Where Is My Research Going

   The general theme of my thesis will be asking how to deal with inconsistent information in common sense reasoning.  So, two areas I will deal with are argumentation and belief revision.  By argumentation, 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.

More specifically, I will focus on argumentation and its connection with belief revision.  So, for the belief revision part, I will use study in argumentation to do justification approach belief revision.  And, if it is within my ability, I also want to view all these things I have mentioned under the framework of epistemic logic.  So, I hope I can also deal with ‘dynamicizing’ justification logic in my thesis. 

two questions about LFI

1. ∃Γ∃A∃B(Γ, A,¬A ⊬ B) and
2. ∀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

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

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

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

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?

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.