I ran across para-consistent logic.

http://en.wikipedia.org/wiki/Paraconsistent_logic

It is interesting to me, I hadn't thought about it before, however I definitely believe that it is possible to reason with inconsistent information or in the absence of certain information, and not just in a "fuzzy" way, and the "Principle of Explosion" or being able to derive anything from a contradiction does seem like a systemic weakness in classical reasoning, especially when I encounter a large "philosophical" work. I always ask: Where is that little subtle contradiction that they used to derive all of "it"?

If you're interested in further reading, I highly recommend Graham Priest's books. In Contradiction would probably be most up your alley.

By the way, most philosophical works (large or not) do not rely on the principle of explosion to derive their conclusions. In fact, I can't even name one where this would be true. Even Graham Priest who has a paraconsistency-oriented narrative of the history of philosophy doesn't make such a claim. Further, since the principle of explosion only works because of the definition of validity and not because of any features of the premises other than their impossible mutual truth, it would be nearly impossible to reason with it without noticing. Employing the principle of explosion is not a natural human inference.

I like the story that Bertrand Russell was challenged to prove that he was the Pope, given the falsehood that 1 = 2. His reply: "The Pope and I are two. Therefore the Pope and I are one. Therefore I am the Pope."

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The Adkins Principle:
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— Winona Adkins

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my favorite logical paradox:
let's start off with the assumption that identical things are identical.
now imagine i make an exact copy of the entire universe.
will i do the same thing in both universes a week from friday?
if yes, then we don't really have free will. that is, our actions are predetermined by that state of the elements in us and around us.
if no, then the identity principle doesn't hold.
so which would you perfer, identity but no free will, or free will but no identity?

That's a really interesting dilemma!

Unfortunately, I think there's some equivocating about identity here. There are two senses of 'universe' in effect here:

1) The universe-at-present.
2) The universe-as-temporally-extended.

If the first is the sense of the word 'universe' we have in mind, then the law of identity will still hold even if we claim we have free-will, since the states we are in next Friday don't affect the identity of the universe.

If the second is the sense of the word 'universe', the main concern I can see is this: just because two things are qualitatively identical doesn't tell us anything else about them. Consider the following parallel argument:

Let's start off with the assumption that identical things are identical.
Now imagine I make an exact copy of the entire universe.
Will I do the same thing in both universes a week from Friday?
If yes, then quantum mechanics is false. That is, our actions are predetermined by that state of the elements in us and around us.
If no, then the identity principle doesn't hold.
So: which would you prefer, identity but no quantum mechanics, or quantum mechanics but no identity?

Now, according to an understanding of the universe under indeterministic laws like those of quantum mechanics there is no genuine dilemma because here we can recognize that the reason the same thing happens a week from Friday is because by stipulation, we have created a cloned universe where the same thing happens at every time definitionally.

But this is consistent with the fact that the same thing happened on Friday in both universes because of mere happenstance: the same results happened despite their being no deterministic relationship between states of the universe.

Making an exact copy U' of the universe U seems already a contradiction in it self. What is going to make that copy? Call it A. A needs to be included in U. So U' must contain a copy A' of A. Is A' now going to make a copy of U or of U'. If both then A must do the same thing. But that is not how A is defined. If A' copies U then U and U' are identical. If A' copies U' then A and A' are not identical because they copy different things unless U and U' are the same thing.

Just to throw a stone in the water.

edit: even the notion of a copy of the universe is paradoxal. If there is such then the universe must contain it.

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I think I am so I think I am.

No free will and no identity?

If your free will ends up producing random results even though you and the surrounding Universe is the same, wouldn't that mean that this free will of yours is essentially just random number generator? If those are not random, what would determine your actions, then? If it's not you or the Universe(both of which you assume to be perfectly identical), where would this free will of yours draw the reason to vary the decisions?

I doubt there is any upside to having a free will that would not, given perfectly identical Universe, always produce the same decision. I like to think that the decisions I make reflect my own character, the idea that the decisions, especially important ethical decisions, would be result of a random process instead of deterministically resulting from my surroundings, does not sound tempting at all.

Yeah, that first paragraph is the common response to people who believe in the combination of indeterminism and free will. They have responses, but none of them have ever seemed intelligible to me.

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In fact they try to avoid it at all costs, either by avoiding contradictions that trigger explosive conclusions or by denying the principle of explosion at all, e.g. by denying disjunctive syllogisms.

Personally I prefer a one-valued logic, e.g. a logic with only the truth-value "true" that basically behaves like an only syntactical logic. It's simple and I'm always right^^.

That sounds like about the simplest option, but why not have fun with a five-valued logic?

http://aeon.co/magazine/world-views/log ... hilosophy/

[The above article is pretty accessible, and it's by Graham Priest who is awesome, so I encourage anyone to check it out!]

@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^^.

I do not see a point yet in those non-classical logics (other than metaphysics). Even Fuzzy-Logic. I cannot understand what a big difference it makes if you say in fuzzy logic that "x is 0.7 true" or if you say in classical logic that "0,7x is true". (Note that fuzzy logic requires a continous function for all values between 0 and 1 and so classical logic could use this function as well to state full truths or falsehoods.)

Gödel would agree with your extra logic values: once you start trying to classify paradoxes and the ineffable, it's turtles all the way down.

And of course, reaching countable infinite value logic is just a stepping stone on the way to logics of larger infinities. =)

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.

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.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

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.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

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?

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.

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.

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



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.



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.



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?

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.