Para-consistent logic

[Bill Spight]

Also, modern higher math uses variables to prove things about infinite sets. E.g. they prove that there are infinite natural numbers, because every number n has a successor n+1, so that there can't be a last one. BUT: That assumes that "n" stands for all possbile natural numbers, infinitely many as we just saw. How can one assume that? How can a single letter stand for 1. a single number but 2. at the same time for all?

In the statement, "every number n has a successor n+1," n does not, repeat, not stand for all natural numbers. There are real problems expressing logic statements in English. Quine suggested this kind of locution, "Whatever n may be, if n is a natural number it has a successor, n+1, which is also a natural number." That makes it obvious that n does not stand for all natural numbers.

[Pippen]

Also, modern higher math uses variables to prove things about infinite sets. E.g. they prove that there are infinite natural numbers, because every number n has a successor n+1, so that there can't be a last one. BUT: That assumes that "n" stands for all possbile natural numbers, infinitely many as we just saw. How can one assume that?

You do not have to assume that.

Suppose that there is a largest natural number. Call it L. Then there is a natural number, L + 1. Call it M. Then M > L, which means that L is not the largest natural number, and our supposition is false.

That would just prove that L is not the largest number, but what about M? You would need to repeat your proof for M since L did stand just for a concrete number (the supposed biggest one). Only if L stands for any number in N then the proof works. But that requires that variables can stand for any number in N and that sounds like weirdo metaphysics. Because first how can a variable stand for infinite numbers? Because by definition infinity is never over, so that a variable could never stand for all numbers in an infinite set. Secondly, how can one imagine this? Obviously a variable contains only space for one number, a variable is not a set. So how can a space for one number contain all infinite numbers somehow? Even if we imagine a Turing machine with infinite power and infinitely fast that can put all numbers in a row into a variable like x then still it wouldn't contain all, because there are infinitely many natural numbers. You cannot say that the infinitely fast turing maschine would beat the infinite many numbers, can't u?

[Monadology]

@Monadology

Thx, that was a hell of a read!!!

I'm still skeptic about those non-classical logics, because they all rely on classical logic in their meta-language (where a logic is ruled and made sense of). Another point: Priest introduces this logic that has the truth values: true, false, truefalse, none, ineffable. What about a sixth truth value of "ineffability of all the previous truth values" and so on and you got an infinite value logic^^.

Well, I'm not sure how you get an infinite set of truth values. If I'm understanding you, you'd only get 9, right?

T
F
TF
0
I
TI
FI
TFI
0I

Truth values like:

TII
or
TTI

just involve redundancies so they're not actually distinct.

The other reason for not getting infinite (or even nine) truth values is presumably that we have no reason to produce those new truth values unless we find statements that qualify for them. So unless there's reason to think that there are ineffable & true propositions, much less an account of how they operate logically, then there's not much cause to include them in our logic.

[SmoothOper]

Also, modern higher math uses variables to prove things about infinite sets. E.g. they prove that there are infinite natural numbers, because every number n has a successor n+1, so that there can't be a last one. BUT: That assumes that "n" stands for all possbile natural numbers, infinitely many as we just saw. How can one assume that?

You do not have to assume that.

Suppose that there is a largest natural number. Call it L. Then there is a natural number, L + 1. Call it M. Then M > L, which means that L is not the largest natural number, and our supposition is false.

That would just prove that L is not the largest number, but what about M? You would need to repeat your proof for M since L did stand just for a concrete number (the supposed biggest one). Only if L stands for any number in N then the proof works. But that requires that variables can stand for any number in N and that sounds like weirdo metaphysics. Because first how can a variable stand for infinite numbers? Because by definition infinity is never over, so that a variable could never stand for all numbers in an infinite set. Secondly, how can one imagine this? Obviously a variable contains only space for one number, a variable is not a set. So how can a space for one number contain all infinite numbers somehow? Even if we imagine a Turing machine with infinite power and infinitely fast that can put all numbers in a row into a variable like x then still it wouldn't contain all, because there are infinitely many natural numbers. You cannot say that the infinitely fast turing maschine would beat the infinite many numbers, can't u?

I am with you Pippen. I am a believer in the axiom of determinancy which essentially disallows infinite proofs, which come to think of it, is how I happened across the para consistent logic article.

All kinds of weird things happen when you allow infinity. Take for example Lindley's paradox http://en.wikipedia.org/wiki/Lindley's_paradox . The paper is quite interesting especially where he proved people have esp, by flipping a coin until the results were significant. Only took a couple million samples.

[Bill Spight]

Also, modern higher math uses variables to prove things about infinite sets. E.g. they prove that there are infinite natural numbers, because every number n has a successor n+1, so that there can't be a last one. BUT: That assumes that "n" stands for all possbile natural numbers, infinitely many as we just saw. How can one assume that?

You do not have to assume that.

Suppose that there is a largest natural number. Call it L. Then there is a natural number, L + 1. Call it M. Then M > L, which means that L is not the largest natural number, and our supposition is false.

That would just prove that L is not the largest number, but what about M?

The only claim about M is that M > L, which is obvious.

You would need to repeat your proof for M since L did stand just for a concrete number (the supposed biggest one). Only if L stands for any number in N then the proof works. But that requires that variables can stand for any number in N and that sounds like weirdo metaphysics.

Why? I can make statements about any person, any tree, any rock, etc. Does that sound like weirdo metaphysics? Besides, L does not stand for any number in N, but for the largest number in N.

Because first how can a variable stand for infinite numbers? Because by definition infinity is never over, so that a variable could never stand for all numbers in an infinite set.

Well, first, if a variable stands for a set, that does not mean that it stands for any member of the set, or for all of its members. For instance, if E stands for the empty set, it cannot stand for any member of the set, because there is none.

You seem to be making a distinction between what the ancient Greeks called (in Greek, OC) absolute infinity and potential infinity. Potential infinity is never reached. Aristotle, IIRC, did not believe in absolute infinity. BTW, the proof that there is no largest natural number does not depend upon absolute infinity.

Obviously a variable contains only space for one number, a variable is not a set.

Let H stand for the set of Honinbo. Let M stand for the set of Meijin Godokoro. What is the intersection of H and M? Do you think that that question has no answer?

[Tryss]

Also, modern higher math uses variables to prove things about infinite sets. E.g. they prove that there are infinite natural numbers, because every number n has a successor n+1, so that there can't be a last one. BUT: That assumes that "n" stands for all possbile natural numbers, infinitely many as we just saw. How can one assume that?

You do not have to assume that.

Suppose that there is a largest natural number. Call it L. Then there is a natural number, L + 1. Call it M. Then M > L, which means that L is not the largest natural number, and our supposition is false.

That would just prove that L is not the largest number, but what about M? You would need to repeat your proof for M since L did stand just for a concrete number (the supposed biggest one).

No, the idea of the proof is :

1) Suppose that there is a largest number.
2) This number is unique (by definition of "the largest")
3) Because he exist and is unique, we can exhibit and name it : L
4) We consider the number M = L+1
5) M>L, so L is not the largest number
6) You have both "L is the largest number" and "L is not the largest number"
7) Point 6) implies that one of these assertions is false, and because our reasoning is valid, that means that "L is not the largest number"


We can argue at different points in this proof :
Points 1), 2), 4), 5) and 6) are not subject to discussion

Point 3) is tricky and is really interesting to be discussed : can we talk about an object that just verify a propriety and exist? Or should we be able to construct it to talk about it? What are the objects we can talk about?

Point 7) can be discussed as well.

But in this proof, nowhere you have infinity

[Mike Novack]

I am with you Pippen. I am a believer in the axiom of determinancy which essentially disallows infinite proofs, which come to think of it, is how I happened across the para consistent logic article.

Not quite. What you perhaps mean is that you don't accept "induction".

The "natural numbers" weren't formalized until Peano by which I mean that he made explicit the axioms defining what the natural numbers were in terms of how mathematics (the majority of it) were using them.

One of these is the axiom of induction "if something being true for m means that it is true for the successor of m and if it is true for the number that isn't the successor of any other number, then it is true for all numbers."

There is an area of mathematics that investigates what might be true without this axiom. But most mathematics considers "the natural numbers" to be entities for which that axiom (of induction) applies.

The concept here is that encapsulated within that axiom is what expresses the extension of the truth of a finite proof over the infinite set of the "countables".

[hyperpape]

I am with you Pippen. I am a believer in the axiom of determinancy which essentially disallows infinite proofs, which come to think of it, is how I happened across the para consistent logic article.

The axiom of determinacy concerns games of infinite length, so apparently you disagree with Pippen, since Pippen is (apparently) doing a bad job of expressing either finitism or ultrafinitism.

[Polama]

Well, I'm not sure how you get an infinite set of truth values. If I'm understanding you, you'd only get 9, right?

...

Truth values like:

TII
or
TTI

just involve redundancies so they're not actually distinct.

You can classify TII as distinct from TI. We started with T and F, and tried to divide the universe of ideas into those two categories. There were gaps, and we labeled some of the gaps ineffable, I. Success.

But then we stared making statements about I, and had to add in TI and FI, for the statements about the ineffable that were true and false. But we, of course, couldn't cover all the ineffable. So there are true statements about the ineffable, false statements about the ineffable, and statements that can't be expressed about the ineffable. Aren't those just ineffable? Well, yes, but there's a distinction here. If we just call them ineffable, we're putting them in the same group as TI and FI. It's reasonable to care about what's ineffable, and what, in particular, is impossible to express. So we add in II. The region of an ineffable idea that makes it ineffable.

And down the rabbit hole we go! Now that we're talking about II, We can actually make TII and FII statements. "The complexity of the Mandlebrot set that eludes any finite description is not a grizzly bear." TII. Also TI, also T. And so on, and so on. Basically, every I added to the logic system lets you classify more finely the truths of the ineffable, but the ineffable always remains, and there are always more gradations of truth you could discern on it's border. Practically, gradations of the ineffable aren't terribly useful, but you can form logic systems with them if you care to.

[Monadology]

Well, I'm not sure how you get an infinite set of truth values. If I'm understanding you, you'd only get 9, right?

...

Truth values like:

TII
or
TTI

just involve redundancies so they're not actually distinct.

You can classify TII as distinct from TI. We started with T and F, and tried to divide the universe of ideas into those two categories. There were gaps, and we labeled some of the gaps ineffable, I. Success.

But then we stared making statements about I, and had to add in TI and FI, for the statements about the ineffable that were true and false.

This is where I don't follow you anymore. The Ineffable truth value is a truth value, it's not there to tell us what a proposition is about. A True statement about the ineffable is just has the truth value 'T'. 'TI' does not designate a proposition that is True and *about* the ineffable. 'TI' designates a proposition that is true *and* ineffable. Thus TII designates a proposition that is true *and* ineffable *and* ineffable. That's a redundancy and it's no more a result of our five-valued logic than classical logic necessitates we have TT, to designated propositions that are True *and True*.

[Pippen]

No, the idea of the proof is :

1) Suppose that there is a largest number.
2) This number is unique (by definition of "the largest")
3) Because he exist and is unique, we can exhibit and name it : L
4) We consider the number M = L+1
5) M>L, so L is not the largest number
6) You have both "L is the largest number" and "L is not the largest number"
7) Point 6) implies that one of these assertions is false, and because our reasoning is valid, that means that "L is not the largest number"

But in this proof, nowhere you have infinity

I disagree, because if L stands for the maximum of N then if L is false that'd mean that N has no maximum and that'd mean it's an infinite set. The problem is: L needs to stand for all members of N, for all natural numbers, not just one. If L stands just for one number, even an arbitrary one, then the proof just shows that this specific but unknown number isn't the largest (but maybe another one). Therefore L needs to stand for all natural numbers and because of L+1 you can prove that every natural number must have a successor and therefore a maximum is impossible in N. But how can L stand for all natural numbers if there are infinitely many ones? That's impossible by definition of infinity and that's where pure speculation (metaphysics) comes in.

@monadology: I agree the five valued logic doesn't imply any more truth values. My quick idea just doesn't work. Semantically I see now a contradiction in the truth value "ineffable", because if something is ineffable then the very same statement is a contradiction by performance like saying: I do not talk now. Also I think this logic confuses logic with epistemology. A statement p can be true or false, but it's a complete different thing if we do know the truth status of p.

[Monadology]

@monadology: I agree the five valued logic doesn't imply any more truth values. My quick idea just doesn't work. Semantically I see now a contradiction in the truth value "ineffable", because if something is ineffable then the very same statement is a contradiction by performance like saying: I do not talk now. Also I think this logic confuses logic with epistemology. A statement p can be true or false, but it's a complete different thing if we do know the truth status of p.

Priest addresses this directly:

Philosophers in the Mahayana traditions hold some things to be ineffable; but they also explain why they are ineffable, in much the way that I did. Now, you can’t explain why something is ineffable without talking about it. That’s a plain contradiction: talking of the ineffable.

[...]

So we have now hit a new problem: the contradiction involved in talking of the ineffable. In a sense, the possibility of a true contradiction is already accommodated by that both option of the catuskoti.

What we have to bear in mind then is that contradictions aren't a problem in our five-valued logic. So showing that there is a contradiction here doesn't amount to a refutation unless we beg the question against our five-valued logic.

In fact though, Priest anticipated our entire dialectic (I am not surprised and should have just reread the article in light of our discussion, philosophers are very good at anticipating objections):

Alas, our contradiction is of a rather special kind. It requires something to take both the values true and ineffable, which, on the understanding at hand, is impossible. Yet the resources of mathematical logic are not so easily exhausted.

In fact, we have met something like this before. We started with two possible values, T and F. In order to allow things to have both of these values, we simply took value of to be a relation, not a function. Now we have five possible values, t, f, b, n and i, and we assumed that value of was a function that took exactly one of these values. Why not make it a relation instead? That would allow it to relate something to any number of those five values (giving us 32 possibilities, if you count). In this construction, something can relate to both t and i: and so one can say something true about something ineffable after all.

Since I'm not familiar with plurivalent logic I don't follow especially well at this point in the article. In particular, Priest glosses over why statements about the ineffable don't just take the truth values TF or 0 (i.e. they are true and false or they are neither). Still it seems like whatever way you want to characterize statements about the ineffable, he's going to have resources to accommodate it.

[Bill Spight]

No, the idea of the proof is :

1) Suppose that there is a largest number.
2) This number is unique (by definition of "the largest")
3) Because he exist and is unique, we can exhibit and name it : L
4) We consider the number M = L+1
5) M>L, so L is not the largest number
6) You have both "L is the largest number" and "L is not the largest number"
7) Point 6) implies that one of these assertions is false, and because our reasoning is valid, that means that "L is not the largest number"

But in this proof, nowhere you have infinity

I disagree, because if L stands for the maximum of N then if L is false that'd mean that N has no maximum and that'd mean it's an infinite set. The problem is: L needs to stand for all members of N, for all natural numbers, not just one. If L stands just for one number, even an arbitrary one, then the proof just shows that this specific but unknown number isn't the largest (but maybe another one).

So you think that a set of numbers may have more than one largest member?

[Pippen]

No, the idea of the proof is :

1) Suppose that there is a largest number.
2) This number is unique (by definition of "the largest")
3) Because he exist and is unique, we can exhibit and name it : L
4) We consider the number M = L+1
5) M>L, so L is not the largest number
6) You have both "L is the largest number" and "L is not the largest number"
7) Point 6) implies that one of these assertions is false, and because our reasoning is valid, that means that "L is not the largest number"

But in this proof, nowhere you have infinity

I disagree, because if L stands for the maximum of N then if L is false that'd mean that N has no maximum and that'd mean it's an infinite set. The problem is: L needs to stand for all members of N, for all natural numbers, not just one. If L stands just for one number, even an arbitrary one, then the proof just shows that this specific but unknown number isn't the largest (but maybe another one).

So you think that a set of numbers may have more than one largest member?

No, but the point of the proof is to assume a largest number L and then find out that it is not, so the assumption was wrong in the first place. If a variable just stands for one number then this proof just shows that the number behind L is not the largest, but maybe L+1 is and there you have a regressus. My point is: to prove something for a set you have to use variables that somehow stand for all objects of the set and that's a problem, if you deal with infinitive sets, because in infinity there is no "all", it's an ever ongoing unclosed process.... (This is just to show you my concern, I am not a math guy, so I can be wrong easily.)

[Pippen]

Philosophers in the Mahayana traditions hold some things to be ineffable; but they also explain why they are ineffable, in much the way that I did. Now, you can’t explain why something is ineffable without talking about it. That’s a plain contradiction: talking of the ineffable.

[...]

So we have now hit a new problem: the contradiction involved in talking of the ineffable. In a sense, the possibility of a true contradiction is already accommodated by that both option of the catuskoti.

That is not good, because it means that such a logic is not only inconsistent, but also inconsistent at its meta-level...and since we just do not understand what 'p and not-p' means it means that this logic just makes no sense. The world might make no sense, but it becomes not better to talk about no sense in no sense^^. If Priest was an esoteric who'd sell books and holy water we'd certainly not take him seriously^^ and let me not even start about guys like Hegel who basically sold gibberish as fundamental philosophy.