Abstract

A critical view of the alleged significance of Belnap four-valued logic for reasoning under inconsistent and incomplete information is provided. The difficulty lies in the confusion between truth-values and information states, when reasoning about Boolean propositions. So our critique is along the lines of previous debates on the relevance of many-valued logics and especially of the extension of the Boolean truth-tables to more than two values as a tool for reasoning about uncertainty. The critique also questions the significance of partial logic.

References

[1]
Aguzzoli
S
Mundici
D
Weierstrass Approximations by Łukasiewicz Formulas with One Quantified Variable
2001
Proc 31st IEEE International Symposium on Multiple-Valued Logic
Warsaw
(pg. 
361
-
366
)
[2]
Avron
A
Lev
I
Non-Deterministic Multiple-valued Structures Journal of Logic and Computation 15
2005
(pg. 
241
-
261
)
[3]
Behounek
L
Cintula
P
From fuzzy logic to fuzzy mathematics: A methodological manifesto
Fuzzy Sets and Systems
 , 
2006
, vol. 
157
 
5
(pg. 
642
-
646
)
[4]
Belnap
ND
Ryle
Gilbert
How a computer should think
Contemporary Aspects of Philosophy
 , 
1977
Oriel Press
(pg. 
30
-
56
)
[5]
Belnap
ND
Dunn
JM
Epstein
G
A useful four-valued logic
Modern Uses of Multiple-Valued Logic.
 , 
1977
, vol. 
837
 
The Netherlands
D. Reidel, Dordrecht
[6]
Benferhat
S
Dubois
D
Prade
H
Some syntactic approaches to the handling of inconsistent knowledge bases: A comparative study Part 1: The flat case
Studia Logica
 , 
1997
, vol. 
58
 (pg. 
17
-
45
)
[7]
Besnard
P
Hunter
A
Reasoning with Actual and Potential Contradictions, Handbook on Defeasible Reasoning and Uncertainty Management Systems Volume 2
1998
Dordrecht, The Netherlands
Kluwer Academic Publ
[8]
Besnard
P
Hunter
A
Quasi-classical Logic: Non-trivializable classical reasoning from inconsistent information
1995
, vol. 
946
 
Symbolic and Quantitative Approaches to Reasoning and Uncertainty, European Conference, ECSQARU’95
Fribourg, Switzerland
(pg. 
44
-
51
Lecture Notes in Computer Science
[9]
Besnard
P
Konieczny
S
Marquis
P
Bipolarity in Bilattice Logics, to appear in Int. J. Intelligent Systems
2008
[10]
S. Blamey Partial Logic
Handbook of Philosophical Logic, Vol. 3
1985
D. Reidel Publishing Company
(pg. 
1
-
70
)
[11]
Bochvar
DA
Ob odnom trechznacnom iscislenii i ego primenenii k analizu paradoksov klassiceskogo rassirennogo funkcional’nogo iscislenija
Matematiceskij Sbornik
 , 
1938
, vol. 
4
 
46
(pg. 
287
-
308
[English translation: Bochvar, D.A., On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus, History and Philosophy of Logic 2, 87-112.]
[12]
Carnap
R
Two concepts of probability, Philosophy and Phenomenological Research
1945
, vol. 
5
 (pg. 
513
-
532
)
[13]
Carnielli
W
Lima
M
Carnielli
Walter A
D’Ottaviano
Itala ML
Marques Society semantics for multiple-valued logics, in Proceedings of the XII EBL- Advances in Contemporary Logic and Computer Science
1999
, vol. 
Volume 235
 (pg. 
33
-
52
American Mathematical Society, Series Contemporary Mathematics
[14]
De
G
Dubois
D
Prade
H
Klement
EP
Cooman From possibilistic information to Kleene's strong multi-valued logics. In Fuzzy Sets, Logics and Reasoning about Knowledge
1999
Dordrecht, The Netherlands
Kluwer Academic Publishers
(pg. 
315
-
323
)
[15]
De
B
Finetti La logique de la probabilit'e, Actes Congrés Int. de Philos
1935
Scient. Paris
 
Hermann et Cie Editions, Paris, IV1- IV9 (1936).
[16]
Dubois
D
Hájek
P
Prade
H
Knowledge-Driven versus data-driven logics
Journal of Logic, Language, and Information
 , 
2000
, vol. 
9
 (pg. 
65
-
89
)
[17]
Dubois
D
Prade
H
Can we enforce full compositionality in uncertainty calculi?
1994
Proc. of the 12th National Conf. on Artificial Intelligence (AAAI’94)
July 31-Aug. 4
Seattle, WA
(pg. 
149
-
154
)
[18]
Dubois
D
Prade
H
Representation and combination of uncertainty with belief functions and possibility measures
Computational Intelligence
 , 
1988
, vol. 
4
 
4
(pg. 
244
-
264
)
[19]
Dubois
D
Prade
H
Possibility theory, probability theory and multiple-valued logics: A clarification
Annals of Mathematics and Artificial Intelligence
 , 
2001
, vol. 
32
 (pg. 
35
-
66
)
[20]
Dunn
JM
Intuitive semantics for first-degree entailment and coupled trees
Philosophical Studies
 , 
1976
, vol. 
29
 (pg. 
149
-
168
)
[21]
Elkan
Ch
The paradoxical success of fuzzy logic
In Proc. AAAI’93.
 , 
1993
Washington, DC
(pg. 
2
-
49
1994. July 11-15, 698-703. Extended version (with discussions), IEEE Expert, 9(4)
[22]
Font
JM
Hájek
P
On Lukasiewicz's Four-Valued Modal Logic
Studia Logica
 , 
2002
, vol. 
70
 
2
(pg. 
157
-
182
)
[23]
Fitting
M
Bilattices and the Semantics of Logic Programming
J. Log. Program
 , 
1991
, vol. 
11
 
1-2
(pg. 
91
-
116
)
[24]
Fitting
M
Many-valued modal logics. Fundam. Inform
1991
(pg. 
235
-
254
Part I: 15(3-4) Part II :17(1-2): 55-73 (1992).
[25]
Fox
J
Motivation and Demotivation of a Four-Valued Logic
Notre Dame Journal of Formal Logic
 , 
1990
, vol. 
31
 
1
(pg. 
76
-
80
)
[26]
Gärdenfors
P
Knowledge in Flux
1988
Cambridge, Mass
MIT Press
[27]
Ginsberg
ML
Multivalued logics: A uniform approach to inference in artificial intelligence
Computational Intelligence
 , 
1992
, vol. 
4
 
3
(pg. 
256
-
316
)
[28]
Godo
L
Hájek
P
Esteva
F
A Fuzzy Modal Logic for Belief Functions. Fundam. Inform
2003
, vol. 
57
 
2-4
(pg. 
127
-
146
)
[29]
Hájek
P
Godo
L
Esteva
F
Fuzzy logic and probability
1995
Proc. 11th Annual Conference on Uncertainty in Artificial Intelligence
Montreal, Morgan Kaufmann
(pg. 
237
-
244
)
[30]
Hähnle
R
Automated Theorem Proving in Multiple Valued Logics
1994
Oxford University Press
[31]
Hähnle
R
Gabbay
D
Guenthner
F
Advanced Multiple Valued Logics. In Handbook of Philosophical Logic
2001
2d. Edition
Kluwer Academic
(pg. 
297
-
395
)
[32]
Hähnle
R
Many-valued logic, partiality, and abstraction in formal specification languages Logic Journal of IGPL
2005
, vol. 
13
 
4
(pg. 
415
-
433
)
[33]
Hájek
P
The Metamathematics of Fuzzy Logics
1998
Kluwer Academic
Dordrecht
[34]
Hintikka
J
Knowledge and Belief
1963
Ithaca
Cornell University Press
[35]
Kleene
SC
Introduction to Metamathematics
1952
North Holland, Amsterdam
[36]
Lang
J
Liberatore
P
Marquis
P
Propositional Independence: Formula-Variable Independence and Forgetting
J. Artif. Intell. Res. (JAIR)
 , 
2003
, vol. 
18
 (pg. 
391
-
443
)
[37]
Łukasiewicz
J
O logice trojwartosciowej (On three-valued logic)
Ruch filozoficzny
 , 
1920
, vol. 
5
 (pg. 
170
-
171
)
[38]
Meyer
J-JCh
Van der Hoek
W
Besnard
P
Hunter
A
Modal logics for representing incoherent knowledge
Reasoning with Actual and Potential Contradictions.
 , 
1998
, vol. 
Vol. 2
 
Dordrecht
Kluwer Academic Publisher
(pg. 
37
-
76
The Handbook of Defeasible Inference and Uncertainty Management Systems
[39]
Łukasiewicz
J
Philosophical remarks on many-valued systems of propositional logic
Reprinted in Selected Works (Borkowski, ed.), Studies in Logic and the Foundations of Mathematics.
 , 
1930
North-Holland, Amsterdam
(pg. 
153
-
179
1970
[40]
Nilsson
NJ
Probabilistic logic
Artificial Intelligence
 , 
1986
, vol. 
28
 (pg. 
71
-
87
)
[41]
Reichenbach
H
The Theory of Probability
1949
University of California Press
[42]
Thijsse
EGC
Partial Logic and Knowledge Representation
PhD thesis.
 , 
1992
University of Tilburg
Delft
[43]
Slaney
J
Relevant logic and paraconsistency
Inconsistency Tolerance, LNAI
 , 
2005
, vol. 
vol. 3300
 (pg. 
270
-
293
)
[44]
Shafer
G
A mathematical theory of evidence
1976
Princeton, N.J
Princeton University Press
[45]
Urquhart
A
Many-Valued Logic. In Dov M. Gabbay and Franz Guenthner, eds. Handbook of Philosophical Logic: Volume III, Alternatives to Classical Logic
1986
Reidel
Dordrecht
(pg. 
71
-
116
)
[46]
van Fraassen
BC
“Singular Terms, Truth -value Gaps, and Free Logic”
Journal of Philosophy
 , 
1966
, vol. 
63
 (pg. 
481
-
495
)
[47]
Zadeh
LA
Fuzzy sets, Information and Control
1965
, vol. 
8
 (pg. 
338
-
353
)
[48]
Zadeh
LA
Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems
1978
, vol. 
1
 (pg. 
3
-
28
)
[49]
Zadeh
LA
Fuzzy Logic and approximate reasoning (In memory of Grigore Moisil), Synthese
1975
, vol. 
30
 (pg. 
407
-
428
)

Author notes

*This paper is based on an invited talk entitled “Some remarks on truth-values and degrees of belief” given at the Workshop “The Challenge of Semantics” Vienna, Austria, July 2004.