers. Although you can show in another way, Cantor's diagonal argument uses negative self-reference, proves by reductio ad absurdum that not all reals simply can be put into 1-to-1 correspondence with the natural numbers (hence it would have the same aleph).
Why isn't it a paradox? Well, if we would assume that all numbers can be well-ordered/put into 1-to-1 correspondence with the natural numbers, then it would be a paradox for us! Yet as we don't take as an axiom that all numbers can be well-ordered, we don't have a problem with this, just like we don't have any problems with irrational numbers.
Yet there is the link to Zeno's paradoxes as they are about infinity.
I'm not sure if I've been able to show the connection to you, but well, all I'll say in the end that paradoxes for me aren't questions to be answered or solved, but more like answers that should be understood.
I like what you said there.
I can't add more black to black and hope to get white!
As for your attempt to show Zeno's paradoxes are self-referential negations, I'm sorry I don't follow. Let's keep it simple, use one of Zeno's many paradoxes (your choice) and demonstratehow it is self-referential negations. Danke!
Curry's paradox does not directly involve negation.
Yablo has a paradox that explicitly does not involve self-reference.
Perhaps it would be proper to say that the set of paradoxes has more paradoxes than just one's with negative self-reference.
The paradox lies in the infinitesimals.
The paradox is that infinitesimals have zero length.
Then why does Wolfram say Quoting Infinitesimal
Because infinitesimals are paradoxical.
Is that a paradox? Don't think so. There has never been a mathematical object as paradoxically as the infinitesimal.
Ok, but they could be described as zeroish. That's the point, oui monsieur?
Could you give an example of a non-logical axiom and explain what makes it non-logical?
No.
Quoting Hillary
No, that's irony.
Still not seeing a paradox.
The paradox lies in the irony. :lol:
No, seriously how can you create a non-zero interval out of length pieces that have zero length? By taking infinitely many? How you do that? You might counter they don't have zero length and it merely approaches zero, but that begs the question. If the distance between two points decreases, the distance will get zero and they touch. The paradox is that they never touch while able to break on through.
But infinitesimals do not have zero length. So that's not what is happening.
I wrote: Quoting Hillary
Why?
How would you round off 0.0001?
Quoting Agent Smith
Life's too short for this. 0.0001 is not an infinitesimal.
But the very definition is paradoxical..
Answer the question instead of beating around the bush.
If 0.0001 rounds off to 0, a fortiori an infinitesimal rounds off to 0, oui? Zeroish.
How, exactly?
Your question makes no sense.
Approaching without being able to touch.
:lol:
Exactly, brother Agent!
Consider two points. There is space between them. For the infinitesimal to exist the can never touch. However close they get.
The thing is that a mathematicians or math education can (and usually does) look at this ahistorically. They just give you the end result, at worst basically an algorithm to use with not much debate about the underlying issues. A proof is given and that's all. Once you understands let's say the notion of limits, for some it looks quite meaningless to ponder about the motion paradoxes of Zeno. These are basically foundational questions about mathematics, and then mathematicians (or philosophers) do understand the question better from that viewpoint.
The paradox lies in the break-up of the continuum in points. If you get on with it a little harder you might see it. The distance between two points gets smaller and smaller. But there will always remain space between them. You can say it's defined like that, but it's non-coherent.
There's nothing paradoxical in that. It's just odd, not paradoxical. A topic for investigation, not @Agent Smith's broken maths or broken logic.
A paradox goes against established opinion.
Why?
The maths works. What more is there?
Quoting Hillary
But infinitesimals are established opinion.
Yes, but the majority of people think the opposite. That establishes the paradox. The twin paradox goes against established opinion (no time delay). Differentials idem dito.
How you put infinite small pieces together to form a meter? How you know the size of your differentials is right, to not arrive at two meter?
:chin: Rounding off a number makes no sense? It's taught in schools, to kids.
"Shut up and calculate" - isn't that the physicist's advice?
:snicker:
It's just a logical paradox. It is thought that you can't have infinitesimal small intervals. And that you can't gather them all together to form a finite interval. But it can be done actuslly.
Not mine. I like to know what is actually computed.
Have a chat to @Metaphysician Undercover. Sounds a bit like you might have common ideas. He has some odd notions concerning instantaneous velocity you might find amusing.
He's right about instantaneous velocity. It's a paradoxical concept. At the fundamental level, the minimum distance is the Planck length.
My thesis is this: If there are true paradoxes,
EITHER
1. We must switch allegiance from classical logic to some version of paraconsistent logic
OR
2. Use Occam's broom and sweep these annoying paradoxes under the rug, ignore them, sequester them in a place where they won't do damage, and the damage can be terrible (vide ex falso quodlibet).
That's all.
Back? I went forward! What is an infinitesimal?
:up:
Can you think of a polygon with an infinite amount of sides?
Can you think of a circle?
An 10 cm diameter polygon with 1 trillion same length sides might look quite like a circle, but still isn't a circle.
[quote=Shakeera]That's the deal my dear.[/quote]
Like I mentioned earlier, if we can't have chocolate, by god we'll have something chocolatish!
Heaven!
Sorry, no can do!
Heavenish then!
Check back in an hour!
You are taking my quote out of context.
Why should they have practical meaning in the first place? But to give a practical example, the twin paradox has significant practical impact to future generations taking a retour trip to Proxima Centauri.
I explained why you are incorrect. You are terribly mixed up and you don't know what you're talking about. And you add additional confusions and misinformation with each post.
Let's select that quote in particular. It is pure misinformation. ZFC, which is the common set theory for mathematics, is formulated so that it does not allow the usual proof that the Russell set exists, and no one has shown that ZFC does prove that the Russell set exists.
The existence of the Russell set was proven using unrestricted comprehension. But ZFC does not have unrestricted comprehension.
You should inform yourself and stop egregiously posting misinformation.
:snicker: A thousand apologies.
The diagonal argument does not require reductio ad absurdum. And 'negative self-reference' needs a definition.
The diagonal argument is constructive and intuitionistically valid. Though it makes use of an ingenious technique, that technique relies on no logical or mathematical principles that are not common elsewhere in mathematics. Indeed, even without the assumption of infinity, the diagonal argument would still go through as couched in terms of "potentially infinite" processes rather than infinite sets.
Quoting ssu
Sure, but it is famously the case that certain mathematicians and philosophers do explain the axioms and give justifications for them.
Quoting ssu
The axiom of choice implies that every set has a well ordering, thus, in particular, the set of real numbers has a well ordering.
As your posting history suggests, your thousand apologies will be followed by a thousand more of your egregiously misinformational posts. One only has to sit back, have a cup of tea, and wait for the next one.
Historically, infinitesimals were not given a rigorous treatment. However eventually non-standard analysis was devised in which infinitesimals are constructed with the ordinary set theoretic axioms.
I'm sorry if you feel that way. What exactly is wrong with what I said?
There's very little original material, which I feel bad about, to critique. You should talk to Russell, Whitehead, Frege, et al. Not to me and since you are accusing me of misinformation, I'm somewhat inclined to believe you don't know what you're talking about.
I said exactly what is wrong with what you said just a few posts ago! And I explained also in posts yesterday. And from months ago I've explained why various of your posts are confusion and outright misinformation.
You're just fooling around. Sorry, not interested, I'm not in the mood to play your silly games. Good day.
Explaining that ZFC does not have unrestricted comprehension that yields the existence of the Russell set is not silly game playing.
Good. You are at your most eloquent with emojis.
Thanks for the compliment. Good day.
Each axiom of group theory is a non-logical axiom.
Quoting Tate
Quoting TonesInDeepFreeze
Classical binary Logic is best used for problems that can be precisely defined with integer numerical values. But human contradictions are seldom concisely defined; instead loosely sketched with inexplicit subjective truth-values.
A formalized version of "paraconsistent logic" (logic of paradox) is the Fuzzy Logic that is used in computer science for complex puzzles that are hard to define numerically, such as human beliefs & intuitions. It is especially useful in Artificial Intelligence and Evolutionary Programming.
We're not aware of our sloppy logic because it is intuitive, so we don't normally examine it with classical rules in mind. That's why divisive emotional issues, such as Abortion & Racism tend to polarize people. And can only be resolved, to some degree, with critical (rational) thinking : to discover the inconsistencies in our beliefs. :cool:
Paraconsistent Logic :
A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" systems of logic which reject the principle of explosion.
https://en.wikipedia.org/wiki/Paraconsistent_logic
Quoting Agent Smith
Actually, most thinkers have an ego-boosting impression of their own reasoning abilities. We find it easier to see the contradictions in other people's ideas than in our own thoughts. Intuition always seems true, even when it aint.
A great many people think they are thinking when they are merely rearranging their prejudices.
— William James.
If I'm not mistaken, there is work in combining formal paraconsistent logic with formal fuzzy logic. But fuzzy logic itself is not a formalization of paraconsistent logic.
The argument only goes to show that the continuum cannot be broken up in points. Leading to confused notions of infinitesimals or differentials.
BTW, brother Smith is not ad confused as you suggest. His contribution is a welcome addition to the body of philosophy. He makes the face of philosophy laugh.
The proof shows that there is no enumeration of the set of real numbers. As formalized, the proof uses only first order logic from the axioms of Z set theory, and it could even be Z without the axiom of infinity by couching the proof with an hypothesis, rather than a given, that there exists an infinite set. Moreover, as I mentioned, the proof can even be couched in constructive mathematics with a notion of a "potential infinity" as a computational process without an upper bound of actions.
And the proof shows no such thing that the set of real numbers is not made of points. The continuum is
Quoting Hillary
Ordinary real analysis does not use infinitesimals. And the non-enumerability of the set of real numbers does not lead to use of infinitesimals.
Quoting Hillary
I demonstrated exactly the manner in which he is confused and misinformational.
But how you glue two points together. How can you construct a continuum with points as building blocks?
You can throw as many points in the bag as you like. It's never filled.
Quoting Hillary
Those are your personal, impressionistic locutions. Real analysis doesn't have such terminology.
Quoting Hillary
The real continuum is constructed in formal axiomatic set theory. We prove that there exists a set and an ordering on that set (which are unique up to isomorphism) such that the ordering has the completeness property.
Your question suggests that you are not familiar with the basics of the subject.
How many points do I have to throw in the bag to fill it?
Did Cantor's original set theory have non-logical axioms?
particle-wave duality is only inconsistent in so far, that it doesn't mesh well with our typical view of the universe. it's however perfectly consistent, experiments will deliver consistently similar results. particle-wave duality is just a name to at least somewhat visualise what is happening in the equations of quantum mechanics
well it will be filled after a certain infinite time, based on the size of each point.
You got some way of putting it, I have to admit! "Impressionistic locutions. Sounds like a new art form.
Points come in sizes?
Gluing schemes in mathematics
I thought I remembered something about gluing in math, so I looked it up. Maybe not precisely what you ask, but interesting nevertheless.
Quoting Hillary
In the real nos, no. But a "point" in a vector space or simply space in math frequently is a function of some kind, like a contour in C, and these might have "sizes".
I can imagine closing an infinitely small hole with a point, like adding one point closes an open interval. But throwing points in a bag...?
That was my point. I was trying to highlight that the only real paradoxes are logical or linguistic, which are trivial. There are no paradoxes in the real world, only stuff we don't understand or that surprises us.
'throw in', 'bag', and 'fill' (in your context) are not mathematical terms, so I can't give you a mathematical answer to your question.
However, the mathematical question "how many 3D-points are in a non-empty volume?" does have the mathematical answer: the cardinality of the set of real numbers.
Cantor didn't have axioms. But of course he did use non-logical principles even if not formalized as axioms.
Except for the pure predicate calculus itself, any mathematical theory (such as formalized in predicate logic) has non-logical axioms.
Does this mean you're sort of stretching the idea of non-logical axioms to address the problems associated with naive set theory?
It appears that axioms were created specifically to block the path to Russell's Paradox. There isn't any logical basis for accepting that blockade.
Sure, where the set theory is not formalized with axioms, we can at least point out that the pre-formal principles it uses are non-logical, at least in the sense that they would be non-logical axioms if they were formalized.
Quoting Tate
It is common for people to put it that way, but it could be misleading to put that way.
Adding axioms cannot block the derivation of statements that are already derivable without the added axioms (i.e. the logic is monotonic). What axiomatic set theory does to avoid Russell's paradox is not add axioms but to refrain from adding the axiom schema of unrestricted comprehension that results in Russell's paradox. Then other axioms are added that permit derivation of the desired mathematical theorems.
Specifically, Zermelo refrained from adding the schema of unrestricted comprehension, so that schema is not available to use to derive Russell's paradox. Then Zermelo does add the schema of separation instead of unrestricted comprehension and the other axioms of Z set theory.
And Fraenkel did the same except he adds the schema of replacement (which is strong than separation but weaker than unrestricted comprehension) rather than the schema of separation. (I think the schema of replacement is important mainly for deriving transfinite recursion, which can't be derived in mere Zermelo set theory.)
In the sense that these principles are untrue in some models? That doesn't make any sense to me. How can a principle be false?
It would be false in some models if it were formalized as a first order sentence, or, for a schema, it would have false instances if the schema were formalized.
The important point for the prior discussion here is that Cantor did not confine himself to merely logical principles. The Cantorian paradoxes are not a failing of logic.
What would that sentence be?
I agree.
The schema of unrestricted comprehension specifies an infinite number of axioms:
If F is a formula and b is a variable that does not occur free in F, then all closures of
EbAy(yeb <-> F)
are axioms.
Every instance of that schema is a non-logical axiom. [Edit: I have to think more closely whether every instance is non-logical. But still, the point prevails that unrestricted comprehension has non-logical instances, and in particular an instance used to derive Russell's paradox is not just non-logical but it is also inconsistent.]
Moreover, the particular instance:
EbAy(yeb <-> ~yey) is not just false in some models but it is false in every model, as we know from Russell's paradox.
Note also that this is not specific to set theory. To derive Russell's paradox from unrestricted comprehension, we don't rely on anything specific about 'e', the membership relation. Rather for any 2-place predicate R whatsoever (whether 'R' stands for 'is a member or' or 'shaves' or 'is a parent of' ...) we derive a contradiction from:
EbAy(Ryb <-> ~Ryy)
What's the model where an instance is false?
EbAy(yeb <-> ~yey) is false in not just some models but it is false in every model.
You lost me.
Assume EbAy(yeb <-> ~yey)
"There exists a set b such that for all sets y, y is a member of b if and only if y is not a member of y".
So beb <-> ~beb
"b is a member of b if and only if b is not a member of b".
So EbAy(yeb <-> ~yey) yields a contradiction.
But any sentence that yields a contradiction is false in every model, so EbAy(yeb <-> ~yey) is false in every model.
I've never understood why this is so. It seems like an artificial definition of "false" Does it make sense to you?
Per the valuation function for truth in models (the Tarski definition by recursion on formulas), every sentence is either true in the model or false in the model but not both. And per that function, the negation ~P of a sentence P is true in the model if and only if P is false in the model; and P is true in the model if and only if ~P is false in the model.
Now, suppose a contradiction P & ~P were true in a model. A conjunction is true in a model if and only if both conjuncts are true the model. So both P and ~P would be true in the model. But then P would be true in the model and false in the model, which is impossible.
This means that if we adopt the method of models, Russell's Paradox is impossible. What are the consequences of not adopting that method?
That is incorrect. No matter about models, if you have inconsistent axioms, then you derive Russell's paradox. Then, it is merely an additional note, not confined to Russell's paradox or unrestricted comprehension, that any inconsistent axiom is perforce a non-logical axiom.
Quoting Tate
The method of models is ubiquitous in mathematical logic. There are many theorems about models (the subject is called 'model theory') so I don't know how to say simply what the consequences are, since there are many consequences.
Three of the most famous consequences are the completeness (and soundness) theorem, the compactness theorem, and Lowenheim-Skolem. Those are sometimes considered to be the "Big Three Pillars" (my term) of first order logic.
[Edit: I overlooked that you said not adopting.]
The crucial consequence of not having a method of models is that we would need to find some other means of providing semantical interpretations for theories. The method of models is the way we say what the formal sentences mean.
I understand, thanks for explaining that.
Also, I overlooked that you said "not adopting". So I added more response accordingly.
Also, aside from providing semantical interpretation, and myriad other result in model theory, we use models for consistency proofs, relative consistency proofs, and independence proofs (the independence of the axiom of choice and the independence of the continuum hypothesis most famously).
And Robinson's non-standard analysis was developed using model theory.
Anyway, languages (not just formal languages) have both syntax and semantics. Models are the ordinary semantics for languages in predicate logic. And logic itself is not just the study of proof but perhaps even more basically the study of entailment. And entailment is semantical in the sense that 'truth' is determined by the model theoretic semantics for a language.
It's not intuitive to me that P & ~P is necessarily a false statement. I think it would be better to say it's a meaningless statement. Could it be that predicate logic is handling meaninglessness by calling it false?
You propose that there are closed well formed formulas that are meaningless (have no interpretation or the valuation function also has meaningless in its range). Mathematical logic does not have that presupposition, so it is not, in its own terms, taking meaninglessness as falsehood. Indeed, the notion that contradictions are false is the ordinary notion through the centuries of the subject of logic. And it facilitates the ordinary notion of entailment.
Of course, one is free to develop a logic in which contradictions are not false but instead valuations include meaningless in addition to true and false, or whatever multi-valued logic one wants to have. I'm only telling you how it happens to work in ordinary predicate logic in which the domain of the valuation function is the set of sentences and the range is {true false}. So every sentence is assigned truth or falsehood and not both per any given model. This is desirable in ordinary predicate logic, especially, as I mentioned, for facilitating the ordinary notions including the ordinary notion of entailment.
Anyway, the context of discussion was the mathematical paradoxes, especially Russell's paradox, and especially the other poster's wildly mistaken notion that classical logic is trivial (proves every formula) because it is inconsistent, unless it eschews the principle of explosion*. In that context, one would ordinarily take it that contradictions are false.
* Contrary to the other poster's foolishness, classical logic is not trivial (it does not prove every formula) and it is not inconsistent, and therefore it does not need to eschew the principle of explosion.
It appears that, from a cursory reading of the Wikipedia article on fuzzy logic, that there are some/many aspects of reality that are spectral in character in terms of how we interact with them. The spectrum in question is divided up into manageable chunks, the boundaries between them are not precise values. This vagueness is what fuzzy logic was developed to tackle. I suppose we could say that fuzzy logic is grey zone logic.
As for the law of noncontradiction (the LNC) and fuzzy logic, my view is that the former holds in the latter. Yes that truth value can be somewhere in between 0 (false) and 1 (true) but it can never be 1 & 0 at the same time, nor is it that a truth value is (say) 0.7 and not 0.7.
Paraconsistent logics, on the other hand, are, as you rightly pointed out, systems in which a proposition has the combined truth value of 0 and 1 (at the same time and in the same respect) i.e. that proposition is a bona fide contradiction.
So, as per my analysis, fuzzy logic isn't a paraconsitent logic. I could be wrong of course, I do hope I'm not though.
Your BothAnd system feels more like paraconsistent logic than fuzzy logic to me. Maybe it's the words "both" and "and" which indicates you want to reconcile thesis (yes/1) AND antithesis (no/0) by approving BOTH.
It could be that I'm getting mixed up between the principle of bivalence and the law of noncontradiction.
It is not difficult.
excluded middle: P or not-P
non-contradiction: not(P and not-P)
bivalence: (P or not-P) and not(P and not-P)
So bivalence is just the conjunction of excluded middle and non-contradiction.
Paradox Lost
If you're talking about the LNC, only one interpretation of it says contradictions are false statements. Another interpretation is that it's impossible to believe a contradiction. But I think we're off topic now. Again, thanks for the explanations.
Is not the superposition of an elementary particle within quantum mechanics an existential paradox?
Is not quantum computing already underway?
Is not Schrödinger's Cat a thought experiment in paradox at the human scale of sensory experience?
Does Schrödinger's Cat have no impact upon the LNC?
Anton Zeilinger, referring to the prototypical example of the double-slit experiment, has elaborated regarding the creation and destruction of quantum superposition:
"[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ -- The Apple Dictionary
The absence of any such information is the essential criterion for quantum interference to appear.
Does not quantum mechanics declare the imprecision is a feature of reality?
Good that you broached the quantum mechanics (QM) topic. QM is frequently used as a source of examples of weirdness (logical hanky panky) in the world.
However, methinks this is misguided because the mathematical descriptions seem not to exhibit any inconsistencies whatsoever.
It's as if a book in language x (math) is perfectly coherent but the moment it's translated into another language (any natural language) it becomes incoherent (full of paradoxes).
That's my analysis of course; I'm neither a mathematician nor a physicist. All I have to offer is what I cobbled together from info I gathered from the internet.
There are true paradoxes e.g. The Grelling-Nelson Paradox and that means the LNC has to go, which in turn implies we need to switch from classical logic to some strain of paraconsistent logic.
The other alternative is to isolate true paradoxes like we do to Covid + people in the ongoing pandemic - damage control that is (vide [I]ex falso quodlibet[/i]).
Am I making any sense?
Perhaps, I should have prefaced that personal opinion with "it seems to me, that . . .". Before Agent mentioned it, I had never heard of "paraconsistent logic". But a quick Wiki review sounded like a description of Fuzzy Logic, which I was already familiar with. For my general purposes, I prefer the more colloquial and less technical-sounding term. From my layman's perspective, both terms seem to reflect the Uncertainty Principle of Quantum Theory, as applied to other fields of investigation. :smile:
My understanding of paraconsistent logic, from Graham Priest, is that things can contradict each other and still be true
Yes. That is the implication of my personal BothAnd Philosophy. Some apparent "paradoxes" result from viewing only one side of the same coin. :smile:
Both/And Principle :
My coinage for the holistic principle of Complementarity, as illustrated in the Yin/Yang symbol. Opposing or contrasting concepts are always part of a greater whole. Conflicts between parts can be reconciled or harmonized by putting them into the context of a whole system.
BothAnd Blog Glossary
This is also Hegel's dialectic. In simple terms, all opposites are defined by a shared property.
Niiiice! :grin:
:up:
I wanna become a "Disciple of Gill"...
[i]A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true. If falsity is assumed to be the truth of negation, a dialetheia is a sentence which is both true and false.
Dialetheism is the view that there are dialetheias...dialetheism amounts to the claim that there are true contradictions.[/i]
Examples include: Russell's paradox, and the liar paradox.
If I present a proof that a certain claim is false, would you say my proof is invalid because it's not also false?
Quoting ucarr
Is not the above statement telling us that no existential paradox can be experienced empirically (i.e. isolated) because even the linguistic concept of paradox collapses the existential expression of paradox?
I have the feeling, mr. Gill, that you and I, despite a considerable difference in age, we're quite alike. You try (tried...) to pull yourself to the top of the boulders by your arms (!), I try a kind of same thing with the boulders of physics. I have the chalk and technique. I know what's on the top. It's a beautiful view. But there are little other people. They rather stay safe down in town. So to speak.
:cool:
There's no paradox in the equations of QM.
There's a paradox when these equations are translates into natural language e.g. English.
Something doesn't add up, oui?
The Grelling-Nelson paradox is a true paradox: G & ~G
1. G & ~G
2. G (1 Simp)
3. G v P (2 Add, P is any god damned statement)
4. ~G (1 Simp)
5. P (3, 4 DS)
QED (ex falso quodlibet)
As you can see, unless we do something to halt the principle of explosion, we're doomed! One solution is to adopt paraconsistent logic, oui? :chin:
Addition/Disjunction Introduction Rule (step 3 vide supra), it doesn't feel right!
I gave you copious explanation why you are wrong about that. You are blatantly wrong about it. I cannot fathom what reward you find in posting blatant misinformation over and over again.
Quoting Agent Smith
Your example is not the Grelling-Nelson paradox.
That sounds like dialetheism. Paraconsistent logic is characterized by the absence of the explosion principle. What Graham Priest text do you refer to?
You can say that again. Someone resurrected it a coupla days ago. I believe the culprit wished to point out flaws in my reasoning.
The takeaway: I need to do a lotta reading.
By the way, what's your opinion on my argument. Do we need to do an overhaul of the logic we're using in this forum and in philosophy as a whole?
You talk of a "true" paradox. I wonder if it will help you if you try to set out explicitly what that might be.
WTF?
1. Grelling-Nelson paradox is a true paradox in the sense both a proposition and its negation is true.
If so,
2. The LNC must be done away with (1 & the LNC are incompatible) as an law of the thought (a counterexample exists).
Then,
3. We have to choose one version of paraconsistent logic (contradiction-tolerant but blocks ex falso quodlibet)
If not,
4. Every and any proposition is true!
The choices are clear. What'll it be?
That's unhelpful. You can't learn if you can't see your errors.
Ok, try this then. Can you set out exactly wha the Grelling-Nelson paradox is, and why it is "true'?
Please go through the literature on the paradox. I'm unable to fathom how you know I'm wrong when you don't know what the Grelling-Nelson paradox is in the first place!
By the way, any (real/true) paradox will do! Do you know any (real/true) paradoxes? If you do, I'm sure you do, follow it to its logical conclusion in re the LNC & paraconsistent logic.
Ok, have it your way. Twit.
Can you explain the Grelling-Nelson paradox, in your own terms, in order to demonstrate that you have understood it? And also then show what it means for the Grelling-Nelson paradox to be "true"?
Because...
Quoting Banno
and...
Quoting Banno
...hence, If you can set it out clearly we will be able to continue the discussion; if not, you may see the nature of the problem with what you take to be an argument.
All I can say for the moment is that it's a true paradox.
Ok,
The word "heterological" describes a word that doesn't describe itself e.g. "long" is a short word and hence, it's heterological.
Question: Is the word "heterological" itself heterological?
If it is heterological then it doesn't describe itself and so it is not heterological.
If it isn't heterological then it describes itself and so it is heterological.
Paradox, oui?
Wouldn't it be simpler to just say that "heterological" is neither heterological nor autological?
Perhaps, but I presented an argument. Is it sound? Does it not lead to a contradiction?
Your "solution" to the paradox is standard technique (like how the liar sentence was banished from the kingdom of propositions). It did cross my mind, but autological and heterological are mutually exclusive and jointly exhaustive i.e. they constitutes the most powerful version of a dilemma. I could be wrong of course, show me where!
You are correct, he calls it dialetheism.
The LNC is the reason we're interested in paradoxes. If you do away with it, we'll just accept contradictions as normal.
Could I be alive and dead at the same time? Of course! The towering human intellect falls in a ditch.
It's better to leave paradoxes in the closets we keep them in. Leave the LNC alone.
Could I be alive and dead at the same time? Of course! The towering human intellect falls in a ditch.
It's better to leave paradoxes in the closets we keep them in. Leave the LNC alone[/quote]
Well, you make complete sense to me: curiosity killed the cat (9 times in a row and now it's dead dead, deader than dead). However, if I'm correct, everyone is drawn to them like a moth to a flame. Consider it self-immolation if you like. I have no explanation for this behavior! It begs for one, wouldn't you say?
As a philosopher, demolition being your primary mission, I'd say paradoxes are right up your alley.
I think it's mainly philosophical types who are drawn to them, some to slay them like dragons and some to peep through them like they're doors to somewhere else.
For there to be a culprit there needs to be a misdeed. It's a bizarre view that the culprit is not the one irresponsibly spreading misinformation but rather the one who corrects that misinformation.
Quoting Agent Smith
You claimed that classical logic, thereby classical mathematics, is devastated by the paradoxes. I gave you fulsome explanation that the paradoxes do not occur in the ordinary mathematical theories. Paradoxes such as you have mentioned are informal. For purposes of formal classical mathematics we are more careful in formulation so that the paradoxes don't occur. I explained specifically, in detail, for the case of Russell's paradox. But you choose to utterly ignore information that is given to you. I said I wondered why someone would prefer to remain ignorant rather than avail themself of information given to them even at no cost. One explanation though is that the person finds themself more charming or fascinating to fancy themself as some kind of of novel thinker rather than to exercise the common humility in recognizing that there are brilliant and wise thinkers in the past who have come up with entire fields of study, such as mathematical logic, in which we find rigorous and brilliant solutions
Yes, and Agent Smith ignores the most obvious choice.
Agent Smith is ignorant of how it actually works in formal mathematics.
What say you to math language holding a homological relationship to the empirical-material world it's modeling? Looking through the lens of a homological relationship between a signifier (math model) & its referent ( material object), can the math model successfully model a self-contradictory material object without containing within itself any contradictory math expressions?
I'm speculating that, if the answer to the above is "yes," then the foundational logic of math need not be overhauled in light of the experimental evidence of QM, but rather should expand its scope to include QM paradoxes.
If the answer is "no," then the foundational logic of math either needs models that, beyond exclusion, preclude the reality of QM paradoxes. If no such models can be fabricated, then foundational logic of math needs reexamination.
:snicker:
:roll:
You just made me realize how powerful paradoxes are. Danke!
I don't fancy myself as anything. My posts will speak for themselves.
You've made plentiful accusations; some may be true, but I get the impression that, at other times, you don't know what you're talking about.
Don't you love a joke every now and then? Doesn't everybody? Laughter, they say, is the best medicine!
I think
So
Well it's a standard paradox, the sort that @TonesInDeepFreeze showed how to deal with earlier. It posits a set and then asks if the set is a member of itself.
So you have a paradox. But your conclusion is that logic is broken. How do you move from the paradox to that conclusion?
Right. And part of that formality is rules for the use of the truth predicate that are artificial. This is why the value of the solution you point out does not extend to the realm of ordinary language, where if a statement can't be asserted, it can't be true or false.
So you have a paradox. But your conclusion is that logic is broken. How do you move from the paradox to that conclusion?[/quote]
You know this stuff. I already explained it in my previoud post.
The long and short of it: The LNC is incompatible with paradoxes; one has to go. I know your choice (you deny that there are true paradoxes).
It's simply a matter of bad axioms, useful but not truthful. If you judge good by usefulness, you'll say the axioms are good. If you judge good by truthfulness you'll say the axioms are bad. To choose the latter may seem like having "odd notions" to you.
Quoting Jackson
This is the root of the problem, and it's demonstrated well in some of Plato's dialogues, like The Sophist and The Parmenides. If we take two things which are categorically different, and set them up as opposites, then we falsely assign a shared property to them. True opposites, in the absolute sense, cannot share any property, or else they are not absolutely opposite.
Absolute opposition requires a separation of category. So for example, if negative and positive are supposed to be opposite in an absolute sense, they cannot share a common property, or else they are not absolutely opposite. And when we allow what you call the "shared property" to be a property of the of the ideas which are opposite, we make a category mistake because opposite is what is assigned to the properties, not to the object itself. Now we produce a property of the property.
So for example, hot and cold are opposite. We can say these two are possible properties of the same thing. But if we look for a "shared property" of hot and cold, we make the category mistake. Hot and cold are not the type of things which themselves have properties. Hot and cold are defined in different terms, terms of activity (becoming), and becoming is not the type of thing which is described through properties, its described by a change in properties.
:up:
If a train of logical reasoning ends on a contradiction (paradox), the following possibilities must be considered
1. Fallacies (mistakes in applying the rules of natural deduction)
and/or
2. One/more false premises (axioms/postulates)
If not 1 and/or 2 then and only then
3. The LNC needs to be scrapped + a version of paraconsistent logic needs to be adopted
Arigato gozaimus Metaphysician Undercover san!
:confused:
Which is why Hegel's dialectic is different from Plato's.
Properties of temperature.
Not so fast. The Law of Non-Contradiction is a good rule of thumb for most contexts. But there is one common circumstance where LNC does not apply : Holism. The reductive methods of science are appropriate for things-in-isolation. But when a thing participates in a larger System, it shares qualities of the system, which compromises some of its own properties. To a reductionist observer such holistic behavior may seem inconsistent and paradoxical.
For example : what Einstein called spooky-action-at-a-distance, Schrodinger called "entanglement". Which implies that some quantum particles in a holistic (waveform) system share some properties with other particles. Apparently, in their waveform state, electrons are connected to all other electrons in the universe, in such a way that a measurement of one instantly affects (e.g. flips the spin of) all similar particles. From that perspective, it's not a contradiction, but a feature of Holism : an emergent property. :smile:
Holism ; Holon :
Philosophically, a whole system is a collection of parts (holons) that possesses properties not found in the parts. That something extra is an Emergent quality that was latent (unmanifest) in the parts. For example, when atoms of hydrogen & oxygen gases combine in a specific ratio, the molecule has properties of water, such as wetness, that are not found in the gases. A Holon is something that is simultaneously a whole and a part — A system of entangled things that has a function in a hierarchy of systems.
BothAnd Blog Glossary
"The opposite of a profound truth is also a profound truth"
___Neils Bohr, baffled by apparent violations of LNC
"We are not only observers. We are participators. In some strange sense, this is a participatory universe. Physics is no longer satisfied with insights only into particles, fields of force, into geometry, or even into time and space." ___John A. Wheeler
SHARING IS PARTICIPATING (parts unite with the whole)
Quoting TonesInDeepFreeze
So there you have my reply. What was your point in asking your question?
(1) I don't know your meaning of 'homological' applied to relationships between a mathematical theory and empirical observation.
(2) There are two senses of 'model'. The first is the formal notion of a model for a language: a state-of-affairs is a model for a mathematical language, and the state-of-affairs is a model of a mathematical theory if and only if the theory is true in the state-of-affairs. The second sense is the exact reverse of the first sense: a mathematical theory "modeling" states-of-affairs such as those observed empirically. Personally, in order to be clear at all times, I prefer to use only the first sense.
Quoting ucarr
If we must use the word 'signifier' here, I would say that the signifier is not a model but rather a theory.
Quoting ucarr
I put it this way: There is no model of a contradictory theory. (That's for classical logic. We may find other things pertain in other kinds of logic.)
Quoting ucarr
It is widely viewed in the study of logic that classical first order logic does not exhaust all the forms of reasoning about many kinds of subject matter.
While I don’t purport to be an expert in applied formal logics, this seems worthwhile to mention:
All apparent paradoxes not accounted for via (1) or (2) might also be accounted for in principle by multi-valued logics, with fuzzy logic as one variant. MVLs can, for example, take into consideration things such as partial truths, or else partial falsehoods (e.g., on which a great deal of spin and misinformation is for example dependent, this in real-life applications of truth-values).
Multi-valued logic does not reject the LNC.
In the sense you mention a 'truth predicate', we actually say a 'truth function'. On the other hand, as to truth predicates, (Tarksi) for an adequately arithmetic theory, there is no truth predicate definable in the theory.
For a language, per a model for that language, in a meta-theory (not in any object theory in the language) a function is induced that maps sentences to truth values. It's a function, so it maps a statement to only one truth value, and the domain of the function is the set of sentences, so any sentence is mapped to a truth value.
And, (same Tarksi result said another way) for a semantic paradox such as the liar paradox, the statement can't be asserted in any arithmetically adequate consistent theory, so it is not mapped to any truth value.
But let's go back to the earlier context in which a poster claimed that the paradoxes ruin classical logic for mathematics:
We must distinguish between informal paradoxes and formal contradictions. We know (the soundness theorem) that classical first order logic alone does not prove contradictions. In the case of the Russell set, first order logic itself proves that there is no such relation such that for all objects x there is an object y such that x bears the relation to y if and only if y does not bear the relation to itself. In the case of the liar paradox, we have the result that the liar sentence is not formalizable in an adequately arithmetic consistent theory. And, as far as we know, and by certain arguments, the mathematical axioms of set theory are consistent.
So if the informal paradoxes motivate us to view them as needing to be allowed formally, then we do wish to allow contradictions in theories but not have them explosive, and then we adopt a paraconsistent logic instead of classical logic. But that is not the ruination of classical logic.
With a paraconsitent logic, one can have both LNC and non-explosiveness. In such a logic, we may have LNC as a theorem (or theorem schema) and also have each conjunct of a contradiction as theorems and also have non-explosiveness. You can look it up yourself.
There are systems with all three: LNC, contradictions, and non-explosiveness. You can look it up yourself; you can educate yourself about this subject on which you are so opinionated yet so ill-informed.
Quoting TonesInDeepFreeze
From Internet Encyclopedia of Philosophy:
“strong paraconsistency includes ideas like:
Some contradictions may not be errors;
classical logic is wrong in principle;
some true theories may actually be inconsistent.”
Sounds like strong ( as opposed to weak) paraconsistency does see contradictions as ruinous not classical logic.
Yes, so?
Sounds like strong ( as opposed to weak) paraconsistency does see contradictions as ruinous to classical logic.
As someone largely ignorant of formalized logics:
The less technical format of the liar paradox (contrasted to "this sentence is false") is that of “someone who is a member of X claims that all Xs are liars”.
-- This can be true in the sense that all members of X have at some point in their lives told lies, thereby being definable as liars. Non-paradox.
-- It can also be true in the sense that all members of X have a larger than normal propensity to tell lies, thereby again being deemed liars. Also non-paradoxical.
-- The claim can only become paradoxical when interpreting that all members of X can only strictly communicate via lies, without any exception. Yet this interpretation is contradictory to the real-life occurrence of liars.
While I don’t know how to translate the aforementioned into formal logic, I do find that the claim “I am a member of humankind, and all humans I know of (including myself) are liars (on account of having lied at some point in their lives)” to hold a truth value.
Again, as someone ignorant in the formalities of the matter, cannot this latter sentence as expressed be mapped to a truth value in formal logics?
Or would one argue that the sentence as expressed (a variant of “someone who is a member of X claims that all Xs are liars”) does not present a valid format of the liar paradox?
Of course, if we wish to have theorems that are contradictions but without explosion, then classical logic doesn't work, and if one wishes to have contradictions without explosion, then one may say that classical logic is thereby wrong.
But the particular argument given by Agent Smith does not vitiate classical logic itself. Classical logic works just fine for a vast amount of the logic for the sciences. You and I are now communicating with computers built by principles of classical logic. And that does not overlook that for arguably paradoxical statements in a wider scientific context, classical logic may be inadequate.
In particular, Russell's paradox does not ruin classical logic, which gives us an immediate and wonderfully simple solution with the theorem that there does not exist such a contradictory relation.
And "traditional" mathematics. But with over 26,000 topics in math it's getting harder to pigeonhole.
Rather than get bogged down in whatever vagaries there might be in the Epimenides paradox, I would suggest the clearer, simpler, mathematically "translatable" simpler and more starkly problematic "This sentence is false".
Quoting javra
No arithmetically adequate and consistent theory can define a truth predicate by which to then formulate a predicate 'is a liar'.
Keep in mind that Tarski's theorem is a claim only about certain kinds of theories (arithmetically adequate and consistent) formulated in classical logic.
However, given predicates Hx for 'x is a human', and Sxy for 'x states y', and Ly for 'y is a lie', and m for you, then we can write:
Ax(Hx -> Ex(Sxy & Ly)) ... "All humans lie sometimes"
Ey(Smy & Ly) ... you lie sometimes.
And those would have truth values.
But so what? It's not in question that we can formalize a lot of non-problematic things. That we can formalize a lot of non-problematic things doesn't refute that "This statement is false" is problematic.
Yes, I didn't write that correctly. What I meant:
Classical logic with added mathematical axioms works just fine for a vast amount of the mathematics for the sciences.
Thanks for the reply!
Quoting TonesInDeepFreeze
I know. But to me this derivative of the Epimenides paradox is as much pure gibberish as is the what might be called paradoxical proposition of, "A square is a circle". To each their own, maybe.
But it's not gibberish. It's syntactical and it talks about the property of truth as pertaining or not to a given sentence, which is a well understood notion. We don't throw out expressions from the language merely because they present logical problems. The expression is well formed; it is only upon further analysis that we find it is problematic. It would be poor analysis to throw out sentences ad hoc only on the basis that de facto they are problematic.
Quoting javra
That's not paradoxical. Rather, with definitions of 'square' and 'circle' and some theorems of mathematics, it simply, without any paradoxical aspects, implies a contradiction.
True. Maybe, more in keeping with "this sentence is false", might be "this square is not a square". But I get the the former has a more of self-referential aspect then the latter.
Still, I'd be grateful to hear of any notion regarding why it should be taken seriously as a proposition. This when something like "this square is not a square" is not. For example, can the proposition be deemed a necessary valid deduction from a grouping of incontestably true premises?
"This square is not a square" is seen as a self-contradiction on its face, and its truth value is falsehood, and there is no contradiction in saying its truth value is falsehood.
"This sentence is false" also implies a self-contradiction, but it is not so easy to say its truth value is falsehood, since if its truth value is falsehood then its truth value is truth and if its truth value is truth then its truth value is falsehood.
I am not familiar with Wittgenstein's views on Moore's paradox.
I keep on overlooking the subtleties. You've pointed them out well.
Here's a better justification for why I find the liars paradox to be gibberish:
TMK, given the LNC, a contradiction between X and Y necessitates one of the following three: a) X is valid but Y is invalid, b) Y is valid but X is invalid, or else c) neither X nor Y are valid. But given the LNC, possibility d), that of both X and Y being valid, will be excluded as impossible.
I'm saying "valid" as shorthand for this applying not only to propositions but also to non-propositional criteria, such as percepts and memories. For example, if event E and event F are mutually exclusive, and if one recalls that one did E at time-interval T and also recalls doing F at T, one could then assume a) having a false memory of E at T but not of F, b) having a false memory of F at T but not of E, or else c) having a false memory of both E at T and of F at T. But one does not conclude that both E and F happened at T.
I'm hoping this makes good enough sense without me needing to engage in more in-depth explanations.
If so, applying this type of general rationale to the self-contradiction of the liar paradox, it can be a) true but not false, b) false but not true, or else c) neither true nor false - but, given the LNC, it cannot d) be both true and false at the same time and in the same way.
We know that if we claim (a) it will also be (b) and that if we claim (b) it will also be (a) - with amounts to (d).
This leaves us with possibility (c): neither true nor false.
If so, this amount to the liar's paradox being syntactically coherent gibberish: a statement devoid of any possible truth value.
I might be somehow wrong in this general perspective - and if you have the time to point out how, that would be appreciated - but it's how I've so far appraised the liar paradox: as being syntactically correct gibberish.
In mathematical logic, 'valid' is used differently from the way you use it. Here's a quick breakdown of the terminology for ordinary first order logic (where 'P' and 'Q' stand for any statements, 'G' for any set of statements, and 'iff' stands for 'if and only if'):
Definitions:
P is true in model M iff [fill in the inductive definition here].
P is false in model M iff it is not the case that P is true in model M.
P is valid iff P is true in every model M.
P is invalid iff P is not valid.
P is contingent iff P is invalid but P is true in at least one model M.
P is a theorem of G iff there is a deduction of P from G.
Meta-theorems:
P is valid iff P is derivable from the logical axioms alone.
P is a theorem of G iff any model in which all the statements in G are true is a model in which P is true.
P is a contradiction iff P is of the form: Q & ~Q. (But sometimes, less formally, we say P is a contradiction iff there is a statement Q & ~Q derivable from {P}).
If P and Q are a contradiction (in the sense that a contradiction is derivable from {P Q}), then there is no model in which both P and Q are true. So, if P and Q are a contradiction, then given any model M, either P is true in M and Q is false in M, or P is false in M and Q is true in M, or P is false in M and Q is false in M.
Quoting javra
Not in classical logic. If M is a model for the language in which P is written, then P is true in M, or false in M, and not both true and false in M, and not neither true nor false in M.
That is, classical logic is 2-valued and the semantics upholds the theorems:
~(P & ~P)
P v ~P
Quoting javra
No. First, your argument is semantical not syntactical. You can't make something not syntactical by a semantical argument. Second, you are overlooking another possibility that you are not seeing but that I have mentioned:
The statement cannot be formulated in the theory such that the interpretation of the statement refers to truth values. That is, as I've said, Tarski's theorem is that an arithmetically adequate and consistent formal theory cannot formulate its own truth predicate. If the theory does formulate its own truth predicate then the theory is inconsistent. An arithmetically adequate and consistent theory lacks axioms (assumptions) that would provide expressing a truth predicate for the language in which the theory is written. This is different from the naive notion that "we can't admit the liar sentence as a legitimate sentence", which is not rigorous because it requires ad hoc and post facto fiats about what is a legitimate sentence. Instead, the syntax is objective, decidable, and unchanging, while we do see that among the syntactical sentences there is none that can be interpreted (semantics) as expressing the truth predicate for an arithmetically adequate and consistent theory.
.
The naive notion is that a set of statements G entails a statement P iff it is impossible that all the members of G are true but P is false.
The rigorous model theoretic notion is that a set of statements G entails a statement P iff there does not exist a model in which all the members of G are true and P is false.
So, instead of the modal notion 'possible', we have the more fundamental notion 'exists'.
In the real world we don't use sentences as truth bearers. I don't think we need to break from ordinary language use in assessing Russell's paradox. I'm just pointing out that the solution you've been talking about is artificial.
Sure we do.
"Provo is in Utah" bears truth.
"Provo is not in Utah" bears falsehood.
See the Introduction in Alonzo Church's 'An Introduction To Mathematical Logic', which is wonderfully cogent, beautifully written, and arguably the very best overview of the subject.
Quoting Tate
Russell's paradox was first presented in context of formal theories. And, at least usually, the interest in Russell's paradox centers around mathematics.
Quoting Tate
I don't know how you evaluate for "artificiality". However, of course, since the subject of mathematical logic is conveyed courtesy of human intellect, I guess it's "artificial" in the same sense that just about any other area of study presented by humans is "artificial". Moreover, even if arguably mathematical logic is especially artificial in some sense, it is an excellent artifact - the product of the great intellect and sagacity of many rigorously critical scholars - that makes rigorous sense of many notions that otherwise would suffer from the vagaries of amphiboly and subjectivity. Wouldn't it be better to learn about the subject rather than glibly dismissing it out of hand as "artificial" before familiarization with even its basics?
Anyway, it's not clear to me that you understand the solution per mathematical logic.
No, we don't. A sentence has to be contextualized by some form of utterance to qualify as a truthbearer.
Quoting TonesInDeepFreeze
I think you'll need more weight than this offers to show that we can't evaluate Russell's paradox using ordinary English rules.
Quoting TonesInDeepFreeze
It's artifical in the sense that we could change it if we wanted to, at least we can imagine doing so.
Quoting TonesInDeepFreeze
I do.
Oh come on, of course we admit that natural language utterances don't have a single definitive unequivocal context. But given some reasonable understanding of given contexts, we do view sufficiently clear sentences as being true or false. When I say "Provo is in Utah" as we both reasonably understand the ordinary context, we agree that that sentence bears truth.
But, indeed, it is mathematical logic itself that rigorously explicates the notion of context by the method of models.
I said that we can evaluate it by formal methods. I didn't say that we must evaluate it only by formal methods.
A sentence is no more than a string of words that conforms to some linguistic rules. Once you add context you have more than just the sentence. You have a statement. The statement can have the property of truth. The string of words can't, not in ordinary language use.
This is pretty standard stuff.
Yes, mathematical logic offers the freedom for anyone to present alternative formulations, definitions, methods, and paradigms. That's a good thing.
In any case, ordinary language and ordinary naive approaches not only can be imagined to change but we know that they do change.
And this makes a world of difference. If we can evaluate it by ordinary standards, the paradox stands.
I'm sure it is good for many purposes. But a solution that's subject to revision is not a strong solution.
Quoting TonesInDeepFreeze
I don't propose that they change anything.
Again, you're not seeing the point among your unnecessarily split hairs.
Sometimes informally we use 'sentence' and 'statement' synonymously. Whether or not to do that is a matter of choice in definition. We don't need to get bogged down in disputes about such choices. Meanwhile, the distinction you mention is usually made in logic as the difference between a sentence and a proposition. And there it becomes a matter of the particular development of the subject whether we say that sentences bear truth values or whether only propositions bear truth values.
The way mathematical logic does it is this: A sentence is a syntactical object. It has no truth value as merely a syntactical object (except valid (i.e. logically true) sentences that are true in every model). However, given a model for the language of the sentence, there is a truth value for the sentence per that model (a model being an "interpretation of the language", i.e. the meanings of the words and then meanings of sentences as they are built from the meanings of the words). So, when I informally say that sentences bear truth, of course, more formally I mean they bear truth per a given model. In the case of "Provo is in Utah" I mean the ordinary interpretation we share of the city we know of and its location in the state we know of.
I have never proposed any argument that it is not paradoxical in ordinary language.
It's subject to revision in the sense that anyone can propose different approaches. Meanwhile, in terms of its ordinary mathematical context, it has proven to be pretty strong as it permits a (presumably*) consistent axiomatization of the mathematics that the system found to suffer from the contradiction also intended to axiomatize.
* Presumably only, because, one can always doubt the axioms or principles used for any consistency proof of even PA.
It's not me splitting the hairs. AP gets very specific about what a sentence is when comparing Tarski's project to ordinary language use. If you wander through the SEP articles touching on the issue you'll get up to speed pretty quickly.
Quoting TonesInDeepFreeze
You're providing a context for the sentence, so it's more than just the string of words. It's a statement.
I think we're broadly in agreement.
And I don't propose any specific changes to the explication of the paradox per mathematical logic. On the other hand, no matter what you propose or do not propose, natural language changes drastically, so if change is your determinant of 'artificiality' then natural language is quite artificial too.
I'm open to being corrected, but I don't think we can imagine changing the rules of natural language the way we can imagine changing a formal system.
Your condescension is belied by comparing our familiarity with the subject.
And you just skipped what I wrote about this. It depends on the author whether 'sentence' and 'statement' are taken as synonymous or whether 'statement' is taken only as 'proposition' or a similar rubric.
I happened to be using 'statement' for 'sentence'. That is not essential. I could just as easily say that, since that usage conflicts with other usage you have come across, then I could confine to 'sentence' and 'proposition' or whatever stipulated uses we choose to agree upon. When reasonable people find an innocent and understandable terminological clash, they may accept from one another that they just happened to have different meanings in mind and then agree to a shared meaning going forward.
Quoting Tate
Please, you purely disregarded what I said about that. I already agreed that sentences do not have truth values without an interpretation. Do you ordinarily go around disputing people when they say things like "He spoke spoke a true sentence when he said 'Provo is in Utah', and you would continue to dispute them even when they granted that some people don't take 'sentence' in the same sense as 'statement' so to take their remark, mutatis mutandis, per whatever agreed upon stipulation so that, of course, one means an utterance or expression in combination with some interpretation of the meaning of the words.
Ordinary language changes in the course of millions of individual choices toward variation but also sometimes in decisive strokes. If you wish to argue that that mitigates that even ordinary language is artificial, then okay I suppose. But then I don't see much persuasiveness in the argument that mathematical (especially mathematical logic) has its explanatory potency diminished by the fact that it always can be augmented in clear, unambiguous, and rigorous ways. Moreover, empirical sciences are always subject to emendation, so we would take them too as "artificial" and dispute their explanatory value while giving a fair amount of weight, at least as far as the criterion of artificiality, to more naive explanations, even superstitions, that have been more stable even if ignorant.
You don't know really anything about the subject of mathematical logic, yet you are persistent to somehow fault it in a quite flimsy way. I wonder why.
You'rr correct! The properties of the whole tend to be inexplicable from a parts point of view (holism: the whole is not just the sum of its parts). The fact that a Unified Theory of Everything has eluded us till now is evidence of that I suppose (the world of the small doesn't quite jibe with the world of the large).
I didn't say anything about its explanatory potency.
Quoting TonesInDeepFreeze
You don't appear to know the basics of the philosophy of truth, so we're even. :razz:
I don't know how you ever came up with the strawman that I don't take the statement as paradoxical in its everyday language context. Indeed, very much to the contrary, I have argued that it is a formalized version that provides not paradox but merely that for any relation R, ~AxEy(Ryx <-> ~Ryy).
You did in so many words. If you object to the paraphrase, then substitute the actual words you used.
Quoting Tate
Appears to you, whose perception is poor.
Moreover, even as I grant that I am not expert in philosophy, I do know some basics, and I point out that I haven't made very much, if anything, in the way of philosophical claims. Mainly I addressed the technical matters that are behind certain claims about logic and mathematics made by another poster. And even though I am not a true expert in those technical matters, I know enough to see outline the nature of his mistakes.
Quoting Tate
No, because I do know at least something about philosophical notions of truth, while you know virtually nothing about the context of mathematical logic that was the context of my remarks regarding the poster's claims about the logic used for mathematics and about paraconsistent logics.
What does 'AP' stand for? And what article online (if it's online) do you refer to in relation to Tarski?
AP: Analytical philosophy.
A fair chunk of AP as it relates to truth revolves around Tarski. I didn't refer to any specific article.
To my understanding neither multivalued logic nor fuzzy logic deny the LNC. They seem to be about truth value, how many of them are there or necessary to make sense of reality to be precise - they both reject the principle of bivalence though.
A system for use with a multi-valued semantics can be paraconsistent or not.
However, as far as I know, a paraconsistent system can't have a classical 2-value semantics.
And a while back I corrected your misconception about LNC and paraconsistent logic. A paraconsistent logic, depending on its formulation, can have both LNC and non-explosiveness.
:ok: An important point!
Also, don't forget that a paraconsistent logic, depending on its formulation, can have both LNC and non-explosiveness.
Suppose there is a person who shaves all and only those who do not shave themselves.
There 'shaves' is in place of 'is a member of'. And a solution works analogous to the mathematical approach:
There is no relation 'x shaves y' such that for all x there is a y such that y is shaved by x if and only if y does not shave y (there is no person such that that person shaves all and only those who do not shave themselves). Just as, in mathematics, there is no membership relation such that for all x there is a y such that y is a member of x if and only if y is not a member of y.
For "This sentence is false" we have something similar: In a consistent theory, there is no sentence that says of itself that it is false.
I'm trying to use homological in a parallel with onomatopoeia as it's used pertaining to verbal language.
Onomatopoeia - a word that sounds like the noise it describes. Examples - boing, gargle, clap, zap, and pitter-patter
Thus a math expression homological to a state-of-affairs, as specified in our example here, expresses contradictory conclusions that are both valid.
Quoting TonesInDeepFreeze
You're telling me that a math expression that asserts a claim is nonetheless considered theoretical?
Quoting TonesInDeepFreeze
You're telling me that all legal permutations of classical logic expressions are devoid of contradictions?
If so, it must be the case that classical logic parameters categorically exclude contradiction.
If so, this is an example of a mathematician modulating axioms to fit a metaphysical principle (LNC).
If so, then, in the wake of QM, a mathematician can re-jigger axioms to admit contradictions, which action, you suggest, has already been taken.
'valid' has a technical meaning. I wonder whether you are using 'valid' in some other sense.
Anyway, a formula is a contradiction if and only if it is of the form "P & ~P" (or sometimes we say it is a contradiction if and only if it proves a formula of the form "P & ~P").
I'm taking classical logic as the context throughout this discussion unless mentioned otherwise:
A contradiction is not true in any model, so, a fortiori, it is not the case that there is a contradiction that is true in every model (i.e. there is no such thing as a contradiction that is valid).
Quoting ucarr
No. 'Theory' in such contexts is has not the same sense as 'theoretical'. A theory is a set of sentences closed under deduction. An interpretation of the language for a theory may be of any kind of entities or states of affairs we may wish to stipulate, not just "theoretical" ones.
Quoting ucarr
'parameters' has technical meanings in mathematical logic and mathematics. I don't know what you mean personally by "logic parameters". And I don't know what you mean by "categorically exclude contradictions". I can tell you that in classical logic:
We can write contradictions. We can put any contradiction as a line in a proof.
There is no model in which a contradiction is true. So, a contradiction entails every sentence (i.e. the semantic version of the principle of explosion ).
A consistent theory never has a contradiction as a member (i.e. no contradiction is a theorem of a consistent theory).
An inconsistent theory has every sentence as a member (i.e. with an inconsistent theory, every sentence is a theorem of that theory, i.e. the principle of explosion), so, a fortiori, an inconsistent theory has every every contradiction as a member (i.e. with an inconsistent theory, every contradiction is a theorem of that theory).
Quoting ucarr
I guess by "moduating axioms" you mean "choosing axioms".
Yes, the axioms of classical logic are chosen so that with them we achieve soundness and completeness:
A sentence P is provable from a set of sentences G if and only if there is no model in which all the members of G are true but P is false.
The logical axioms are true in every model, and there is no model in which a sentence and its negation are both true. So, it is not the case that there is a sentence and its negation that are both provable from the logical axioms alone.
Quoting ucarr
One can depart from classical logic and allow that a system can prove contradictions but without explosion. However, the semantics for such a paraconsistent framework may be more complicated than the 2-value semantics for classical logic (which for the portion that is the sentential calculus alone, are quite simple). So I don't recall for paraconsistent systems, how, if at all, one specifies the difference between a logical axiom and a non-logical axiom or non-logical premise.
That's true! I cannot understand why more people don't lie awake at night about it! It requires a global response. We need the World Bank, the World Heath Organization, the International Court of Justice, Interpol, the United Nations Security Council, and the entire cast of 'Glee' on this!
Either that, or Agent Smith just doesn't know the meaning of 'existential threat'.
That's a paradox itself. :cool:
:snicker:
Paradoxes are an existential threat to epistemology (truth) & logic. When these two are assaulted (successfully), our world comes crashing down around our ears!
:cool:
How so?
There have been controversial puzzles in epistemology for centuries. I don't see any crashing down of the world related to this. What do you think will happen, the media will announce that philosophers still haven't reached agreement on solutions to the logical and linguistic paradoxes and then the financial markets will all collapse, followed by all the populations lapsing into chaos and war?
Like the tv show Lost in Space. The robot flashing lights, that does not compute, that does not compute!
It's hard to say what'll actually happen - chaos is inherently unpredictable! All I can say, with a fair amount of certainty, is we would be utterly baffled by everything, aren't we already?