Para-consistent logic

Tryss · **#21**

Pippen wrote:

Bill Spight wrote:

Pippen wrote:

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 · **#22**

SmoothOper wrote:

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 · **#23**

SmoothOper wrote:

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 · **#24**

Monadology wrote:

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 · **#25**

Polama wrote:

Monadology wrote:

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 · **#26**

Tryss wrote:

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 · **#27**

Pippen wrote:

@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:

Quote:

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):

Quote:

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 · **#28**

Pippen wrote:

Tryss wrote:

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 · **#29**

Bill Spight wrote:

Pippen wrote:

Tryss wrote:

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 · **#30**

Quote:

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.

Bill Spight · **#31**

Tryss wrote:

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

Pippen wrote:

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).

Bill Spight wrote:

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

Pippen wrote:

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

No, L was the largest, by assumption. In that case there is no natural number, L+1 (or otherwise), which could also be the largest.

Pippen wrote:

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.)

So you do not believe that all natural numbers are greater than zero, because you cannot prove that for each natural number, one at a time?

Edited for emphasis and clarity.

paK0 · **#32**

Bill Spight wrote:

Tryss wrote:

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

Pippen wrote:

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).

Bill Spight wrote:

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

Pippen wrote:

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

No, L was the largest, by assumption. In that case there is no natural number, L+1, which could be the largest.

L being the largest was in fact an assumption, which is pretty much the point of the proof. An assumption can be verified or falsified, in this case the assumption that L is the largest natural number was falsified

Pippen wrote:

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.)

So you do not believe that all natural numbers are greater than zero, because you cannot prove that for each natural number, one at a time?

From a mathematical standpoint its actually quite easy to show (some) things for infinite numbers, as natural numbers are defined by the peano axioms.

The relevant ones for this example would be:
- 0 is part of the natural numbers
- Every natural number has a successor (which implies: S(m) = n => m < n)
- 0 is not the successor of any natural number

So you basically get:
x < 0
S(x) = 0 which is forbidden, leading to the conclusion that there is no natural number smaller than 0.

SmoothOper · **#33**

hyperpape wrote:

SmoothOper wrote:

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.

No actually the axiom of determinancy says that there are no games of infinite length. IE simply stating I can continuously generate larger numbers than you is not sufficient, at some point you have to stop generating larger numbers, all arguments must be of finite length.

Bill Spight · **#34**

SmoothOper wrote:

actually the axiom of determinancy says that there are no games of infinite length.

So no need for a ko or superko rule, right?

Monadology · **#35**

Pippen wrote:

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^^.

Priest is a dialetheist. He's advocating for inconsistency and it's begging the question in this matter to criticize him or his logic for allowing inconsistencies.

EDIT: There's also no reason to think that we don't know what 'p and not-p' means. Here's an example: "This sentence is false."

Here's another example, in the form of a representation of a situation in which a contradiction occurs.

One way to handle those examples is to suggest that the appearance of a contradiction is illusory. The other way is to take seriously that a contradiction is occurring in some form. But argument has to be made either way (and Priest, in his published works elsewhere, makes arguments for why the Liar Sentence should be taken to be an example of a genuine contradiction). So I don't see why, prima facie, we should assume that we don't know what 'p and not-p' means, given that we can have debates about purported examples.

Quote:

If Priest was an esoteric who'd sell books and holy water we'd certainly not take him seriously^^

You're right. Fortunately, he's a very intelligent logician and philosopher who has written extensively on these technical issues in a rigorous and thoroughgoing manner.

Quote:

let me not even start about guys like Hegel who basically sold gibberish as fundamental philosophy.

Hegel is not gibberish. Appropriately (given your opinion on Priest), Priest takes Hegel to be a crucial and important turning point in the history of philosophy. But setting Priest aside, there is enough clear-headed exposition of Hegel going on these days (For example, Robert Pippin, Terry Pinkard and Robert Brandom) that there is a substantial burden of evidence on anyone claiming Hegel is gibberish. I can understand why someone would walk away with that impression after a first attempt at the Phenomenology of Spirit, but it's an impression that is false. I'll leave it at that since this thread isn't about Hegel anyway.

SmoothOper · **#36**

Monadology wrote:

Pippen wrote:

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^^.

P vs NP was always a little perplexing to me because the size of the input in terms of the numbers of symbols including logical statements, always seemed to grow exponentially relative to the number of variables. In fact the input could be exponential, and only have two or three variables. If you limit the total number of arguments then I suspect the size of the truth tables should be quite manageable.

lemmata · **#37**

Pippen wrote:

I should mention I am very skeptic of infinities also. I do not think we can prove them, we can just prove that "it goes on and on and on and on [but we do not know if there is an end in fantasiciollion years]". I find it paradoxical and inconsistent to talk about infinite sets, because since this set has infinite objects it is never finished and stable. Every proof about this set has to be incomplete.

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? And on top of that there are no rules/axioms about that, it's just pure assumption and practice.

Therefore I like the "only what we can acutally calculate (even with a computer)"-math. Anything else is metaphysics in disguise.

People usually do not object to statements like "each child has a biological father" by proclaiming that there is no way that the word "child" can stand for every child on earth. This is essentially the objection you are raising about the statement in the form "every child c has a biological father f". However, latter is just the mathematical style of expressing the former.

SmoothOper · **#38**

lemmata wrote:

Pippen wrote:

I should mention I am very skeptic of infinities also. I do not think we can prove them, we can just prove that "it goes on and on and on and on [but we do not know if there is an end in fantasiciollion years]". I find it paradoxical and inconsistent to talk about infinite sets, because since this set has infinite objects it is never finished and stable. Every proof about this set has to be incomplete.

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? And on top of that there are no rules/axioms about that, it's just pure assumption and practice.

Therefore I like the "only what we can acutally calculate (even with a computer)"-math. Anything else is metaphysics in disguise.

People usually do not object to statements like "each child has a biological father" by proclaiming that there is no way that the word "child" can stand for every child on earth. This is essentially the objection you are raising about the statement in the form "every child c has a biological father f". However, latter is just the mathematical style of expressing the former.

Actually, I didn't read that carefully enough. I think it is really annoying when mathematicians fail to work at set level. For example everyone knows that the integers I are a subset of the rationals R therefore there must be more rationals than the integers. However there is a cockamamie proof where mathematicians make a mapping from integers to IxI then claim that there are as many I as R. Though in my opinion, it is impossible to do division on all integers without a mapping to the rationals R in which the integers I are a strict subset of the rationals. Though I could see how you could go for that kind of proof if you don't have a concept where you can't refer to an infinite set of things.

Bill Spight · **#39**

SmoothOper wrote:

Actually, I didn't read that carefully enough. I think it is really annoying when mathematicians fail to work at set level. For example everyone knows that the integers I are a subset of the rationals R therefore there must be more rationals than the integers.

As is often the case, what "everybody knows" is not so.

For one thing, integers are not a subset of rationals, strictly speaking. A rational is an ordered pair of integers. True, there are rationals that are equal to integers, but that is not exactly the same thing.

Quote:

However there is a cockamamie proof where mathematicians make a mapping from integers to IxI then claim that there are as many I as R.

What does it mean to say that two infinities are equal, or that one infinity is greater than another? If you do not admit absolute infinities, then the question is meaningless. Fine. But then you do not get to say that there are more rationals than integers.

SmoothOper · **#40**

Bill Spight wrote:

SmoothOper wrote:

Actually, I didn't read that carefully enough. I think it is really annoying when mathematicians fail to work at set level. For example everyone knows that the integers I are a subset of the rationals R therefore there must be more rationals than the integers.

As is often the case, what "everybody knows" is not so.

For one thing, integers are not a subset of rationals, strictly speaking. A rational is an ordered pair of integers. True, there are rationals that are equal to integers, but that is not exactly the same thing.

Quote:

However there is a cockamamie proof where mathematicians make a mapping from integers to IxI then claim that there are as many I as R.

What does it mean to say that two infinities are equal, or that one infinity is greater than another? If you do not admit absolute infinities, then the question is meaningless. Fine. But then you do not get to say that there are more rationals than integers.

Someone doesn't understand the axiom of determinancy. 1:2 is not a rational number (1,2) is not a rational number, 1/2 is a rational number given the operation of division and equivalency classes, however if you have a suitable set of equivalency classes, then you can see that there are more rationals than integers, since integers are a subset. If you don't have the equivalency classes you can't define division, thus your sets of tuples aren't rational numbers. Some people who believe in the axiom of choice will sort of choose to do the mapping implicitly to make the proof work, other people won't be so kind as to go along.

Para-consistent logic

Who is online