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On proof-theoretic approaches to the paradoxes : problems of undergeneration and overgeneration in the Prawitz-Tennant analysis

On proof-theoretic approaches to the paradoxes : problems of undergeneration and overgeneration in the Prawitz-Tennant analysis

자료유형
학위논문
개인저자
최승락
서명 / 저자사항
On proof-theoretic approaches to the paradoxes : problems of undergeneration and overgeneration in the Prawitz-Tennant analysis / Seungrak Choi
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서울 :   고려대학교 대학원,   2019  
형태사항
ii, 173 p. ; 26 cm
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On proof-theoretic approaches to the paradoxes   (DCOLL211009)000000085017  
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참고문헌: p. 169-173
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260 ▼a 서울 : ▼b 고려대학교 대학원, ▼c 2019
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900 1 0 ▼a Choi, Seung-rak, ▼e
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On proof-theoretic approaches to the paradoxes : problems of undergeneration and overgeneration in the Prawitz-Tennant analysis (120회 열람)
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컨텐츠정보

초록

When a doctor finds a patient, he diagnoses what the illness the patient has and prescribes it
in accordance with his diagnosis. Likewise, when a logician faces a problematic argument
(or proof), he characterizes the problem and solves it on the basis of his characterization.
It is often believed that solutions to the paradoxes are closely tied with the characterization
of the paradoxes. For instance, an informal characterization of a paradox proposed by
Sainsbury (2009, p. 1) says that it is an unacceptable conclusion elicited from the acceptable
premises via acceptable reasoning. A diagnosis of the paradoxes through Sainsbury’s
characterization can be that it is a trouble that acceptability leads to unacceptability. Thus,
from the diagnosis with the characterization, three responses to the paradoxes can be proposed
such that either the premises or the reasoning is not in fact acceptable, or else the
conclusion is acceptable. We shall call the first response the premise-rejection, the second
the reasoning-rejection, and the last the conclusion-acceptance. Of course, it is not to say
that traditional characterizations of and solutions to the informal notion of a ‘paradox’ are in
full conformity with Sainsbury’s definition. However, his informal definition is the simplest
way to understand ‘paradox’ and the solution to it. In this dissertation, we presume that the
traditional understandings of ‘paradox’ are quite coherent with Sainsbury’s definition.
It seems to be that a proof-theoretic solution to the paradoxes relies on how we characterize
the informal notion of a ‘paradox’ in a proof-theoretic fashion. In this regard, it is
possible that the proof-theoretic criterion for and the solution to the paradoxes differ from
Sainsbury’s definition. The present dissertation aims to investigate the proof-theoretic criterion
for and the solution to the paradoxes from the perspectives on the Prawitz-Tennant
analysis of the paradoxes.

First of all, we will mainly deal with the set-theoretic/semantic paradoxes which were
primarily discussed in the late 19th to the early 20th century for the foundation of mathematics.
In other words, we will center on paradoxes, often called self-referential paradoxes.
This dissertation consists of five chapters. Chapter 1 will summarize the traditional
approaches to the paradoxes by dividing the cases into the set-theoretic paradox and
the semantic paradox. The traditional approaches consist of three types of responses: the
premise-rejection, the reasoning-rejection, and the conclusion-acceptance. Traditional approaches
to the paradoxes have some aspects that a constructivist can hardly accept. Those
approaches use a model-theoretic method which often applies constructively invalid inferences,
such as classical reductio. Also, the proof-theoretic investigation of the paradoxes
may offer the uniform solution to the set-theoretic and semantic paradoxes on the perspectives
of constructivism.

In the last part of Chapter 1, Section 1.3, we will introduce the Prawitz-Tennant analysis
of the paradoxes. While investigating Russell’s paradox in natural deduction, Prawitz
(1965, p. 95) first remarks, ‘the set-theoretic paradoxes are ruled out by the requirement
that the [derivations] shall be in normal.’ His derivation formalizing Russell’s paradox falls
into a non-terminating reduction sequence and so is not reducible to a normal derivation.
The requirement of a normal derivation may be a promising proof-theoretic solution to the
paradoxes and it can be interpreted as below.

The Requirement of a (Full) Normal Derivation(RND): For any derivation D in natural
deduction, D is acceptable only if D is (in principle) convertible into a (full) normal
derivation.

Neil Tennant (1982, 1995, 2016, 2017) regards the non-terminating reduction sequence
as the primary feature of genuine paradoxes and proposes his criterion for paradoxicality
(TCP).

Tennant’s Criterion for Paradoxicality:(TCP) LetDbe any derivation of a given natural
deduction system S. D is a T-paradox iff
(i) D is a (closed or open) derivation of ⊥,
(ii) id est inferences (or rules) are used in D,
(iii) a reduction procedure of D generates a non-terminating reduction sequence, such as a
reduction loop.

When he first introduces his criterion, Tennant (1982, p. 268) wants to regard the criterion
as the conjecture for genuine paradoxes that for any derivation D, D formalizes a genuine
paradox iff D is a T-paradox. if his conjecture is true, any derivation of a genuine paradox
is T-paradox. Also, since a T-paradox is unable to be reduced to a normal derivation, RND
can block the T-paradox and becomes to be a proof-theoretic solution to the paradoxes.
In this dissertation, we shall investigate whether TCP can be a correct criterion for genuine
paradoxes and whether RND can be a proof-theoretic solution to the paradoxes. Tennant’s
criterion has two types of counterexamples. The one is a case which raises the problem
of overgeneration that TCP makes a paradoxical derivation non-paradoxical. The other
is one which generates the problem of undergeneration that TCP renders a non-paradoxical
derivation paradoxical. Chapter 2 deals with the problem of undergeneration and Chapter
3 concerns the problem of overgeneration. Chapter 2 discusses that Tenant’s diagnosis of
the counterexample which applies CR−rule and causes the undergeneration problem is not
correct and presents a solution to the problem of undergeneration. Chapter 3 argues that
Tennant’s diagnosis of the counterexample raising the overgeneration problem is wrong and
provides a solution to the problem. Finally, Chapter 4 addresses what should be explicated
in order for RND to be a proof-theoretic solution to the paradoxes. The contents of Chapter
2–4 are summarized as follows:

Abstract of Chapter 2. In order to solve the problem of undergeneration raised by Rogersontype
counterexamples, Tennant (2015) seems to presume that the application of Classical
Reductio, i.e. CR−rule, is the culprit of the trouble that it disguises the main
feature of paradoxicality, such as a non-terminating reduction sequence. Tennant may
not take the problem of undergeneration seriously. We will claim that the undergeneration
problem is not solved by simply accusing CR−rule of the trouble. In order to
show that the occurrence of a non-terminating reduction sequence is independent of
the use of CR−rule. We suggest two examples of the Liar paradox. First, we suggest
derivations of the Liar paradox and Curry’s paradox which neither use CR−rule nor
generate a non-terminating reduction sequence. In addition, we provide derivations
of the Liar paradox in which the non-terminating reduction sequence is produced
even though the CR−rule is used. After we diagnose the culprit of preventing a
non-terminating reduction sequence, it will be discussed that the problem of undergeneration
will be solved by adding the condition to TCP that only harmonious rules
are to be used.

Abstract of Chapter 3. Tennant(2016) asserts that if all elimination rules are stated in
generalized form, the problem of overgeneration can be solved. However, we claim
that the mere choice of generalized elimination rules fails to solve the problem because
there exist Ekman-type reductions which are stated in generalized form and
produce a non-terminating reduction sequence. Thus, we claim that the real issue is
which set of reductions is proper. In order to find a criterion for a proper reduction,
we shall investigate Schroeder-Heister and Tranchini’s Triviality test and argue that
Triviality test does not block every Ekman-type reduction procedure since it works
relative to a system. At last, we will propose an alternative way to evaluate a proper
reduction, called Translation test.

Abstract of Chapter 4. In order for RND to be a proof-theoretic solution, there are three
things to be explicated: (i) ‘which paradoxes are genuine paradoxes?’, (ii) ‘why
should we accept only a normalizable derivation?’, and (iii) ‘should we consider only
⊥ as an unacceptable conclusion?’ With regard to the first question (i), we will discuss
that Tennant does not have a clear standard for genuine paradoxes. In addition,
with respect to the second question (ii), if proof-theoretic validity implies normalizability,
then RND can be the proof-theoretic solution. However, it will be noted that
the relation should be extended to a general case. Moreover, it will additionally discussed
that if RND could be a proof-theoretic solution, it would be a different type of
solution rather than a reasoning-rejection solution which constrains a particular inference
rule. Lastly, with the third question (iii), we shall consider a normal derivation
of ¬ϕ ∧ϕ which seems to be a paradoxical derivation and argue that if any formula
having the form ¬ϕ ∧ϕ is regarded as an unacceptable conclusion, since RND fails
to block the normal derivation of ¬ϕ ∧ϕ, it cannot be the proof-theoretic solution to
the paradoxes. Hence, it should be explicated why ⊥ should be the only unacceptable
conclusion in proof theory.

More precisely, in chapter 2, we will introduce counterexamples proposed by Rogerson
(2006) which raises the problem of undergeneration. Rogerson’s derivation formalizes
Curry’s paradox that Tennant may regard it as a genuine one but it does not generate a
non-terminating reduction sequence by using the rule for Classical Reductio, i.e. CR−rule.
In other words, in spite of the fact that her derivation formalizes the genuine paradox, it is
not a T-paradox and shows that TCPE undergenerates. Section 2.1 introduces preliminary
notations, rules, and the harmony relation between introduction and elimination rules.
Section 2.2 introduces Tennant’s diagnosis to the problem of undergeneration occurred
by the example of using the rule for classical reductio, CR−rule, and argues that his diagnosis
is not correct. Perhaps he seems to assume that the CR−rule not only produce a
normal derivation of ⊥, but it also masks the key feature of a paradoxical derivation. He
explains this phenomenon and expresses it as the ‘classical rub.’ Also, in the direction of
avoiding the phenomenon, he presents the Methodological Conjecture that ‘genuine paradoxes
are never classical.’ Even if his methodological conjecture is correct, it needs to
discover the fact that which causes the problem of undergeneration. Tennant may believe
that the CR−rule has the problem of causing a normal derivation of ⊥ and concealing a
non-terminating reduction sequence, i.e. a primary feature of the paradoxes. Section 2.3
provides derivations which cause the problem of undergeneration but do not use CR−rule.
That is, CR−rule is not the culprit of the undergeneration problem. To find a solution to
the problem, Section 2.4 diagnoses what preventing the occurrence of a non-terminating
reduction sequence. With some observations, we propose a possible diagnosis that a nonterminating
reduction sequence does not occur if a derivation in question includes (i) a
major premise which has no reduction process to eliminate it or (ii) a formula having a
principal constant which has no reduction procedure to get rid of it. Then, we suggest an
additional condition to TCP that a derivation formalizing a genuine paradox only uses harmonious
rules. If the suggested condition is acceptable, the condition can solve the problem
of undergeneration.

Chapter 3 will cover the problem of overgeneration. In particular, Ekman’s paradox
presented by Schroeder-Heister and Tranchini (2017) will be introduced. Ekman’s paradox
is not to be considered a genuine paradox because it involves an inadequate reduction process,
and so it causes the problem of overgeneration because it is a T-paradox with respect
to Tennant’s criterion. To begin with, we will see the response of Tennant (2016) to the
Ekman’s paradox. He argues that if all elimination rules are stated in generalized form,
then the problem of overgeneration will be solved. However, in Section 3.2, we will argue
that Tennant’s response is inappropriate and that the problem of overgeneration will still
occur, even if only generalized elimination rules are used. Furthermore, it will be discussed
that Tennant’s criterion needs to have an additional condition of which reduction procedure
is proper. Section 3.3 introduces Triviality Test of Schroeder-Heister and Tranchini (2017)
for appropriate reduction procedures. We shall argue that their Triviality test appears to be
unsuitable for the evaluation of standard reduction procedures and it is inappropriate to test
a reduction precess independently of a system. Then, Section 3.4 will present Translation
test. According to Translation test, Ekmann-type reduction procedures are not proper because
it is a detour-making process, and Translation test will have the advantage of being
able to test the reduction procedure itself compared to Triviality test.

In Chapter 4, we will examine whether the requirement of a normal derivation(RND)
can be a solution to the paradoxes. To this end, we will consider three questions of (i)
which paradox is a genuine paradox and which formalization is legitimate for the genuine
paradox, (ii) why the only normalizable derivations are acceptable, and (iii) why the only
propositional constant ⊥ for absurdity is an unacceptable conclusion. If RND is the solution
to genuine paradoxes, it needs to be answered what genuine paradox is. Furthermore, even
if RND could prevent paradoxical derivation, RND would not be justified to be a prooftheoretic
solution to the paradoxes, unless we had reason to use only normal derivations.
Also, if there is a derivation of a genuine paradox which is in normal form and leads to an
unacceptable conclusion, RND fails to prevent the derivation. In this case, too, RND would
not be a proof-theoretic solution to the paradoxes.

In Section 4.1, we shall introduce his argument on why Russell’s paradox is not a genuine
paradox, and argue that by following his argument, if Russell’s paradox is not a genuine
paradox, neither is the Liar paradox. Tennant has no standard for genuine paradoxes.
Our discussion comes into a question of which formalization is legitimate for the genuine
paradox. RND only blocks non-normalizable derivations, such as T-paradoxes. If RND
is regarded as a promising proof-theoretic solution to genuine paradoxes, it should be answered
to the first question of which paradoxes are genuine paradoxes.

In Section 4.2, we will explore the second question of why it is desirable only to use
normal derivations. One possibility is that proof-theoretic validity implies normalizability.
In other words, if a paradoxical derivation is not normalizable, it can be ruled out by
RND because it is not a proof-theoretically valid derivation, RND can be a solution to the
paradoxes. Section 4.2 will establish that in a particular system, proof-theoretic validity
implies normalizability. However, in order for RND to be a proof-theoretic solution, the
result should be extended to a general case. Section 4.3 discusses that the requirement of a
normal derivation is different from the reasoning-rejection solution commonly considered
as a restriction of a particular inference rule. If a reasoning-rejection solution is regarded
as a solution to constrains a certain inference rule, RND will not be the reasoning-rejection
solution because it constrains every derivation in an intended system. Section 4.4 introduces
a normal derivation of ¬ϕ ∧ϕ presented by Petrolo and Pistone (2018) and argues
that RND cannot be a proof-theoretic solution if we accept a formula of the form ¬ϕ ∧ϕ
as well as ⊥ as an unacceptable conclusion. All in all, only when proof-theoretic validity
generally implies normalizability and any formula having the form ¬ϕ ∧ϕ is not regarded
as an unacceptable conclusion, RND can be a proof-theoretic solution to the paradoxes.

목차

Abstract
Contents 
1 Introduction: A Proof-Theoretic Criterion of and Solution to the Paradoxes 
1.1 Traditional Responses to the Paradoxes and Dialetheism 
1.1.1 Traditional Approaches to Russell’s and the Liar Paradox 
1.1.2 Problems of Traditional Responses and Dialetheism 
1.2 Preliminaries 
1.3 Tennant’s Criterion for Paradoxicality (TCP) and the Requirement of a Normal Derivation (RND)
2 Classical Reductio and A Problem of Undergeneration
2.1 Preliminaries: Generalized Elimination Rules and Harmony Relation
2.1.1 Generalized Elimination Rules
2.1.2 An Intrinsic and a GE-Harmony Relation Between Introduction and Elimination rules
2.2 The Methodological Conjecture and the Problem of Undergeneration
2.3 The Undergeneration Problem without CR−Rule
2.4 Diagnosis
2.5 Conclusion
2.A Appendix 2.A: Tennant’s id est Rules for a Liar Sentence and a T-Paradox Using CR−rule
2.B Appendix 2.B: Forms of Permutation Conversions in Natural Deduction
3 A Problem of Overgeneration: Ekman and Crabbé Cases
3.1 Ekman’s Paradox
3.2 The Later Version of Tennant’s Criterion for Paradoxicality
3.2.1 Tennant’s Solution to the Overgeneration
3.2.2 A Problem of Tennant’s Solution
3.3 Schroeder-Heister and Tranchini’s Triviality Test
3.3.1 Triviality Test
3.3.2 Problems of Triviality Test
3.4 Translation Test and Crabbé’s case 
3.4.1 An Ekman-Type Reduction as a Detour-Making Process
3.4.2 Does Crabbé Reduction Overgenerate? 
3.5 Conclusion
4 Can the Requirement of a Normal Derivation be a Solution to the Paradoxes? 
4.1 Which Paradoxes Are Genuine Paradoxes? 
4.2 Why Should We Accept Only a Normalizable Derivation? 
4.3 Is RND a Reasoning-Rejection Solution?
4.4 Should We Consider Only ⊥ as an Unacceptable Conclusion? 
4.5 Summary 
5 Conclusion
Bibliography
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