Well, logical nihilism is not the position that true and false are always relative, it's the position that nothing follows from anything else. It is certainly easier to argue for it if truth is relative, but it's the claim that truth cannot be inferred. You could presumably claim that there are absolute truths, just not that there is anyway to go from one truth to another. — Count Timothy von Icarus

Pretty sure that's just a conclusion some would assert about it. Saying there's no general rule that universally ties evidence to truth is a bit different than, no logic. And I disagree, if I'm arguing there are multiple routes to a true conclusion then I'm discussing a relativistic system. If I'm just wrong by definition then it's business as usual I suppose, but those sound like secondary assumptions.

I was intrigued by this:

Deletion is shorthand for considering different sets - or using the set division operation. The sets I'm referring to were $\mathbb{R}$ and $\mathbb{R}^2/\{0\}$. — fdrake

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. — fdrake

is denying mathematician the right to write $\mathbb{R}^2/\{0\}$, and hence deny them the right to think about $\mathbb{R}^2/\{0\}$. I think this an interesting case study in what we have been discussing. Monism would have it that "you can't think that".

In terms of a puzzle analogy, this seems more like claiming the pieces don't fit together, in which case it doesn't even seem like a puzzle any more. — Count Timothy von Icarus

What if there were several puzzles mixed up? Then sometimes, some pieces would not fit together, being from different puzzles. But that does nto make the puzzles unsolvable. (Nice analogy,

You say the point at the center of a circle can be "deleted" and I say it can't, but you presuppose that there is no way of adjudicating this question. — Leontiskos

Of course there is no way of adjudicating this question. Removing the centre point is a stipulation, of the sort that mathematicians and logicians do as a matter of course. "What happens if we consider $\mathbb{R}^2/\{0\}$? Well, then we have a whole, cool new puzzled to play with..."

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.

[

This seems like a useful clarification of terms. Where I have seen the term used, and how it is used in the papers we have been discussing, the idea is that there is no logical consequence relationship. It is not that there is no general consequence relationship that obtains in all cases. The idea that there are truth-preserving rules of logical consequence but that they might vary is called logical pluralism.

This is why deflationism is question begging. You can set up the argument like so:

1. Truth is defined relative to different formalisms.
2. Different formalisms each delete some supposed "laws of logic," such that there are no laws that hold across all formalisms.
3. The aforementioned formalisms each have their own definition of truth and their systems preserve their version of truth.
C: There are no laws vis-á-vis inference from true premises to true conclusions.

A deflationary pluralist could well say this equivocates between "truth tout court" (which doesn't exist) and qualified truth relative to some system, and that the nihilist is just a deflationary pluralist with an edgy name.

The non-deflationist of any variety can say the entire argument hinges on the premise of deflation and that we are only speaking of "correct logics," which preserve truth qua truth, not a stipulated truth condition that is defined arbitrarily.

What if there were several puzzles mixed up?

Sounds like pluralism. You need to find the structure of each discrete puzzle.

Nihilism seems more to me like we all have wood blocks and jigsaws and we can cut out whatever we please. Which, as an analogy for "how does one derive conclusions from true premises," seems like a poor one if one has any notion that truth is not some sort of post-modern "creative act."

This is simply using unclear terms. It's "P is true in L iff P is true in L." Whereas "P is true it and only if P," would simply be meaningless or ambiguous. — Count Timothy von Icarus

I don't follow this, and I don't think it is only becasue you appear to have left out a few quote marks. So let's make it clearer.

"P is true in L iff P is true in L" is a simple tautology, and nothing like the sort fo thing Tarski used. The sort of thing he would have said is more like "'P' is true in L iff S" where S is a sentence in a language other than L, carefully defined so that the S is satisfied only when P is satisfied. That's what that long bit in Tarski's paper that no one reads does - it matches the names in the object language with new names in the meta language.

Quoting oneself is becoming de rigueur...

Designation and Satisfaction
So we have, as a general form for any theory of truth, what Tarski called "Material adequacy",

For any sentence p, p is true if and only if ϕ

And we want to understand what ϕ is.

And we have that in order to avoid the Liar Paradox, we avoid having a language that can talk about itself. Instead, we employ a second language, and use it to talk about the truth of our sentences. We call this the metalanguage, and it talks about the object language. Our sentence "For any sentence p, p is true if and only if ϕ" is a part of the metalanguage, referring to any sentence p of the object language and ϕ is a sentence in the metalanguage

So what is ϕ?

The obvious solution is that ϕ and p are the same. ϕ=p.

But the problem here is that ϕ and p are in different languages. In the metalanguage, p is effectively a name for a sentence in the object language.

Tarski worked around this by introducing terms in his metalanguage that refer to the same thing as terms in the object language; the notion of designation; and then using this to define truth in terms of satisfaction.

Suppose we restrict the object language to being about a group of people, Adam, Bob and Carol...

And in the metalanguage we can have a definition of "designates":

A name n designates an object o if and only if (( n = "Adam" and o = Adam) or ( n = "Bob" and o = Bob) or( n = "Carol" and o = Carol)...

Doubtless this looks cumbersome, despite my having skipped several steps, but it gives us
a metalanguage and and object language both talking about the same objects, Adam, Bob and Carol..., and a way to use the same name in both languages.

We want to add predication. To do this, Tarski developed satisfaction. Suppose we have two nationalities in our object language, English and French. We need a way of talking aobut those nationalities in the metalanguage. We can define "satisfaction":

An object o satisfies a predicate f if and only if ((f="is english" and o is English) or (f="is french" and o is french)

And so, in a cumbersome way, we have the object language and the metalanguage talking about the same predicates and objects.

Here I've used finite lists, but it is possible to construct similar definitions for designation and satisfaction for infinite objects and predicates, and for n-tuple predicates. I'm just not going to do it here. — Banno

A name n designates an object o if and only if (( n = "Adam" and o = Adam) or ( n = "Bob" and o = Bob) or( n = "Carol" and o = Carol)...

There's no "unclear terms" here - indeed, it is clear to the point of being pernickety. Hence the improt of the paper.

I believe that Tarski did not say that truth was nonsense in natural languages, but that it was indefinable. That would be a natural consequence of his theorem that a language cannot contain it's own definition of truth.

Kripke subsequently showed that a language can contain it's own definition of truth, provided one makes use of paraconsistent logic.

So with Tarski we have truth in layers of language, each one talking about the one below it. This is, speaking roughly, what is used in the iterative conception of set theory.

Speaking generally, on the one hand we have clean and clear definitions of truth within formal systems and in terms of satisfaction, and on the other hand we have a broad, ill-defined notion of truth that is supposed to be useful in adjudicating between differing logics as well as in natural languages.

1. Truth is defined relative to different formalisms.
2. Different formalisms each delete some supposed "laws of logic," such that there are no laws that hold across all formalisms.
3. The aforementioned formalisms each have their own definition of truth and their systems preserve their version of truth.
C: There are no laws vis-á-vis inference from true premises to true conclusions. — Count Timothy von Icarus

Is that conclusion supposed to follow? That there are no universal laws does not deny that there are laws specific to each logic.

It is maybe worth pointing out that if someone proposes a new logic, they are obliged to set it out for us to see it, and we can judge it's consistency within itself, as well as its applicability to various situations in comparison to other logics.

Or if you like, why is it false, whatever "it" is supposed to be? How do we know that it is false? Is it because you said so? But you saying so does not make a thing false, so that's a dead end. Even Wittgenstein understood that a sentence cannot prove or show its own truth or falsity. — Leontiskos

Suppose that the liar's sentence is false. Then the liar's sentence is true because it says that it is false.

Suppose that the liar's sentence is true. Then the sentence is false because it says that it's false and we're saying it is true.

In either case you end up with the circuit of evaluation which yields both "...is true" and "...is false" regardless of its starting truth value.

Though I can see you're not having it.

Do you at least agree that paraconsistent logic is different enough to count as pluralism?

You haven't managed to address the argument. Let's set it out again:

The clause "...is false" presupposes an assertion or claim.
"This sentence" is not an assertion or claim.
Therefore, "This sentence is false," does not supply "...is false" with an assertion or claim.

Now here's what you have to do to address the argument. You have to argue against one of the premises or the inference. So pick one and have a go. — Leontiskos

I'll start with your first premise. "...is false" presupposes no such thing as an assertion or claim -- like I noted earlier "This duck is false" could mean "This duck is fake", right?

So it follows that the meaning of a clause depends upon the name and the predicate -- "...is false", outside of everyday, has no meaning.

Note too that, "This sentence is false," is different from, "This sentence is false is false," or more clearly, " 'This sentence is false' is false. " Be clear on what you are trying to say, if you really think you are saying something intelligible at all. Be clear about what you think is false.

I agree that "This sentence is false" differs from "This sentence is false is false" -- I think once we introduce substitution we're no longer in everyday reasoning, but it works at any level from what I can tell.

"This sentence is false" is all I need. It's a nefarious sentence. Or a purposefully chosen set that play with the notion of true and false and self-reference.

Also, even if we introduce subsitution the liar's works -- it's the extended liar's sentence. (the "strengthened" liar's sentence is what convinced me that it cannot be assigned some third value, as in many-valued logics)

Actually that's another example that I'm wondering about with respect to pluralism -- do logics with more than 2 values count as plural logics, or no?
***

Also I can just drop this point here. We're starting to getting into liar's paradox points and if it's something that doesn't really jive with you then there's no point in continuing here since the point isn't the liar's sentence but pluralism.

That's a brilliant, thoughtful and charitable post. Well done.

Thank you!

Actually that's another example that I'm wondering about with respect to pluralism -- do logics with more than 2 values count as plural logics, or no? — Moliere

Pretty much. Even including infinite-valued logics.

The liar is clear, in the way you have argued. Rejecting it as a "nonsense" is a failing of nerve, rather than an act of rationality. There are three ways of dealing with it that I think worth considering. Tarski would say that it is a mistake to assign truth values to sentences within the same language, but permissible between languages, so the problem with the liar is that it tries to say something about the falsity of a sentence within it's own language. Kripke would say that we can assign truth values within one language, but that we shouldn't assign them to every sentence, the liar being an example of a sentence to which we cannot assign a truth value. Revision theories would have us say "this sentence is true" is true on the first iteration, false and the second, true on the third... and so on.

Here we have three examples of how accepting and facing the liar enables the development of new and interesting approaches, of creativity. Whereas simply rejecting it as a nonsense closes of such play.

Perhaps that's a nice example of the methodological difference between pluralism and monism. I don't actually think this is quite right, but at the least it shows a difference in approach.

Perhaps that's a nice example of the methodological difference between pluralism and monism. I don't actually think this is quite right, but at the least it shows a difference in approach. — Banno

Interesting. I like this approach of defining the difference as a matter of method.

The liar is clear, in the way you have argued. Rejecting it as a "nonsense" is a failing of nerve, rather than an act of rationality. There are three ways of dealing with it that I think worth considering. Tarski would say that it is a mistake to assign truth values to sentences within the same language, but permissible between languages, so the problem with the liar is that it tries to say something about the falsity of a sentence within it's own language. Kripke would say that we can assign truth values within one language, but that we shouldn't assign them to every sentence, the liar being an example of a sentence to which we cannot assign a truth value. Revision theories would have us say "this sentence is true" is true on the first iteration, false and the second, true on the third... and so on. — Banno

I notice a distinct lack of dialetheism in your approach ;)

The way I understand Tarski's attempt to deal with it is the distinction between meta- and object- language. I think that's the neatest way to deal with it, but upon reading Priest I've reconsidered.

I'm not sure I understand the difference between Tarski and Kripke, though. By your sentences they look the same to me, so I'm missing something.

Revision theories sound like they can't make a decision. Not that I'd know anything about that ;D

I'm not sure I understand the difference between Tarski and Kripke, though. By your sentences they look the same to me, so I'm missing something. — Moliere

Try this:

Tarski's ideas lead to a hierarchy of languages that, like Russian Dolls, each give the truth of the language that they enclose.

Can a language contain its own truth predicate? Various theories do manage this trick. The one I'd like to bowdlerise next derives from a paper by Kripke. The trick, as mentioned earlier, is avoiding the liar paradox: "This sentence is false".

Again, suppose we restrict the language to being about a group of people, Adam, Bob and Carol... and their respective nationalities, English, French... We can construct any number of sentences from these: Adam is English", "Bob is English", "Adam and Bob are french"...

We start by adopting three truth values instead of two. So as well as assigning "true" and "false" to the statements of our language, we add a third value, pictured as sitting in between - not true and not false. (a Kleen evaluation)

Let's call this third value "meh"

We assign "meh" to all the statements of our language.

Then we can give an interpretation to the language, and assign "true" or "false" to these as appropriate; so "Adam is English" is true, and "Adam is French" is false, and so on.

Notice that so far any sentence that contains the term "true" will still have the truth value "meh". So "'Adam is English' is true" is neither truth nor false.

We then start to permit sentences that contain "true" or "false" to be assigned values other than "meh", but under strict conditions. So:

If "Adam is English" is true, then we allow that "'Adam is English' is true" is also true.
If "Adam is French" is false, then we allow that "'Adam is French"' is false" is true.
And so on. Generally, if p is true, then "p is true" is true, and '"p is true" is true' is true, and so on; if p is false, then "p is false" is true, and '"p is false" is true' is true, and so on.

But notice that in this construction, we never get to assigning a truth value to the sentence "this sentence is false". So it remains with the truth value "meh" - neither true nor false. — Banno

So you are quite right that they both use the notion that if "Adam is English" is true, then so is '"Adam is English" is true'. But whereas Tarski uses layered languages, Kripke gives a methodical way to assign truths and avoid liars in the same language.

Is that conclusion supposed to follow? That there are no universal laws does not deny that there are laws specific to each logic.

Yes, that's the pluralist response. Like I said, I think they can accuse the nihilist of equivocating here to the extent that their argument relies on assuming deflation. But nihilism ultimately has to be about a broader notion of truth preservation across all correct logics, else it is demonstrably false. LNC holds "generally" if we only look within one context for very many contexts, etc.

Hence my example, statements like "propositions must be either true or false" are ambiguous in a deflationary context. The answer is: "it depends, LEM and bivalance aren't universal." It's like saying "marijuana is legal," without specifying a jurisdiction, and then equivocating on the relevant context.

I don't know how to respond to the rest of what I wrote because you keep on responding to things that obviously are not what I'm saying, e.g. "This paper uses 'exists' univocally" for "I don't think logic has existential quantifiers."

I point out that STT allows for relativity in the context of discussing a paper that is almost entirely using examples of such relativity and you suppose that I am confused and referring to the level where it isn't relative.

Suffice to say, STT can be interpreted in a deflationary manner and was developed with that in mind. If the point in question the existence of a general logical consequence relationship applicable to truth preservation vis-á-vis science or to metaphysical truth it is question begging to assume deflation.

It is maybe worth pointing out that if someone proposes a new logic, they are obliged to set it out for us to see it, and we can judge it's consistency within itself, as well as its applicability to various situations in comparison to other logics.

Either all logics are correct logics, in which case nihilism is "true" but truth becomes essentially meaningless or there are just some correct logics. Since many people are not willing to embrace the former (full deflation, truth is arbitrary) they need some criteria for deciding which logics are correct. So, we are back to ambiguous definitions anyhow, we've just obfuscated this fact.

This is quite odd. As if we are talking about different things. That's why the detail is so important. I'll have a think and a re-read and see if i can make some sense of it.

Do us a favour, and read the first few paragraphs here.

Notice the bit that says

Given that, together with the fact that he took the instances of (T) to be contingent, his theory does not qualify as deflationary.

Now what do you make of this? I've understood you as saying Tarski is unavoidably deflationary, and that this is a bad thing.

For my part, talking off the top of my head, I agree with it, and add that deflation is pretty much the only description of truth generally, inflationary accounts only be of use in somewhat special cases.

THis by way of looking for common ground.

How can you insist that one is more correct than another? — fdrake

I think I've been pretty clear that I don't think one is more than correct than another, at least in the face of a skepticism or a univocity like your own. For instance:

If they are different theories then they define different things, i.e. different "circles." The monist can have Euclidean circles and non-Euclidean circles. He is in no way forced to say that the token "circle" can be attached to only one concept. — Leontiskos

In common usage there are no square circles, but if we redefine either one then there could be. I've said this many times now.

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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. — fdrake

I don't know where you're getting these ideas. This started with an offhand comment to frank about "square circles lurking just around the corner," and then you launched into an extended argument in favor of square circles. Early on I asked about your motivations, and you said something in favor of "shit-testing" and then tried to repair that idea in favor of "counterexamples based on accurate close reading." But it is not coincidental that shit-testing is something like the opposite of close reading, and that your posts haven't engaged in much close reading at all.

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."

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. — fdrake

It is petitio principii to simply insist that, say, an inclined plane is not reducible to a Euclidean plane qua circles. You haven't offered anything more than arguments from your own authority for such premises. Beyond that, I see misreading, not close reading. I have said things like this many times:

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

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I would say that someone correctly understands a mathematical object when they can tell you roughly... — fdrake

But how do you know that when I talk about a circle I am restricting myself to a very strictly interpreted Euclidean conception, such that an inclined plane is not reducible to a Euclidean plane? You are the one who is insisting that there is a right answer to questions like these, not me.

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. — fdrake

But it's odd to talk about an "object" here. As you go on to say, you don't even know if the "object" exists. You're just attempting to solve a problem or create a model.

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. — fdrake

J's new thread seems on point.


The interesting question I see here is something like, "Why should we disagree?" What is a sufficient reason to disagree with someone? You seem to have fallen into the odd trap of claiming that mathematics is all arbitrary and that I have nevertheless committed some grievous sin by supposing that an inclined plane can be reduced to a Euclidean plane. If all mathematics is arbitrary, then there are no grievous sins. There is just ignorance of stipulations (such as the "great circle"). So then perhaps I am ignorant of the precise properties of a commonly-known stipulation in the math world (i.e. a "great circle"). But is that really a problem? Does someone really need to have a Masters in mathematics and understand the stipulated metaproperties of great circles in order to claim that there are no square circles lurking around the corner? I really doubt it.

Granted, I realize you think some mathematical constructs are more applicable than others, but I won't press you on that unless you somehow think that it bears on this question of the great circle.

Though I can see you're not having it. — Moliere

I'm not having it because you keep begging the question. You say there is a sentence/claim but you won't say what the sentence is.

It's not much different to say, "Suppose there is a sentence that is true and false. Therefore the PNC fails."

Or else, "Suppose there is a sentence that is true if it is false and false if it is true. Therefore the PNC fails." But that's not an argument. It's, "Suppose the PNC fails; therefore the PNC fails." In order to make an argument you would actually have to identify such a sentence, and I have already pointed out the problems with the "Liar's sentence."

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I'll start with your first premise. "...is false" presupposes no such thing as an assertion or claim -- like I noted earlier "This duck is false" could mean "This duck is fake", right? — Moliere

"If false doesn't mean 'false', but instead means 'fake', then <This duck is false> succeeds even though 'this duck' is not an assertion or claim."

Do you see how silly this is? You redefined falsity as something other than falsity in order to try to make a substantive point about falsity. Do you see why I feel that I am wasting my time? These are the sort of moves that so-called "Dialetheists" routinely engage in, at least on TPF.

Note too that, "This sentence is false," is different from, "This sentence is false is false," or more clearly, " 'This sentence is false' is false. " Be clear on what you are trying to say, if you really think you are saying something intelligible at all. Be clear about what you think is false. — Leontiskos

"This sentence is false" is all I need. — Moliere

So what do you think is false? <This sentence>, or <This sentence is false>? "This sentence" cannot refer to both at the same time. You have to pick one.

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

- There's no point in continuing if it is the same thing over and over. I have tried to move it away from the great circle into questions about disagreement in general, but if you only want to keep bringing it back to the great circle without introducing any new arguments regarding the great circle, then it will be the same thing over and over. In that case I agree that we should not continue.

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

- I was trying to use your own verbiage there, as I had been using the word "contains." For example:

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. — fdrake

So suppose we are talking about the cross-section of a sphere, which is what I originally thought you were pointing at. Is that something like a circumscribed inclined plane? It is certainly a set of coplanar points. Now you say, "The incline plane can be reduced to a flat plane without distortion." This captures what I said by, "an inclined plane is [...] reducible to a Euclidean plane." "Qua circles," meant to indicate the idea that an inclined cross-section of a sphere could be reduced to a Euclidean circle or else a flat circle." Or to use my own language, the inclined cross-section of a sphere "contains" a Euclidean circle.

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:).

- You have often ignored my inquiry about whether it is possible to delete a point as rhetorical or unworthy, but I don't think it is. In Aristotelian terms you are conflating a description with a definition. There are different ways to describe a circle, but where each description overlaps the object in question is identical, at least according to an Aristotelian frame. That is to say, whether we draw a circle with a compass or with Aristotle's method, we still arrive at a circle. The method of drawing is not itself the definition of a circle.

You seem to identify different mathematical representations with the definition of a circle in a curious way. This strikes me as odd, but I don't mean to imply that a consensus of mathematicians would favor my view. So to nail it down a bit:

• EC (Euclid's Circle): The set of points equidistant from a single point.
• AC (Aristotle's Circle): "The locus of points formed by taking lines in a given ratio (not 1 : 1) from two given points constitute a circle."

(We are implicitly talking about a plane figure.)

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?

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.

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 is thus subsets of it are not 'planar figure's in some sense. — fdrake

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.

However, for a clarified definition of plane that lets you treat a plane that is at an incline as a standard flat 0 gradient plane, the "clearly a circle" thing you draw in it would be a circle. — fdrake

But here too, I would say that you are confusing a "flat" plane with a Euclidean plane. A Euclidean plane is not a "0 gradient plane," it is a plane without any gradient dimension whatsoever. I have been overlooking these sorts of errors, but if you are going to be persnickety about what you see as my errors then I suppose I should return the favor, especially given that you haven't shown interest in trying to mete out the question of why/when we should disagree.

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. — fdrake

Yep, I sympathize with him.

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. — fdrake

People really will say that they have four of the same tires.

But the same question about Euclid's Circle vs. Aristotle's Circle is arising here. If there is no right answer to these questions then there are no real questions, and in that case I don't know why we're arguing.

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. — fdrake

I'm being flippant, but not "just." :wink:

But no, I take it that your "correctly assertible" means something like "justifiably assertible," and on that reading I think it is correctly assertible that the cross-section contains a Euclidean circle. At the same time, I think the phrase "correctly assertible" is only a placeholder for further explication, because justification doesn't have food to eat unless there is a truth of the matter, at least on the horizon.

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?

Do you see why I feel that I am wasting my time? — Leontiskos

I believe that I do, and I'm happy that you continue to respond in spite of the frustration.

Gonna call it for tonight and rethink stuff, though obviously not in your favor :D

I'd appreciate you answering my question about whether or not paraconsistent logic would count as a plural logic insofar that we accept both paraconsistent logic and classical logic.

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. — fdrake

Okay, so R^3 is a Euclidean space and R^2 is the place where all of Euclid's mathematics lives. I mean, your early insistence on locating Euclidean circles in R^2 is why I am thinking of R^2 as Euclidean space. Apparently you are making the "...ean" of Euclidean do a lot of work here.

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?

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. — fdrake

None of this matters much to me. I only took Euclid's definition as a point of departure or something I would be comfortable with. But I view Euclid's definition as describing a relative property of a continuous curved line that forms an enclosed shape, which is probably why I don't think the center can be "deleted."

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? — fdrake

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

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.

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

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

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...".