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
Deletion is shorthand for considering different sets - or using the set division operation. The sets I'm referring to were and . — 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
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,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
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 ? Well, then we have a whole, cool new puzzled to play with..."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
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'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
Maybe "propositional logic" is as slippery as "circle." — Leontiskos
If I had the theorems I should find the proofs easily enough. — Riemann
What does it mean to "correctly understand a stipulated object"? — Leontiskos
What if there were several puzzles mixed up?
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.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
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)...
Is that conclusion supposed to follow? That there are no universal laws does not deny that there are laws specific to each logic.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
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
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
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.
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
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
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'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
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
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.
Given that, together with the fact that he took the instances of (T) to be contingent, his theory does not qualify as deflationary.
How can you insist that one is more correct than another? — fdrake
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
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
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
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
I would say that someone correctly understands a mathematical object when they can tell you roughly... — fdrake
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
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
Though I can see you're not having it. — Moliere
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
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
It is petitio principii to simply insist that, say, an inclined plane is not reducible to a Euclidean plane qua circles. — Leontiskos
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
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 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
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
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
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
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
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
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 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
Yep, I sympathize with him. — Leontiskos
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
Do you see why I feel that I am wasting my time? — 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. — fdrake
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
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? — Leontiskos
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? — Leontiskos
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