機會率哲學 2.1

Posted on November 5, 2012 by rpflee

The problem of induction 1.1 | 西瓜 6.1

這段改編自 2010 年 4 月 3 日的對話。

大學二年級時，我們曾經旁聽李教授的「哲學分析」科。其中一課是講「歸納法問題」。「歸納法」的意思是，過往重複發生過很多次的事件，我們估計將來都會發生。

但是，「歸納法」是沒有「必然性」的。例如，中學甲於過去十年，每年公開試英文科的合格率，都達九成以上。但是，你最多只能「預測」，而不能百分百「保證」，來年都是那樣。

又例如，根據你的電視機例子，這個牌子這個型號，來自同一條生產線的電視機，即使之前十萬部的壽命，都超過三年，售貨員最多只能「預測」，而不能百分百「保證」，你買的那一部，都是那樣。

過去會發生的事情，即使已重複發生了很多次，也不代表，將來會發生。那就引發了，你所追問的「歸納法問題」。「歸納法問題」的意思是，既然我們運用「歸納法」所得的結論，是沒有保障的，為何我們要接受「歸納法」呢？又或者說，既然我們運用「歸納法」所預測的將來，不一定是正確的，「歸納法」還符合「理性」嗎？

其中一個答法是，人類利用「歸納法」，過往無論是在日常生活、科學研究，還是科技發展，都取得了鉅大的成功，所以「歸納法」是可信可用的。

但是，這個說法正正是利用了「歸納法」本身，去辯護「歸納法」，循環論證也。

— Me@2012.11.05

Why does the universe exist? 4

Posted on November 4, 2012 by rpflee

Existence, 8

The sentence “the universe exists” can be meaningful if you interpret it as “something exists“.

The universe exists

= Something exists

= There is something

And then you interpret the “universe” as the container of that “something”. In other words, you define the “universe” as the “there”.

However, it is an abuse of the word “universe”. The meaning of the word “universe” is “everything” or “all the things“, not just “something”. Also, the sentence “something exists” is useless. At most, it is just a meaningful nonsense.

— Me@2012.10.15

機會率哲學 1.6

Posted on November 3, 2012 by rpflee

這段改編自 2010 年 4 月 3 日的對話。

In fact, the spectrum of interpretations in quantum mechanics has a close analogue in probability theory. The “wave function is real” view is analogous to the “frequentist” view of probability theory where probabilities describe “random pheonomena” like rolling dice or radioactive decays and the “wave function represents what you know about the system” view is analogous to the Bayesian view where probability is just a consistent way of assigning [likelihoods] to propositions independent of whether they have anything to do with a “random process.”

— Bayesian Probability Theory and Quantum Mechanics

— John Baez

（安：但是，我又可以這樣追問。「這一部」電視機在第一年內，要麼會損壞，要麼不會損壞。

你說「這一部」電視機，在第一年內故障的機會率是「三千分之一」，究竟是什麼意思呢？難道「這一部」電視機在第一年內，會有三千分之一的部分會損壞嗎？）

你的意思是，既然是討論「個別單一事件」，理應用不上「統計資料」，因為「統計」是眾多案例的歸納。亦即是話，你正在變相追問「機會率」的哲學涵義。有什麼理論基礎，令到我們可以利用「過往眾多事件」的統計資料，來判定「特定事件」的機會率？而那個「機會率」數字，又代表什麼呢？

「機會率」的詮釋問題，其實是對應於「量子力學」的詮釋問題。換句話說，如果你可以搞清「機會率」的真正意義，你就可以搞清「量子力學」的背後原理，反之亦然。

可惜，無論是「機會率詮釋」，還是「量子力學詮釋」，學術界仍然未有終極結論。所以，你的問題走得太遠，已經走到人類現時的知識邊緣。

— Me@2012.11.03

致讀者：我於去年（2011）已經搞清了「機會率」的真正意義。如果你想知道，請參閱本網誌 quantum probability （量子機率）和 single-world interpretation（單重宇宙）類的文章。你將會得到部分答案。

其中一個核心要點是，「現實世界」是「所有」「可能世界」的疊加。

— Me@2012.11.03

Existence 7

Posted on November 2, 2012 by rpflee

For a horse, wings do not exist

= A horse does not have wings

The non-existence of the wings is not their own property, but the horse’s. An object’s existence or non-existence is not a property of the object itself, but a property of its owner.

— Me@2012.10.15

Existence 6

Posted on October 31, 2012 by rpflee

Your existence is not your own property, but the universe’s. Same for non-existence.

— Me@2012.10.15

The Beginning of Time

Posted on October 29, 2012 by rpflee

Existence, 5 | Why does the universe exist? 3

The sentence “there is nothing in the north of the North Pole” is inaccurate, because it assumes that there a place in the north of the North Pole, although that place has nothing in it. Instead, we should say

The North Pole has no “north”.

The word “north” is meaningless at the North Pole.

— Me@2012.10.15

There 1.3

Posted on October 27, 2012 by rpflee

Existence, 3.6 | Why does the universe exist? 1.6

“There” is “那裡” in Chinese. Literally,

here = 這裡 = this inside

there = 那裡 = that inside

For example,

A dog exists

= There is a dog

= A dog is in there

“There” is a container. That is why both the sentence “the universe exists” and the sentence “the universe does not exist” have no meanings.

— Me@2012.10.15

There 1.2

Posted on October 25, 2012 by rpflee

Existence, 3.5

Why does the universe exist? 1.5

To specify something exists or not, you need a “there”, range for searching. To prove something exists, you may not need to search all over “there”. But to prove something does not exist, you need to.

— Me@2012.10.15

Paradox 5

Posted on October 24, 2012 by rpflee

Meta-time 4 | Cumulative concept of time, 13

The grandfather paradox has this structure:

Your-2012-self = [ …, your-2002-self, … ]

B = [ …, A, … ]

If your-2012-self can go back to, such as, year 2001,

Your-2002-self = [ …, your-2012-self, … ]

A = [ …, B, … ]

Unless A = B, it is logical impossible to have both “A is part of B” and “B is part of A“.

— Me@2012.04.02

Existence, 4

Posted on October 23, 2012 by rpflee

Why does the universe exist? 2

The sentence “the universe does not exist” is meaningless. However, its limited version “there are nothing” or “the universe has nothing” may be meaningful.

If the universe is finite in space, in principle, you can search all over the space to confirm that there are really nothing. So it seems that the sentence “the universe has nothing” does not violate the confirmation principle. However, there are three problems.

First, spacetime is also a “thing”, provided that the definition of the word “thing” is not limited to “object” or “matter”. Second, “spacetime” has no meaning if there are no matter and no energy. Moreover, you, as an observer, is also a “thing”.

After all, “the universe has nothing” is meaningless, in the sense that it violates the confirmation principle.

— Me@2012.10.15

There

Posted on October 20, 2012 by rpflee

Existence, 3.4

Why does the universe exist? 1.4

Dogs do not exist

= There are no dogs

The non-existence is not a property of the dogs. Instead, the “non-existence of dogs” is a property of “there”, the system containing the dogs, such as a room. In other words, “there” is a location, an address, or an environment.

The universe exists = There is a universe

The universe does not exist = There is no universe

These two sentences are both meaningless, because the definition of the word “universe” is “everything”. The universe has no “outside”. The universe has no “there”. The question “Where is the universe?” makes no sense.

— Me@2012.10.15

Existence, 3.3

Posted on October 19, 2012 by rpflee

Why does the universe exist? 1.3

In “dogs do not exist in this room“, the non-existence is not a property of the dogs.

Instead, the “non-existence of dogs” is a property of the room.

— Me@2012.10.15

Why does the universe exist? 1.2

Posted on October 18, 2012 by rpflee

Existence, 3.2

Verification principle, 3

The sentence “the universe does not exist” has no meanings, because it violates the confirmation principle. When we say that “dogs do not exist in this room“, we can search all over this room to prove the non-existence of dogs. However, the definition of the word “universe” is “everything”. So the universe has no “outside”. The universe is not contained within a bigger system. So when we say that “the universe does not exist“, we cannot search all over some bigger environment to prove the non-existence of the universe, even in principle.

The sentence “the universe exists” has the same problem. It also violates the confirmation principle. When we say that “a dog exists in this room“, as long as we can find a dog within the room, we prove the existence of the dog. However, the definition of the word “universe” is “everything”. So the universe has no “outside”. The universe is not contained within a bigger system. So when we say that “the universe exists“, we cannot “find” the universe, even in principle.

Whatever we find, such as a dog, a room, a house, a city, etc., is only part of the universe. “Part of the universe exists” does not imply “the universe exists“. For example, I have part of one million dollars, such as 500 thousand dollars, doesn’t mean that I have one million dollars.

— Me@2012.10.15

— Me@2012.10.18

Why does the universe exist?

Posted on October 16, 2012 by rpflee

Existence, 3.1

The sentence “the universe exists” and the sentence “the universe does not exist” have no meanings, because the “existence” of something is not a property of that thing, but a property of a bigger system. But the definition of the word “universe” is “everything”. So the universe has no “outside”. The universe is not contained within a bigger system.

— Me@2012.10.15

Paradox 4.3

Posted on October 15, 2012 by rpflee

According to the definition/usage of the word “true”:

A = “A” is true

“A” is true = A

e.g. The sentence “Snow is white” is true = Snow is white

Not-A = “A” is false

“A” is false = Not-A

e.g. The sentence “Snow is white” is false = Snow is not white

Paradox is due to the mixing of para-level (meta-level) and original level. As long as we do not allow mixing levels, there are no paradoxes. Every sentence should only be allowed to describe sentences which have lower levels. So the sentence “this sentence is false” should not be allowed.

this sentence is false = “this sentence is false” is (in ordinary level) true

this sentence is false = “this sentence is false” is (in meta level) false

In detail, according to the definition of the word “true”:

this sentence is false = “this sentence is false” is true

But according to the meaning of the sentence itself:

this sentence is false = “this sentence is false” is false

So there is a contradiction.

— Me@2012.04.01

— Me@2012.10.05

— Me@2012.10.15

Curry’s paradox 1.2

Posted on October 14, 2012 by rpflee

Paradox 4.2

The resolution of Curry’s paradox is a contentious issue because resolutions (apart from trivial ones such as disallowing X directly) are difficult and not intuitive. Logicians are undecided whether such sentences are somehow impermissible (and if so, how to banish them), or meaningless, or whether they are correct and reveal problems with the concept of truth itself (and if so, whether we should reject the concept of truth, or change it), or whether they can be rendered benign by a suitable account of their meanings.

— Wikipedia on Curry’s paradox

Regarding Curry’s paradox is an unresolved important question is silly.

— Me@2012-10-13 02:41:50 PM

Curry’s paradox 1.1

Posted on October 13, 2012 by rpflee

Paradox 4.1

When you equal the meta sentence level and the sentence level, it is nothing special about the paradox, for it is just a contradiction between the sentence and the meta-sentence.

In Curry’s paradox, “X -> Y” is what you want to prove. So you try “if X”, to see if you can get Y.

But by letting “X = X -> Y” and if X, and X = X -> Y, then Y. You are actually doing circular proof.

Effectively you use X to prove X, which is a useless proof.

— Me@2012.04.01

— Me@2012.10.05

— Me@2012.10.13

Existence, 2

Posted on October 12, 2012 by rpflee

Existence is not a property of an object. It is a property of properties.

For example, when we say “X does not exist”, we do not mean that “there is an X and X has a property of non-existence.” Instead, existence is a property of the collection of other objects.

Existence is a property of the system that X is in.

X exists

= the system has the component X

= the system has the property of “having X”

X doe not exist

= no systems have the component X

= no systems have the property of “having X”

— Me@2012.10.11

Existence

Posted on October 8, 2012 by rpflee

Properties of objects are called first-order properties.

Properties of first-order properties are called second-order properties.

Existence is not an property of an object. It is a property of properties.

For example, when we say “X does not exist”, we do not mean that “there is an X and X has a property of non-existence.” Instead,

X does not exist

= the property of being X is useless

= the set which contains X’s has no members

— Me@2012.10.07

— Me@2012.10.08

Many-valued logic

Posted on October 7, 2012 by rpflee

Relation to classical logic

Logics are usually systems intended to codify rules for preserving some semantic property of propositions across transformations. In classical logic, this property is “truth.” In a valid argument, the truth of the derived proposition is guaranteed if the premises are jointly true, because the application of valid steps preserves the property. However, that property doesn’t have to be that of “truth”; instead, it can be some other concept.

Multi-valued logics are intended to preserve the property of designationhood (or being designated). Since there are more than two truth values, rules of inference may be intended to preserve more than just whichever corresponds (in the relevant sense) to truth. For example, in a three-valued logic, sometimes the two greatest truth-values (when they are represented as e.g. positive integers) are designated and the rules of inference preserve these values. Precisely, a valid argument will be such that the value of the premises taken jointly will always be less than or equal to the conclusion.

For example, the preserved property could be justification, the foundational concept of intuitionistic logic. Thus, a proposition is not true or false; instead, it is justified or flawed. A key difference between justification and truth, in this case, is that the law of excluded middle doesn’t hold: a proposition that is not flawed is not necessarily justified; instead, it’s only not proven that it’s flawed. The key difference is the determinacy of the preserved property: One may prove that P is justified, that P is flawed, or be unable to prove either. A valid argument preserves justification across transformations, so a proposition derived from justified propositions is still justified. However, there are proofs in classical logic that depend upon the law of excluded middle; since that law is not usable under this scheme, there are propositions that cannot be proven that way.

— Wikipedia on Many-valued logic

2012.10.07 Sunday ACHK

Physics Town

continues

Category Archives: mathematical logic

機會率哲學 2.1

Why does the universe exist? 4

機會率哲學 1.6

Existence 7

Existence 6

The Beginning of Time

There 1.3

There 1.2

Paradox 5

Existence, 4

There

Existence, 3.3

Why does the universe exist? 1.2

Why does the universe exist?

Paradox 4.3

Curry’s paradox 1.2

Curry’s paradox 1.1

Existence, 2

Existence

Many-valued logic