My Rough Ideas

Where Is My Research Going

2011-10-15T12:48:00.000+08:00

The general theme of my thesis will be asking how to deal withinconsistent information in common sense reasoning. So, two areas I will deal with areargumentation and belief revision. Byargumentation, I mean works by Anthony Hunter, John Pollock, Henry Prakken,Phan Minh Dung and many other authors. It is more within the area of AI.

Morespecifically, I will focus on argumentation and its connection with beliefrevision. So, for the belief revisionpart, I will use study in argumentation to do justification approach beliefrevision. And, if it is within myability, I also want to view all these things I have mentioned under theframework of epistemic logic. So, I hopeI can also deal with ‘dynamicizing’ justification logic in my thesis.

two questions about LFI

2011-09-27T16:04:00.000+08:00

∃Γ∃A∃B(Γ, A,¬A ⊬ B) and
∀Γ∀A∀B(Γ, oA, A,¬A ⊢ B

Q1: From a. and b., can we read out that ‘o’ mean ‘consistency’?

Q2: Let ‘o’ means consistency. Do we have the intuition that b. holds?

[ZO] An Intuition

2011-08-24T15:37:00.001+08:00

Let all other things be equal.
Then, the more inconsistent the information is,
the less conclusions we should draw from the information.

A Possible Structure of my Thesis

2011-08-21T16:26:00.001+08:00

Thesis Title: Reasoning under Inconsistent Information

CH0 Introduction

CH1 Using consistent subsets
(1) Connect Preservationism and Hunter’s Argumentation systems and…
(2) Use modal logic to reformulate the approach
(3) The limit of this approach

CH2 Using inconsistent models
(1) Adaptive logics, relevant logic and LFI
(2) philosophical discussion of how to make sense of inconsistent models
(3) mathematical part?

CH3 Comparing two approaches

CH4 Practical applications
(1) paraconsistent belief revision
(a) iterative belief revision
(b) measures of inconsistent information
(c) justification approach of belief revision
(2) argumentation

CH5 Philosophical applications?
(1) Inconsistent theory of truth?

CH6 Conclusion

Meta-philosophy questions

2011-04-28T19:27:00.003+08:00

If we take philosophy as a discourse,
(1) what is the semantics for the philosophical language?
(2) what is the proper truth theory of the philosophical language?
(3) what is the logic that governs the philosophical thinking?

Some naive questions about paraconsistent logic

2011-03-13T18:38:00.002+08:00

Let X be a machine that can do inference according to some classical proof system. However, there is a slight difference between X and usual classical machines:

X has the capacity to recognize whether a proof is identical with the following proof Π:

1. A premise

2. not-A premise

3. A or B By 1

4. B By 2, 3

Machine X is designed to never carry out any proof that is identical with Π.

(Proofs obtained by adding redundant steps into Π will be counted as identical with Π.)

Question 1: Is X explosive or not? This depends on how many proofs from {A, not-A} to B there are. Can we refute that proof Π is the only one proof form {A, not-A} to B, (given a proper definition of proof identity)?

Remark. A diagnosis of where goes wrong with explosion might be problematic, if the diagnosis only focuses on proof Π without refuting the possibility that there are other different classical proofs from {A, not-A } to B.

Question 2: If X is not explosive, does X model the correct way to reason under inconsistent information? (If X is not explosive, X somehow represents the minimal change of classical reasoning.) In general, among all non-explosive logics, what is the criterion to decide which of them are good logics for inconsistent information?

Question 3: If X is not explosive, in long term will X get closer and closer to truth?

Why Not Contradiction Implies Nothing?

2010-05-14T19:17:00.000+08:00

If the discipline we investigate in does not allow true contradictions, what goes wrong in Lewis’ argument for explosion is the step of &-elimination.

[Claim] (A & not-A) implies nothing (except that at least one of the assumptions is false).
[Argument]
Assume that (i) we are under a specific area of some discipline; (ii) M is our intended model; (iii) no contradiction is true of M and no contradiction is possible to be true in this area.

1. If we know that A is false for intended model M, that is, M is not as what A describes, then if the purpose of our investigation is to find out what is true of M (and what is not), then it is strange to derive any B from A and think B is true of M.

For example, I know it is false that there is a largest prime. It follows that it will be strange to derive some statement B from the proposition that there is a largest prime and think B is true of the intended model of arithmetic.

However, before I know it is false that there is a largest prime, it was an epistemic possibility for me that there is a largest prime, so I was/might be interested in what follows from this proposition. But, the point is: once I know it is false, it is no longer an epistemic possibility, so I was no longer interested in what follows from it.

2. (A & not-A) is impossible to be true of M, that is, it must be false, and we know this.

3. It follows that it is strange that we derive A from (A & not-A) and think A is true of M.

Ps. Of course, it is correct to derive the negation of one of the assumptions, based on the derivation from this set of assumptions to (A & not-A).

[Claim] It is reasonable not to derive a further conclusion from (A & not-A) (except that at least one of the assumptions is false).
[Argument`]
Let X be a set of our assumptions. Assume that from X, we derive a contradiction (A & not-A). We know that (A & not-A) can not be true, so we know that it is impossible that all sentences in X are true, given our rules are truth-preserving. In other words, we know the actual situation can not be the case that X describes, that is, the case that X describes is not an epistemic possibility of the actual situation. It follows that we are not interested in what follows from X, after we derive that contradiction from X. Therefore, we do not keep doing inference under the set X of assumptions, (once we derive a contradiction from X).

Final Remark.
Bolzano takes the position that nothing follows from inconsistent premises.

A Gap between the Kripke Semantics and the BHK-Interpretation

2010-04-15T19:00:00.002+08:00

I would like to argue that the Kripke semantics for intuitionistic logic does not accurately capture the BHK-interpretation.

Let us consider the logical constant, ‘for all’. According to the BHK-interpretation, the intended meaning of “w╟ for all x, P(x)” in the Kripke semantics should be:

(*) at stage w, the ideal mathematician S has had “a general method of construction which , if any element a [of the given domain ] is chosen, yields by specialization the construction of P(a)” (Heyting).

In the Kripke semantics, “w╟ for all x, P(x)” is formally defined as follows:

(**) for all w’≧w, for all a available at w’, P(a).

In the following, I would like to give an example to show that (**) does not imply (*), so the Kripke semantics does not accurately capture the BHK-interpretation.

Consider an ordered set , where M={t_0, t_1, t_2, …} and t_n ≦ t_m, if n≦m. Assume:
At t_0, the ideal mathematician S constructs natural number 0 and prove that P(0).
At t_1, the ideal mathematician S constructs natural number 1 and prove that P(1).
At t_2, the ideal mathematician S constructs natural number 2 and prove that P(2).
And so on.

Also assume that for any n, m, the proof D for P(n) and the proof D’ for P(m)are very different. That is, we do not have a unified proof for every P(n) (,that is, we do not have “a general method of construction”.)

In such a case, according the Kripke semantics, it holds that t_0╟ (for all x, P(x)). However, at t_0, the ideal mathematician S does not have a proof for the statement that for x, P(x).

What is a formal system?

2010-02-17T06:14:00.001+08:00

Heyting –

“We may consider the formal system itself as an extremely simple mathematical structure;
its entities (the signs of the system) are associated with other, often very complicated, mathematical structures.
In this way formalizations might be carried out inside mathematics,
and it becomes a powerful mathematical tool.”

Locating My Future Research (The Top-down Version)

2009-11-04T11:56:00.000+08:00

[Abstract] I would like to locate my future research in substructural logics, paraconsistent logic or belief revision. The following is the explanation why I would like to do so.

Introduction
One piece of good advice about doing research I was told goes as follows:
(1) Start with a small project/issue.
(2) However, this small project should be able to be put under a significant global program/general direction.
This short article is about (2). Three options (three directions) for my future research will be reached. They are substructural logics, paraconsistent logic and belief revision.

Why Substructural Logics? Plurality of Logics and General Insights Beyond Formal Systems
We have so many logics, today! Finding a way of unifying this plurality of logics might help us reach ‘general insights beyond formal systems’, mentioned by van Benthem. I think, study of substructural logics could contribute to unifying this plurality. So, I take substructural logics as one option for my future research.

Remark - Limitation Theorems
In study of logic, the most attractive part of this discipline for me could be limitation theorems. Striking examples of these theorems are Gödel’s incompleteness theorems, which draw the limitation of mathematical reasoning. When we go from a specific logic to a theory unifying a class of logics, it is exciting to see whether this general theory unifying the plurality of logics can bring us limitation theorems in a general setting. For example, it might answer in general when a logic has the fixed point lemma. (These limitation theorems in a general setting might draw the limitation of rational reasoning in general.)

Why Paraconsistent Logic or Belief Revision? Global/Real-life Phenomena
I am interested in global/real-life phenomena that can be dealt with by logic. Examples are non-monotonicity and combining logics.
(a) Inconsistent Information
One global phenomenon attracting me is that we can live with paradoxes and inconsistencies quite well (at least to some degree). In other words, a paradox/inconsistency could not make the whole system of knowledge collapse. It seems to me that our rationality has a mechanism to block/minimize the ill impact of paradoxes and inconsistencies. I would like to investigate this mechanism.

(b) Belief Revision
Besides, in van Benthem’s ‘Logic in Philosophy’, he wrote down the following striking lines, which might be relevant to what I just said:
“What seems crucial about us is not the use of infallible methods, but reasoning with whatever means we have, plus an amazing facility for belief revision, i.e., coping with problems as they arise.”
“Logic is not some vaccination campaign eradicating all diseases once for all. It is rather the immune system of the mind!”

So, paraconsistent logic or belief revision could be an option for me. However, right now I am not sure which one is what I really want.

Remark – Methodological Principles
In dealing with inconsistent information, what happens when we approach in different methodological principles? By methodological principles, I mean the starting points of doing a theory. In logic, one example could be compositionality. The model-theoretic definition of logical consequence could be another; so is the proof-theoretic one. For me, sometimes, it might be less interesting to argue whether a methodological principle is right or wrong. As to dealing with inconsistent information, I am more interested in investigating what the pros and cons of different methodologies in analyzing the same phenomenon are.

The Next Step
So, which specific small project should I begin with? (Strategy: simplification is the mother of invention.)

Where is Logic Going, and Should It?

2009-10-16T13:36:00.004+08:00

This is a summery on van Benthem’s ‘Where is Logic Going, and Should It?’.

[A broader view about Logic]
van Benthem says:
Logic is not only about deductive mathematical reasoning,
but also about any other “process that transforms and transmits information: (non-mathematical) reasoning, computation, questioning, announcing, or learning”.

The ultimate goal of this view could be to formalize/to make explicit/to fully understand the Rationality of Mankind.

------------
The following are some general issues that van Benthem uses to illustrate this broader view of Logic.

1. General Insights Beyond Formal Systems
“Can we cast logical insights in a formalism-independent format (, just as major laws of physics can be understood without buying into elaborate formal languages and axioms systems)?”

The following is one instance of such insights:
Gains in expressive power are lost in higher complexity [in model checking for truth, consistency maintenance, or valid inference].

2. Plurality
We have so many logics, today! To make sense of this ‘plurality of logical systems’ is an important issue:
“Could there be one over-arching theory of reasoning behind all of this?”
“Is there a way of unifying this multiplicity of logics?” [Quoted from UniLog 2010 webpage.]

3. Information Carriers
In addition to language, pictures can also be information carriers. Furthermore, in principle, “every physical system with enough regularity in the environment” can be an information carrier.
When information carriers become non-linguistic, will we have a different logic? How are language-orientated logic and non-language-orientated one to be integrated?

4. Dynamics, Many agents and Interaction
This article is written by van Benthem. So, it is no surprise that the dynamic aspect of logic is highlighted. He addresses one observation: logical activities, such as asking, answering and telling, usually involve more than one agent. So, he suggests that the general logic of interaction among many agents in a logical activity is an interesting domain to discover.

5. Time scales and Probability
A single step inference is just one logical action. Conversation or games take longer stretch of time. However, when we bring in probability, we can deal with much longer term of logical activities.

An interesting instance of this trend of research is the Zero-One Law for predicate logic – saying that in the long run , on finite models, first order statement are either true of false with probability 1. The discovery of ‘emergent’ statistical properties of automated proof systems is only starting.

矛盾是否蘊含任意一句話?

2009-09-25T11:48:00.011+08:00

A&not-A implies B, for any B?

The argument for YES goes as follows:

1. A&not-A [premise]
2. A [by 1]
3. A or B [by 2]
4. not-A [by 1]
5. B [by 3 and 4 - Disjunction Syllogism]

HOWEVER!
When we take/commit Disjunction Syllogism, we presuppose that A and not-A can't both be ture.
["A or B" means that at least one of A and B must be true.
Under the presupposition just mentioned, the truth of "not-A" rules out the possibility of A's being ture.
Therefore, under the presupposition, we can infer B from (A or B) and not-A.]

However, when we assume premise 1, we presuppose that A and not-A can both be ture, so Disjunction Syllogism is not applicable here.

Roughly speaking, Disjunction Syllogism is only applicable to subjects that are consistent.
However, when assuming premise 1, we simply do not deal with such kind of subjects.

In short, when we take a contradiction as a premise,
intuitively, Disjunction Syllogism is not a good/proper inference rule.

邏輯課的第一個困惑

2009-07-18T11:58:00.004+08:00

從大一上基礎邏輯課開始, 一直有個問題困擾我: 為何有效論證是定義成前提為真結論必然為真? 以下用問答方式闡述我現在對這個問題的回答.

Z: 為何有效論證是定義成前提為真結論必然為真?
O: 這只是定義啊! 定義本來就隨便人家定.
Z: 我認為定義有兩種. 一種是如你所說的是任意性的. 例如, 定義函數f(x)=x+1. 另一種定義其實是在做概念分析, 不能任意定. 例如, 將知識定義成被證成的真信念. 而, 對有效論證的定義應該是後面這種.
O: 所以你其實想問的是: 為何有效論證要分析成前提為真結論必然為真的論證?
Z: Yes!
O: 直覺上, 所謂的有效論證其實只是指好/正確的論證. 好論證的必要條件之ㄧ是前提為真結論必然為真的. 其他的必要條件可能還包括相關性. 但相關性是啥沒辦法清楚定義, 故而有點任意地就取前提為真結論必然為真為好論證的充要條件.
Z: 但, 歸納論證在經驗科學的脈落下一樣算是好論證啊, 但歸納論證並不是前提為真結論必然為真.
O: 這個嘛~ 讓我們請問一下chasist兄! 呵!

chasist: Hello!兄弟! 喔~你們的問題是這樣的啊.
chasist: O弟會認為好論證的必要條件之ㄧ是前提為真結論必然為真的似乎是先預設的演譯論證(deductive)與歸納論證(inductive)的區分然後侷限在只討論演譯論證.
O:恩~
chasist: 在這樣的預設下, Z的問題其實是空洞的. 因為區分演譯和歸納的性質即為是否前提為真結論必然為真. 在這樣的預設下, Z的問題形同在問為何前提為真結論必然為真的論證要分析成前提為真結論必然為真的論證?
chasist: 一種理解整個事情的方式為下. 在我們的實務(practice)中, 如O弟所說, 好論證可以區分演藝的和歸納的 (,即前提為真結論必然為真的好論證和並非如此的好論證). 我們叫那些前提為真結論必然為真的好論證(演譯論證)為有效論證. 而邏輯主要是在討論這種好論證.
Z: 照你這麼說, 有效論證的定義其實是比較任意性的那種定義.
chasist: 是的.
Z: 恩~讓我再好好想想.

-----------------------
[Abstract]
The following question might be vacuous/trivial:

“Why a good/correct step of deductive inference is identified as a truth-preserving step of inference?”

At the first sight, this question seems meaningful. However, it might not, in fact!

When asking this question, if we presuppose the distinction of deductive inference and inductive inference, the question might be a vacuous one. It is because the difference between deductive inference and inductive inference is just that the former is truth-preserving but the later is not. So, if we presuppose the distinction, asking the question is asking why a truth-preserving step of inference is truth-preserving!

The Proof Idea (Master Thesis)

2009-05-20T19:28:00.002+08:00

My Formal Training & Research Interests

2009-05-12T13:14:00.005+08:00

(I) Formal Training during My Master Study
In my master research, I studied two important proof-theoretic methods – cut-elimination and normalization. I did so, partially because I hoped to have some solid training in mathematical logic, and partially because my advisor’s research interest during that period was in proof theory. My study started from some simple proof systems, where it is much easier to see the ideas[1] of cut-elimination and normalization. In my master thesis, applying what I learned, I showed that a more complicated natural deduction system has normalization theorem and a weaker[2] version of subformula property. Applying this weaker version of subformula property, I showed that this system tolerates classical inconsistencies[3].

(II) Research Interests
Generally speaking, I am interested in mathematical logic, philosophical logic, philosophy of logic, and philosophy of mathematics. I plan to focus my doctorial research on philosophical logic or philosophy of logic. In these two areas, I am interested in intuitionistic logic and issues on logical consequence, respectively. However, I do not mean that my future research must follow my current interests.

(i) The Interest based on My Master Research

Cut-elimination and normalization are two powerful poof-theoretic methods in studying minimal, intuitionistic, and classical logic[4]. I would like to try to extend these two methods to other logics, such as some specific substructural logic that is of considerable interest in its own right. It is interesting to see whether these two methods can also bring us profound results in other logics. Furthermore, I am interested in the following general direction[5]: as the algebraic method gives us interesting general results about classes of modal logics, we might use proof-theoretic methods to build general results of some interesting class of logics. (Besides, in the second paragraph of the footnote 5, there is a question I asked based on my master research.)

(ii) Intuitionistic Logic

I have been interested in mathematical intuitionism by Brouwer, Heyting, and Dummett. This is one reason why intuitionistic logic interests me. Furthermore, I am interested in using intuitionistic logic as the metalogic to study intuitionistic formal systems. By doing so, I will really practice constructive mathematics; that helps me better understand intuitionism. And, one aspect of intuitionistic metamathematics that interests me is that in intuitionistic metamathematics, we have difficulty obtaining ‘completeness for full IQC for the notion of intuitionistic validity’[6]. Besides, I am also interested in the Curry-Howard Isomorphism. The isomorphism connects the natural deduction for intuitionistic logic and type theory. My interest in proof theory leads me to this specific interest.

------------------
[1] I believe, the underlying ideas of many meta-theorems can be understood in a very simple/intuitive way. For me, overcoming the surface complexities and grasping underlying simple ideas of formal things is one of the most interesting things in doing research.

[2] Here, I would like to explain why I have difficulty proving the full-strength version of subformula property holds for the system. Roughly speaking, it is because the system has the falsum rule ⊥/A but disallows the inference rule B/(A->B) (that is, ‘B infers AàB’). To prove subformula property, usually we need to be able to convert any prooftree D into a prooftree D’ in which the consequence of every application of the falsum rule is atomic. To do this conversion, we might need the inference rule B/(A->B) to convert ⊥/C->D into ⊥/D/C->D. Therefore, disallowing the inference rule B/(A->B) leads to difficulty.

[3] A proof system tolerates classical inconsistencies iff some classically inconsistent sets of formulas are consistent under the system.

[4] These two methods bring us many interesting meta-theorems, such as the interpolation theorems of minimal, intuitionistic, and classical logic and the disjunction property of minimal and intuitionistic logic.

[5] Maybe, we can say the ultimate goal of this direction is to gain ‘general logical insights in a formalism-independent format’. (I borrow Johan van Benthem’s words in Where is Logic Going, and Should It?, 2005.)

In addition to the system I dealt with in my thesis, I also worked out proofs to show that two other natural deduction systems, which both disallow the inference B/(A->B), tolerate classical inconsistencies. With these three cases, the following question came to my mind: is it the case that all consistent extensions of implication logic disallowing the inference B/(A->B) tolerate classical inconsistencies? This question might not be boring, if we treat answering the question as a tiny step toward the goal of gaining ‘general logical insights in a formalism-independent format’.

[6] Here, I borrow van Dalen and Troelstra’s words in the page 694 of Constructivism in Mathematics, volume 2.

Toward Abstract Proof Theory

2009-04-24T16:41:00.005+08:00

I. 透過做碩士論文所受的formal訓練
碩班時, 透過做碩士論文, 我希望給自己在數理邏輯的某個基本領域有一些扎實的訓練. 而正好那時我的指導教授研究興趣在證明論, 故而我碩士論文做證明論的東西. 我把注意力集中在structural proof theory. Cut elimination 和normalization 是 structural proof theory 很重要的兩個方法. 故而, 我做碩士論文時把重心放在 study 這兩個東西上. 我從較簡單的系統出發, 這使得我較容易抓到cut elimination 和 normalization 的精神[1].

論文中, 我應用我所學, 證明了一個較複雜的自然演繹法系統的normalization theorem和此系統的一個較弱版本[2]的subformula property. 運用這個弱版本的subformula property, 我證明了這個系統tolerates classical inconsistencies[3].

II. 碩士論文所引發的興趣
在我碩士論文的訓練之後, 我對以下這件事感興趣: 將cut elimination和normalization這兩個技術extend 到 minimal logic, intuitionistic logic,與classical logic以外的邏輯. 在study前面所提到的三個邏輯上, cut elimination 和normalization是很powerful的proof theoretic方法. 應用這兩個方法我們可得到許多有意思的後設定理, 例如這三個邏輯的interpolation theorems, 與 minimal logic和直觀邏輯的disjunction property . 故而, 會想要嘗試extend這兩個技術到其他邏輯上, 看看是否profound的結果. 這樣的extension是放在做'abstract proof theory'這個更大picture之下. 換句話說, apply到其他邏輯是做'abstract proof theory'的前置準備工作(做抽象理論先要有具體經驗).

III. Abstract Proof Theory
A.[Plurality] There are so many interesting logic systems, now. These systems can be divided into families, such as modal, relevant, paraconsistent,...

B.[Abstract Proof Theory]Use proof theoretic methods to 'organize' families of logics.

C. [Goal of Abstract Proof Theory]
First, we will get general results about families of logics.
Second, through these results in 'abstract proof theory', we intend to reach general insights beyond formal systems.

-----------------------------
[1] I believe, the underlying ideas of many meta-theorems could be understood in a very simple/intuitive way. For me, overcoming the surface complexities and grasping underlying simple ideas of formal things is one of the most interesting things in doing research.

[2] 在此我想解釋為何我有困難去證明這系統full strength的subformula property. 粗略地說, 這是因為此系統有the falsum rule, ⊥/A但卻不允許B/AB這個推論. 要證明subformula property我們通常需要能夠把任意一個prooftree Π 轉換成另一個prooftree Σ in which the consequence of every application of the falsum rule is atomic. 為了完成這個轉換, 我們需要B/AB這個推論幫我們把⊥/CD轉換成⊥/D/CD.

[3] A proof system tolerates classical inconsistencies iff some classically inconsistent sets of formulas are consistent under the system.

Study Notes On Heyting's 'Disputation'

2009-03-28T15:10:00.002+08:00

Heyting對古典數學的態度是批評性的
而非認為: 古典數學和直觀數學只是兩種作數學的方式; 沒有哪個是有問題的.
Heyting認為做數學不應該參雜形上學預設/claim.
而古典數學參雜形上學預設/claim, 故而古典數學是有問題的.
(如果有人並不贊同Heyting這種去形上學的態度, 那對於那個人古典數學是沒有問題)

為何古典數學參雜形上學預設/claim? 論證如下:

古典數學有排中律
假設 ExAx是個open question, 亦即我們沒有ExA(x)的證明和否證
根據排中律ExA(x) or not-ExAx成立
故而, either (ExA(x)成立且我們沒有ExA(x)的証明) or (not-ExAx成立且我們沒有not-ExAx證明).
故而, 這個使得A(x)成立的x的存在或不存在是獨立於我們的知識.
故而, 古典數學預設 (weak) knowledge independence (定義在下面).
故而, 古典數學參雜形上學預設/claim.

[ Definition of (Weak) Knowledge-independence]
An object is (weakly) independent of our knowledge, iff it can exist even if we don’t have any proof for its existence and it might not exist even if we don’t have any proof for its non-existence.

-------------------------------------------
Heyting有認為做數學不應該參雜形上學預設/claim嗎?

Heyting說做數學時我們不應該去問形上學問題.
故而我認為Heyting也會讚同: 做數學時我們不應該commit某個形上學的問題的答案. (不該問當然更不該回答.)
而commit某個形上學問題的答案, 就是commit某個形上學預設/claim.

如果 “是否數學物件的存在或不存在是獨立於我們的知識?” 是個形上學問題, 而且做數學時我們不應該去問形上學問題,
則作數學時, 我們不應該commit數學物件的存在或不存在是獨立於我們的知識, 亦不該commit數學物件的存在或不存在不是獨立於我們的知識.

-----------------------------------------
對Heyting而言, (weak) knowledge independence就算形上學預設.

Can our everyday math practice have an idealism interpretation?

2009-01-31T11:14:00.005+08:00

[Abstract] Can our everyday math pratice have an idealism interpretation? That is, can we interpret 'numbers in our everyday math practice' as mental entites? It seems to me that intuitionism claims NO! At this point, is intuitionism right?

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To my best understanding, intuitionism claims that our everyday math practice presupposes platonism, because our everyday math practice presupposes the law of excluded middle. In other words, accepting the law of excluded middle will force us to commit platonism. Therefore, according to intuitionism, if we want to be an antirealist, we should revise our everyday math practice, that is, drop the law of excluded middle.

I have the following three considerations that are unfavor of intuitionism.
(1) My intuition is that our everyday math practice is metaphysics-free, that is, it is compatible with realism and antirealism/ you can interpret it in either a realism way or an antirealism way.
(2) The revision required by intuitionism is too huge, because dropping the law makes our math practice too difficult.
(3) Fictionalism, a version of antirealism, endorses classical logic and asks no revision. In other words, fictionalists say that you can accept the law of excluded middle without committing platonism.

Brouwer rejects (1). Fictionalist agrees with (1). Furthermore, unlike a fictionalist, who thinks that numbers doesn't exit, can our everyday math practice/ classical practice have an idealism interpretation, that is, can we interpret 'numbers in our everyday math practice' as mental entities? It seems to me that Brouwer says no. Is Brouwer right?

The Law of Excluded Middle and Fictionalism

2009-01-25T16:21:00.003+08:00

I checked SEP today and read this exciting item of fictionalism by Mark Balaguer.

As a reader interested in intuitionism, I would like to ask a question concerning the law of excluded middle and fictionalism. The paragraph after the question is about my motivation of asking it.

First of all, please allow me to quote the line in the item - “he argues not just that it's acceptable for a philosophy of mathematics to allow for the possibility of bivalence failures but that it's desirable.” Here is my question: does author's intention-based version of fictionalism withdraw the law of the excluded middle? In other words, does accepting author's version of fictionalism imply dropping the law of excluded middle?

Fictionalism attracts me, for it is a version of antirealism that doesn’t require revision of mathematical practice like intuitionism. (Hope I don’t misunderstand fictionalism.) But, if we allow the possibility of bivalence failures, it seems to me that we don’t take the law of excluded middle to be valid any more and we need to revise our mathematical practice.

Or, allowing the possibility doesn’t imply the revision of our mathematical practice?

Two Kinds of Challenges to a Logic

2008-12-26T14:13:00.006+08:00

[Abstract] Accepting intuitionism’s challenge to classical logic is much different from accepting the challenge to the truth table of classical logic’s “if”.
It is because the former involves changing our way to do the informal reasoning, but the latter doesn’t.

[中文摘要] 接受直觀主義對古典邏輯排中律的批評意味著:
我們的(informal)推理模式整個要做改變.
但接受一般對於古典邏輯的if的truth table的批評,
我們無須改變我們的推理模式.
故而這兩種批評是很不同的.

-------------------------------------------------------------

What is a logic? A (formal) logic (system) is a mathematical structure used to model/represent the correct informal reasoning of some discourse, such as mathematics, philosophy, and so on.

Therefore, there are at least two kinds of challenges to a logic. An example of the first kind of challenges I want to mention is the usual criticism to the truth table of classical logic’s “if”. An example of the second kind of challenges is intuitionism’s challenge to the law of the excluded third of classical logic.

What the first kind of challenges focus on is that the logic doesn’t accurately model a part of our actual informal reasoning, which is also thought as correct. One example is the usual criticism to the truth table for ‘if’ in classical logic. Another is relevant logic’s challenge to the explosion of classical logic.

The second kind of challenges is about (a part of) our actual informal reasoning (that the logic well models). More specifically, the second kind of challenges argues that although the logic accurately models our actual informal reasoning, our actual informal reasoning is not correct, in fact. For example, the intuitionism challenges the law of excluded third of classical logic, which is really used in our actual informal reasoning and is taken to be correct.

The first kind of challenges leads to a worry that because of the inaccurateness of modeling, the meta theorems of the logic might not really reflect the actual properties of the informal correct reasoning. However, the logic challenged can be defended without modifying its modeling, by arguing that those bugs in modeling do not influence that much.

Accepting the second kind of challenges means that we should change our informal reasoning way. This change is very big. However, accepting the first kind of challenges does not necessarily lead to the change of our way to do the informal reasoning.

A Criticism for (All) Non-classical Logics

2008-11-19T21:16:00.008+08:00

[Abstract] Is it bad to use classical logic as the metalogic of a non-classical one? In other words, is it bad to use classical logic to prove the meta theorems of a non-classical one?

Yes, if you intend the non-classical one to model the deduction in mathematics. No, if you don’t intend the non-classical logic but classical logic to model the deduction in mathematics.

[中文摘要] 當我們在研究非古典邏輯時
我們會證一些關於這些非古典邏輯系統的定理
例如直觀邏輯的完備性定理
我們進行這些證明的動作是在meta level
當我們在meta level做推論時我們一定也在使用某個邏輯
通常為古典邏輯
但做非古典邏輯不就是在反對古典邏輯嗎? 為何又使用古典邏輯呢 ?
Therefore, here comes the problem.

(當我們在證明任意一個非古典邏輯系統的後設定理時, 我們所用邏輯往往是用古典邏輯. 這樣做是不合理的嗎? 如果你認為這個非古典邏輯系統是用來model數學推理, 這樣做是不合理的. 如果你認為model數學推理的正確邏輯是古典邏輯而非這個非古典邏輯, 這樣做不會不合理.)

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A criticism for (all) non-classical logics goes as following: usually, the metalogic used for doing the metatheory of any non-classical logic is still classical logic. In other words, usually, people use classical logic to prove the meta theorems of non-classical logics. Therefore, classical logicians might challenge their non-classical fellows by the following reasoning: if you think the correct logic is your favorite non-classical logic, why do you use classical one when doing your metatheory?

Here is a response to this criticism. I read this idea from David Makinson’s book Bridges from Classical to Nonmonotonic Logic (page 3).

When we are doing the metatheory of some logic, which is either classical or non-classical, we are doing mathematics, that is, the very thinking we are applying is mathematical thinking. Therefore, what logic we should use in doing the metatheory of any logic is the correct logic for mathematics.

In other words, given a non-classical logic X, if the very thinking you intend X to model is mathematical thinking, you should use X to do the metatheory of X. If the thinking you intend X to model is not mathematical thinking, you have no duty to use X to do X's metatheory .

[Remark] Here are Makinson’s original lines:

“Indeed, logic…was developed in order to obtain a deeper understanding of
deduction in mathematics. The resulting system is known as classical logic.”
“Although our subject is nonmonotonic reasoning, we will constantly make use
of classical logic.”
“As our enterprise is mathematical in nature, our own
reasoning will be carried out in accord with the principles of classical
deduction.”

A Methodology Question Concerning Meaning

2008-09-21T12:12:00.004+08:00

[中文摘要]
給定一個句子, 什麼是它的意義? 對於這個問題, 我們通常所見的回答都不脫以下這樣的形式: 一個句子的意義為它的XX條件. 但, 我們一定要這樣去回答問題嗎?

Göran Sundholm在他的Proof Theory and Meaning這篇文章(第一個section), 給了理由為什麼對於這個問題的答案一定要是這樣的形式.

(Proof Theory and Meaning是Göran Sundholm在Handbook of Philosophical Logic(第一版第三冊)的一篇文章. 用Google也可以找的到這篇文章.)

[Abstract]
What is the meaning of a given sentence? To this question, answers are usually of the following form: the meaning of a given sentence is its X-condition. One very cool thing in Göran Sundholm’s paper, Proof Theory and Meaning is that he gives reasons why the answer should be always of this form.

P.s. The paper is in first edition of Handbook of Philosophical Logic III. You can also use Google to find its pdf file.
------------------------------------------------------------------

What is the meaning of a sentence? To this question, here come two popular positions:
[Davidson] The meaning (content) of a sentence is its truth condition.
[Dummett] The meaning (content) of a sentence is its proof condition.

These are two competing positions. However, they share the same methodology:
(1) Each formulates its theory in terms of one key concept- Davidson ‘truth,’ Dummett ‘proof.’
(2) Each uses its own key concept to give the meaning (content) of a given sentence.

WHY should we formulate our meaning theory in this way? That is, to the question what the meaning of a given sentence is, WHY should the answers of this question be always of the following form:
the meaning of a given sentence is its X-condition.

In the first section of Göran Sundholm’s paper, he gives reasons to support this methodology. Before reading his paper, I just take the methodology for granted without any doubt. Hence, his paper is so cool, because (1) he makes it explicit that there is such a fundamental methodology, (2) he reminds us that the methodology could be doubted and (3) he gives reasons to justify the methodology.

I think, it is fun to think about your own reason for (or against) the methodology. I would like to show his argument for the next time!!!

Amazing! 韓國信仰基督教人口有55%!!!

2008-01-11T13:01:00.000+08:00

昨天和同學吃飯
聽他提到
韓國現在有過半的人信仰基督教
今天網作查證
真的是如此!!!
Amazing!!!
這個link有簡介

南韓：６０％（５０％）
這是目前的比例，還在快速上升中。韓國教會的信徒對於信仰非常火熱，全天禱告、徹夜不眠禱告都是常有的事，全民皆兵、人人佈道是他們快速成長的秘訣，派出大量宣教士向全球工場進軍也是他們的異象。，目前全球各地韓僑信主比例亦頗高

上面這段文字則是copy自以下連結
當前東亞各國基督徒人口比例列表

為何南韓會有這麼多人信基督教呢?
和我媽討論的結果,
我媽根據她看綜多歷史韓劇的印象
她猜測有可能是因為韓國人一直想在文化上和中國切割

到底原因為何?
呵我已經去奇摩知識申請開通
希望之後問有人回:D

'What is truth?' 為何要擔心這個問題? (II)

2007-09-14T19:23:00.000+08:00

[Claim] (似乎) 解消liar paradox並不是一個去對'truth'這個概念做哲學探討的充足理由.

[Argument]
1. 哲學的最終目標是了解世界.
2. 解消liar paradox只是在做概念系統的調整.
3. 我還看不太出來, 解消liar paradox這個概念系統調整的工作如何能幫助我們去'了解世界.'
4. 所以, 我還看不出來, 解消liar paradox可以是一個去對'truth'這個概念做哲學探討的充足理由.

[Explanation]
(a)關於'哲學的最終目標是了解世界'
這邊是武斷地直接take我在Shapiro的Thinking about Mathematics讀到的立場.

(b)我贊同概念系統的調整(即概念分析)是重要的哲學工作.
但其重要性來自於做這樣的概念調整(/概念分析)是有助於我們去了解世界的.

(c)對我而言, 有趣的似乎不是解消liar paradox, 而是我們居然可以和liar paradox和平共處!
我的意思是說, liar paradox並不會影響到其他知識的穩固性.
我們不會用liar paradox來質疑羅爾斯的正義兩原則.
我們也不會用liar paradox來質疑愛因斯坦的相對論.
這好似, 我們的概念系統有辦法block住liar paradox的影響!
講到這, 我似乎快要自打嘴吧了! 哈!
要解開為何我們可以和liar paradox和平共處這個謎, 難道不需要對'truth'這個概念做哲學探討嗎?
恩~我需要再想想.

不過這裡的討論似乎告訴我們: paraconsistency logic是值得做的邏輯.

[Next Step]
我是順著Stephen Read的Thinking about Logic在想問題.
Read有交代他覺得為什麼需要做真理論.
因為還沒讀到, 所以等讀完再向像大家報告.
似乎是和形上學的考量有關.

Two Senses of Realism

2007-08-29T14:20:00.000+08:00

[摘要]
下面這個區分提示了一個有趣的方向:
要堅持數學的客觀性似乎不必然要接受數學物件(數字, 集合...等)是客觀存在的.

[區分]
Realism in ontology: at least some mathematical objects exist objectively, independent of the mathematician.

Realism in truth-value: mathematical statements have objective truth values, independent of the minds, languages, conventions, and so on of mathematician.

[簡短說明]
Realism in truth-value 用數學陳述真假的客觀性來刻畫了一個意義的數學客觀性.
這是在Shapiro的Thinking about Mathematics上讀到的區別.
書中的這段文字我覺得很有趣:
這個區別"shifting the focus from the existence of math objects to the objectivity of math discourse."

[應用]
不知道這個區別應用在道德哲學上, 會不會有有趣的發展?
堅持道德有客觀性但不用說有道德性質存在- it sounds good!

'What is truth?' 為何要擔心這個問題?

2007-08-24T11:54:00.000+08:00

We have no trouble in using the concept, truth. So, why worry about the question 'what is truht'?

1. 我們在使用truth這個概念上, 並不會遇到任何困難.

男人對女人發誓他講的是真的, 會被質疑的是他講的內容而不是'什麼是真.' 在數學研討會, 我們可能會爭議某個數學陳述是否為真, 但我們並不擔心 '真' 在這裡的指什麼. 在法庭上, 我們爭論真相(truth)為何. 科學家在探究這個物理世界的真相(truth). 歷史學家, 根據史料在推敲歷史事件的真相(truth).

2. 雖然在使用上沒有困難, 但當要進一步問'what is truth'時, 我們卻很難回答.

3. '不會癢就不要騷': 難回答的問題不見得就是值得回答的問題. 如果我們並沒有需求去進一步對truth這個概念做分析, 去分析truth這個概念是很空洞的.

So, why should we do the philosophy of 'truth'?

[整理自Stephen Read的Think about Logic的第一章的開頭. 不過有些小出入.]

[Question] Are logical relations parts of the world?

2006-11-19T13:21:00.000+08:00

That(1) ALL MEN ARE MORTAL is an aspect of the world.
That(2) I AM MORTAL is another aspect of the world.

It is trivial that (1) implies (2).

MY QUESTION: Is the implying relation between fact (1) and fact (2) a part of the world?

(Is this question meaningful?)

道德上的自然主義不能滿足我 --- WHY?

2006-11-07T13:21:00.000+08:00

道德上的自然主義似乎想把所有的東西化成物理的
但這如何可能會是我所想要的?
當我們感受到有價值的東西時
如果這ㄧ切都可以化約成物理的
那我們感受到的價值似乎不過是種幻像
我如何可能有意義的活在這樣的幻象中?

How do we explain away 'SHOULD', such a very very...very basic concept??

2006-11-06T11:29:00.000+08:00

There are three ways to explain away a concept.
(1) concept reduction: eg. we use 'truth' to define 'logical consequence.'
(2) introduction rules and elimination rules approach: Brandom's idea
(3) use AXIOMS to characterize a concept: eg. set theory

My opinion is that for 'SHOULD,'
IT IS VERY LIKELY THAT CONCEPT REDUCTION WILL FAIL.
FOR 'SHOULD' IS TOO BASIC TO REDUCE.

So, in METAethics, we should use (2) or (3) to explain away 'SHOULD'?

The Continuous and the Infinitesimal - in mathematics and philosophy

2006-02-06T17:57:00.000+08:00

An interesting book by John L. Bell

One Arnon Avron's very interesting Paper

2006-01-13T10:13:00.000+08:00

It is named "Gentzen-Type Systems, Resolution and Tableaux"!
In this paper, he tries to explain the relation among Gentzen-Type Systems, Resolution and Tableaux. Resolution which is the main technique in Automated Reasoning is just a form of a cut, while cut-elimination is the principal tool in proof theory! He writes, "we show that both resolution and tableaux are based on attempts to exploit the power of cut-elimination theorems in Gentzen-tye calculi and explain in which cases each should be preferred."

You can get this artile in this link Arnon Avron - Online Available Papers. This article falls in to the category of Proof Theory, Automated Reasoning, and Foundations of Logic.

Why I can't understand the claim that MEANING IS USE

2005-12-31T14:40:00.000+08:00

Let us first make a convention that in the following discussion we will use "mean" and "meaning" this way:

(1)"John" means John.
(2)The meaning of "John" is John.

Because these two are just convention for fixing the meaning of our theoretic terms "mean" and "mean", to follow this convention won't force us to commit to any particular meaning theory(e.g. use theory of meaning, Frege's theory of meaning).

Under this convection, to say the meaning of "John" is the usage of "John" is to say that John is the usage of "John".
This is obvious false. Therefore, the word "meaning" occuring in the famous use theory thesis that meaning is use is not used in the same way as the convection. That thesis must means differently by "meaning".

I would like to ask, under this convection how to formulate the famous thesis of use theory that 'meaning' is use, where we use "meaning" occuring in this thesis in a way differnt from the way convented by the convection?

Pro.鄭凱元 suggests a version of use theory's 'meaning', say meaning*:
(1)A word has meaning*
(2)and it is the meaning* that makes that there are correct usages and wrong usages of a word. In other words, having meaning* makes a word different from just a string of symbols.
Therefore, MEANING IS USE iff meaning* is use iff what makes that there are correct usages and wrong usages for a word is the use of a word.

The reason why I couldn't understand the thesis that MEANING IS USE is that I always used the word "meaning" in the way suggested by the convection.

[ZO] Reason, Original position, Nonmonotonic logic

2005-07-08T15:10:00.000+08:00

1. I am interested in the project of formalizing the concept, 'reason'.
It is probably fun to consider the logic underlying our inferences about (practical) reason (實踐理由).
And formalizing may be the first step to consider the underlying logic of inferences about reason.
One question comes to mind: is this logic nonmonotonic?

2. I think Original position(OP) can be formalized in terms of 'reason'.
OP can be treated as a theory of reason on 'public affairs'.
If we can formalize the concept, 'reason' and can give a good account about what the underlying logic of inferences about reason,
then it is possible that we can understand Rawls' theory better and can say more about his theory.

[Study Note]One Difference b.t.w. Semantics and Proof Theory

2005-01-25T11:24:00.000+08:00

In the semantics of 1st order logic,
we can treat the ∀ item of Tarski's truth defnitions as the semantical version ω-rule.
From this point of view, the difference between semantics and proof theory is that
semantics has ω-rule, but proof system doesn't.

[Studying Note] Restall's idea

2005-01-16T22:15:00.000+08:00

Maybe it is his point:
Some nececcary relation can't be saved/restorted into 'form'

Two Interesting Papers in Stanford Encyclopedia of Philosophy

2005-01-15T18:02:00.000+08:00

Logical Consequence
By Greg Restall and JC. Beall

Tarski's Truth Definitions
By Wilfrid Hodges

A Way to Understand Filter

2004-12-28T11:25:00.000+08:00

[Definition of filter]:
Let I≠Φ. A filter on I is a collection F∈P(I) s.t.
(a) not(Φ∈F)
(b) for all A,B∈F, A∩B∈F.
(c) for all A∈F and B∈I, if A∈B, then B∈F

First, understand ∩ as an operation which outputs {x: x∈A and x∈B}, when inputed A,B.

Then, we can understand (b) as saying that F is closed under ∩.
F is closed under ∩ iff input any A,B∈F into ∩, the output A∩B is always s.t. A∩B∈F.

About Gupta's revision theory of truth

2004-12-16T20:15:00.000+08:00

After listening to Professor Wang's talk on Gupta's revision theoy of truth, I had the following two observations/ideas:

(1)Gupta wants to argue that self-reference has nothing to do with the paradoxical feature of the liar. I am highly interested in the 'has nothing to do with' relation, here. Which sense of 'has nothing to do with' Gupta intends to mean? [Sense1.]'has nothing to do with'='doesn't imply' or [sense2.]'has nothing to do with'='is not determined by'?

(2)In Gupta's revision theory, the extension of a concept is no longer a collection but a series of collections. I think it is the phenomenon of concept expansion. What I mean by "concept expansion" is that we extend the content of a concpet. Here, Gupta extends the very concept, 'extension' from 'a collection' to 'a series of collections'. In another words, it is not 'a collection' but 'a series of collections' that is part of the content of the very concept, 'extension'. Here is the followup question I want to ask: is this concept expansion justified?

Here is an interesting book that talks about 'concept expansion':
The logic of concept expansion
Meir Buzaglo
Cambridge, U.K. ;Cambridge University Press,2002.New York

It is available in our library. Hope someday I will have time to read it.