I agree, but that's why I would not say that an incline plane in a Euclidean space is definitely a Euclidean plane. I don't see that there are incline planes in Euclidean space. — Leontiskos

Then we're using Euclidean space differently. To me a Euclidean space is a space like R^3, or R^2. If you push me, I might also say that their interpoint distances must obey the Euclidean metric too. Neither of these are Euclid's definition of the plane. "A surface which lies evenly with straight lines upon itself" - R^2 isn't exactly a surface, it's an infinite expanse... But it's nice to think of it as the place all of Euclid's maths lives in. R^3 definitely is not a surface, but it is a Euclidean space.

Yep, I sympathize with him. — Leontiskos

You also disagree with him strongly if you like Euclid or Aristotle's definition of a circle. I actually prefer his, since you can think of the car wheels as its own manifold, and the one he would give works for the great circle on a hollow sphere too. I think in that respect the one he would give is the best circle definition I know. Even though it individuates circles differently from Aristotle and Euclid.

Do Euclid and Aristotle disagree on what a circle is? That sort of question is what I think lurks behind much of our disagreement, such as the deletion of points. If two people draw something differently, can they both have drawn a circle? — Leontiskos

I'm not familiar with Aristotle's definition of a circle at all. I might not even understand it. Though, if I understand it, I think the two definitions are equivalent in the plane. So there's no disagreement between them. Which one's right? Well, is it right to pronounce tomato as tomato or tomato?

Now does such a cross-section really contain a Euclidean circle? Trying to gain a great deal of precision on the answer to this question seems futile, but it seems to me that it is "correctly assertible" that it does (whatever your "correctly assertible" is exactly meant to mean :razz:). — Leontiskos

I think it contains a circle. It's just that the contraption you use to show that it contains a circle also means you need to go beyond Euclid's definition. An incline plane in a Euclidean space is definitely a Euclidean plane. An incline plane can't contain a circle just rawdogging Euclid's definition of a circle, since an incline plane is in a relevant sense 3D object - it varies over x and y and z coordinates - and thus subsets of it are not 'planar figure's in some sense. However, for a clarified definition of plane that lets you treat a plane that is at an incline as a standard flat 0 gradient 2D plane, the "clearly a circle" thing you draw in it would be a circle.

I mean, what would a university math professor think if they saw someone arguing that they can delete the point in the center of a circle and make it a non-circle? I think they would call it sophistry. They might say something like, "Technically one can redefine the set of points in the domain under consideration, but doing this in an ad hoc manner to try to score points in an argument is really just sophistry, not mathematics." — Leontiskos

I have had a similar experience to this. It was a discussion about rotating an object 90 degrees in space, and having to consider it as a different object in some respects because it is described by a different equation. One of the people I spoke about it with got quite frustrated, rightly, because their conception of shape was based on intrinsic properties in differential geometry. I believe their exact words were "they're only different if you've not gotten rid of the ridiculous idea of an embedding space". IE, this mathematician was so ascended that everything they imagine to be an object is defined without reference to coordinates. So for him, circles didn't even need centres. If you drop a hoop on the ground in the NW corner of a room, or the SE, they're the same circle, since they'd be the same hoop, even though they have different centres.

Which might mean that a car has a single wheel, since shapes aren't individuated if they are isomorphic, but what do I know. Perhaps the set of four identical wheels is a different, nonconnected, manifold.

Now does such a cross-section really contain a Euclidean circle? Trying to gain a great deal of precision on the answer to this question seems futile, but it seems to me that it is "correctly assertible" that it does (whatever your "correctly assertible" is exactly meant to mean :razz:). — Leontiskos

I can't tell if you're just being flippant here (which is fine, I enjoyed the razz), or if you actually believe that something really being the case is impossible to demonstrate in maths (or logic). Because that would go against how I've been reading you all thread.

It is petitio principii to simply insist that, say, an inclined plane is not reducible to a Euclidean plane qua circles. — Leontiskos

Can you give me a lot more words on the phrase "an incline plane is reducible to a Euclidean plane qua circles"? I'd really like to understand the predicate:

X is reducible to Y qua Z

There's not much point continuing this if you feel like it's the same thing over and over.

Thank you!

So are there rational norms or aren't there? What does it mean to "correctly understand a stipulated object"? One minute you're all about sublanguages and quantification requiring formal contexts, and the next minute you are strongly implying that there is some reason to reject some sublanguages and accept others. I suggest ironing that out. — Leontiskos

I'm saying that one can understand a language without being committed to whether it is a "correct language", and be able to say whether a given statement in it is correct or incorrect. Because the norms of the sublanguage are fixed. Like all the statements in propositional logic are bivalent, the LEM holds etc.

Where this breaks down is the intuition that propositional logic "ought" apply to all meaningful sentences. Hence the Liar and indeterminate truth values now serving as "counterexamples" in this context. They can be understood as counterexamples when one expects propositional logic to work for all meaningful sentences. This was analogised with our circle discussion.

We were talking about circles as a concept, and they have associated formalisms, we've now seen that there are different formalisms for it in different contexts, and sometimes they disagree. How can you insist that one is more correct than another? Which one is baked in the metaphysics? I don't really need you to know the final answer on it, I just want to know how you'd go about deciding it even in principle.

I've had plenty of university math. You strike me as someone who is so sunk in axiomatic stipulations that you can no longer tell left from right, and when you realize that you've left yourself no rational recourse, you resort to mockery in lieu of argument. — Leontiskos

Alright. It just surprises me that you survived all of these different things to do with maths concepts with a strong intuition remaining that there's ultimately one right way of doing things in maths and in logic, and that understanding is baked right into the true metaphysics of the world. And also seem to align this understanding with Aristotle?

Maybe "propositional logic" is as slippery as "circle." — Leontiskos

Neither of them is particularly slippery. The slippery thing is a pretheoretical conception of logic, or circles, which might be better exemplified in some ways by some theories and in other ways by others. There's wide agreement on what the theorems are in propositional logic, how it's used etc. I don't believe it makes sense to say something is slippery when the norms of its use are so well enshrined that it's taught to people the world over.

Neither of us disagree on what Euclidean, taxicab or great circles are at this point, I think. So they're not "slippery", their norms of use are well understood. The thing which is not understood is how they relate to the, well I suppose your, intuition of a circle. I seem to have a spectrum of intuitions about circles that apply in different contexts. Maybe you don't?

I am getting the impression that you have quite an all or nothing perspective on this - either there is a single unified objective system or there is a sea of unrestrained relativism and mere subjectivity over what theorems are provable in what circumstances. I would suggest that people can agree on what theorems are provable in what circumstances without an opinion about whether they're the "right" theorems. It seems to be knowing what theorems something should satisfy and having the right formalism to prove them are inextricably related in mathematical creativity and reasoning - eg:

If I had the theorems I should find the proofs easily enough. — Riemann

Which brings us onto understanding a stipulated object.

What does it mean to "correctly understand a stipulated object"? — Leontiskos

I would say that someone correctly understands a mathematical object when they can tell you roughly what theorems it should satisfy, give some examples of it, and has ideas about proof sketches for theorems about it. That means they know how it behaves and what contexts it dwells in. They know how it ought to be written down and how to write it. They know how what they imagine is captured by how they write it down, and that what's written down captures all it should capture about the object.

That's also quite contextually demarcated - eg I would say I understand differentiable bijections in terms of real analysis objects but my understanding of their role in differential geometry is much much worse, despite their major role in the latter context.

There's a bit of graph theory I work on in my spare time, regarding random fields on graphs with an associated collection of quotient graphs, and I have an idea of what I want that contraption to do, but I've yet to find a good formalism for it. Every time I've come up with one it ends up either proving something which is insane, and I reject it, or I realise that the formalism doesn't have enough in it to prove what I need to. Occasionally I've had the misfortunate of making assumptions so silly I can prove a contradiction, then have to go back to almost square one. I wouldn't say I understand the object well yet, nor what theorems it needs to satisfy, but I have a series of mental images and operations which I'm trying to be able to capture with a formalism. I would call this object "slippery", but that's because I haven't put it in a cage of the right shape yet. Because I don't have the words or the insight yet. Perhaps I never will!

Terence Tao has a blogpost on stages of mathematical comprehension in a domain of competence, if you're interested I can dig it up.

I also don't want to say that all objects are "merely" stipulated, like a differential equation has a physical interpretation, so some objects seem to have a privileged flavour of relation to how things are, even if there's no unique way of writing that down and generating predictions. I had an old thread on that, which was not engaged with due to poor writing and technical detail, called "Quantitative Skepticism and Mixtures". It's just a recipe for making largely useless models that produce the same predictions as useful ones, but have pathological properties. And the empirics aren't going to distinguish them if you choose the numbers right.

A final comment I have is that we should probably talk about the development of formalism also changing what counts as a pretheoretical intuition - cf the way of reading general relativity that undermines Kant's transcendental aesthetic, since noneuclidean geometries aren't just intelligible, they're baked into the reality of things. Also people who overdose on topology come out changed.

We seem to think about mathematics very differently. You think that a point can be deleted; that a set of coplanar points might not lie on a plane, etc. Those strike me as the more crucial disagreements. — Leontiskos

I don't know what to tell you other than you learn that stuff in final year highschool or first year university maths. If you're not willing to take that you can do those things for granted I don't know if we're even talking about maths.

A set of coplanar points could have a plane drawn through them if you had the ability to form a set in that space which was a plane... and contained them. So they wouldn't even be coplanar if you couldn't draw the plane, no? Like how would coplanarity even work if you've just got three points {1,2,3}, {4,5,6} and {7,8,9} embedded in no space.

Maybe we're talking about Leontiskos-maths, a new system. How does this one work? :P

At the heart of this thread seems to be the question of whether we can actually say that someone is wrong. — Leontiskos

Of course you can. If someone tells you that modus ponens doesn't work in propositional logic, they're wrong.

our notion of "correctly assertible" seems to be something like a subjective consistency condition, in the sense that it only examines whether someone is subjectively consistent with their own views and intentions. — Leontiskos

More normative. It's not correct to assert that modus ponens fails in propositional logic because how propositional logic works has been established. And modus ponens works in it.

Okay, but I still don't understand why you are calling this "shit testing." Why does it have that name? It sounds like you want to give counterexamples that highlight subjective inconsistencies. Fine, but why is it called "shit testing?" — Leontiskos

I used it as a joke and then ran with it. And they aren't subjective inconsistencies, they're norms of comprehension, and intimately tied up with what it means to correctly understand those objects.

Counterexamples that I've been giving don't just refute stuff, they mark sites for theoretical innovation and clarification.

will have had the understanding that the surface of a sphere has nothing like a working concept of a "planar figure" applicable to it at all. — fdrake

That isn't strictly speaking true, it's just that the generalisation of the concept of planar figure which applies to circles is so vast it doesn't resemble Euclid's one at all. You can associate planes with infinitely small regions of the sphere - the tangent plane just touching the sphere surface at a point. And your proofs about sphere properties can include vanishingly small planar figures so long as they're confined to the same vanishingly small region around a point.

Edit: or alternatively I guess you could think of shapes on a sphere's surface, but they have much different properties than those on the plane. Like triangle angles adding up to more than 180, the analogue of lines being great circles, and thus there's no parallel lines on the sphere surface.

Someone who was familiar with that, eg Srap Tasmaner, will have seen the highlighted great circle, said something like "goddamnit, yeah", and understood that the intention of presenting the image in the context of your reference to Euclid was to reference only the circle on its surface, since they will have had the understanding that the surface of a sphere has nothing like a working concept of a "planar figure" in it at all. — fdrake

Ironically enough this is similar to one of Lakatos' quips in Proofs and Refutations. I can't remember the exact wording, but he pokes fun at mathematicians for the amount of assumed knowledge supposedly self contained and fully rigorous proofs they write have. Which is also unavoidable when building on top of theories.

(I suppose you might have meant, "Testing an idea to see if it is shit," except that that is much too far away from the quibbling that I complained of.) — Leontiskos

I meant it as two complementary aspects - treating a definition exactly at its word to see what it entails. Sometimes this will entail something that seems very pathological. Eg here's an example of a curve which is discontinuous but you could draw without lifting your pen off a piece of paper or instantaneously changing the angle you're drawing at. Shit testing allows you to distinguish concepts, in the case of that curve, it provides an example that distinguishes continuity from the intermediate value property, by finding a curve which is not continuous but has the intermediate value property.

Since counterexamples like that let you distinguish concepts engendered by formalisations, they also let you try to distinguish what concept a collection of definitions are trying to capture from what concept they actually capture.

Philosophy has analogues, like Gettier cases exemplify shit testing of the justified true belief theory of knowledge. The concept "a rock a being cannot lift" is an attempted counterexample to an unrestricted concept of omnipotence. Lord of the Rings might serve as a counterexample to a strictly coherentist view of truth, since it may satisfy the definition of a self consistent and expansive set of propositions which nevertheless is not the one we live in. There is no Walmart in Middle Earth.

What I was calling shit testing is the process of finding good counterexamples. And a good counterexample derives from a thorough understanding of a theory. It can sharpen your understanding of a theory by demarcating its content - like the great circle counterexample serves to distinguish Euclid's theory of circles from generic circles. Counterexamples of this form have a modus tollens impact on the equivalence of a target concept from concepts in terms of a theory targeted at that concept understood at face value in its stated terms.

I don't think the sphere cross section's circumference is a "good" counterexample like that, since the thing cutting the sphere to make a cross section definitely is a plane, some Euclid fan will be able to talk about "enclosing space" like the disk the cross section whose boundary is the great circle is is, or the fact the circle lays in a plane, but just an incline one. But the circles you make on the surface of a sphere alone are a good counter example in that sense, because there's no centre point and no enclosed space.

I switched counterexamples mid explanation because it became evident you weren't familiar with the difference in geometry between sphere surfaces and planes, in virtue of reading the great circle as the boundary of a cross section of the sphere. And also weren't comfortable playing around with weird subsets of the plane. Those latter examples were attempts to make similar flavour counterexamples without the... nuclear levels of maths... that help you distinguish the surface of a sphere from flat space.

The incline plane does let you see something important though, you might need to supplement Euclid's theory with something that tells you whether the object you're on is a plane. Which is similar to something from Russell's paper... "For all bivalent...", vs "For any geometry which can be reduced to a plane somehow without distortion...". The incline plane can be reduced to a flat plane without distortion, the surface of the sphere can't - so I chose the incline plane as another counterexample since it would have had the same endpoint. But you get at it through "repairs" rather than marking the "exterior" of the concept of Euclid's circles. Understanding from within rather than without.

Someone who was familiar with the weirdness of sphere surfaces, eg @Srap Tasmaner, will have seen the highlighted great circle, said something like "goddamnit, yeah", and understood that the intention of presenting the image in the context of your reference to Euclid was to reference only the circle on its surface, since they will have had the understanding that the surface of a sphere has nothing like a working concept of a "planar figure" applicable to it at all.

A little too creepy for me. — T Clark

Makes sense. Almost every panel has a palpable sense of wrongness.

I learned that Blood on the Tracks published its final volume last September, and I binge read it this evening. Hands down that's the most disturbing thing I've read/engaged with in any medium.

I won't participate, probably, but I will watch the thread like a hawk to ensure it stays textual if you like @Antony Nickles. If you're leading the group and want assistance keeping this on topic, please PM me regarding any poster who isn't being sufficiently textual/exegetical and I'll come in and examine. Any actions I take to keep things on topic wouldn't be seen as formal mod actions or warnings etc, it would just be to keep something that could be very excellent indeed very on topic.

The answers seem to be, respectively, "Who knows?" and "The other guy!" :lol: — Leontiskos

I have invented a logic in which there is no other guy and no one knows who they are.

@Leontiskos - 's linked paper here seems to interface with your position much more explicitly than Russell's paper. The argument is quite sequential and not modular so skim reading would be difficult, there is nothing particularly maths or logic technical in it, but the discussion regarding whether there is a privileged logic for metaphysics - and what that would even mean - are far closer to what I think you want this discussion to be. Also @Srap Tasmaner, assuming you're interested in pursuing the thread of argument regarding formalism, "the true rules" and metaphysics earlier. Also @Joshs, because the paper has a rare Rorty vibe while being very much from the mathematics and logic flavour analytic philosophy branch.

I did enjoy it. It is also written in a very entertaining way. I would need to read it a few more times to follow the argument though.

Does the circumference of a (Euclidean) circle encircle space? Yes, two-dimensional space. — Leontiskos

You forgot that Euclid specifies a circle as a plane figure. I realise you're not going to accept that a great circle is not a Euclid circle, or that a circle in a plane at an angle isn't a Euclid circle without a repair of his definition - but please, trust someone who's wishy washy on logic that you're just wrong that Euclid's definition encompasses all circles.

I've been using the word "verbatim" to try to mean a couple of things:
A ) At face value.
B ) Using only the resources at hand in a symbolic system.

Thus Euclid's definition of a circle, verbatim, would exclude the great circle. And I keep bringing that up because it neatly illustrates the interplay between formalism and intuition and also a pluralism vs monism point.

But I never assented to any of these sorts of interpretations. — Leontiskos

And if you want to just talk about your intuitions without recourse to formalism, I don't know if this topic of debate is even something you should concern yourself with. You might not even be a logical monist in the OP's sense, since the kind of logic it's talking about is formal?

So you are ("perhaps") willing to say that there are circumstances in which one can correctly assert that there are square circles, but you won't commit yourself to there being square circles. This is odd. — Leontiskos

If you actually want my perspective on things, rather than trying to illustrate points from the paper: I'm very pragmatist toward truth. I prefer correct assertion as a concept over truth (in most circumstances) because different styles of description tend to evaluate claims differently. As a practical example, when I used to work studying people's eye movements, I would look at a pattern of fixation points on an image - places people were recorded to have rested their eyes for some time, and I would think "they saw this", and it would be correctly assertible. But I would also know that some subjects would not have had the focus of their vision on some single fixation points that I'd studied, and instead would have formed a coherent image over multiple ones, in which case they would not have "seen" the area associated with the fixation point principally, they would've seen some synthesis of it and neighbouring (in space and time) areas associated with fixation points (and other eye movements). So did they see it or didn't they?

So I like correctly assertible because it connotes there being norms to truth-telling, rather than truth being something the world just rawdogs into sentences regardless of how they're made. "There are 20kg of dust total in my house's carpet"... the world has apparently decided whether that's true or false already, and I find that odd. Because it's like I'm gambling when I whip that sentence out.

I apply the same kind of thought to maths objects, though they're far easier to build fortresses around because you can formalise the buggers. I'm gambling a lot less.

The idea behind this sort of thinking seems to be that every utterance is limited by an implicit context, and that there are no context-independent utterances. There is no unrestricted quantification. There is no metaphysics. I take it that this is not an uncontroversial theory. Here is an example of a statement with no implicit formal context, "There are no Euclidean square circles." You would presumably agree. But then to be wary of the claim that there are no square circles, you are apparently only wary of ambiguity in the terms. You might say, "Well, maybe someone would say that without thinking of Euclidean geometry." But we both know that there is no verbatim meaning of "square" and "circle," at least when subjected to this level of skepticism. . — Leontiskos

I would agree that every quantification is into a domain, and I don't think there are context independent utterances. I do not think it follows that there is no metaphysics. I'm rather fond of it in fact, but the perspective I take on it is more like modelling than spelling out the Truth of Being. I think of metaphysics as, roughly, a manner of producing narratives that has the same relation to nonfiction that writing fanfiction has to fiction. You say stuff to get a better understanding of how things work in the abstract. That might be by clarifying how mental states work, how social structures work, or doing weird concept engineering like Deleuze does. It could even include coming up with systems that relate lots of ideas together into coherent wholes! Which it does in practice obv.

I do also agree that there are no square circles in Euclidean geometries as the terms are usually understood.

But we both know that there is no verbatim meaning of "square" and "circle," at least when subjected to this level of skepticism.

I think this goes too far, you can do your best to interpret someone accurately and what they say can still be too restrictive or too expansive. Good shit testing requires accurate close reading. This is how you come up with genuine counterexamples.

This is a nominal dispute, but it won't touch on things like logical pluralism, for that question has to do with concepts and not just names. A new definition of "circle" will not move the needle one way or another with respect to the question of logical pluralism. As noted, the taxicab case involves equivocation, not substantial contradiction

I would have thought it clear how it relates to logical pluralism. If you model circles in Euclid's geometry, you don't see the great circle. But if you look for models of the statement "a collection of all coplanar points equidistant around a chosen point", you'll see great circles on balls (ie spheres, if you don't limit your entire geometry to the points on the sphere surface). They thus disagree on whether the great circles on balls are circles.

If you agree that both are adequate formalisations of circlehood in different circumstances, this is a clear case of logical pluralism.

As noted, the taxicab case involves equivocation, not substantial contradiction. — Leontiskos

The taxicab example is designed as a counterexample to the circle definition "a collection of all coplanar points equidistant around a chosen point", since the points on the edge of the square in Euclidean space are equidistant in the taxicab metric on that Euclidean space. It isn't so much an equivocation as highlighting an inherent ambiguity in a definition. And mathematicians can, and do, call those taxicab squares circles when they need to.

You can side with the thing as stated, or refine it to mean "a collection of all coplanar points Euclidean equidistant around a chosen point". Which would still fall pray to the great circle on the hollow sphere considered as is own object, since the point they're equidistant about is no longer part of the space.

The point isn't to say that we don't know what a circle is - that's sophistical - the point is to show that there are mutually contradictory but fruitful understandings of what a circle is. Which is a pluralist point par excellence.

Even going by @Count Timothy von Icarus's excellent reference:

We define logical pluralism more precisely as the claim that at least two logics provide extensionally different but equally acceptable accounts of consequence between meaningful statements. Logical monism, in contrast, claims that a single logic provides this account

The extensional difference between all of these different formalisms are the scope of what counts as a circle. A pluralist could claim that some definitions work for some purposes but not others, a monist could not.

To put it in super blunt terms, Euclid's theory would have as a consequence that the great circle on a ball is not a circle. The equidistant coplanar criterion would prove that the great circle on a ball is a circle. Those are two different theories - consequence sets - of meaningful statements. A pluralist would get to go "wow, cool!" and choose whatever suits their purposes, a monist would not.

Absolutely crystal quote, thank you.

there are square circles. — Leontiskos

My position was that there are circumstances in which it makes sense to say there are square circles, perhaps even that there are circumstances in which one can correctly assert that there are square circles, not "there are square circles" with an unrestricted quantification in "there are". Quantifying into an undifferentiated, uncircumscribed domain is a loaded move in this game. I do not imagine myself hacking into the mainframe of being to view the source code.

only one logic, — Banno

Only one type of logical law, all systems provide instances of? Only one type of truth, all systems provide instances of? I don't like it, or believe it, but it's possible.

Thinking of all systems as univocal would appear to be putting unnecessary restrictions on the development of logic. — Banno

I suppose there's a distinction between "having the same underlying concepts of truth and meaning and law" and "having different laws", maybe all the systems we've created, despite proving different theorems, have proof and truth as analogous family-resemblance style concepts in them. Maybe they have a discoverable essence.

Not that I'm persuaded.

fdrake, what is the confusion here, do you think? Is it to do with the commensurability of differing logical systems? If logical monism is the view that all logical systems are commensurable, then there is presumably some notion of translation that works between them all. I find that difficult to picture. Perhaps all logics might be found to be variations on Lambda Calculus or some other "foundational" logic, in which case there would be one true logic, begging for some wit to find a logic that is not based on that foundation. — Banno

I think it's a confusion regarding the connection of meaning to truth, and about truth. It might not be a confusion, it could be an insistence on a unified metalanguage having a single truth concept in it which sublanguages, formal or informal, necessarily ape.

It's quite suspicious that you can talk about "for all bivalent phi" in Russell's paper but also "for all phi which are true, false or neither" in natural language, and the reader will understand some birthing of new context and propagate their understanding into that context. As if there's some big Understanding Truth Machine that gazes through the eyes as soon as you see someone write down a new system of axioms.

All the while you know there's a wealth of intended objects for the symbols to capture.

begging for some wit to find a logic that is not based on that foundation. — Banno

They're always going to need semantics, too. I've no idea how to specify the connection between a syntax and a semantics without using some informal metalanguage, so there will always be some unformalised remainder I think!

I suppose the question is whether you read the necessity of that unformalised remainder as a sign that all systems should be thought of univocally, or whether you can erect little fortresses of axioms and interpretations amid the sea of chaos whose waves are one voice.

To take a few, you haven't defined the operations, commutativity relations, numbers, variables, etc. — Leontiskos

Understand them as you usually would. + and times are spelled out in the field axioms (see classical definitions). Add that subtraction of a is equivalent to adding -a. IE x-a=x+(-a)

There is an interesting question about the great circle, but the method which outright denies that the great circle is a circle can outright deny anything it likes. It is the floodgate to infinite skepticism. I think we need to be a bit more careful about the skeptical tools we are using. They backfire much more easily than one is led to suppose. — Leontiskos

To be clear you would have been compelled to deny the great circle was a circle by only using Euclid's definition of it verbatim, I would not have!

Can you? There is an idea that floats around, according to which one can give quibble-proof arguments. I don't think this is right. I'd say the idea that there is some quibble-proof level of exactness won't cash out. — Leontiskos

Sure. Here is a quibble proof argument.

Let x belong to the field of real numbers.
Stipulate that x+1=2
therefore x=2-1
therefore x=1

Where's the issue?

Has it been fixed? The "sophist" would say no, and can quibble endlessly. They might ask you to specify what exactly "I am Gillian" means; what 'I' means; what a name is; what the predication of amness means (all difficult questions). They might splice (1) and (2) into different contexts, pointing out that (1) is a third-person description and (2) is a first-person description, and that it is not clear that these two discrete contexts can produce a conclusion that bridges them. "Shit-testing" seems to have no limits and no measure. — Leontiskos

Those are quite different I believe. There's no attempt to change the verbatim meanings of argument terms in @Count Timothy von Icarus's repair, in fact there's an insistence on representing the conceptual content of what's said in spite of the means of its representation (predicate logic vs "I"). In effect, Timothy's takes the truth of the argument for granted and treats the inability of the verbatim machinery of propositional logic to reflect that truth as a failing of the logic... thus repairing the argument by explicitly spelling out the context sensitivity of "I".

Whereas your examples do not insist on taking the conceptual content of what's said for granted, indeed they're attempting to distort it. Allegorically, the logic of shit testing is that of a particularly sadistic genie - taking someone at their word but exactly at their word, using whatever pretheoretical concepts they have. The logic of your sophist is closer to doubting the presuppositions which are necessary for the original problem to be stated to begin with.

Our dispute was similar to the former - we both have the same pretheoretical intuitions about what a circle is. Agreeing on Euclid's and on the great circle's satisfaction of it. And we'd probably agree on the weird examples containing deleted points too, they would not be circles even though if you drew them they'd look exactly like circles. The issue we were having is that Euclid's definition clearly did not accurately represent our (mostly) shared pretheoretical intuition regarding what a circle was - what it looked like -, and I kept asking you to repair it.

Remember even Euclid saw fit to define a circle axiomatically. And his works exactly as planned in the plane. Just circles also live outside the plane, and thus are not bound by Euclid's plane figure definition of them verbatim.

"For all circles in the plane... (Euclid's theorems follow)" - another example which could've been in Russell's paper.

So a logical nihilist might say "Aha, "this sentence is false" disproves LEM!, we cannot use propositional logic". and Russell invites us to say: "I'm going to use propositional logic only for sentences we know satisfy LEM". The latter constrains the range of stuff you can sensibly throw into the collection of models of the logic, and so you end up filling up the semantic entailment relation again in the system by artfully removing the counterexamples.

In effect the nihilist doubt machine gets going by noticing that there's arbitrary degrees of contextual variation, and throws every available piece of crap against the expectations of logical form a universalist has (like @Leontiskos and I's discussion earlier), when ultimately only the universalist need read the nihilist doubt machine as nihilist - it's just a doubt machine, you can tell it to sod off by specifying the exact mess you're in.

Per Russell it is "the claim that there are no laws of logic, i.e., no pairs of premise sets and conclusions such that premises logically entail the conclusion." — Count Timothy von Icarus

But... P & P => Q entails Q in propositional logic, who is denying this? It does not seem Russell is:

like thinning, cut, and the sequent forms of conjunction elimination. The
reason is this: a natural interpretation of the claim that there is no logic is that
the extension of the relation of logical consequence is empty; there is no pairing
of premises and conclusion such that the second is a logical consequence of the
first. This would make any claim of the form Γ |= φ false, but it would not
prevent there from being correct conditional principles.10

And footnote ten:

A note about vocabulary: arguments are often said to be neither true nor false, but
rather valid or invalid. This is correct as far as it goes, but a principle containing a turnstile
as its main predicate can be regarded as a sentence making claim about the relevant argument.
Such a claim will be true if the argument is valid, false if it is not. Hence the nihilist can be
said to believe that there are no true atomic claims attributing logical consequence.

The logical consequence relation is preserved, even if the intended objects it's supposed to refer to can be taken as counter models. Like "This sentence is false" might be taken as a countermodel for the law of excluded middle, or the great circle might be taken as a countermodel for Euclid's definition of a circle.

Consider Russell's proof and refinement of LEM:

Either φ is true in a model M, or it is false. In the first case, φ∨¬φ is true in M because of the truth-clauses for ∨. In the second case, ¬φ is true in M because of the truth-clause for negation, and
so again φ ∨ ¬φ is true in M. So either way it is true in the model, and—since M was arbitrary—it is true in all models. So φ ∨ ¬φ is a logical truth...

So we examine our simple proof and realise that our assumption that the sentence could only be true or false is violated by the monster*. Hence our culprit is the assumption that sentences can
only be true and false. Still, perhaps there are some sentences which can only be true or false—sentences in the language of arithmetic might be like—and our result would hold for these. Our new theorem reads: for any φ which can only be true or false, φ ∨ ¬φ is a logical truth. Just as the geometry teacher dubs polyhedra which satisfy the stretchability lemma simple, so we could give a name to sentences which meet our assumption. Perhaps bivalent would be suitable. Then we can retain the proof above as a proof of:

For all bivalent φ, φ=>φv~φ

I underlined "bivalent" in the final bit, since you produced a similar repair to the argument:

1 ) Gillian is in Banf
2) Therefore, I am in Banf.

by understanding "I" as "Gillian", then adding this as a specification in the argument:

1 ) Gillian is in Banf
2 ) I am Gillian.
3 ) Therefore, I am in Banf.

Your repair could well have read "For all I-s who are Gillian", just like Russell's repair of LEM reads "for all bivalent φ".

It's also worth noting that Russell's countermodels, monsters and context specifying information (eg "for all bivalent") aren't necessarily in the object language in question. EG propositional logic just
assumes bivalent φ, so LEM applies, so you couldn't formulate a "neither" valued statement in its standard operation.

And since her countermodel of a statement which evaluates to "neither" does not have an interpretation in terms of standard propositional logic, she expands what ought to clearly be the scope of any logic of propositions to include that statement, goes "bleh, any logic worth its salt should account for this...". marks down on the page "eh, propositional logic as is works fine for bivalent φ" and then moves onto new pastures of polyvalent φ.

Russell's approach is largely telling logical nihilists not to throw the baby out with the bathwater, just because they expect logical laws to behave like The One Law To Rule Them All, a kind of context invariant divine providence.... and when they don't, why not just say they work when they work and find out where they work?

That makes sense. Equivalence classes of pre-images of projections under some relation seems like a cool idea.

Sphere, not ball. The surface. The 2-manifold. — Srap Tasmaner

With you. Yeah.

Am I getting something wrong here? — Srap Tasmaner

Nah it looks fine. I'm just confused, it's doing away with the centre by providing an equivalent construction of the centre. Which is also fine, I just want to see what you're seeing in it.

those three points determine a unique plane, — Srap Tasmaner

Are those points in the interior of the sphere or on its surface?

Do they? Isn't the question one of the questions at issue whether anything follows from anything else? — Count Timothy von Icarus

I don't think that's an issue at stake at all. If no principle holds in complete generality, they may still hold in certain well understood and well demarcated cases and contexts. Such as modus ponens in propositional logic.

The idea that (nothing follows from anything else in virtue of a valid argument) if (there are no principles which hold in complete generality) is ultimately not a premise in the OP's argument. It could be, and I believe Gillian Russel lectures as if, there are valid arguments even if there are no principles which hold in complete generality. Because she specifies what context she's speaking in. It then remains to be seen if a sense of complete generality can be restored by supplanting restricted statements - like Euclid's definition of a circle - with disambiguating phrases - like "in plane geometry".

Do they? Isn't the question one of the questions at issue whether anything follows from anything else? — Count Timothy von Icarus

Does everyone who understands a system and a proof in it believe the conclusion if the proof is correct and understood? Yes. Everyone agrees that P & P=>Q allows you to derive Q in propositional calculus. That's less about there being rules which cover everything, and more about there being followable rules. That's a followable, derivable rule.

However, it seems to be something quite different to claim that all[/ claims are true only relative to stipulated systems and that none are more true than any other. — Count Timothy von Icarus

Are any of these more true than any other? — Count Timothy von Icarus

Indeed. It doesn't seem meaningful to claim that the axiomatic systems are true or false in toto. But nevertheless, if there is a single unifying, bivalent truth concept, and two systems have incompatible theories, we should be able to say which is true and which is false. If we did not need to, we'd have to suspend that some claims are not evaluable as true or false in principle - and thus jettison bivalence by destroying the assignment mechanism of statements to truth values. And if we did need to, we'd have to claim that some systems are... false.... somehow, even when they seem to adequately represent concepts in precisely the same manner as others, just different concepts.

claims are true only relative to stipulated systems and that none are more true than any other. — Count Timothy von Icarus

I don't think logical pluralists are committed to that. Everyone agrees what follows from what stipulations. So it's true to say that "not every group is abelian". You can think of stipulations as disambiguations - which is what lemma incorporation works like.

The underlying issue seems to be that everyone can agree that eg groups have certain properties, but if you stipulate the definitions differently you change the properties. But you don't change the properties of the intended object when that object is the group, you perhaps change what the intended object is tout court.

I think that is in the direction of the intended thread topic. Because the ability to stipulate lemmas that make an axiomatic system better track an intended object's properties thereby lets you make more universal judgements about more precisely demarcated structures. Everyone will agree that Euclid's definition of circle captures plane circles, but not all pre-theoretically intuited circles are plane circles.

In effect this is a way of massaging the "complete generality" predicate in the OP's argument. You can restore a sense of "complete generality" by using lemmas, by speaking about something ultra specific and formalised you can guarantee that it works in that way for that system, the latter applies without exception. Applies without exception in the sense that "fdrake is sitting drinking tea now" is true at time of writing, and thus applies at that time without exception forever. Only "now" for those refined systems is a new lemma, allowing them to better specify their intended conceptual content.

If it is the case that different "correct (truth preserving) logics" contradict one another, what exactly are they preserving? — Count Timothy von Icarus

Jack: I don't know. We know a member when we see one... except lots of people disagree about membership. — Count Timothy von Icarus

It breaks down ambiguities in a concept, attempts to clarify and resolve them, and if the resolutions contradict each other they are presented with their merits and drawbacks. That seems like standard flavour "conceptual analysis" to me. There's just no presupposition that there's one right way of doing things, even if there is a presupposition that people can come to understand the same things with sufficient thought and chatting.

And regarding truth, truth as a concept applies to both.

Gillian is in New York
Therefore, I am in New York.

will have true premises and conclusion when and only when "I" and "Gillian" refer to the same entity. It's thus not a valid argument in the standard sense, as it can be false (the author need not be understood to be Gillian).

vs

Gillian is in New York
I am Gillian.
Therefore, I am in New York.

will be valid, as you've plugged the hole in the previous argument.

If it is the case that different "correct (truth preserving) logics" contradict one another, what exactly are they preserving? — Count Timothy von Icarus

Referencing the above, what they preserve is truth of conclusion given true premises. That is just what truth preservation means. Stipulate what you like, see what follows from it.

Whether you have true premises is a different issue. When you stipulate axioms, you treat them as true. Are they true? Upon what basis can they be considered as such?

Whether you have a true axiomatic system is a different issue again, and I don't really know what it means. How would you compare Peano Arithmetic and Robinson Arithmetic, for example? Which one is true? Is one "more true" than another? What about propositional logic and predicate calculus? These aren't rhetorical questions btw.

I would posit that axioms can be considered to be correct when they entail the intended theorems about the object you've conceived. That is, when they reflect the imagination. For example with @Leontiskos using Euclid's characterisation of circles as a plane figure, it would entail that a great circle on a sphere surface is not a circle... whereas it seems to be "contained in the intended concept" (scarequotes) of a circle. Which might lead you to reject the axioms, or insist upon them... Hence the method adopted in the paper and my dialogue with Leontiskos.

And there is a formalistic definition of truth, a statement is true in a theory when that statement holds in every model of that theory. Like "swans are birds" is true because there are no swans which are not birds, but "swans are white" is false because there are swans which are not white. Every collection of swans is a model of the term "swan", and all you need is one collection with a black swan in it to show the latter is false. Similarly if you wrote down the axioms of a group, something would be true of groups when it is true of every model of the theory induced by group axioms - the sets the groups are made of, and the set operations the group mappings use.

You can think of the latter as related to my dialogue with Leon in the following way - the intuition of a circle makes you want to put the great circle into every theory of circles, everything which describes what circles are, so if you think it should be in the theory, you have to reject (or repair) Euclid's definition.

Edit: or, my preferred option, acknowledge that "circle" is an imprecise concept in natural language and also that there are lots of different useful ways of fleshing it out.

If I'm doing something dumb, it's okay to just say that. — Srap Tasmaner

It's just a question of understanding the detail for me.

And you might then think of the center of the circle as a projection of the center of the sphere. And it is, but it's entirely optional. That projection comes after we already have the circle. It's the canonical projection alright, but you could also project that point to any point on the plane, because this projection is just a thing you're doing ― the circle doesn't need it, isn't waiting for this projection, you see? — Srap Tasmaner

1) So I pick a point A in 3 space A={0,0,10} as {x,y,z} coords.
2) I place a plane cutting the point {0,0,0} with unit normal vector {0,0,1} (that's the xy plane). The axel is parallel to the z-axis, it points in the direction of the unit normal.
4) I then pick a circle in the x-y plane, let's just say it's centred at the origin O={0,0,0} with radius r=sqrt(10), which I think is the appropriate distance to make your construction with the cone work.

The center circle O is equivalently determined by the distance sqrt(10), the point A and the choice of the x-y plane.

That connotes a more general construction.

1) I form the sphere of radius R around A.
2) I pick a projection P and a point A. I constrain the projection P that it projects onto a plane whose normal vector is parallel to the sphere radius and that norm(PA)<=R. Intuitively, you travel along a sphere radius and blow up a plane orthogonal to the radius from a point on the radius.
3) I apply P to A, producing PA.
4) I collect the points this plane intersects the sphere's surface together, this will be a circle of radius... sqrt(d(A,PA)^2 + d(PA,intersection point of plane with sphere surface)^2)
5) I have more than enough degrees of freedom in the distance expression in 4, when I can pick A and P and R, to define any circle centred at any point.

Was that the construction?

Or does the truth and validity depend on the system being used? — Count Timothy von Icarus

The validity of Russel's argument depends upon the interpretation mechanism you apply to the sentences in it, and their terms. The first formalisation of it is:

1) Gillian is in Banf,
2) Therefore, I am in Banf.

In standard predicate logic, there would be nothing saying that Gillian=I, because all you can do is assign symbols based on what's in the argument. "I" and "Gillian" are distinct referential symbols, therefore they must be parsed as different entities. In standard predicate logic, something being red does not imply that it is coloured.

If you're thinking "that's nuts", because the argument clearly is "valid" in some sense, you need to come up with a reason why. And you did just that, you mapped the argument as presented to another argument:

1) Gillian is in Banf
2) I am Gillian
3) Therefore, I am in Banf — Count Timothy von Icarus

Which is clearly valid in the original predicate logic. However the mapping between the arguments:

1) Gillian is in Banf.
2) Therefore, I am in Banf.

to

1) Gillian is in Banf
2) I am Gillian
3) Therefore, I am in Banf

is not an operation available to you in original predicate logic. It's an extra logical operation to map argument to argument like that, through the means of natural language comprehension. In effect you've supplemented the original predicate logic with an extra rule, in which you resolve coreference classes of each denoting term in the argument's sentences prior to evaluating whether the premises can be true and the conclusion nevertheless false.

You could then prove a meta-theorem that states that any argument of the first form is valid so long as it's valid in your new logic that resolves the coreference classes - any one where "I" and "Gillian" co-denote.

That is, you stipulate a set of equivalent denoting terms prior to evaluating it - in this case, you would stipulate that "I" would denote the same entity as "Gillian", which makes sense since Gillian was understood to be author.

In another interpretation of that same argument, the argument would be invalid, since when fdrake writes the argument, fdrake is the author, so we don't belong in the same coreference class.

There's a considerable ambiguity in natural language terms and concepts, which gives them a kind of cohesion through fuzzy boundaries, which can then be interpreted as a coherent unity, which seems to be @Leontiskos's method of argument in this thread, to my reckoning.

This is still circular logic. What makes one collection of cells and protoplasm a member of the human species? It is not merely the presence of a particular set of genes/chromosomes - there must be something else. — EricH

@Bob Ross

I want to "yes, and" Eric's comment. Even if you grant that a being is a member of the human species, that does not mean they count as a person or as a moral agent.

Everyday intuitions about moral agency are also limited by the status of a person. Children (especially) and young adults are treated with more lenience for behaving in a socially unacceptable manner and for committing moral wrongs, children's legal status is also different. People's status as an agent may change if they go into a permanent coma, we have next of kin rules, waivers, and even (arguably) the ability to extend our capacity for consent after our death with organ donation and wills. Moreover, unfertilised gametes and severed limbs are recognisably of the species homo sapiens and are not treated as moral persons - unless one is willing to admit that shagging, the normal functioning of fertilised ovums, menstruation and masturbation are each a peculiar brand of industrial slaughter.



To summarise, each of those entities counts as a member of the species homo sapiens, but they are not a moral agent. Some of them count as persons and have restricted or removed moral agency, some of them don't count as persons but nevertheless are treated as moral agents.

Straight lines on spheres? That's interesting too. — creativesoul

Yep! It turned out a property that uniquely characterised straight lines in our normal kind of space also applied to spheres, and it makes great circles. It's the taxicab circle thing again. Straight lines are only the things we expect in Euclidean ("flat") space. But that's an artificial restriction.

Edit: even flatness. The volume in the room you're in is flat.