Proof and truth [Elektronische Ressource] : an anti-realist perspective / vorgelegt von Luca Tranchini

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• Philosophie

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• Publié par: eberhard_karls_universitat_tubingen
• Publié le: 01 janvier 2010
• Nombre de lectures: 42
• Langue: English

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PROOF AND TRUTH
An anti-realist perspective
Inaugural-Dissertation
zur Erlangung des Grades eines
Doktors der Philosophie
der Fakultät für Philosophie und Geschichte
der Eberhard Karls Universität Tübingen
vorgelegt
von
Luca Tranchini
aus
Vittorio Veneto
(Italien)
im Juli 2010Hauptberichtstatter: Prof. Dr. Peter Schroeder-Heister
(Universität Tübingen)
Mitberichtstatter: Prof. Gabriele Usberti
(Universität Siena)
Dekan: Prof. Dr. Georg Schild
Tag der mündlichen Prüfung: 13. September 2010
gedruckt mit Genehmigung der
Philosophischen Fakultät
der Universität TübingenFor ChiaraivContents
Acknowledgments ix
Summary xiii
1 Realism 1
1.1 Tarski’s truth-definitions . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 The propositional case (I) . . . . . . . . . . . . . . . . . . . . 2
1.1.2 The first-order case (I) . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Dummett’s semantic theories . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 The propositional case (II) . . . . . . . . . . . . . . . . . . . . 6
1.2.2 The first-order case (II) . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Truth and assertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 The propositional case (III) . . . . . . . . . . . . . . . . . . . 12
1.3.2 The first-order case (III) . . . . . . . . . . . . . . . . . . . . . 15
1.4 Slipping into realism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Exegesis of Dummett’s exegesis of Frege . . . . . . . . . . . . . . . 17
1.5.1 Frege on the category of predicates . . . . . . . . . . . . . . 18
1.5.2 Concepts: the semantic of predicates . . . . . . . . . . . . . 24
1.6 Dummett’s qualms about realism . . . . . . . . . . . . . . . . . . . . 33
2 Proof-theoretic Semantics 35
2.1 Atomic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Canonical argumentations . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.1 Validity and assertion: a first approximation . . . . . . . . 41
2.3 Closed and open argumentations . . . . . . . . . . . . . . . . . . . . 43
vvi CONTENTS
2.3.1 The gap between validity and assertion . . . . . . . . . . . 44
2.4 Dummett’s fear of realism . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5 ‘anti-realist’ move . . . . . . . . . . . . . . . . . . . . . . 48
2.5.1 What kind of reduction is possible? . . . . . . . . . . . . . . 48
2.5.2 The semantic status of open argumentations . . . . . . . . 50
2.6 The ‘extensions business’ . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.7 A constructive semantics . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7.1 Proof-theoretic semantics and intuitionistic logic . . . . . . 56
2.7.2 Constructive methods vs open argumentations . . . . . . . 59
3 Anti-realist truth 63
3.1 The paradox of deduction . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 Two unsatisfactory solutions . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1 Inferences as definitions . . . . . . . . . . . . . . . . . . . . . 66
3.2.2 A Wittgensteinian perspective . . . . . . . . . . . . . . . . . 67
3.3 From Holism to Molecularity . . . . . . . . . . . . . . . . . . . . . . 69
3.3.1 Static and dynamic holism . . . . . . . . . . . . . . . . . . . . 69
3.3.2 Molecularism: the golden mean . . . . . . . . . . . . . . . . 72
3.4 A proof-theoretic solution to the paradox . . . . . . . . . . . . . . . 75
3.4.1 Direct evidence and objects: a Fregean point . . . . . . . . 76
3.4.2 Direct and indirect evidence . . . . . . . . . . . . . . . . . . 79
3.5 Truth and its recognition . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.1 Truth versus truth-recognition preservation . . . . . . . . . 81
3.5.2 An exegetical remark . . . . . . . . . . . . . . . . . . . . . . . 84
3.6 Dummett on truth and assertion . . . . . . . . . . . . . . . . . . . . 85
3.6.1 The need of a gap . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.6.2 Dummett’s solution . . . . . . . . . . . . . . . . . . . . . . . . 89
3.7 An alternative way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.7.1 Tarskian worries . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.7.2 Challenging assertions . . . . . . . . . . . . . . . . . . . . . . 93
3.8 Open and non-canonical argumentations . . . . . . . . . . . . . . . 97
3.8.1 Predicates and inferences . . . . . . . . . . . . . . . . . . . . 97
3.8.2 Assertion, open validity and truth . . . . . . . . . . . . . . . 100
3.9 A tentative evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 105CONTENTS vii
4 Falsificationism 109
4.1 Refutations: an informal adequacy condition . . . . . . . . . . . . . 110
4.2 Refutations as open derivations . . . . . . . . . . . . . . . . . . . . . 111
4.2.1 Intuitionistic negation . . . . . . . . . . . . . . . . . . . . . . 111
4.2.2 Refutations in NJ . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2.3 and assumptions . . . . . . . . . . . . . . . . . . 115
4.2.4 Thesis 1 and Lopez-Escobar’s clauses . . . . . . . . . . . . . 117
4.2.5 A syntactic notion . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.3 Falsificationism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.3.1 An alternative interpretation of deduction . . . . . . . . . . 119
4.3.2 Dual-intuitionistic logic . . . . . . . . . . . . . . . . . . . . . 123
4.3.3 Harmony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.3.4 F-validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.3.5 Negation in Falsificationism . . . . . . . . . . . . . . . . . . . 131
4.3.6 Proofs in NDJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.3.7 A remark on classical logic . . . . . . . . . . . . . . . . . . . 134
4.3.8 A direct notion of refutation . . . . . . . . . . . . . . . . . . . 137
4.4 Refutations and direct negation . . . . . . . . . . . . . . . . . . . . . 141
4.4.1 An asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.4.2 Refutation through direct negation . . . . . . . . . . . . . . 143
4.4.3 Direct negation vs dual-intuitionism . . . . . . . . . . . . . 146
4.5 Final considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5 Concluding Remarks 151
5.1 Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.3 Dummett’s ‘anti-realist’ reason . . . . . . . . . . . . . . . . . . . . . 154
5.4 Rejecting the priority of the categorical notion . . . . . . . . . . . . 155
5.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.6 Hypothetical First . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Bibliography 159viii CONTENTSAcknowledgments
The present work is the result of my doctoral studies, carried out partially at
Siena University and partially at Tübingen University.
In fall 2007, I was enrolled as a Ph.D. student at Siena University. The re-
search proposal submitted for the admission was based on my master thesis,
‘Negation. A meaning-theoretical investigation’, which was written under the
supervision of Gabriele Usberti and discussed in June 2007 at Siena University.
The proposal was based on two ideas which appeared, in a rather confused
way, in my master thesis. One was that the need of a notion of truth in Dum-
mett and Prawitz’ anti-realist perspective is due to the role played by implica-
tion in the definition of validity. The other was that, in a proof-theoretic setting,
the notions of proof and refutation are connected with the roles of sentences as
(respectively) assumptions and conclusions in derivations.
Although in the research proposal I was stating the will of developing an
alternative and fully-fledged proof-theoretic semantic picture, what I achieved
in these three years is, if anything, a clarification of these two ideas.
Short before discussing my master thesis, in spring 2007, I had the oppor-
tunity to meet Peter Schroeder-Heister, who had been invited to give two lec-
tures in Pisa, by Enrico Moriconi. It turned out that the work on negation I
was carrying out was of some interest to him. One year later, I was moving to
Tübingen.
The work with Peter and his ‘team’ contributed in a unique way to my
growth. With Peter, Thomas Piecha and Bartosz Wie¸ckowski I always had the
possibility of presenting and discussing (most of the times in English) my on-
going research, even when my ideas were not completely clear to myself. Pe-
ter’s lectures ‘on the fly’—no need of slides (sometimes not even of notes),
ixx ACKNOWLEDGMENTS
just a marker for the whiteboard—will be an example to which I will try to
attain. Tübingen has been (and still is) an extremely stimulating environment
into which I could find myself. If I were not in Tübingen, I could not achieve
the few certainties I now have about the ground concepts of proof-theoretic
semantics (for the uncertainties I will never menage to clarify I am the only to
be blamed). I am glad of having had the possibility of sharing with Peter the
excitement in discussing the subtleties of the definition of validity.
I tribute the most significance influence on the