Right, so what's with complete generality? Why not say all logics. — Cheshire

The impression I got was that "complete generality" doesn't commit you to quantifying over logics. A principle holding in complete generality, being understood as the entailment relation being the same for all logics, would need to contend with the fact that you can arbitrarily make systems that prove a claim and corollary systems that prove its negation when they share the same set of symbols.

So you can come up with a logic where modus ponens holds, and come up with a logic where modus ponens does not hold. Which would mean that if you wanted to find The Logic Of All and Only Common Principles (tm), you'd need to jettison modus ponens. Since it is not a common principle, since two logics disagree on whether it is a theorem.

The paper gives lots of strategies for coming up with schematic counter examples to many, many things. You can come up with scenarios where even elementary things like "A & B... lets you derive A" don't hold. So much would need to be jettisoned, thus, if The Logic Of All and Only Common Principles was taken exactly at its word, in the sense of intersecting the theorems proved by different logics.

And that's kind of a knock down argument, when you consider X is true in system Y extensionally at any rate (which is AFAIK the standard thing to do)

Phrasing it in terms of "complete generality" thus gives a whole lot of wiggle room regarding what it would mean for a principle to hold in complete generality, like you might be able to insist somehow that any logic worth its salt must have LEM, or any logic worth its salt must have modus ponens as a theorem. So that sense of "complete generality" (NOT completeness of a logic) might mean "in every style of reasoning", so it would let you think of some logics as styles of reasoning and some as not.

You also have the opportunity to think of informal logical principles as holding in "complete generality", as eg if someone believes that the No True Scotsman fallacy is fallacious in some sense, an argument definitively establishing its fallaciousness might be considered a theorem of The Logic of All and Only Common Principles, even though No True Scotsman doesn't admit an easy formalisation. Next paragraph is just extra detail supporting that it doesn't have an easy formalisation.

Just for extra detail, No True Scotsman doesn't admit of an easy formalisation in terms of predicate logic because deductively it kind of works. If x is always p( x ), and someone provides an example of x such that ~p( x ), it should be taken as a refutation. But the fallacy corresponds to interpreting the person providing the counterexample as instead providing an example of someone for whom some distinct property q( x ) holds where q( x ) != ~p( x ). Which isn't exactly a fallacy, it's a reinterpretation, and sometimes it's a good thing to do when arguing - sometimes people make bad counterexamples. But what makes it a fallacy is somehow that the suggested q( x ) only has irrelevant distinctions from ~p( x ), like a true Scotsman is just a Scotsman. You could also read it like the the asserter that x is always p( x ), upon receiving the counterexample, clarifies their position to some predicate q( x ) such that the counterexample given does not apply to it while still using the same predicate label ("Scotsman"). In that case the fallacy consists of revising the content of the claim to "just" exclude counterexample for no other reason, which deductively is without problem, but provides another irrelevant distinction. In either case, the sense of irrelevance of distinction is the thing which is so norm ladened and contextually situated that you're not going to be able to put it into a logic without (unknown to me) profound insight about logical form in natural language.

So if you wanted to have the fact that No True Scotsman is a fallacy as a "theorem" of The Logic Of All and Only Common Principles, maybe your whole logic needs to be informal to begin with.

and are you willing to say that the fanfiction can be good or bad? — Leontiskos

Yes. Harry Potter and the Methods of Rationality is definitely better written than My Immortal.

I think such things are useful, but I also think that at some point we have to venture out beyond the bay and into the open sea. — Leontiskos

I do enjoy the open sea, I just tend to think its openness is necessary. If you'll forgive me the excess of portraying metaphysical intuition through vagueness.

There is also a really odd thing that happens constantly on TPF (and it usually happens with SEP). Someone will champion a position like logical pluralism or dialetheism or something like that, but when it comes down to the question of what exactly they are promoting they are at a loss for words. They don't have any clear definition of, say, logical pluralism. — Leontiskos

I would be pretty happy to defend logical nihilism as set out in Russell's paper.

You are the one who brought up Euclid in the first place, but I really don't see the two descriptions as competing. — Leontiskos

Ah. That's unfortunate. Euclid's definition makes the great circle not a circle. The closed curve one makes it a circle.

Pretheoretical or intuitive reasoning need not be quantified, does it? In making that comment I was making the point that pretheoretical reasoning represents the same basic idea as the calculus definition you gave. "...In calculus [consistent roundness] cashes out as a derivative, but folks do not need calculus to understand circles. Calculus just provides one way of conceptualizing a circle." — Leontiskos

It's the same basic idea, yeah. When understood in the context of a circle. You can think of curvature as a more general concept than roundness, since curvature's also "pinchiness" and "pointiness". and "flatness" etc all rolled into one. So it's sort of like roundness is to curvature as apples are to fruit.

Or rather, producing thing that can produce discursive knowledge. And knowing a true logical system is a kind of knowledge, which is probably discursive. I think that's right. But they are prepackaged in a very relevant sense, particularly for those of us who are not their inventors. — Leontiskos

It's both innit. Getting the definitions right is one thing - yay, you have found the commonality between circles. Using the definitions to produce even more knowledge is another.

But I also don't think a logic like Frege's is merely a model, nor that it could be. To invent a logical system is to attempt to capture a (or the?) bridge to discursive knowledge, and I don't know that any success or failure is complete.

I don't think any of the examples we've discussed so far is "merely" a model, since the different frameworks place much different commitments and demands on the behaviour of people that use them.

One of the great things about producing formalisms is that they're coordinative. If you and I operated on the constant curvature definition, we'd be committed to the same beliefs about circles. The same with the Euclid one. When you add that to our ability to mathematise abstractions expressively in a common language, you end up being able to write down the mathematical rules one must follow when dealing with an abstraction - just in case you have successfully defined it in the symbols. At that point, whether it is the right abstraction for the job seems a different issue.

I certainly wouldn't tell my students that a circle is a closed curve of constant curvature, I'd show them examples of circles and just say "like this". Roll them about. Measure them. I wouldn't even show them Euclid, or try to define the shape. For a lot of things you can get an okay idea of what they are without a formalism, but that loses its charm when you need to explore things that have less straightforward intuitions associated with them.

Like the example I gave of continuous functions vs Darboux functions (functions with the intermediate value property). Mathematicians thought those were equivalent for a long time based on pretheoretical notions.

We could say that a circle is a [closed] figure whose roundness is perfectly consistent.* There is no part of it which is more or less round than any other. — Leontiskos

That reads disingenuously to me. Your use of "roundness" previously read as a completely discursive+pretheoretical notion. If you would've said "I think of a circle as a closed curve of constant curvature" when prompted for a definition, and didn't give Euclid's inequivalent definition, we would've had a much different discussion. I just don't get why you'd throw out Euclid's if you actually thought of the intrinsic curvature definition... It seems much more likely to me that you're equating the definition with your previous thought now that you've seen it.

The latter of which is fair, but that isn't a point in the favour of pretheoretical reasoning, because constant roundness isn't a concept applicable to a circle in Euclid's geometry, is it? Roundness isn't quantified...

Presumably it is an intuitive concept, and are intuitive concepts mathematical formalisms? — Leontiskos

Mathematical concepts tend to be expressible as mathematical formalisms, yeah. And if they can't, it's odd to even think of them as mathematical concepts. It would be like thinking of addition without the possibility of representing it as +.

not a prepackaged formalism. — Leontiskos

And therein lies a relevant distinction. Formalisms aren't prepackaged at all. In fact I believe you can think of producing formalisms as producing discursive knowledge!

Edit: And why can't a quibbler say that R^3 and even R^2 spaces are not Euclidean? What's to stop him? When is a disagreement more than a quibble? — Leontiskos

Oh. Because the definition of a Euclidean space, in the modern sense, includes both. They're infinite expanses of points whose interpoint distances are given by straight line distance. In the old sense, in Euclid's sense, only R^2 could be, since R^3 isn't a surface.

"If the center was deleted—per impossibile—then there would only be an Aristotelian Circle." — Leontiskos

It's interesting really. Since deleting the point from the plane impacts lots of possible circles. There will be Euclid circles in that space which are not Aristotle circles too, I believe. Though I'm not totally convinced.

But the deeper issue is that I don't see you driving anywhere. I don't particularly care whether the great circle is a Euclidean circle. If you have some property in your mind, some definition of "great circle" which excludes Euclidean circles, then your definition of a great circle excludes Euclidean circles. Who cares? Where is this getting us? — Leontiskos

The discussion about capturing the intended concept is relevant here. The interplay between coming up with formal criteria to count as a circle and ensuring that the criteria created count the right things as the circle. That will tell us what a circle is - or in my terms, what's correctly assertible of circles (simpliciter).

That's the kind of quibble we've been having, right? Which of these definitions captures the intended object of a circle... And honestly none of the ones we've talked about work generically. I believe "A closed curve of constant positive curvature" is the one the differential geometry man from above would've said, but that doesn't let you tell "placements" of the circle apart - which might be a feature rather than a bug.

Suppose the quibbler has "deleted" the center, and therefore it can only be shown to be an Aristotle Circle? — Leontiskos

Interesting. But yes.

You stipulated that we've got to understand them in the plane in Euclid's sense, which I'll assume is R^2, and that has every point in it. So the "deletion" doesn't provide a counter model, this is similar to the "for all bivalent" thing from the paper. If we understand the definitions both to apply to the whole of R^2, if you deleted a point from R^2 we're just not dealing with R^2.

If you take the definitions and apply them on arbitrary sets, they can disagree. So, you'd begin the proof of their equivalence like "In R^2, consider...".

But why couldn't a quibbler say that their definitions disagree on account of the formal differences between them? — Leontiskos

Because every Aristotle Circle can be shown to be a Euclid Circle and vice versa.

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.