Thursday, February 28, 2008

The Law Of Non-Contradiction

In logic, the law of non-contradiction is an important principle. This law states that no instance in the same respect can be in two or more contradicting states. Contradicting states can be defined as mutually exclusive states, or disjoint states.

In boolean logic, the not operator, -also called the negation operator- is defined as the operator which evaluates true as false, and false as true, where true and false are by definition -in the context of boolean logic- two contradicting states.

The law of non-contradiction is a concept that holds several types of logic from dissolving into trivial systems. Even higher forms of logical systems like set theory and mathematics would dissolve into trivial systems if the law of non-contradiction is not taken as part of its axioms.

Now I will try to demonstrate why the law of non-contradiction is important. First in logic. And for people who are not familiar with logic notation, I will then demonstrate it using mathematics. And finally, the importance of the law of non-contradiction for language and communication of ideas in general.

In Logic:
~: The logical negation
^: Logical AND
v: Logical OR

The law of non-contradiction states:
p: any statement
p ^ ~p = False

Denying the law of non-contradiction:
j: is a statement that defies the law of non-contradiction
j ^ ~j = True

The law of non-contradiction prohibits the existence of such statement as j. Lets see what happens if a statement like j actually existed:
= p v False
= p v ~True
= p v ~(j ^ ~j) ; Denying the law of non-contradiction
= p v (~j v j) ; By DeMorgan's rule
= j v True ; Tautology
= True ; Tautology

= ~p v False
= ~p v ~True
= ~p v ~(j ^ ~j)
= ~p v (~j OR j)
= ~p v True
= True

The result is that all logical statements are evaluated as True. Even a statement and its contradiction became both true.

In Mathematics:
The law of non-contradiction would say that two different numbers are different.
IF a is always not equal to b THEN a is never equal to b

Lets say that there exists only one exception to this rule. Lets say: "1=2". Denying the law of non-contradiction would mean that a statement like "1=2" can be true. The law of non-contradiction prohibits such statements.

Lets now deny the law of non-contradiction and say: "1=2".... Now we can simply prove that any number is equal to any other number!
Consider this:
1=2 ; premise

= 2 + 3
= 1 + 3 ; premise
= 4 ; Result 1 (5=4)

= 5 + 2 + 13
= 4 + 2 + 13 ; Result 1 (5=4)
= 4 + 1 + 13 ; premise
= 18 ; Result 2 (20=18)
= 2 + 16
= 1 + 16 ; premise
= 17 ; Result 3 (20=17) and (18=17)

= 1 - 1
= 2 - 1
= 1 ; Result 4 (0=1)

Practically, by having ONLY ONE such an exception to the law of non-contradiction, as "1=2"... We can show that ALL other numbers are equal to one another! Not only for integer, but also for any other number. Observe:
For rational numbers:
= 10 / 20

We can prove that 10=200, and that 20=13

So we get:
= 10 / 20
= 200 / 13

For irrational numbers:
SQR: Square root
SQR 20
= SQR 50
= SQR 901

Now someone might say, that the operator of addition (+) is the problem, and that saying "20 = 1 + 19" is the problem. Fine, if we can't do it with addition, we do it with some other operators. Observe:
Using multiplication:
= 2 * 3
= 1 * 3 ; premise
= 3

Using power:
^: The power operator
= 4 ^ 2
= 4 ^ 1 ; premise
= 4

The end result, that to protect mathematics from dissolving by one such instance where the law of non-contradiction does not hold, you must deny numerous other operators that can be used to dissolve the system, including -among others-: Addition, subtraction, multiplication, division, powers, logarithms... etc.

In short, the whole concept of algebraic manipulation needs to be abandoned to hold the system from dissolving into a trivial system.

This would mean heaven for lazy students who hate math and science. Because this means that any answer they put in their exams is necessarily correct!! All students now get 20 out of 20. And its no problem if you get 0 out of 20, because 20 equals 0! So if you are a lazy student, I suggest you start a campaign with the slogan "Say NO to the law of non-contradiction!"...

Communication and Language:
Denying the law of non-contradiction makes communication of ideas very difficult. To deny the law of non-contradiction is to say that: "To say something, and to negate that same something are two compatible views"...

Funny dialogs can result from this assertion. Consider this dialog, between two people: A and B... Hegel is a philosopher who -is said to have- denied the law of non-contradiction, and this fictional dialog was assembled to demonstrate how problematic this denial is.

A: Are you still a follower of Hegel?

B: Of course! I believe everything he wrote. Since he denied the law of noncontradiction, I deny this too. On my view, P is entirely compatible with not-P.

A: I'm a fan of Hegel myself. But he didn't deny the law of noncontradiction! You read the wrong commentators!

B: You're wrong, he did deny this! Let me get my copy of The Science of Logic.

A: Don't get so upset! You said that he did deny the law, and I said that he didn't. Aren't these compatible on your view? After all, you think that P is compatible with not-P.

B: Yes, I guess they're compatible.

A: No they aren't!

B: Yes they are!

A: Don't get so upset! You said that they are compatible, and I said that they aren't. Aren't these two compatible on your view? Recall that you think that P is compatible with not-P.

B: Yes, I guess they're compatible. I'm getting confused.

A: And you're also not getting confused, right?

source: Philosophy, et cetera Blog (link)

Thursday, February 21, 2008

Quantum Mechanics And Classical Physics: The Limit Relation

I am not qualified to discuss neither classical physics nor modern physics (aka quantum mechanics) but I have some knowledge that I would like to share that should make things easier to understand to make things clearer to an "average Joe".

In short: Classical Physics is the limit to infinity of Quanta in Quantum Mechanics. Thats in a nutshell.

For people who are not familiar with the mathematical concept of limits consider this example:
Say, F(X)=1-5*10^-X (one minus five times ten to the power of minus X). In this example:

This is the concept of a limit, notice that for small values of X, say between 0 and 3 the value of the function changes dramatically. But from the values of 4 and above the change is negligible. Consider how much F(4) is close to one!!

So in Quantum Mechanics the formulas are more complex than Classical Physics, but when the number of Quanta is large, certain parts of the equation can be discarded since they are negligible. Just like how in "F(X)=1-5*10^-X" we can discard the "-5*10^-X" since its almost zero for large values of X. The concept of limits apply when moving from Quantum Mechanics to Classical Physics.

Another hot topic is the probabilistic model of Quantum Mechanics. While we cannot be certain of the outcome of Quanta in overly simplistic scenarios, this does not apply to large numbers of Quanta. Its the same concept of limits.

Consider the following scenario:
Say X is a variable that initially has the value of 0. We throw a die, and we add the value of the die to X. So lets see what happens to X after 3 throws.
Say, the outcome of the die was: 3 / 3 / 6
Another experiment has this outcome: 4 / 1 / 2
Yet a third with this outcome: 5 / 6 / 4

It seems that X is different for each of these trials... This is an analogy to what happens with Quanta. Each Quanta produces a different stochastic (random) behavior. And the behavior of a system of Quanta is the sum of all the behavior of Quanta at that moment.

Consider now the above example with a billion throws. While the outcome of 3 trials might have large differences in each experiment, as the number of trials increases the error margin decreases. The the most basic property of probability. The more the trials the more stable results you will get.

In the above example, I can say that the average outcome of the die is (1+2+3+4+5+6)/6 = 3.5
So in one billion trials, the outcome should be very close to 3.5 billions.

The conclusion is that although when we consider Quanta in isolation they might produce diverse results, studying large numbers of Quanta is much easier and easily predictable with high confidence.

It is worth noting that one proton contains: 2.26*10^23 Quanta!! As you can see, even a single subatomic part contains a huge number of Quanta!! So even for a single atom assuming that the number of Quanta in it is infinite is a reasonable approximation in limit equations!

Thursday, February 14, 2008

Friday, February 08, 2008

Games Of Life - Part 2: Equilibrium

Equilibrium is the state where all players have equal benefits and losses. There are several ways to achieve equilibrium. Assuming perfect play, a game reaches equilibrium if the game is balanced. A "balanced game" is the game which when played perfectly reaches equilibrium. Almost all popular games are balanced games. Poker is a balanced game. Card games like Hearts, Bridge, and Tricks are all balanced games. Chess is believed (although not proven) to be a balanced game!

In the case of card games, this does not mean that every round or game will end in a draw!! This means that if the game was repeated a large number of time, the scores will be drawish. The idea is, since cards are shuffled and given to players randomly, then making large number of games will be drawish. The concept is same as rules of chances in experiments like throwing dice. Everyone knows that for a "fair dice" each outcome from 1-6 will appear almost equally given a large number of trials. Thats why to reach balance we need repetitions to form statistically valid results.

Is life a balanced game? Assuming each player is unique from his parents, and considering the way things work, I would think that life cannot be considered a balanced game. Different amounts of effort are required by different players to reach equivalent statuses. This may not be true. It might be that if all the players played "perfectly" all the humans would reach a state of equality (even though they might not start with equal statuses). If we think of life as a balanced game and assume all players are playing perfectly, it should be that the rich get poorer and the poor get richer, until all players reach a state of equality. This may or may not be true. It is observed that the rich gets richer, and the poor gets poorer. But this outcome is not based on perfect play, and so it is not indicative of what would happen in the case of perfect play!

The concept of a "fair" game is different from a "balanced" game. A balanced game is defined as the behavior of the game on the long run. But not all balanced games are necessarily fair games. A fair game is a balanced game that requires equal effort to reach the state of equilibrium. If we assume that life is a balanced game, it probably is not a fair game. Maybe everyone can reach an equal state as other players, but this does not constitute equal effort.

Concepts of evolution suggest that life is not a fair game. A term like "survival of the fittest", suggests that certain individuals are more fit for the game of life than others. Thats why biologists model life as a game with bias towards certain traits that are more adaptive to the environment.

In this series:
Games Of Life - Part 1: The Broad Lines
Games Of Life - Part 2: Equilibrium
Next: Games Of Life - Part 3: Games Of Non-Zero-Sum
Next: Games Of Life - Part 4: Advanced Insight Into Games Of Perfect Information
Next: Games Of Life - Part 5: Gaming Theory And Decision-Making Theory

Thursday, February 07, 2008

Facebook: For The Sheeple of Earth

Reading the "Terms of Use" of Facebook, I found this article:
By posting User Content to any part of the Site, you automatically grant, and you represent and warrant that you have the right to grant, to the Company an irrevocable, perpetual, non-exclusive, transferable, fully paid, worldwide license (with the right to sublicense) to use, copy, publicly perform, publicly display, reformat, translate, excerpt (in whole or in part) and distribute such User Content for any purpose, commercial, advertising, or otherwise, on or in connection with the Site or the promotion thereof, to prepare derivative works of, or incorporate into other works, such User Content, and to grant and authorize sublicenses of the foregoing. You may remove your User Content from the Site at any time. If you choose to remove your User Content, the license granted above will automatically expire, however you acknowledge that the Company may retain archived copies of your User Content.

source: Facebook - Terms of Use

SHEEPLE WAKE UP BEFORE YOU FIND YOURSELVES SELLING GUM AT THE STREET LIGHTS!!! Facebook is blackmailing all its users, and they are like sheep: Happy, singing "Maaa" ("I Agree")!

And for the last two sentences... There is a catch... Well, you can only pity the not-so-smart ones! (Hint: I underlined the clues)

This sin is not exclusive to Facebook only, but it is God-damned one of the most dangerous Terms of Use on the surface of this planet!! This post is a word for the wise...

Disclaimer: I am not a lawyer, and I don't have any degrees in the Law, nor do I claim proficiency in any Legal matters. You may wish to consult a lawyer.

Monday, February 04, 2008

Games Of Life - Part 1: The Broad Lines

Gaming theory is an interesting field of study that has been at numerous time applied to human life. Several gaming types where applied on real-life with different degrees of success. For example, Poker (a gambling game of cards) has been used several times to study tactical operations in war for military purposes. Yet, later it was proved that Poker is not a good model for war tactics regardless of how strong the analogy might be.

Poker obviously cannot be applied to war tactics (or any other aspect of life) due it being a zero-sum game: Life, and especially war, is not a zero-sum game, and modeling it as such is necessarily flawed logic! Sure you might ask what does "zero-sum game" mean...

Games can be divided into several categories according to certain properties. Some games are zero-sum others are non-zero-sum. Some games are games of perfect-information others are games of imperfect information.

Zero-Sum Games:
A zero-sum game is a game where one's gain is another's loss, and someone's loss is another's gain. Consider a game of chess: If one side wins, the other must lose. They cannot both win! There is no win-win situation in a game of chess! Same goes for poker, if someone wins 10$, then someone else must have lost 10$!! If we consider winning 10$ as +10, and losing 10$ as -10, then the sum of it all is ZERO... Thats why they call it a zero-sum game!
This applies to two-player games as well as multi-player games where applicable. Games of zero-sum always adds up to zero for all the players involved.
Life is obviously not a zero-sum game. There are a few win-win situations, but the most dominant situation is loss-loss. Consider the case of a war: Say a country is worth 1 million points, and each person is worth 1 point. Now assume that country A attacked country B, and a war happened that cause 1000 deaths in country A and 2000 deaths in country B, and finally country A took country B. In this hypothesized situation, country A has won 1 million points and lost 1000 points, giving a net of (+999000). Obviously country A has won something. On the other hand, country B has lost 1 million points and 2000 points, giving a net of (-1001000). Adding up the points of both countries the sum is (-3000). It did not add up to ZERO, because certain losses have irrecoverably vanished. So this hypothesized war is a losing game, or in other words a negative-sum game. On the other hand, sex can be though of as a positive-sum game, as both players in that game are satisfied.

Games of Perfect Information:
Chess is a classical example of a game of perfect information. The state of the game is well-known to both players. No player has anymore information than the other. Everything is laid-out in the clear. All possibilities are deductible.
(Most) Card games are games of imperfect information. In a typical four-players card games, one player has information about his set of cards, while having absolutely no idea about the cards of the other three players. This obviously turns the game into a game of imperfect information.
Life can be modeled as a game of imperfect information. Assuming that humans are players in the game of life, we can find that most of the time a player has no knowledge of all the variables that influence the outcome of the game.The element of surprise is dominant.

Games of Chance:
Monopoly is a classical example of a game of chance. Unlike card games, all the players have equal knowledge of the state of the game. No-information is hidden from any of the players, except the output of the dice, which none of the players know in advance. The game consists of what game theory experts call "chance nodes" and "decision nodes"... A chance node is simply the moment of throwing the dice, whose outcome is probabilistically modeled. A decision node is the moment one (or more) of the players have to make a decision that will change the state of the game.
Also, unlike chess, although players have full knowledge of the state of the game, they have no power to influence the outcome of the game assuming perfect play by all the players.

Games of chance might include the element of imperfect information. Texas-Holdem poker is such an example. In Texas Holdem poker, each player has information about his own cards and not other players just like games of imperfect information. But in addition to that, there are chance nodes where cards are revealed in a probabilistic manner. Bets made by players are the decision nodes.

Life can be modeled as a game of chance with imperfect information. So all-in all, life can be modeled as a game of chance with imperfect information and non-zero-sum! Numerous other factors can be taken into consideration to refine the model of life as a game.

In this series:
Games Of Life - Part 1: The Broad Lines
Games Of Life - Part 2: Equilibrium
Next: Games Of Life - Part 3: Games Of Non-Zero-Sum
Next: Games Of Life - Part 4: Advanced Insight Into Games Of Perfect Information
Next: Games Of Life - Part 5: Gaming Theory And Decision-Making Theory