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The principle of non-contradiction.
What if (P & ~P) is true?

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The principle of non-contradiction.
xzJoel
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Posted 07/01/09 - 06:16 AM:
Subject: Questioning the principle of non-contradiction. 		quote post
#1
This topic has some background in my non-bivalence thread, but I wanted to make a clean break from it. Background discussion follows the direct questions.

What do you all make of Dialetheism? On the face of it, accepting contradictions as true seems absurd, but the more I consider it, the happier I am with the possibilities. Why should we accept the law of non-contradiction? If it is useful to deny the law of non-contradiction, should we do so?

I will happily discuss the background material if you have questions about that.


_________________________________

Classical logic has a few rules/laws, but one of the most important (sometimes called Aristotle's First Principle) is that of non-contradiction. http://plato.stanford.edu/entries/aristotle-noncontradiction/

Semi-formally, it looks likes this:

~ (P & ~P)

Which is to say that any proposition (or even fact) cannot be both true and not true. (Hereinafter "LNC" or "PNC")

A dialetheia is a sentence, A, such that both it and its negation, ~A, are true.

Dialetheism is a philosophy that believes that there are some dialetheia that are true.
http://plato.stanford.edu/entries/dialetheism/

One example of a dialethia is related to the recent discussion of Achilles and a Tortoise. These characters are references to the characters used by Zeno in one of his famous paradoxes.
http://plato.stanford.edu/entries/paradox-zeno/#AchTor (Specifically)
http://plato.stanford.edu/entries/paradox-zeno/ (Generally)

One argument of Dialetheism directs itself towards the paradox of the arrow.
http://plato.stanford.edu/entries/paradox-zeno/#Arr

In this paradox, one is to consider a flying arrow. In a moment of time, a flying arrow travels no distance and is, therefore, stationary. Because time consists of the addition of moments, there is no time at which the arrow is moving.

It is not dreadfully important to understand the paradox fully or its possible solutions, but it is useful to think of one possible solution to the paradox as that of an arrow both being where it is and not being where it is. That is to say, even in a discrete moment in time an arrow is in motion, it is not stopped. This solution, the arrow is both moving and not moving is a dialethia.
http://plato.stanford.edu/entries/dialetheism/#3.3

A modern use of a dialethia may even be that of "instantaneous change."
http://plato.stanford.edu/entries/change/#deny
http://plato.stanford.edu/entries/change/#inst

I am tempted to go into a history of calculus and the like, but I will not. Rather, I encourage you to look into it for further background.
http://en.wikipedia.org/wiki/Derivative
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Incision
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Posted 07/01/09 - 09:53 AM:
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#2
We may have work to do in defining the LNC. In "What is a Contradiction?" Grim identified "on the most conservative of estimates, [. . .] some 240 senses of contradiction." In particular he suggests they can be grouped into four families,

semantic, such as "contradictions are pairs of propositions, of which one must be true and one must be false,"

syntactic, such as "contradictions are statements of the form 'P & ~P,' "

pragmatic, such as "contradictions are joint assertions of propositions and their denials," and

ontological, such as "if a contradiction holds, then some x is F and ~F."

And if the LNC "forbids" contradictions, then for each sense of "contradiction" there will be as many LNCs as there are senses of "forbid." In particular, say the LNC is a formula of SL, "~(P & ~P)." Then dialetheists should not have any more problem with the LNC than with any formula, say, "P -> Q," since dialetheists may believe that sometimes ~(P & ~P), just as sometimes P -> Q. Even if it's a sort of metalogical rule, "|= ~(P & ~P)," then they may not have a problem with it, since, I suppose, for dialetheists, the mere fact that something always holds isn't necessarily an obstacle to its not holding.
xzJoel
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Posted 07/01/09 - 10:09 AM:
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#3
Incision wrote:
We may have work to do in defining the LNC. In "What is a Contradiction?" Grim identified "on the most conservative of estimates, [. . .] some 240 senses of contradiction." In particular he suggests they can be grouped into four families,

semantic, such as "contradictions are pairs of propositions, of which one must be true and one must be false,"

syntactic, such as "contradictions are statements of the form 'P & ~P,' "

pragmatic, such as "contradictions are joint assertions of propositions and their denials," and

ontological, such as "if a contradiction holds, then some x is F and ~F."

And if the LNC "forbids" contradictions, then for each sense of "contradiction" there will be as many LNCs as there are senses of "forbid." In particular, say the LNC is a formula of SL, "~(P & ~P)." Then dialetheists should not have any more problem with the LNC than with any formula, say, "P -> Q," since dialetheists may believe that sometimes ~(P & ~P), just as sometimes P -> Q. Even if it's a sort of metalogical rule, "|= ~(P & ~P)," then they may not have a problem with it, since, I suppose, for dialetheists, the mere fact that something always holds isn't necessarily an obstacle to its not holding.


I don't think the LNC forbids anything, rather the LNC is taken to be something that is necessarily true with no further information. One might even argue that it is impossible to refute the LNC because to do so would require a proof that invokes the LNC for support.

In any event, pick your poison. Is there some definition of "contradiction" that you feel comfortable discussing?

Chapter 3 (an Article by Patrick Grim) is about 20 pages long.
http://books.google.com/books?id=v..._result&ct=result&resnum=1

P.S. I'll try to give the meanign of acronyms as I see them in the article.
PG. 51 "WFF" is a well formed formula. http://en.wikipedia.org/wiki/Well-formed_formula#Predicate_logic


P.P.S. Notice on pg. 60 that Banno's reference to Lewis Carol and "What the Tortoise Said to Achilles" is discussed.


Edited by xzJoel on 07/01/09 - 10:36 AM
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yasseford
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Posted 07/01/09 - 02:50 PM:
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#4
I think with general relativity (a topic I have recently decided to read into), the idea of dialetheism seems more intriguing. If you are a complete idiot when it comes to physics and higher math, then I highly recommend Hawking's A Briefer History of Time (that's right BriefER, for those of whom the original was still too complicated).

Essentially, time and space can only be measured subjectively. For example, imagine that you are riding on a train, on a car that is clearly visible to the outside. You bounce a ball. To you, the ball simply goes straight down and then straight up. However, to an observer who is standing beside the train tracks, when the ball bounced, it traveled 100mph in a particular direction. Einstein says that neither of these people are mistaken, and their judgments of the location and movement of the ball are equally valid. If an alien with incredible powers of perception was looking from space down at the ball, he would see it moving at tens of thousands of miles per hour, as per the rotation of the Earth and its revolution around the Sun. These are all eqaually valid judgments, and yet they all appear to contradict each other.

Another example, dealing with time:
Imagine a rocket ship that is so tall that it takes exactly one second for light to travel from one end of the rocket to the other. Now, imagine that there is a person on either end of the rocket, and they both are carrying super-accurate clocks that tick at the exact same time. Suppose that whenever the clocks tick, the person at the tip of the rocket shines a light down to the person at the bottom. Assuming this rocket is in space, the flashes of light will correspond to the ticking of the clocks. However, if the rocket is sitting on a launchpad, the bottom person will receive the signals of light at shorter intervals, due to the fact that the person at the bottom is deeper in Earth's gravitational field and thus time passes slower. In this case, the light is both flashing at the interval of a second and not a second, and both are true.

Yasseford
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Posted 07/03/09 - 07:36 AM:
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#5
The principles of logic are, like anything else, open to doubt and criticism. I just haven't seen any good criticism of the law of contradiction. The arrow paradox can be solved in other ways too - anyway, as far as I can see it simply relies on using a semantic trick to describe the situation in a way that intuitively seems paradoxical.

yasseford wrote:

Essentially, time and space can only be measured subjectively.


You know, I've always been puzzled by the fact that many people (sometimes unconsciously) take this to imply that time and space are subjective. Even if they are relative and can only be measured subjectively, they can be objective features of the world. Just like, say, the amount of food on a plate. All measurement is subjective anyway (after all, it's us doing the measuring), and the amount of food is relative to the time of measurement. But that does not mean the food is not really there, or that it is "subjective".

But I am no expert in physics, so there might be something I am missing.
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Posted 07/03/09 - 05:30 PM:
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xzJoel wrote:
Is there some definition of "contradiction" that you feel comfortable discussing?

Heh, I wouldn't describe myself as comfortable with any part of the dialetheism debate, since it's so technically and conceptually difficult. But let's imagine that the LNC = "~(P & ~P)."

Then here's an argument for it. A peculiar feature of statements is that they are truth-apt, so each statement is either true or false -- inclusive, let's say, to be conservative -- A is true iff ~A is false; A & B is true iff A is true and B is true. Then " is always true.

P | ~ P & ~ P
T | T T F F T
B | B B B B B
F | T F F T F

(Fanfare.) Except that as far as logic is concerned, maybe pigs do and don't fly. So is it really the LNC?

The point I want to make is that dialetheism doesn't have a problem with that theorem. What is has a problem with is (maybe?) some metalogical statement like "contradictions are always false" or (maybe?) some epistemological statement like "rational people don't believe contradictions." Maybe bivalence is really the issue. So as far as that's concerned, I'm afraid all I can do is point to some possible problems in framing the issue and some possible directions for improvement.

Here's another. . .issue. Dialetheism is closely connected, though not identical, to a rejection of classical logic. If you buy classical logic, statements of the form "~(P & ~P)" are always false, and more importantly, asserting any one commits you to silly things like Priest's "I'm a fried egg." So we'll either need a new logic on the table, or do without one.
aufbau87
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Posted 07/04/09 - 11:22 PM:
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For any sentence, it and its denial cannot be true. If we deny this, it is apparent that we are changing the meaning of "is true". This is essentially Quine's verdict on variant logics that are said to "deny" the LNC, in Philosophy of Logic (1970). Basically, I see it as one not literally denying the LNC -- they are rather changing the logical vocabulary.
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Posted 07/05/09 - 02:00 PM:
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sqeecoo wrote:

You know, I've always been puzzled by the fact that many people (sometimes unconsciously) take this to imply that time and space are subjective. Even if they are relative and can only be measured subjectively, they can be objective features of the world. Just like, say, the amount of food on a plate. All measurement is subjective anyway (after all, it's us doing the measuring), and the amount of food is relative to the time of measurement. But that does not mean the food is not really there, or that it is "subjective".

But I am no expert in physics, so there might be something I am missing.



Well if I'm not mistaken, Einstein would say that both observers in this case are objectively correct. In the case of the rocket ship, both observers are using the exact same method of measurement - if they stand beside eachother, they will see their clocks ticking at the exact same time. It's not as if they are using their own interpretation of a "second."
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xzJoel
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Posted 07/29/09 - 01:56 PM:
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Did you ever start a thread, somehow miss a few responses, and then find yourself a month later thinking, "Crap! I wanted to talk about this!"?

I will have to recompose myself and write something nice. Anyone want to add anything before I do?
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Posted 07/29/09 - 02:50 PM:
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xzJoel wrote:
Did you ever start a thread, somehow miss a few responses, and then find yourself a month later thinking, "Crap! I wanted to talk about this!"?


All the bloody time. I had put it down to age.

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Russel Morris: There's a meaning there, but the meaning there doesn't really mean a thing...
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