Alfred Tarski, 3

The undefinability theorem shows that this encoding cannot be done for semantic concepts such as truth. It shows that no sufficiently rich interpreted language can represent its own semantics. A corollary is that any metalanguage capable of expressing the semantics of some object language must have expressive power exceeding that of the object language. The metalanguage includes primitive notions, axioms, and rules absent from the object language, so that there are theorems provable in the metalanguage not provable in the object language.

— Wikipedia on Tarski’s undefinability theorem


Tarski’s 1969 “Truth and proof” considered both Gödel’s incompleteness theorems and Tarski’s undefinability theorem, and mulled over their consequences for the axiomatic method in mathematics.

— Wikipedia on Alfred Tarski



2019.07.20 Saturday ACHK

Confirmation principle

Verification principle, 2.2 | The problem of induction 4


The statements “statements are meaningless unless they can be empirically verified” and “statements are meaningless unless they can be empirically falsified” are both claimed to be self-refuting on the basis that they can neither be empirically verified nor falsified.

— Wikipedia on Self-refuting idea


In 1936, Carnap sought a switch from verification to confirmation. Carnap’s confirmability criterion (confirmationism) would not require conclusive verification (thus accommodating for universal generalizations) but allow for partial testability to establish “degrees of confirmation” on a probabilistic basis.

— Wikipedia on Verificationism


Confirmation principle should not be applied to itself because it is an analytic statement which defines synthetic statements.


Even if it does, it is not self-defeating, because confirmation principle, unlike verification principle, does not requires a statement to be proven with 100% certainty.

So in a sense, replacing verification principle by confirmation principle can avoid infinite regress.


Accepting confirmation principle is equivalent to accepting induction.

“This is everything to win but nothing to lose.”

— Me@2012.04.17



2019.04.06 Saturday (c) All rights reserved by ACHK

The problem of induction 3.3

“Everything has no patterns” (or “there are no laws”) creates a paradox.


If “there are 100% no first order laws”, then it is itself a second order law (the law of no first-order laws), allowing you to use probability theory.

In this sense, probability theory is a second order law: the law of “there are 100% no first order laws”.

In this sense, probability theory is not for a single event, but statistical, for a meta-event: a collection of events.

Using meta-event patterns to predict the next single event, that is induction.


Induction is a kind of risk minimization.

— Me@2012-11-05 12:23:24 PM



2018.12.28 Friday (c) All rights reserved by ACHK

The problem of induction 3.1.2

Square of opposition


“everything has a pattern”?

“everything follows some pattern” –> no paradox

“everything follows no pattern” –> paradox

— Me@2012.11.05


My above statements are meaningless, because they lack a precise meaning of the word “pattern”. In other words, whether each statement is correct or not, depends on the meaning of “pattern”.

In common usage, “pattern” has two possible meanings:

1. “X has a pattern” can mean that “X has repeated data“.

Since the data set X has repeated data, we can simplify X’s description.

For example, there is a die. You throw it a thousand times. The result is always 2. Then you do not have to record a thousand 2’s. Instead, you can just record “the result is always 2”.

2. “X has a pattern” can mean that “X’s are totally random, in the sense that individual result cannot be precisely predicted“.

Since the data set X is totally random, we can simplify the description using probabilistic terms.

For example, there is a die. You throw it a thousand times. The die lands on any of the 6 faces 1/6 of the times. Then you do not have to record those thousand results. Instead, you can just record “the result is random” or “the die is fair”.

— Me@2018-12-18 12:34:58 PM



2018.12.18 Tuesday (c) All rights reserved by ACHK

The problem of induction 3.2

The meaning of induction is that

we regard, for example, that

“AAAAA –> the sixth is also A”

is more likely than

“AA –> the second is also A”


We use induction to find “patterns”. However, the induced results might not be true. Then, why do we use induction at all?

There is everything to win but nothing to lose.

— Hans Reichenbach

If the universe has some patterns, we can use induction to find those patterns.

But if the universe has no patterns at all, then we cannot use any methods, induction or else, to find any patterns.


However, to find patterns, besides induction, what are the other methods?

What is meaning of “pattern-finding methods other than induction”?

— Me@2012.11.05

— Me@2018.12.10



2018.12.10 Monday (c) All rights reserved by ACHK

The problem of induction 3


In a sense (of the word “pattern”), there is always a pattern.


Where if there are no patterns, everything is random?

Then we have a meta-pattern; we can use probability laws:

In that case, every (microscopic) case is equally probable. Then by counting the possible number of microstates of each macrostate, we can deduce that which macrostate is the most probable.


Where if not all microstates are equally probable?

Then it has patterns directly.

For example, we can deduce that which microstate is the most probable.

— Me@2012.11.05



2018.11.19 Monday (c) All rights reserved by ACHK

defmacro, 2

Defining the defmacro function using only LISP primitives?


McCarthy’s Elementary S-functions and predicates were

atom, eq, car, cdr, cons


He then went on to add to his basic notation, to enable writing what he called S-functions:

quote, cond, lambda, label


On that basis, we’ll call these “the LISP primitives”…

How would you define the defmacro function using only these primitives in the LISP of your choice?

edited Aug 21 ’10 at 2:47

asked Aug 21 ’10 at 2:02


Every macro in Lisp is just a symbol bound to a lambda with a little flag set somewhere, somehow, that eval checks and that, if set, causes eval to call the lambda at macro expansion time and substitute the form with its return value. If you look at the defmacro macro itself, you can see that all it’s doing is rearranging things so you get a def of a var to have a fn as its value, and then a call to .setMacro on that var, just like core.clj is doing on defmacro itself, manually, since it doesn’t have defmacro to use to define defmacro yet.

– dreish Aug 22 ’10 at 1:40



2018.11.17 Saturday (c) All rights reserved by ACHK




Alt + Up/Down

Switch between the editor and the REPL

— Me@2018-11-07 05:57:54 AM




(defmacro our-expander (name) `(get ,name 'expander))

(defmacro our-defmacro (name parms &body body)
  (let ((g (gensym)))
       (setf (our-expander ',name)
	     #'(lambda (,g)
		 (block ,name
		   (destructuring-bind ,parms (cdr ,g)

(defun our-macroexpand-1 (expr)
  (if (and (consp expr) (our-expander (car expr)))
      (funcall (our-expander (car expr)) expr)


A formal description of what macros do would be long and confusing. Experienced programmers do not carry such a description in their heads anyway. It’s more convenient to remember what defmacro does by imagining how it would be defined.

The definition in Figure 7.6 gives a fairly accurate impression of what macros do, but like any sketch it is incomplete. It wouldn’t handle the &whole keyword properly. And what defmacro really stores as the macro-function of its first argument is a function of two arguments: the macro call, and the lexical environment in which it occurs.

— p.95


— On Lisp

— Paul Graham


(our-defmacro sq (x)
  `(* ,x ,x))

After using our-defmacro to define the macro sq, if we use it directly,

(sq 2)

we will get an error.

The function COMMON-LISP-USER::SQ is undefined.
[Condition of type UNDEFINED-FUNCTION]

Instead, we should use (eval (our-macroexpand-1 ':

(eval (our-macroexpand-1 '(sq 2)))

— Me@2018-11-07 02:12:47 PM



2018.11.07 Wednesday (c) All rights reserved by ACHK

Existence and Description

Bertrand Russell, “Existence and Description”


§1 General Propositions and Existence

“Now when you come to ask what really is asserted in a general proposition, such as ‘All Greeks are men’ for instance, you find that what is asserted is the truth of all values of what I call a propositional function. A propositional function is simply any expression containing an undetermined constituent, or several undetermined constituents, and becoming a proposition as soon as the undetermined constituents are determined.” (24a)

“Much false philosophy has arisen out of confusing propositional functions and propositions.” (24b)

A propositional function can be necessary (when it is always true), possible (when it is sometimes true), and impossible (when it is never true).

“Propositions can only be true or false, but propositional functions have these three possibilities.” (24b)

“When you take any propositional function and assert of it that it is possible, that it is sometimes true, that gives you the fundamental meaning ‘existence’…. Existence is essentially a property of a propositional function. It means that the propositional function is true in at least one instance.” (25a)

— Brandon C. Look

— University Research Professor and Chair



2018.10.07 Sunday ACHK

The Sixth Sense, 3

Mirror selves, 2 | Anatta 3.2 | 無我 3.2


You cannot feel your own existence or non-existence. You can feel the existence or non-existence of (such as) your hair, your hands, etc.

But you cannot feel the existence or non-existence of _you_.

— Me@2018-03-17 5:12 PM


Only OTHER people or beings can feel your existence or non-existence.

— Me@2018-04-30 11:29:08 AM



2018.04.30 Monday (c) All rights reserved by ACHK

機遇再生論 1.6





假設,你現在手中,有一副樸克牌,存在於某一個排列 A 。洗牌一次之後,排列仍然是 A 的機會極微。

一副完整的撲克牌,共有 N = 52! \approx 8.07 \times 10^{67} 個,可能的排列。亦即是話,洗牌後仍然是排列 A 的機會率,只有 \frac{1}{N}

由於分母 N 太大(相當於 8 之後,還有 67 個位),洗牌後,理應變成另外一個排列 B 。

P(A) = \frac{1}{N}

P(\text{not} A) = 1 - \frac{1}{N}

洗了一次牌後,發覺排列是 B 不是 A 後,我們可以再問,如果再洗一次牌,「是 A」和「不是 A」的機會,分別是多少?


由於,機會率只是與未知的事情有關,或者說,已知的事件,發生的機會率必為 1;所以,如果發生了第一次洗牌,而你又知道其結果的情況下,問「如果再洗一次牌,『是 A』和『不是 A』的機會,分別是多少」,第二次洗牌各個可能結果,發生的機會率,與第一次洗牌的結果無關。

第二次洗牌結果為組合 A 的機會率,仍然是

P(A) = \frac{1}{N}

不是組合 A 的機會率,仍然是

P(\text{not} A) = 1 - \frac{1}{N}





「如果洗牌兩次,起碼一次洗到原本排列 A 的機會率是多少?」

把該事件標示為 A_2

A_2 = 兩次洗牌的結果,起碼一次洗到原本排列 A

再把該事件的機會率,標示為 P(A_2)

由於 P(A_2) 相對麻煩,我們可以先行運算其「互補事件」的機會率。

A_2 的互補事件為「不是 A_2」:

不是 A_2

= 兩次洗牌的結果,不是起碼一次洗到原本排列 A

= 兩次洗牌的結果,都不是排列 A


P(\text{not} A_2) = (1 - \frac{1}{N})^2


= 1 - P(\text{not} A_2)
= 1 - (1 - \frac{1}{N})^2



如果洗牌 m 次,起碼一次洗到原本排列 A 的機會率是多少?


P(A_m)= 1 - (1 - \frac{1}{N})^m

留意,N = 52! \approx 8.07 \times 10^{67},非常之大,導致 (1 - \frac{1}{N}) 極端接近 1。在一般情況,m 的數值還是正常時, P(A_m) 會仍然極端接近 0。

例如,你將會連續洗一千萬次牌(m = 10,000,000),起碼有一次,回到原本排列 A 的機會是:

= 1 - (1 - \frac{1}{N})^m
= 1 - (1 - \frac{1}{52!})^{10,000,000}

你用一般手提計算機的話,它會給你 0。你用電腦的話,它會給你

1.239799930857148592 \times 10^{-61}

— Me@2018-01-25 12:38:39 PM



2018.02.13 Tuesday (c) All rights reserved by ACHK

機遇再生論 1.5



是句「科學句」(經驗句),因為你知道在什麼情境下,可以否證到它 —— 如果你在甲過身後,等了一千億年,甲還未重生的話,那句就為之錯。













假設,你現在手中,有一副樸克牌,存在於某一個排列 A 。洗牌一次之後,排列仍然是 A 的機會極微。

一副完整的撲克牌,共有 N = 54! = 2.3 \times 10^{71} 個,可能的排列。亦即是話,洗牌後仍然是排列 A 的機會率,只有 \frac{1}{N}

由於分母 N 太大(相當於 2 之後,還有 71 個位),洗牌後,理應變成另外一個排列 B 。

P(A) = \frac{1}{N}

P(not A) = 1 - \frac{1}{N}

— Me@2017-12-18 02:51:11 PM
2017.12.18 Monday (c) All rights reserved by ACHK

Determined by what?

If you say “an event is determined”, in order to be meaningful, you have to specify, explicitly or by context, that the event is determined by whom.

Similarly, if you say something is free, you have to specify “free from what” or “free with respect to what”. 

free ~ independent of

Without a grammatical object, the phrase “independent of” is meaningless, unless the context has implied what that grammatical object is.

— Me@2015-05-23

free [without an object] ~ free from everything

is meaningless, because the word “everything” is meaningful only if it has a context.

— Me@2017-07-20

2017.07.29 Saturday (c) All rights reserved by ACHK


注定外傳 2.3.4 | Can it be Otherwise? 2.3.4









— Me@2017-02-03 04:15:54 PM

2017.02.03 Friday (c) All rights reserved by ACHK

注定外傳 2.3.3

Can it be Otherwise? 2.3.3


彷彿是「神的旨意」一樣 — 即使有「神的旨意」,它並不能指引你,去做最佳的決定。













— Me@2016-12-30 03:37:35 PM

2017.01.01 Sunday (c) All rights reserved by ACHK

注定外傳 2.3.2

Can it be Otherwise? 2.3.2


其中一個版本是,針對外在因素 — 有時,你想選擇一條路,但是,因為受制於外在因素,唯有違反原本意圖,改為選擇另一選項。多數人也會把這個情況,歸類為「沒有自由意志」。少數人會把這個情況,標籤為「仍然有自由意志」,因為你仍然可以思考,有自己的意向;你只是「沒有自由行動」,而不是「沒有自由意志」。

另一個版本,則針對內在因素 — 有時,連你那個選擇想法本身,根本也沒有自由;換句話說,你的原本意圖是什麼,其實也是受制於各項因素。 這個情況是,絕對的「沒有自由意志」。


— Me@2016-10-15 06:10:12 AM

2016.10.15 Saturday (c) All rights reserved by ACHK

注定外傳 4.0

Can it be Otherwise? 4.0

One of the major difficulties of free-will-VS-determinism problem is its “always-meta” nature.

— Me@2016-08-19 09:00:14 AM

You can will to act, but not will to will.

Man can do what he wants, but he cannot will what he wants.

You can do what you will, but in any given moment of your life you can will only one definite thing and absolutely nothing other than that one thing. 

— Schopenhauer


— Me@2016-01-06 06:50:56 PM

By definition, will is a first cause. So you cannot control it.

— Me@2016-01-06 06:55:13 PM

2016.08.22 Monday (c) All rights reserved by ACHK

注定外傳 3.0

Can it be Otherwise? 3.0







1. 人有自由;

2. 因為一切皆注定,人沒有自由。












無論你有什麼行動,或者有什麼態度 ,你既可以解釋成,因為「自己有自由」;亦可以解釋成,因為「一切皆注定」。



— Me@2016-07-04 11:21:49 PM

2016.07.04 Monday (c) All rights reserved by ACHK

注定外傳 2.6

Can it be Otherwise? 2.6 | The Beginning of Time, 7.3











— Me@2016-05-18 11:40:31 AM

2016.05.18 Wednesday (c) All rights reserved by ACHK

注定外傳 2.5

Can it be Otherwise? 2.5 | The Beginning of Time, 7.2



4. 即使可以追溯到「時間的起點」(第一因),所謂的「可以」,只是宏觀而言,決不會細節到可以推斷到,你有沒有自由,明天七時起牀。


















— Me@2016-03-15 08:43:58 AM

2016.03.31 Thursday (c) All rights reserved by ACHK