Interactions of metaphysical and epistemic concepts - Alexandre

210 Pages·2007·1.19 MB·English

Interactions of
metaphysical and epistemic concepts
Th`ese pr´esent´ee `a la Facult´e des lettres et sciences humaines
Universit´e de Neuchaˆtel - CH
Pour l’obtention du grade de docteur `es lettres
Par
Alexandre Fernandes Batista Costa Leite
Accept´ee sur proposition du jury:
Prof. Jean-Yves B´eziau, Universit´e de Neuchaˆtel, directeur de th`ese
Prof. Pascal Engel, Universit´e de Gen`eve, rapporteur
Prof. Paul Gochet, Universit´e de Li`ege, rapporteur
Prof. Arnold Koslow, The City University of New York, rapporteur
Soutenue le 03 Juillet 2007
Universit´e de Neuchˆatel
2007
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châte;l M. PascalE ngel,p rofesseuàr I'Universitdée Genève; M. Paul Gochet,p rofesseuàr
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Abstract
This work sets out the results of research on topics at the intersection
of logic and philosophy. It shows how methods for combining logics can be
applied to the study of epistemology and metaphysics. In a broader per-
spective, it investigates interactions of modal concepts in order to create a
bridge between metaphysics and epistemology. The first study of such in-
teractions offers an analysis of a paradox in the limits of knowledge from
the viewpoint of fusions and products of modal logics. We next examine the
formal combination of an epistemic logic of knowledge with a contingency
logic in order to determine whether contingent propositions can be known.
The third example investigates a combination of logics for belief and con-
tingency logics to determine in what sense the world is an object of belief.
The work also studies contradictions in the context of modal operators and
their relations with paraconsistent logics. In the final section, some gen-
eral approaches to modalities are studied and some directions are presented.
The concept of paraconsistentization of theories and logics, as well the idea
of philosophical categorification are presented from an abstract viewpoint.
The work presents some tools to deal with modalities and proposes some
concepts for exploring problems in philosophy and logic.
Mot cl´es en francais: combinaisons, connaissance, croyance, contin-
gence, paraconsistentization, categorification philosophique
Mot cl´es en anglais: combinations, knowledge, belief, contingency,
paraconsistentization, philosophical categorification
PhD Thesis:
Interactions of metaphysical and epistemic concepts
Research Areas: Philosophy and Logic
Author: Alexandre Fernandes Batista Costa Leite
Supervisor: Jean-Yves B´eziau
Facult´e des lettres et sciences humaines
Universit´e de Neuchˆatel
July 2007
Work supported by a grant of the Swiss National Science Foundation
To
Confoederatio Helvetica
CH
Contents
Introduction 1
1 Plug-ins: possibility and knowledge 28
1.1 Fusions of modal logics and Fitch’s paradox . . . . . . . . . . 35
1.1.1 Church or Fitch? . . . . . . . . . . . . . . . . . . . . . 36
1.1.2 The verification principle . . . . . . . . . . . . . . . . 36
1.1.3 The collapse principle . . . . . . . . . . . . . . . . . . 38
1.1.4 The nature of Fitch’s paradox . . . . . . . . . . . . . . 38
1.1.5 So many people . . . . . . . . . . . . . . . . . . . . . . 45
1.2 Other formulations . . . . . . . . . . . . . . . . . . . . . . . . 47
1.2.1 Philosophical foundations of the paradox . . . . . . . 48
1.3 Products of modal logics and the limits of knowledge . . . . . 50
1.3.1 The logic for knowability . . . . . . . . . . . . . . . . 52
1.3.2 Combined modality? . . . . . . . . . . . . . . . . . . . 57
1.4 Imagination and possible knowledge . . . . . . . . . . . . . . 58
2 Interplays: knowledge and contingency 63
2.1 Logical skepticism . . . . . . . . . . . . . . . . . . . . . . . . 65
2.2 Combining knowledge and contingency . . . . . . . . . . . . . 70
2.2.1 Two notions of contingency . . . . . . . . . . . . . . . 71
2.2.2 Contingency logics . . . . . . . . . . . . . . . . . . . . 74
2.2.3 Mixing knowledge and contingency . . . . . . . . . . . 78
2.3 Logical skepticism revisited . . . . . . . . . . . . . . . . . . . 90
2.3.1 Knowledge and reality . . . . . . . . . . . . . . . . . . 90
2.3.2 Two basic criteria . . . . . . . . . . . . . . . . . . . . 91
2.4 Fusions, skepticism and the border of concepts . . . . . . . . 98
3 Connections: belief and contingency 100
3.1 Shaking knowledge and belief . . . . . . . . . . . . . . . . . . 102
ii
3.2 Different conceptions of belief . . . . . . . . . . . . . . . . . . 105
3.3 Interactions of knowledge and belief . . . . . . . . . . . . . . 110
3.3.1 The knowledge-belief reduction . . . . . . . . . . . . . 115
3.4 Combining belief and contingency . . . . . . . . . . . . . . . . 118
3.5 Fusions, skepticism and the border of concepts II . . . . . . . 123
4 Combinations: contradictions, contingency, knowledge and
belief 125
4.1 Can contradictions be object of knowledge and belief? . . . . 130
4.2 Paraconsistentization of logics . . . . . . . . . . . . . . . . . . 134
4.2.1 The case of modal logics . . . . . . . . . . . . . . . . . 137
4.2.2 The problem of a paraconsistent negation . . . . . . . 141
4.2.3 Desparaconsistentization of logics? . . . . . . . . . . . 143
4.3 Paraconsistentization of classical propositional logic . . . . . 143
4.3.1 Fusions and paraconsistentization . . . . . . . . . . . . 149
4.4 Paraconsistentizing fusions. . . . . . . . . . . . . . . . . . . . 151
5 Network: a universal approach to concepts 154
5.1 Philosophical categorification . . . . . . . . . . . . . . . . . . 157
5.1.1 A categorial approach . . . . . . . . . . . . . . . . . . 157
5.1.2 Logic applied to philosophy . . . . . . . . . . . . . . . 158
5.1.3 Category theory and philosophy . . . . . . . . . . . . 162
5.1.4 Categories and logics . . . . . . . . . . . . . . . . . . . 163
5.1.5 Categories and philosophy . . . . . . . . . . . . . . . . 165
5.1.6 A system of philosophy based on category theory? . . 167
5.2 The universal approach to modalities . . . . . . . . . . . . . . 167
5.2.1 Structuralist theory of modalities . . . . . . . . . . . . 168
5.2.2 The n-dimensional modal logics . . . . . . . . . . . . . 168
5.2.3 The n-opposition theory . . . . . . . . . . . . . . . . . 169
5.3 The duality of reality . . . . . . . . . . . . . . . . . . . . . . . 173
Conclusion 176
History and Interview 182
Bibliography 192
iii
Acknowledgements
This work reflects actions of a great variety of people and my interactions
with everybody. Writing a PhD thesis is difficult; we need original ideas and
some basic results.
I would like to thank professor Jean-Yves B´eziau who invited me to
Switzerland, and who has helped me in the development of my ideas for
a long time. His philosophy and way of thinking have been important to
my own developments, andI am especially happybecause Jean-Yves B´eziau
allowed me to write this document developing my own ideas and contribu-
tions. I have no doubt that he is one of the most important researchers in
logic and philosophy working nowadays; he is deeply contributing to both
areas. To Jean-Yves B´eziau, my eternal respect and gratitude.
ThegrantoftheSwiss National ScienceFoundation hasbeenanecessary
condition for the development of this work. Without this grant, the ideas
developed here would not exist.
Thanks to the members of the Jury: Prof. Arnold Koslow, Prof. Paul
Gochet andProf. Pascal Engel. Theyhelpedmea lot withtheirsuggestions
and corrections.
I also would like to thanks all my Logic teachers: Wagner de Campos
Sanz, Adriano Naves de Brito, Marcelo Coniglio, Walter Carnielli and Itala
D’Ottavianno. With them, I learned the exact value of logic in philosophy.
This is a list of people that I will be forever indebt for helping me in
many different aspects: Saul Kripke (for coming to UNILOG’05), Carlos
Caleiro (for accepting me in Lisbon), Maarten Marx (for accepting me in
Amsterdam), Charles Alunni and Andrei Rodin (for accepting me in Paris),
Eva Glu¨ck, C´ecile Guex-Joris (my beloved girlfriend), and people from the
Institute of Logic at the University of Neuchaˆtel: Prof. Denis Mi´eville,
Nadine Gessler, Alessandro Facchini, C´edric D´egrange, Pierre Joray and
Christiane Tripet. Thanks also to Prof. Anne-Nelly Perret-Clermont, from
the Institute of Psychology at UniNE for her useful help in a very difficult
moment. Another list of people who I am indebted can be found in the
iv
section History and Interview.
ThanksalsotomanyfriendsthatIknewwhilewritingthisthesis, friends
who will never be forgotten: Dietrich Choffat (my Swiss-Brazilian friend in
Neuchaˆtel), Ale Mercha´n (el poeta maldito), Fabien Schang (who wrote a
book where I am a character) and Alessio Moretti.
Thankstomylogically-mathematically-philosophically-oriented colleagues
Michael Wright (Mike), Mark Jago and Casey McGinnis for helping in the
improvement of the English language of this book. It is not easy to write a
book in a language different from one’s native language.
Thanks to my family and God, for always protecting me against the
flowing of reality.
v
Introduction
“All men by nature desire to know. An indication of this is the
delightwetakeinoursenses; forevenapartfromtheir usefulness they
are loved for themselves...” (ARISTOTLE in [2])
This book deals with concepts. It is a study on the interactions of
metaphysical and epistemic modalities (i.e a particular family of concepts),
andtherelationsbetweenthenotionsofknowledge andbelief,andthefurther
connection of these concepts withthe notions of possibility and contingency.
Thus it deals with one aspect of the relations between metaphysics and
epistemology. Belowthereaderfindsomeexplanationsofitsmainobjectives
and purposes. At the end of this opening section, a general theory of the
combination of logics is proposed. This serves to introduce the main topic
of the work.
Interactions
Thisthesisstartedasanattempttodefendskepticism. Thattheworldisnot
anobjectofknowledgebutjusttheobjectofbeliefisafamiliarphilosophical
thesis. This work aims to provide logical foundations for an account of the
interactions of different modalities. The study of such interactions starts
in old problems such as induction, the contingency of the world, and the
epistemic status of agents in relation to such contingency. Inconsistencies
in the borders of knowledge and belief are investigated, as is a universal
logical framework for dealing with modalities. The concept of interaction,
which appears in the title of the work, refers here to a logical formalism
in which non-interdefinable modalities appear in the same formula. The
interactions of two or more modalities can appear in different ways. People
studying the combination of logics have employed the notion of interaction
especially in the analysis of implications (See for instance the work [44]).
But an interaction can also be viewed as a formula in which different non-
interdefinable modalities appear.
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