Nuel Belnap
(Some) papers and publications

List of Publications 



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2. “A modification of Ackermann’s ‘rigorous implication’[by A. R. Anderson and NDB] (abstract), Journal of symbolic logic, vol. 23 (1958), pp. 457-458. 

4.“Modalities in Ackermann’s ‘rigorous implication’[by A. R. Anderson and NDB],  Journal of symbolic logic, vol. 24 (1959), pp. 107-111. 

5.“Pure rigorous implication as a ‘Sequenzen-kalkul(abstract), Journal of symbolic logic, vol. 24 (1959), pp. 282-283.

6.“A proof of the Loewenheim-Skolem theorem” [by A. R. Anderson and NDB] (abstract), Journal of symbolic logic, vol. 24 (1959), pp. 285-286.

9. Book note: Axiomatic set theory (New York, 1960) by P. Suppes, Review of metaphysics, vol. 14 (1960-61), p. 175. 

11.“Tautological entailments(abstract), Journal of symbolic logic, vol. 24 (1959), p. 316.

12.“A simple treatment of truth functions[by A. R. Anderson and NDB], Journal of symbolic logic, vol. 24 (1959), pp. 301-302.

13. “A simple proof of Goedel’s completeness theorem” [by A. R. Anderson and NDB] (abstract), Journal of symbolic logic, vol. 24 (1959), pp. 320-321.

15. Book note: Leviathan: a simulation of behavioral systems, to operate dynamically on a digital computer (Santa Monica, 1959) by Beatrice K. Rome and Sydney C. Rome, Review of metaphysics, vol. 15 (1961-62), p. 195. 

18.  Review of “Existential presuppositions and existential commitments” (Journal of philosophy, 1959) by J. Hintikka, Journal of symbolic logic, vol. 25 (1960), p. 88. 

19. Review of “Nondesignating singular terms” (Philosophical review, 1959) by H. Leblanc and T. Hailperin, Journal of symbolic logic, vol. 25 (1960), pp. 87-88.

20. Review of “Towards a theory of definite descriptions” (Analysis, 1959) by J. J. Hintikka, Journal of symbolic logic, vol. 25 (1960), pp. 88-89. 

21.“Entailment and relevance,Journal of symbolic logic, vol. 25 (1960), pp. 144-146.

22.“Enthymemes[by A. R. Anderson and NDB], Journal of philosophy, vol. 58 (1961), pp. 713-723.

23.“Tautological entailments[by A. R. Anderson and NDB], Philosophical Studies, vol. 13 (1962), pp. 9-24.

24.“Tonk, Plonk and Plink,” Analysis, vol. 22 (1961-62), pp. 130-134.

26. “First degree formulas” (abstract),  Journal of symbolic logic, vol. 25 (1960), pp. 388-389.

27.“First degree entailments [by A. R. Anderson and NDB], Mathematische annalen, vol. 149 (1963),  pp. 302-319.A slightly revised version of item 10.

28. Review of Computers and common sense (Columbia University Press, 1961) by Mortimer Taube, Modern uses of logic in law, March 1963, pp. 34-38.

30.“The pure calculus of entailment [by A. R. Anderson and NDB], Journal of symbolic logic, vol. 27 (1962), pp. 19-52.

32. Review of Natural deduction (Wadsworth, 1962) by John M. Anderson and Henry W. Johnstone, Jr., American mathematical monthly, 1963.

35. “On not strengthening intuitionistic logic [by NDB, H. Leblanc and R. H. Thomason], Notre Dame journal of formal logic, vol. 4 (1963), pp. 313-320.

36.“On not strengthening intuitionistic logic [by NDB, H. Leblanc and R. H. Thomason] (abstract), read at December 1963 meeting of the Association for Symbolic Logic, Journal of symbolic logic, vol. 28 (1963), p. 297. 

37. Review of “A logic of questions and answers” (Philosophy of Science, 1961), Communication: A logical model (The M.I.T. Press, 1963), and “A model for applying information and utility functions”(Philosophy of Science, 1963) by David Harrah, Journal of symbolic logic, vol. 29 (1964), pp. 136-138. 

39. Review of “A measure of subjective information” by Rulon Wells, Structure of language and its mathematical aspects, Proceedings of symposia in applied mathematics, vol. 12, American Mathematical Society, 1961, pp. 237-244, and “Comments” by J. D. Sable and R. Wells, ibid., pp. 267-268.Journal of symbolic logic, vol. 30 (1965), pp. 244-245. 

42.“Questions, answers, and presuppositions,” Journal of philosophy, vol. 63 (1966), pp. 609-611.Abstract of a paper read at the 1966 meeting of the Eastern Division of the American Philosophical Association.

43. Combined review of three articles by B. Sobocinski and six articles by Ivo Thomas, all on Lewis-like modal logics, Journal of symbolic logic, vol. 31 (1966), pp. 498-500.

44.“Intensionally complemented distributive lattices [by NDB and Joel Spencer], Portugaliae Mathematica, vol. 25 (1966), pp. 99-104.

45.“Intensional models for first degree formulas,” Journal of symbolic logic, vol. 32 (1967), pp. 1-22.

46.“Special cases of the decision problem for entailment and relevant implication,” Journal of symbolic logic, vol. 32 (1967), pp. 431-432. Abstract of a paper read at the 1967 meeting of the Association for Symbolic Logic. 

47.“Homomorphisms of intentionally complemented distributive lattices [by J. Michael Dunn and NDB] (abstract), Journal of symbolic logic, vol. 32 (1967), p. 446.

50.“Homomorphisms of intensionally complemented distributive lattices [by J. M. Dunn and NDB],  Mathematische annalen, vol. 176 (1968), pp. 28-38. 

52.“The substitution interpretation of the quantifiers[by J. M. Dunn and NDB], Noûs, vol. 2 (1968), pp. 177-185.

55.“Every functionally complete m-valued logic has a Post-complete axiomatization [by NDB and Storrs McCall], Notre Dame journal of formal logic, vol. 11 (1970), p. 106

56.“Conditional assertion and restricted quantification,” Noûs, vol. 4 (1970), p. 1-13.

57. Review of “A propositional logic with subjunctive conditionals (Journal of symbolic logic, 1962) by R. B. Angell, Journal of symbolic logic, vol. 35 (1970), pp. 464-465.

58.“S-P interrogatives, Journal of philosophical logic, vol. 1 (1972), pp. 331-346. 

63.“A prosentential theory of truth [by D. L. Grover, J. L. Camp, Jr. and NDB], Philosophical Studies,  vol. 27 (1975), pp. 73-125. 

76.“Rescher’s hypothetical reasoning:an amendment,” The philosophy of Nicholas Rescher:discussion and replies, ed. E. Sosa, D. Reidel, 1979, pp. 19-28.

77.“Relevant analytic tableaux[by M. A. McRobbie and NDB], Studia Logica, vol. 38 (1979), pp. 187-200.

78.“A consecution calculus for positive relevant implication with necessity [by NDB, Anil Gupta, and J. Michael Dunn],  Journal of philosophical logic, vol. 9 (1980), pp. 343-362. (By error the title appeared as “A consecutive… .”) 

87.“Display logic,” Journal of philosophical logic, vol. 11 (1982), pp. 375-417.

88.Gupta’s rule of revision theory of truth,”Journal of philosophical logic, vol. 11 (1982), pp. 103-116.

92.“Display logic (abstract), Journal of symbolic logic, vol. 48 (1983), p. 907. 

95.“A note on extension, intension, and truth” [by A. Gupta and NDB], Journal of philosophy, vol. 84 (1987) pp. 168-174.

102. “Semantic holism[by NDB and G. J. Massey], Studia logica, vol. 49 (1990) pp. 67-82.

103. “Declaratives are not enough,” Philosophical studies, vol. 59 (1990) pp. 1-30.

104. “Linear logic displayed,” Notre Dame journal of formal logic, vol. 31 (1990) pp. 14-25. 

106. “Before refraining: concepts for agency,” Erkenntnis, vol. 34 (1991) pp. 137-169.

107. “In the realm of agents [by NDB and M. Perloff], DEON ‘91:First International Workshop on Deontic Logic in Computer Science, J.-J. Ch. Meyer and R. J. Wieringa (eds.), Amsterdam, pp. 107-126. 

108. “Backwards and forwards in the modal logic of agency,” Philosophy and phenomenological research, vol. 51 (1991) pp. 777-807.

110. “Branching space-time,” Synthese, vol. 92 (1992) pp. 385-434.

111. “The way of the agent [by NDB and M. Perloff], Studia logica, vol. 51 (1992) pp. 463-484.

114. “On rigorous definitions,” Philosophical studies, ed. Marian David, vol. 72 (1993) p. 115-146.

115. “Indeterminism and the thin red line[by NDB and Mitchell Green], Philosophical perspectives, vol. 8, Logic and language, James Tomberlin (ed.), Ridgeview Publishing Co. (1994), pp. 365-388.

117.“The deliberative stit: a study of action, omission, ability, and obligation [by John F. Horty and NDB], Journal of Philosophical Logic, vol. 24 (1995), no. 6, pp. 583-644

119.“Reply to Robert Koons [by A. Gupta and NDB], Notre Dame journal of formal logic,  vol. 35 no. 4 (1995), pp. 632-636. 

121.“Branching space-time analysis of the GHZ theorem [by Laszlo Szabo and NDB], Foundations of Physics, vol. 26, no. 8, (1996), pp.989-1002.

128.  “Concrete transitions,” in Actions, norms, values:discussions with George Henrik von Wright, George Meggle (ed.), Walter de Gruyter, 1997. (1999, pdf format)

Following von Wright, ``transitions'' are needed for understanding agency. I indicate how von Wright's account of transitions should be adapted to take account of objective indeterminism, using the idea of branching space-time. The essential point is the need to locate  transitions not merely in space-time, but concretely amid  the indeterministic, causally structured possibilities of our (only) world. (This is a ``postprint'' of Belnap 1999, as cited in the paper. The page numbers do not, of course, match those of the original.)

133.  EPR-like funny business in the theory of branching space-times Non-locality and Modality, T. Placek and J.
Butterfield (eds.), Kluwer Academic Publishers, 2002, pp. 293-315.(2002, pdf format.)

EPR-like phenomena are (presumably) indeterministic, but they furthermore suggest that our world involves seeming-strange ``funny business.'' Without invoking any heavy mathematics, the theory of branching space-times offers two apparently quite different ways in which EPR-like funny business goes beyond simple indeterminism. (1) The first is a modal version of a Bell-like correlation: There exist two space-like separated indeterministic initial events whose families of outcomes are nevertheless modally correlated. That is, although the occurrence of each outcome of each of the two space-like separated initial events is separately possible, some joint occurrence of their outcomes (one from each) is impossible. (2) The second sounds like superluminal causation: A certain initial event can bear a cause-like relation to a certain without being in the causal past of that outcome. The two accounts of EPR-like funny business are proved equivalent, a result that supports the claim of each as useful to mark the line between mere indeterminism and EPR-like funny business. (This is a ``postprint'' .  Numbers 135 and 139 below are closely related.)

134.   "Double time references:   Speech-act reports as modalities in an indeterministic setting"  in Advances in Modal Logic, F. Wolter, H. Wansing, M. De Rijke, and M. Zakharyaschev (eds.) , 2002.(2001, pdf format)

The background is a theory that describes agents and their choices in the branching-time represention of indeterminism, and that describes a language appropriate for use in such a setting.  The problem of this paper is to clarify the meaning, in the context of indeterminism, of speech acts such as promising, ordering, advising, asserting, and betting.  Such speech acts often have sentence that serves as a "declarative core," whose truth-conditional semantics in branching time contributes to the meaning of the speech act.  Direct-discourse reports of such speech acts can be treated as a kind of "modal" connective, e.g. "a1 promised a2 to make 'A' true." A key feature of the contribution of the declarative core is made in terms of its (now) being definitely settled at a later moment (in the branching tree) that the declarative core of the speech act was true (then) at the moment of the speech act.  That is the "double time reference" that needs clarification.

135.  No-common-cause EPR-like funny business in branching space-times (in Philosophical Studies 2003, pdf format.)

There is “no EPR-like funny business” if (contrary to apparent fact) our world is as indeterministic as you wish, but is free from the EPR-like quantum mechanical phenomena such as is sometimes described in terms of superluminal causation or correlation between distant events. The theory of branching spacetimes can be used to sharpen the theoretical dichotomy between “EPR-like funny business” and “no EPR-like funny business”. Belnap (2002) offered two analyses of the dichotomy, and proved them equivalent. This essay adds two more, both connected with Reichenbach’s “principle of the common cause”, the principle that sends us hunting for a common-causal explanation of distant correlations. The two previous ideas of funny business and the two ideas introduced in this essay are proved to be all equivalent, which increases one’s confidence in the stability of (and helpfulness of) the BST analysis of the dichotomy between EPR-like funny business and its absence.

Branching histories approach to indeterminism and free will (a pre-print of 141B. “Branching Histories Approach to Indeterminism and Free Will,” Truth and Probability Essays in Honour of Hugues Leblanc. Bryson Brown and Francois Lepage, eds. 2005. pp. 197—211.)

An informal sketch is offered of some chief ideas of the (formal) ``branching histories'' theory of objective possibility, free will and indeterminism. Reference is made to ``branching time'' and to ``branching space-times,'' with emphasis on a theme that they share: Objective possibilities are in Our World, organized by
the relation of causal order.

136. "Agents in branching space-times," Journal of Sun Yatsen University, Social Science Edition, vol. 43, 2003, pp. 147--166.

The aim of this essay is to make some brief suggestions on the beginnings of a theory of agents and agency in branching space-times.  The thought is to combine the ideas of agency as developed against the relatively simple background of branching time with the richer notions of indeterminism as structured in the theory of branching space-times.  My plan is to say a little about agency in branching time and a little about branching space-times, and then ask how the two can be brought together.  At the end there is an appendix, extracted from Belnap, Perloff and Xu 2001 (Facing the future), listing in a convenient form all the main ideas about agents and their choices in branching time.

137.  "Some non-classical logics seen from a variety of perspectives,"Journal of Sun Yatsen University, Social Science Edition, vol. 43, 2003, pp. 167--179.

Logicians have worked with so many different logical systems that it is not possible even to estimate the number.  Of these, many are best seen as extensions of classical logic, including both those of interest to mathematics and those of interest to philosophy and computer science.  (Henceforth I will use the term "intelligent systems theory" for the common ground of philosophical logic and that part of computer science that concerns itself with activities plausibly taken to embody intelligence in some degree.) On the mathematical side are, of course, higher order logics, set theories, systems of arithmetic, and so forth.  On the intelligent systems side, useful extensions of classical logic include modal logic, deontic logic, epistemic logic, tense logic, indexical logic, and so forth.

This essay, however, does not deal with those logics; instead, it concerns itself with non-classical logics of interest to intelligent-systems theory.  There are doubtless hundreds of non-classical logics, and I consider only a few.  Chiefly I will talk about "relevance" logics and some close cousins.  Sometimes these are called "substructural logics" for reasons that will emerge.  Concerning these logics, I wish to emphasize the very large number of approaches to them that have proved enlightening and useful.

139.  A theory of causation:  Causae causantes (originating causes) as inus conditions in branching space-times  (in British Journal of the Philosophy of Science. vol. 56, 2005, pp. 221-253)  

140.  "Under Carnap's Lamp:  Flat Pre-Semantics,"  (in Philosophical Studies, vol. 80, No. 1, June 2005, pp. 1--28)

Branching space-time, postprint January, 2003 (2003, pdf format.)

141.  "How Causal Probabilities Might Fit into Our Objectively Indeterministic World," w/ Matt Weiner (in Synthese, Volume 149, March 2006, pp. 1--36.)

143.  “Prosentence, Revision, Truth and Paradox, “Philosophy and Phenomenological Research”, Vol. LXXIII No. 3, November, 2006, pp. 705—712.

144.  “Propensities and probabilities,” Studies in History and Philosophy of Modern Physics, Volume 38, 2007, pp.  593—625.  DOI:  10.1016/j.hpsb.2006.09.003

144x.  "Propensities and probabilities,"  (2010 corrected postprint of 144).

145.  "From Newtonian determinism to branching-space-time indeterminism," Logik, Begriffe, Prinzipien des Handelns (Logic, Concepts, Principles of Action).  Thomas Müller/ Albert Newen (eds.), mentis Verlag GmbII, 2007, pp. 13--31.

145-revised.  "From Newtonian determinism to branching-space-time indeterminism," revision of 145, forthcoming in Synthese.

147.  "Funny business in branching space-times:  infinite modal correlations,"  w/ Thomas Mueller and Kohei Kishida (in Synthese (2008) 164:  pp. 141--159.

148.  "Branching with Uncertain Semantics: Discussion Note on Saunders and Wallace, ‘Branching and Uncertainty’," [by NB and Thomas Mueller] Brit. J. Phil. Sci. (2010), pp. 1–16.

 Future Contingents and the Battle Tomorrow w/ Michael Perloff forthcoming in Review of Metaphysics.

Notes on the Art of Logic 2009 (2009, pdf format, Unpublished)

Notes on the Science of Logic 2009 (2009, pdf format, Unpublished)

 How a computer should think (from Entailment II)

 Facing the Future (Chapters 7 and 8)

Indeterminism is a modal notion:  branching spacetimes and Earman's pruning(Placek and Belnap 2010) Forthcoming Synthese

Phil 2505, S 2011, Indeterminism, BT, and BST.

Prolegomenon to norms in BST