Bibliography

I list only those works which are cited in the text or elsewhere in the bibliography. Page references are to the most recent English reference listed unless noted otherwise.

Quotation marks and logical notation in all quotations have been changed to conform with the conventions of this book (see p. 5 for the use of quotation marks).

ANDERSON, Alan R. and Nuel D. BELNAP, Jr. 1975 Entailment

Princeton Univ. Press. ARRUDA, Ayda I.

1980 A survey of paraconsistent logic Mathematical Logic in Latin America, ed. A. Arruda, R. Chuaqui, and N. C.A.da Costa, North-Holland.

198? Aspects of the historical development of paraconsistent logics Typescript.

AUNE, Bruce 1976 Possibility

In Edwards, 1967, Vol.6, pp.419-424. BENNETT, Jonathan

1969 Entailment Phil. Rev., vol. 78, pp.197-235.

BERNAYS, Paul 1926 Axiomatische Untersuchung des Aussagen-Kalkiils der

Principia mathematica Mathematische Zeitschrift, vol. 25, pp. 305-320.

BLOK, W. J. and K6HLER, P. 1983 Algebraic semantics for quasi-classical modal logics

The Journal of Symbolic Logic, vol. 48, no.4, pp. 941-963. BLOK, W. J. and PIGOZZI, D.

1982 On the structure of varieties with equationally definable principal congruences I Algebra Universalis, vol.15, pp.195-227.

1989 Algebraizable Logics Memoirs of the American Mathematical Society, no. 396.

198? The deduction theorem in algebraic logic Typescript.

BOCHENSKI, I. M. 1970 A History of Formal Logic

Chelsea. A revision and translation of the German F ormale Logik, Verlag Karl Alber, Freiburg, 1956.

351

352 BIBLIOGRAPHY

BOOLOS, George 1979 The unprovability of consistency

Cambridge Univ. Press. 1980 A Provability, truth, and modal logic

J. Phil. Logic, vol. 9, no.1, pp.1-7. 1980 B On systems of modal logic with provability interpretations

Theoria, vol.46, no.1, pp. 7-18 BROUWER, L. E. J.

1907 Over de grondslagen der wiskunde Dissertation, Amsterdam. Translated as 'On the foundations of mathematics' in Brouwer, 1975, pp.11-101

1908 De onbetrouwbaarheid der logische principes Tijdschrift voor wijsbegeerte, vol. 2, pp. 152-158. Translated as 'The unreliability of the logical principles' in Brouwer, 1975, pp.107-111.

1912 lntuitionisme en formalisme Inaugural address, Univ. of Amsterdam. Translated as 'Intuitionism and formalism' in Bulletin of the American Math. Soc., vol. 20 (Nov. 1913), pp. 81-96, and reprinted in Philosophy of Mathematics, ed. P. Benacerraf and H. Putnam, Prentice-Hall, Englewood Cliffs N.J. 2nd edition, 1983, Cambridge Univ. Press, pp. 77-89.

1928 Intuitionistische Betrachtungen iiber den Formalismus Sitzungsberichte der Preussischen Akademie der Wissenschaften, Phys.-math. Kl., pp. 48-52. Translated as 'Intuitionistic reflections on formalism' in van Heijenoort, 1967, pp.490-492.

1975 The collected works of L. E. J. Brouwer ed. A. Heyting, North-Holland.

CANTOR, Georg 1883 Grundlagen einer allgemeinen Mannigfaltigkeitslehre

Teubner, Leipzig. The translation in the text comes from Georg Cantor, J. Dauben, Harvard Univ. Press, 1979, pp.128-129.

CARNIELLI, Walter A. 1987 A Methods of proof for relatedness and dependence logic

Reports on Mathematical Logic, vol. 21, pp. 35-46. 1987 B Systematization of finite many-valued logics through the method

of tableaux The Journal of Symbolic Logic, vol.52, pp. 473-493.

CHELLAS, Brian 1980 Modal Logic

Cambridge Univ. Press. CHRISTENSEN, Niels Egmont

1973 Is there a "logic" or formal system based on the concept of a truth determinant? Danish Yearbook of Philosophy, vol.10, pp. 77-85.

CLEAVE, J. P. 197 4 The notion of logical consequence in the logic of inexact predicates

Zeit. Math. Logik und Grundlagen, vol. 20, no. 4, pp. 307-324.

COPELAND, B. J. 1978 Entailment, the Formalisation of Inference

Doctor of Philosophy Thesis, Oxford. 1984 Horseshoe, hook, and relevance

Theoria, vol. L, pp.148-164. DACOSTA, Newton C.A.

BIBUOGRAPHY 353

1963 Calculs propositionnels pour les systemes formels inconsistents Comptes Rendus de /' Academie des Sciences de Paris, Serle A, vol. 257, pp. 3790--3792.

1974 On the theory of inconsistent formal systems Notre Dame Journal of Formal Logic, XV, no. 4, pp.497-510.

Seealso D'OTTAVIANOandDACOSTA DACOSTA, Newton C.A. and Diego MARCONI

198? An overview ofparaconsistent logic in the 80's To appear inLogicaNova, Akademie-Verlag.

DE MORGAN, Augustus 1847 Formal Logic or the Calculus of Inferences Necessary and

Probable London. Reprinted by Open Court, 1926.

DE SWART, H. 1977 An intuitionistically plausible interpretation of intuitionist logic

Journal of Symbolic Logic, vol.42, no. 4, pp. 564-578. D'OTT A VIANO, Itala M. L.

1985 A The completeness and compactness of a three-valued flrst-order logic Revista Colombiana de Materruiticas, XIX, 1-2, pp. 31-42.

1985 B The model-extension theorems for J3-theories Methods in Mathematical Logic, ed. C. A. Di Prisco, Lecture Notes in Mathematics, no. 1130, Springer-Verlag.

1987 Deflnability and quantifier elimination for J 3-theories Studia Logica, XL VI, 1, pp. 37-54.

D'OTTAVIANO, ItalaM.L. and Newton C.A. daCOSTA 1970 Sur un probleme de Jaskowski

C. R. Acad. Sc. Paris, 270, Serle A, pp.1349-1353. DREBEN, Burton and Jean van HEUENOORT

1986 Note to Gooel's dissertation In Godel,1986, pp.44-59.

DUMMETT, Michael 1959 A propositional calculus with denumerable matrix

The Journal of Symbolic Logic, vol. 24, pp. 97-106. 1973 The philosophical basis of intuitionistic logic

In Logic Colloquium '73, ed. H. E. Rose and J.C. Shepherdson, North-Holland, Amsterdam. Reprinted in Dummett, Truth and Other Enigmas, Harvard Univ. Press, 1978.

1977 Elements of Intuitionism Clarendon Press, Oxford.

DUNN, J. Michael 1972 A modification of Parry's Analytic Implication

Notre Dame J. of Formal Logic, vol.13, no. 2, pp.l95-205.

354 BIBUOORAPHY

EDWARDS, Paul (ed.) 1967 The Encyclopedia of Philosophy

Macmillan and The Free Press. EPSTEIN, Richard L.

1979 Relatedness and implication Phil. Studies, vol. 36, no.2, pp.137-173.

1980 A (ed.) Relatedness and Dependence in Propositional Logics Research Report of the Iowa State Univ. Logic Group.

1980 B Relatedness and dependence in propositional logics Abstract, The Journal of Symbolic Logic, vol. 46, no. 1, p. 202.

1985 Truth is beauty History and Phil. of Logic, vol. 6, pp. 117-125.

1987 The algebra of dependence logic Reports on Mathematical Logic, vol.21, pp.19-34

1988 A general framework for semantics for propositional logics In Methods and Applications of Mathematical Logic, Proceedings of the VII Latin American Symposium on Mathematical Logic, ed. W. Camielli and L. P. de Alcantara, Contemporary Mathematics, American Math. Soc., no. 69.

198? A theory of truth based on a medieval solution to the liar paradox Typescript.

EPSTEIN, Richard L. and Walter A. CARNIELLI 1989 Computability

Wadsworth & Brooks/Cole. EPSTEIN, Richard L. and Roger D. MADDUX

1981 The algebraic nature of set assignments In Epstein, 1980 A.

FINE, Kit 1979 Analytic implication

Notre Dame J. of Formal Logic, vol.27, no.2, pp.169-179. FITTING, M. C.

1969 lntuitionistic Logic, Model Theory and Forcing North-Holland.

FREGE, Gottlob 1879 Begriffschrift

L. Nebert, Halle. Translated as Begriffschrift, a formula language, modeled upon that of arithmetic ,for pure thought, in van Heijenoort, 1967, pp.1-82.

1892 iiber Sinn und Bedeutung Zeit.fur Phi/osophie undphi/osophische Kritik, vol.100, pp.25-50. Translated as 'On sense and reference' in Translations from the Philosophical Writings of Gottlob Frege, ed. M. Black and P. Geach, Basil Blackwell, 1970, pp. 56-78.

1918 Der Gedanke: eine logische Untersuchung Betriige zur Philosophie des deutschen ldealismus, pp.58-77. Translated by A. and M. Quinton as 'The thought: a logical inquiry' in Mind, (new series) vol. 65, pp. 289-311, and reprinted in Philosophical Logic, ed. P.F.Strawson, Oxford Univ. Press, 1967, pp.17-38.

1980 Philosophical and Mathematical Correspondence Univ. of Chicago Press.

GENTZEN, Gerhard 1936 Die Widerspruchsfreiheit der reinen Zahlentheorie

Mathematische Annalen, vol. 112, pp. 493-565. GLIVENK.O, V.

1929 Sur quelques points de Ia logique de M. Brouwer

BIBliOGRAPHY 355

Academie Royale de Belgique, Bulletins de Ia classe des sciences, ser. 5, vol.15, pp.183-188.

GODEL, Kurt 1932 Zum intuitionistischen Aussagenkalkfil

Akademie der Wissenschaften in Wien, Math.-natur. Klasse, vol.69, pp. 6~. Translated as 'On the intuitionistic propositional calculus' in Glide/, 1986, pp. 223-225.

1933 A Zur intuitionistischen Arithmetik und Zahlentheorie Ergebnisse eines mathematischen Kolloquiums, vol.4 (1931-32), pp.34-38. Translated as 'On intutionistic arithmetic and number theory' in The Undecidable, ed. M. Davis, Raven Press, New York, 1965, pp. 75-81, and in Godel, 1986, pp. 287-295.

1933 B Eine Interpretation des intuitionistischen Aussagenkalkiils Ergebnisse eines mathematischen Kolloquiums, vol. 4 (1931-32), pp. 39-40. Translated as 'An interpretation of the intuitionistic sentential logic', in The Philosophy of Mathematics, ed. J. Hintik:ka, Oxford Univ. Press, 1969, pp.128-129, and as 'An interpretation ofthe intuitionistic propositional calculus' in Godel, 1986, pp. 301-303.

1933 C Uber Unabhiingigkeitsbeweise in Aussagenkalkiil Ergebnisse eines mathematischen Kolloquiums, vol. 4 (for 1931-32), pp. 9-10. Translated as 'On independence proofs in the propositional calculus' in Godel, 1986, pp. 269-271.

1986 Collected Works, Volume 1 ed. Feferman et al., Oxford Univ. Press.

GOLDBLATT, Rob 1978 Arithmetical necessity, provability and intuitionistic logic

Theoria, vol. 44, pp. 3~ 1979 Topoi, the categorial analysis of logic

North-Holland. GOLDFARB, Warren D.

1979 Logic in the twenties: the nature of the quantifier The Journal of Symbolic Logic, vol.44, pp. 351-369.

GRZEGORCZYK,Anmrej 1967 Some relational systems and the associated topological spaces

Fundamenta mathematicae, vol. 60, pp. 223-231. HAACK, Susan

1974 Deviant Logic Cambridge Univ. Press.

1978 Philosophy of Logics Cambridge Univ. Press.

HANSON, William H. 1980 First-degree entailments and information

Notre Dame J. of Formal Logic, vol. 21, no.4, pp.659-671.

356 BIBLIOGRAPHY

HENKIN, Leon 1954 Boolean representation through propositional calculus

Fundamenta Mathematicae, vol.41, pp. 89-96. HEYTING, Arend

1930 Die formalen Regeln der intuitionistischen Logik Sitzungsberichte der Preussischen (Berlin) Akademie der Wissenschaften, Phys.-Math. Kl, pp.42-56. The quotations in the text are fromBochenski, 1970, pp. 293-294.

HUGHES, G. E. and M. J. CRESSWELL 1968 An Introduction to Modal Logic

Methuen. 2nd printing with corrections, 1971. 1984 A Companion to Modal Logic

Methuen and Co. ISEMINGER, Gary

1986 Relatedness logic and entailment The Journal of Non-classical Logic, vol. 3, no.1, pp. 5-23.

JASKOWSKI, S. 1948 Rachunek zdarl dla system6w dedukcyjnych sprzecznych

Studia Societatis Scientiarum Torunensis, Section A, vol. 1, no. 5, pp.57-77. Translated as 'Propositional calculus for contradictory deductive systems', Studia Logica, XXIV, 1969, pp.143-157.

JOHANSSON, Ingebrigt 1936 Der minimalkalkiil, ein reduzierter intutionistischer Formalismus

Compositio mathematica, vol.4, pp.119-136. The translation in the text is by D. Steiner.

J6NSSON, B. and Alfred TARSKI 1951 Boolean algebras with operators, Part I

Amer. J. Math., vol. 73, pp. 891-939. KALMAR, Laszl6

1935 Uber die Axiomatisierbarkeit des Aussagenkalkiils Acta Scientiarum Mathematicarum, vol. 7, pp. 222-243.

KIELKOPF, Charles F. 1977 Formal Sentential Entailment

University Press of American, Washington D.C. KLEENE, Stephen Cole

1952 Introduction to Metamathematics North-Holland. Sixth reprint with corrections, 1971.

KNEALE, William and Martha 1962 The Development of Logic

Clarendon Press, Oxford. KOLMOGOROFF, A. N.

1925 Sur le principe de tertium non datur Matematiceskij Sbornik, vol. 32, pp. 646-667. Translated as 'On the principle of excluded middle', in van Heijenoort, pp.416-437.

1932 Zur Deutung der intuitionistischen Logik Mathematische Zeit., vol. 35, pp. 58-65

K6RNER, Stephan 1976 Philosophy of Logic

Univ. of California Press. KRAJEWSKI, Stanislaw

1986 Relatedness logic Reports on Mathematical Logic, vol. 20, pp. 7-14.

KRIPKE, Saul A. 1959 A completeness theorem in modal logic

1965 The Journal of Symbolic Logic, vol. 24, pp.l-14. Semantical analysis of intuitionistic logic, I

BIBUOGRAPHY 357

1972

In Formal Systems and Recursive Functions, ed. J. N. Crossley and M.A. E. Dummett, North-Holland, Amsterdam, pp. 92-130. Naming and necessity

1975

In Semantics of Natural Language, ed. D. Davidson and G. Harman, pp. 253-355. Outline of a theory of truth J. of Philosophy, vol. 72, pp.690-716.

LEIVANT, Daniel 1985 Syntactic translations and provably recursive functions

Journal of Symbolic Logic, vol. 50, no. 3, pp.682-688. LEMMON, E. J.

1977 An Introduction to Modal Logic

LEWIS, C. I. 1912

In collaboration with Dana Scott, edited by K. Segerberg, American Philosophical Quarterly, Monograph 11, Basil Blackwell, Oxford.

Implication and the algebra of logic Mind, vol.21 (new series), pp.522-531.

LEWIS, C. I. and C. H. LANGFORD 1932 Symbolic Logic

The Century Company. 2nd edition with corrections, Dover, 1959. LEWIS, David K.

1973 Counterfactuals

LOS,Jerzy 1951

Harvard Univ. Press.

An algebraic proof of completeness for the two-valued propositional calculus Colloquium Mathematicum, vol. 2, pp. 236-240.

LUKASIEWICZ, Jan 1920 0 logice tr6jwarto§ciowej

1922

Ruch Filozoficzny, vol.5, pp.170-171. Translated as 'On three-valued logic' in Lukasiewicz, 1970, pp. 87-88, and inMcCall,l967, pp.16-18. On determinism Translation of the original Polish lecture, in Lukasiewicz,J970, pp.110-128, and in McCall, pp.19-39.

1930 Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalkiils

358 BIBliOGRAPHY

Comptes Rendus des Seances de Ia Societe des Sciences et des Lenres de Varsovie, vol.23, cl.iii, pp.51-77. Translated as 'Philosophical remarks on many-valued systems of propositional logic' in Lukasiewicz,J970, pp.153-178, and in McCal/,1967, pp.40-65.

1952 On the intuitionistic theory of deduction Konikl. Nederl. Akademie van Wetenschappen, Proceedings, Series A, no. 3, pp.202-212. Reprinted in Lukasiewicz, 1970, pp.325-335.

1953 A system of modal logic J. Computing Systems, vol.1, pp.111-149. Reprinted in Lukasiewicz,J970, pp. 352-390.

1970 Selected Works ed. L Borkowski, North-Holland.

LUKASIEWICZ, Jan and Alfred TARSKI 1930 Untersuchungen fiber den Aussagenkalldil

Comptes Rendus des Seances de Ia Societe des Sciences et des Lenres de Varsovie, vol.23, cl.iii, pp. 39-50. Translated as 'Investigations into the sentential calculus' in Lukasiewicz,J970, pp.131-152, and in Tarski,J956, pp. 38-59. References in the text are to the latter.

MARCISZEWSKI, Witold (ed.) 1981 Dictionary of Logic

Martinus Nijhoff. MATES, Benson

1953 Stoic Logic Univ. of California Publications in Philosophy, Vol.26. Reprinted by the Univ. of California Press, 1961.

1986 The Philosophy of Leibniz Oxford Univ. Press.

McCALL, Storrs (ed.) 1967 Polish Logic

Oxford Univ. Press. McCARTY, Charles

1983 Intuitionism: an introduction to a seminar Journal of Philosophical Logic, vol. 12, pp.l05-149.

McKINSEY, J. C. C. 1939 Proof of the independence of the primitive symbols of Heyting' s

calculus of propositions Journal of Symbolic Logic, vol.4, pp.155-158.

McKINSEY, J. C. C. and Alfred T ARSKI 1948 Some theorems about the sentential calculi of Lewis and Heyting

Journal of Symbolic Logic, vol.13, no.1, pp.1-15. MONTEIRO, A.

1967 Construction des algebres de Lukasiewicz trivalentes dans les algebres de Boole monadiques, I Math. Japonicae, vol.12, pp.l-23.

PARRY, William Tuthill 1933 Ein Axiomensystem fiir eine neue Art von Implikation (analytische

Implikation) Ergebnisse eines mathematischen Kolloquiums, vol. 4, pp. 5-6.

BIBliOGRAPHY 359

1971 Comparison of entailment theories Typescript of an address to the Association of Symbolic Logic, an abstract of which appears in Journal of Symbolic Logic, vol. 37 (1972), pp. 441-442.

197? Analytic implication: its history, justification, and varieties Typescript of an address to the International Conference on Relevance Logics.

PERZANOWSKI, Jerzy 1973 The deduction theorem for the modal propositional calculi formalized after the

mannerofLemmon,Partl Reports on Mathematical Logic, vol.1, pp.1-12.

PIGOZZI, Donald. See BLOK and PIGOZZI PORTE, Jean

1982 Fifty years of deduction theorems In Proceedings of the Herbrand Symposium Logic Colloquium '81, ed. J. Stern, North-Holland, pp. 243-250.

POST, Emil L. 1921 Introduction to a general theory of elementary propositions

Amer. Journal of Math., vol.43, pp. 163-185. Reprinted in van Heijenoort, 1967, pp. 264-283

PRIOR, Arthur 1948 Facts, propositions and entailment

Mind, (new series) vol. 57, pp. 62-68 1955 Formal Logic

2nd edition with corrections, 1963, Clarendon Press, Oxford. 1960 The autonomy of ethics

Australasian J. of Phil., vol. 38, pp.199-206. Reprinted in Prior, 1976, pp.88-96.

1964 Conjunction and contonktion revisited Analysis vol. 24, pp.191-195. Reprinted in Prior,1976, pp.159-164.

1967 Many-valued logic In Edwards,1967, vol. 5, pp.1-5.

1976 Papers in Logic and Ethics ed. P. T. Geach and A. J.P. Kenny, Univ. of Massachusetts Press.

PUTNAM, Hilary 1975 Mind, Language and Reality

Cambridge Univ. Press. QUINE, Willard Van Orman

1950 Methods of Logic 4th edition, Harvard Univ. Press.

1970 Philosophy of Logic Prentice-Hall.

RASIOWA, Helena 1974 An Algebraic Approach to Non-classical Logics

North-Holland. RESCHER, Nicholas

1968 Many-valued logic In Topics in Philosophical Logic, N. Rescher, D. Reidel.

1969 Many-valued Logics McGraw-Hill.

360 BIBUOGRAPHY

ROSSER, J. Barkley 1953 Logic for Mathematicians

McGraw-Hill. ROSSER, J. Barkley and Atwell R. TURQUETTE

1952 Many-valued Logics North-Holland.

RUSSELL, Bertrand See WHITEHEAD and RUSSELL

SCOTI, Theodore Kermit 1966 John Buridan: Sophisms on Meaning and Truth

Appleton-Century-Crofts, New York. SEARLE, John R.

1970 Speech Acts Cambridge Univ. Press.

1983 Intentionality Cambridge Univ. Press.

SEGERBERG, Krister 1968 Propositional logics related to Heyting's and Johansson's

Theoria, vol.34, pp.26--61. 1971 An Essay in Classical Modal Logic

Filosofiska Studier, no. 13, Uppsala Univ. SHA W-KWEI, MOH

1954 Logical paradoxes for many-valued logics Journal of Symbolic Logic, vol.19, no.1, pp. 37-40.

SILVER, Charles 1980 A simple strong completeness proof for sentential logic

Notre DameJ. of Formal Logic, XXI, pp.179-181. SLUGA, Hans

1987 Semantic content and cognitive sense In Frege Synthesized, ed. L Haaparanta and J. Hintikka, D. Reidel.

SMILEY, T. J. 1959 Entailment and deducibility

Proc. Aristotelian Soc., vol. 59, pp. 233-254. 1976 Comment on 'Does many-valued logic have any use?' by D. Scott, in

Korner, 1976, pp. 74-88. SMULL Y AN, Raymond

1978 What is the name of this book? Prentice-Hall.

SPECKER, Ernst 1960 Die Logik Nicht Gleichzeitig Entscheidbarer Aussagen

Dialectica, vol. 14, pp. 239-246. Translated as, 'The logic of propositions which are not simultaneously decidable', in The Logico-Algebraic Approach to Quantum Mechanics, vol.1, ed. C. A. Hooker, D. Reidel, 1975, pp.135-140.

SURMA, Stanislaw J. 1973 A (ed.) Studies in the History of Mathematical Logic

Polish Academy of Sciences, Warsaw.

BIBUOGRAPHY 361

1973 B A history of the significant methods of proving Post's theorem about the completeness of the classical propositional calculus In Surma, 1973 A, pp.19-32.

TARSKI, Alfred 1930 Uber einige fundamentale Begriffe der Metamathematik

Comptes Rendus des seances de Ia Societe des Sciences et des Lettres de Varsovie, cl. iii, vol. 23, pp. 22-29. Translated as 'On some fundamental concepts of metamathematics' in Tarski,1956, pp. 30--37.

1936 The concept of truth in formalized languages Reprinted in Tarski, 1956, pp.152-278, where a detailed publication history of it is given.

1956 Logic, Semantics, Metamathematics 2nd edition with corrections, 1983, edited by J. Corcoran, Hackett Publ., Indianapolis.

See also J6NSSON and TARSKI, LUKASIEWICZ and TARSKI, McKINSEY and T ARSKI

TROELSTRA, A. S. and VAN DALEN, Dirk 1988 Constructivism in Mathematics

North-Holland. TURQUETTE, Atwell R.

1959 Review of papers by Rose and Rosser, Meredith, and Chang Journal of Symbolic Logic, vol. 24, pp. 248-249.

See also ROSSER and TURQUETTE VAN FRAASSEN, Bas C.

1967 Meaning relations among predicates Nous, vo1.1, no.2, pp.161-179.

VAN HEIJENOORT, Jean (ed.) 1967 From Frege to Giidel: A source book in mathematical logic 1879-1931

Harvard Univ. Press. W AJSBERG, Mordechaj

1931 Aksjomatyzacja tr6jwartosciowego rachunku zdan Comptes Rendus des seances de Ia Societe des Sciences et des Lettres de Varsovie, cl.iii, vol. 24, pp.126-145. Translated as 'Axiomatization of the three-valued propositional calculus' in McCall, 1967, pp. 264-284.

WALTON, Douglas N. 1979 Relatedness in intensional action chains

Phil. Studies, vol.36, no.2, pp.175-225. 1982 Topical Relevance in Argumentation

John Benjamins, Philadelphia. 1985 Arguer's Position

Greenwood Press, London. WHITEHEAD, Alfred North and Bertrand RUSSELL

1910--13 Principia Mathematica Cambridge Univ. Press.

WILLIAMSON, Colwyn 1968 Propositions and abstract propositions

In Studies in Logical Theory, ed. N. Rescher, American Philosophical Quarterly, Monograph no. 2, Basil Blackwell, Oxford.

362 BIBUOGRAPHY

W6JCICKI, Ryszard Theory of Logical Calculi D. Reidel.

WRIGLEY, Michael 1980 Wittgenstein on inconsistency

Philosophy, vol.55, pp.471-484. Y ABLO, Stephen

1985 Truth and reflection J. Phil. Logic, vol. 14, pp. 297-349.

Glossary of Notation

General

§ iff

• ~.~

indicates a section of this book 'if and only if' end of proof direction of proof

Formal Languages

L(p0, Pp ... l, -7, "· v) a formal language, 17, 29, 108

p0, pi, . . . propositional variables of the formal language p, q, qo, ql' ...

metavariables ranging over {p0, Pi, ... } PV the collection of all propositional variables p0 , pi' ... , 27, 93 A, B, C, A0, AI' ...

metavariables ranging over wffs of the formal language or propositions, 16, 29

Wffs the collection of all wffs of the formal language, 29, 93, 108 r, L, d, ... collections of propositions or wffs, 20,31 L, M logics

=ner

-7-wff C(A)

A(q) PV(A) [A]

the formal language for logic L the class of all formal languages for the general framework, 108 language L with connective a deleted, 312 a formal connective equivalent by definition, used for introducing defined connectives

or abbreviations a wff with principal connective -7 , 30 A is a subformula of C, 31 q is a variable appearing in A, 31 the collection of all propositional variables appearing in A, 31 the Godel number of A, 179

363

364 GLOSSARYofNOTATION

Connectives and Abbreviations in the Formal Language

1

A

v ~

H

:::>

0

<>

_l

I J,

"*' w R(A,B)

D(A,B)

E(A,B)

M(A,B)

formalization of 'not', 8 formalization of 'and', 8 formalization of 'or', 8 formalization of 'if ... then .. .', 'implies', etc., 8 formalization of 'if and only if', defined as (A~ B) A (B ~A), 24 the material conditional, dependent on truth-values only;

defined as 1(AA 1B), 57,151 material equivalence, defined as (A :::> B) A (B :::> A), 162 necessity operator

in classical modallogics, 1A~A or A~(A~A), 154,161 inL3 , 1<>1A, 237 inJ3 , 1(A~(AA 1A)), 270

possibility operator in classical modal logics, 101A or 1(A~1A), 154,155 inL3 , 1A~A.237

inJ3 , -o-A, 267

formalization of 'implies' in classical modal logics; in this text '~' is used, 147

weak negation in J 3 , 266 falsity, a propositional constant, 109, 225 Sheffer stroke, 'nand', 34, 79 formalization of 'neither ... nor ... ', 34 conjunction of all the wffs in the collection, 137,272 disjunction of all the wffs in the collection, 247, 254, 299 relatedness abbreviation, true in a model

ofR or S iff R(A,B), 77-78 of S iff s(A) 11 s(B) '# 0, 73

dependence abbreviation true in a model ofD iff s(A);;;;;! s(B), 125 ofDuaiD iff s(A)!;; s(B),134

equivalence of contents abbreviation for Eq , true in a model iff s(A) = s(B), 136

abbreviations for A~(B~B)

abbreviation in certain classical modal logics for (A vB)~B. true in model iff s(A)!;; s(B), 159

1 repeated n times, 202 abbreviation for A~(A~B) for Deduction Theorem for L 3 , 238 indeterminacy abbreviation for L 3 , AH1A, 236 abbreviation for J 3 indicating that the wff has a classical (absolute)

truth-value, 1(<>A A <>-A), 269 wff used to determine whether a logic has n-valued semantics, 254

Semantic Terms

T F u M M* v

<v,s> s S,U, T,C SubS 0 s(A) s(A) B,C,A,N <v,R> <v,D> <W,R,e,w>

w

R e

w <W,R> <W,R,Q,e> Q

true, 13 false, 13 unknown,undefined,250 a model, 31,92-93,109-110

GLOSSARYofNOTATION 365

the model correlated to M by translation * , 304-305 a valuation, 92, 93, 109, 110 a model for classical logic (PC), 18, 31 a set-assignment model, 90-92, 109 a set-assignment, 109 content sets the collection of subsets of S, 93 the empty set the content set assigned to A the complement of s(A) relations governing the tables for ~, A, v, 1 respectively, 90-92 a model for S, 72; a model for R, 74 a model for D, 122 a Kripke possible-worlds model for a classical modal logic, 151 a Kripke model for the intuitionistic logic Int, or a Kripke tree, 200 a collection of possible worlds, 151 a collection of states of information, 201 an accessibility relation, 151 an evaluation for: a classical modal logic, 151-152

the intuitionistic logic Int, 200-201 Johansson's minimal calculus J, 225 a many-valued logic, 233

a possible world, 151 a frame, 152, 200 a model for Johansson's minimal calculus J, 225 a collection of inconsistent states of information, 225

Syntactic Consequence Relations

1-A A is a theorem, 41 L 1-A A is a syntactic consequence of L , 41 A 1-B B is a syntactic consequence of {A} L, A 1- B abbreviation for L u {A} 1-B Th(L) the theory of L (the collection of all syntactic consequences of L) 42 1-L 0 a modal logic consequence relation using the rule of necessitation, 192 1-J3 ·¢ the consequence relation for J 3 where ¢ is primitive, 276 1-J3 •1 the consequence relation for J 3 where 1 is primitive, 279

366 GLOSSARY of NOTATION

Semantic Consequence Relations

I= A A is valid, A is a tautology, 20, 96 r I= B B is a semantic consequence of r (every model which validates

AI=B vi= A <v,s>I=A <v,B>I=A MI=A wi=A

<W,R,e>I=A <W,R> !=U>, I=* M

SR

RB

C(SR) C(RB)

Translations

L~M

L-»M A* r* M*

every wff in r also validates B), 20, 96 every wff in ll is a semantic consequence of r, 20 B is a semantic consequence of A, 19, 31, 96 A is true in the classical model v , i.e., v(A) = T A is true in model <v, s > , i.e., v(A) = T A is true in model <v, B > , i.e., v(A) = T A is true in model M, 109 A is true in (at) world w, 151 the state of information w verifies A, 200-201 A is true in model <W,R,e>, 151,200 frame <W,R> validates A, 152 special definitions of validity for G*, 182-183 a formal semantic structure, 109 a class of models, 304 the class of all formal set-assignment semantic structures for all formal

languages in L, 109 the class of all formal relation based semantic structures for all formal

languages in L, 110 the collection of all consequence relations for SR, 110 the collection of all consequence relations for RB, 110

there is a translation from logic L to logic M, 290 there is a grammatical translation from logic L to logic M, 291 the translation by * of A the collection of translations of all wffs in r the model correlated to M by translation *, 304-305

Index

Italic page numbers indicate a definition, theorem, or quotation.

All page numbers greater than 323 refer to the Summary of Logics.

absolute truth (-value), 266,269 abstract proposition, 5-7

and logical form, 22, 25 abstracting, xxi, 6, 27, 107, 315,320 Abstraction

Classical, 12 Oassical, Fully General, 27 Classical Modal,149 Dependence,119, 121 Dependence, Fully General,122 DPC, Fully General, 142 Eq, Fully General, 136 Finitistic Fully General, 94 Fully General, 93-94, 95, 98 Int, Fully General, 202 Kripke Semantics, Fully General,152 Possible Worlds,148 Relatedness, 64,10 Relatedness, Fully General, 73 Set-Assignment Semantics for Qassical

Modal Logics, Fully General, 157

accessibility relation, 150,151,332

actions, 133, 321

adjunction, rule of, 40, 126, 199, 328 agreement, xxi, 2-3, 7, 87, 94, 315-316 algebraic semantics, xxii, 103, 105, 130, 168

alternation. See disjunction

ambiguity, 2-4, 6, 9, 22, 60,264,315

analytic implication, 118--119

analytic tableaux, 80, 116, 140, 244

analyticity, 115-116. See also necessity

'and', 8, 87, 105. See also conjunction;

connectives, English

Anderson, Alan R., 141

antecedent, 9 anti-reflexive relation, 180 anti-symmetric relation, 150,171 anti-tautology,19

universal, 113-114 appearance of a variable, 31 argument

enthymematic, 301 valid,20

Aristotle, 39,234-235,237,248 Arruda, Ayda 1., 265 aspect of a proposition, xx-xxi, 86, 95, 106, 146,

315,319 assertibility (assertion), 197,219-220,224,228 Associativity of Conjunction, 36 Associativity of Disjunction, 36 atomic proposition,13,17, 72, 89,273

choice of, 22 Aune, Bruce, 146, 149 axiom,41 axiom system, 41 axiomatization

complete, 44 independent, 55,258--261 sound,44 strongly complete, 44 See also name of logic, axiomatization

axiomatizing, description of, 45

B (modallogic),J74-175, 184-193,254,295, 331-335

-set-assignment, 174, 334 bachelor, 118--119

background, xxi, 315-321

basal intuition, 196

367

368 INDEX

'because', 23 Belnap, Nuel D., 141 Bennett, Jonathan, 59, 74, 83 Bernays, Paul, 52 Beth trees, 221 Bew,179 biconditional, 24 binding of connectives, order of, 16 bivalence, xx, 3, 65, 86-87,231-232,234,

237,318,320 in intuitionism, 219-222 See also Excluded Middle; trivalence

Blok, W. J., xxii, 101, 103, 177 Bochenski, I. M., 58, 197, 198 Boolean algebra, 222, 313 Boolean operation of set theory, 98-99 Boolos, George, 146, 171,178-184,332 Brouwer, L. E. J., 196,191 Buridan, Jean, 69 'but', 23

calculus, a, 19 canonical model

forint, 207 for a modal logic, 186-187, 191

Cantor, Georg,196 Carnielli, Walter, 28, 30, 35, 40, 80, 108,

116,135,140,233,244,254 certainty, 231. See also ambiguity characteristic matrix, 233 Chellas, Brian, 146, 164, 184 chess, 316 chimpanzees, 266 Christensen, Niels Egmont, 130, 131 Classical Abstraction,12, 14 classical connective, 15,109 classical logic. See PC Classical Modal Logic Abstraction, 149 classical necessity. See necessity, classical classical way of thinking, 318

Classically Dependent Logic. See DPC Cleave, J.P., 252 closure under

R,200 a rule, 42 subformulas, 189, 208

closure, transitive, 153

communication, 315,317,318 Commutativity of Conjunction, 36

Commutativity of Disjunction, 36

compact decidability. See decidability, compact compactness

and nonconstructive reasoning, 44 of semantic consequence, 50,101 of syntactic consequence, 43, 101,291-292

compatible proposition, 156 complete collection of wffs, 46, 273

for classical modal logics, 186 classically (PC), 46, 107, 148, 185,208,274,

280 forD,126 relative to J 3, 274, 276, 277 relative to L3, 241

complete and consistent theory, 43, 274 for classical logic, 46, 107, 148, 208, 274,

280 for a classical modallogic,191 as description of the world, 43, 273, 275 is an endpoint of a Kripke model for Int, 208 is a possible world, 148, 185 relative to J 3, 274 relative to L 3 , 241

complete frame, 188 complete theory, 42 completeness proofs

general description of, 45, 164 nonconstructive vs. constructive, 44-45,

49-50,52,129,239 completeness theorem, 44

finite strong, 45, 53,239 strong, 44, 101 See also name of logic, (Strong)

Completeness Theorem complex proposition. See compound proposition

compossible propositions,156 compound proposition,J7, 89, 318 conceptual (contextual) framework of a proposition, ll7-ll9, 135 conditional9, 87, 90, 96, 108, 116-118, 147

classical14,15, 58-60 content of, ll9-120 dependence style, 132

dual dependence-style,J33, 136 and equality of contents, 135-136

general table for, 90 in J 3, 267-268 reasonableness of classical, 58-60 subject matter related, 71 subjunctive,59,92, 112,157 transitivity of, 37, 76-77, 124 truth-default, 286 weak table for, 92-93, 157 See also connection of meanings;

'if ... then .. .'; implication; material implication

congruence relation, 103, 105,282 conjunct,9 conjunction,9, 18,91,105,108

classical, 13,15,109 classical laws of, 36 definable in D, 125 definable in Dual D, 134-135 definable in Eq?, 140 definable in S, 78 general table for, 91 parentheses and, 22 truth-default table, 286 See also name of logic, truth­

conditions (truth-tables) conjunctive normal form, 38. See also

normal form of a proposition connection of meaning, 59, 90, 95, 99,318 connective(s), 8-9, 87-88, 108

binary, 9 classical, 15, 109 content of, 119-120 conventions on formalizing English,

23-24 definable, 102,269 and definition of 'proposition', 86 deviant, 105-106 English, 8, 15,23-24, 87, 101, 125 finite number of only, 108 in general framework, 87-88, 108-114,

318,320 impredicative explanation of, 223 independence of in Int, 213, 261 intensional,JJ2, 273,283 main,30 order of binding, 16

principal, 30

INDEX 369

Quine on, 105-106 strong (K3), 251,344 syncategorematic, 66, 118 translation of, 302, 306-307 truth-and-content functional, 90, 102 truth-functional, 32,109 truth-and-relatedness functional, 78 truth-and-relation functional, 98,102 unary, 9 uniformly interpreted,110-111 See also definability of connectives;

functional completeness of connectives; propositional constant

consequence relation, semantic,19-20, 31, 96, 97, 101,109, 232

compact, 44, 50,101,292 for classical modal logics, 163,191-194 determines content in DPC, 142-143 formal semantic structure, 109 K 3 is only, 252 many-valued logic, alternate, 234 many-valued logic, meaning of, 232 properties, list of, 31 reduced to tautologies, 101 for S, alternate, 83-84 and translations, 290-292 See also name of logic

consequence relation, syntactic, 41, 95-91, 101 compact, 43-44, 101, 291-292 properties, list of, 43 for S, alternate, 83-84 and translations, 290-292 See also proof

consequent of a conditional, 9 contained in the antecedent, 116-118

consistency, 42, 47, 107, 273 classical (PC), 46, 107,273-274,280 forD,127 forInt, 205-206 relative to J 3, 274, 276, 277 relative to L 3 , 241 and paraconsistency, 273-275 A- ,128,186 See also complete and consistent theory;

contradiction; information, inconsistent constant, propositional, 108

constituent, immediate, 30

370 INDEX

content, 88-89,95,99-100, 215, 224,232,

318 as consequences in PC,J42-143 constructive, 222 and dependence-style semantics, 131-133 and form, 89-90 mathematical, 220, 222 of a paradoxical sentence, 246 referential, 100, 115-120, 135

of a tautology, 66, 84, 120 vs. truth, 118-120 See also set-assignment

contextual (conceptual) framework of a proposition, 117-119, 135

contradiction, 19, 42,273 assertion of, 224, 228 classical (PC), 42, 273 in intuitionism, 198, 224, 225 inJ3 ,274 and paraconsistency, 263-265,273-275 See also consistency

contrapositive, 37 convention, 315 converse, 24 Copeland, B. J., 68, 155 counterfactual, 24, 59.

See also conditional, subjunctive Cresswell, M. J., 59, 130, 146, 159-160,

164, 165, 174, 191 Cut Rule, 31, 43

D (Dependence Logic), §V.A,J22, 325, 328

Abstraction, 119 algebraic semantics for, 103, 130

axiomatization,J26, 328 complete and consistent theory,

126-127 completeness proof, 127-129

conjunction is definable in, 125

consequence relation,J22

decidability, 123-124 Deduction Theorem, 129-130

in discussion of general framework, 89

disjunction in, 121, 124 and Dual D, 133-135, 140, 141, 293,

307-308,312

and Eq, 136, 140, 141, 293,307 finite-valued semantics, has no, 254-255 functional completeness of connectives, 125 and J (motivation), 228 and modal logics, 130 -mode1,120-122 normal form theorem, 125, 129 and PC, 123-125, 141 relation governing table for --+,122-123,

138,328 same logic as Dual D?, 312 set-assignments for, 120-121 simple presentation, 98, 100 Strong Completeness Theorem, 129 tautology, 122, 141 translations, 130,133, 136, 292-293,295,

298,301,307,312 truth-conditions, 120 variable sharing criteria for tautologies, 141 See also dependence-style semantics

da Costa, Newton C. A., 83, 263, 264, 265, 266, 268,271

decidability, 35, 59,103-104 and arithmetical predicates, 250-251 classical modal logics, 165, 189-191 compact, 103-104, 123-124, 134, 136

D, 123-124 DualD,134 Eq,136 Int, 209 many-valued logics, 234 PC, 35-36, 301 R, S, 75,301

deducibility, 41, 59, 82, 131, 159-160, 163, 178. See also consequence, syntactic

deduction, valid, 20. See also consequence,

semantic Deduction Theorem, Semantic, 96,100-1 OJ, 103

for classical modal logics, 192-194 forEq, 140 for finite consequences, 1 OJ for J3 , 269 forL 3 , 238 material implication form, 82, 140, 192

forPC,32 for R, S, 82-84

The,JOO

Deduction Theorem, Syntactic,101 forD, 128-130 for finite consequences,101 for lnt, 204-205 for J, 225 for J 3 , 276, 280 forL 3 ,239,241 material implication form, 82, 130, 140 forPC,47 The,101

Deduction Theorems and translations, 291,302

default truth-value, 285-286 definability of connectives, 102

in lnt, 213, 261 in J3 , 269-271 inS, 77-78 strong, 312 See also functional completeness of

connectives; name of logic, functional completeness

degree of truth, falsity, 266 De Morgan, Augustus, 230 De Morgan's Laws, 37 Dependence Logic. See D dependence relation,122-123, 138,328 dependence-style semantics, 132 dependence truth-conditions, 132 dependent implication,120, 142, 222 Dependent PC. See DPC dependent proposition, 118 derivability, 40. See also proof derivation. See proof derived rule, 42, 43, 55

in PC, 50-51 See also name of rule, e.g.,

substitution, rule of designated element. See designated

truth-value designated subset, 98, 173-115 designated truth-value, 232-233, 246, 266

designated world, 151, 332 De Swart, H., 221

detachment, principle of, 37, 125. See also material detachment;

modus ponens deviant arithmetic, 317-318

deviant logic, 105-107 dialectics, 264 Diodorus, 147, 156 disjunct, 9 disjunction, 9, 108

classical,J3-15, 109 in classical modal logics, 162 exclusive vs. inclusive, 13,24 general table for, 91 intuitionistic proof of, 197, 203 in J3 , 237

INDEX 371

in relatedness logic, 65,71-72,76, 80 strongly dependent,l21, 124 truth-default table for, 286 weakly dependent,121, 124

disjunctive normal form, 38, 52 Disjunctive Syllogism, 37, 76, 124 distinguished truth-value (element).

See designated truth-value Distribution Laws for A, v, 37 distribution schema (axiom),163-164 Dostoevsky, Fyodor, 62 D'Ottaviano, ltala M. L., 263, 265-266, 269,

271 double induction, 67 Double Negation, law of, 36, 197,202-203,

257 double negation translation of PC

into lnt, 211-212,294, 310 into J,226

doubt factored into a logic, 231 DPC (Classically Dependent Logic,

Dependent PC), 89,142-143 DPCn,l43 Dreben, Burton, 51,52 dual Boolean equation, 132 Dual D (Dual Dependence Logic),J33-135,

325,329 axiomatization,134, 329 conjunction is definable in, 134-135 andD, 133-135,140,141,293,307-308,312 dual to D, 133-134 and Eq, 140, 141, 293, 307 and PC, 141 -relation, 329 same logic as D?, 312 Strong Completeness Theorem, 134

372 INDEX

Dual D (continued) translations, 133, 292-293,295, 307,

312 truth-conditions, 132-133, 157 variable sharing criteria for tautologies,

141 Dual Dependence Logic. See Dual D dual dependence-style semantics, 135 dual of a wff, 38 duality, principle of, 38 dualization of semantics, 135 Dummett, Michael, 195, 197, 198, 200,

201,209-210,211,215,218-222, 220,221,228,234,246,254,257, 336

Dunn, J. Michael, 131

Eggenberger, Peter, 316 elementary equivalent models, 304 Eliot, T. S., 313 endpoint, 208 English,5,88,93 English, formalized.

See language, semi-formal English connectives, 8, 15,23-24, 87, 101,

125 entailment, 20, 59, 77, 96, 117-118,

130-131, 160, 163 first degree, 141 See also implication

enthymematic argument, 301 Epstein, Richard L., 4, 28, 30, 35, 40, 74,

80,103,112,123,130,251,285 Eq (a logic of equality of contents), §V.D,

325,329-330 axiomatization, 139-140, 330

conditional in, 135-136 conjunction definable in?, 140 decidability, 136

disjunction in, 136 and D, Dual D, 136, 140, 141,293,307

in discussion of general framework, 90, 98 finite-valued semantics, has no,

254-255 Fully General Abstraction, 136 -model, 135-136 and PC, 140, 141

-relation, 136-139 Strong Completeness Theorem, 140 translations, 136, 293, 295, 298, 307 variable sharing criteria for tautologies, 141

equiform word (sentence, proposition), 3-4 equivalence frame, 169 equivalence, semantic, 20, 102,269-276.

See also extensionally equivalent propositions; proposition, same

euclidean relation, 185 evaluation. See name of logic (e.g., L3 -), or

type (e.g., modal) evaluation eventually true wff, 182-183 exact vs. inexact concepts, 265 Excluded Fourth, 236 Excluded Middle (Excluded Third,

tertium non datur), 36 added to Int yields PC, 215 Brouwer's analysis of, 197 fails in Int, 202 fails in L 3 , 236 inJ3 ,272 Lukasiewicz on, 234,237 a weak form of, 239 See also bivalence; trivalence

experience, 316-318. See also perception Exportation, 37, 76, 84, 125, 140, 141 Extensionality Consideration

and classical logic (PC), 15 in the general framework, 89, 90, 98, 105 and relatedness logic, 70

extensionally equivalent propositions, 104-105, 130, 282. See also equivalence, semantic; proposition, same

external dualization, 135

fact(s), 87, 117,201 falsity, 2-4, 65, 86-87

degree of, 266 in a model, 31 vs. not-true, 230 See also truth-value

falsity-default semantics, 112-114, 286 falsity-weighted table, 287

Fermat's Last Theorem, 63, 116 Fine, Kit, 120, 131, 141 finite consequence translation, 292, 294, 309

Finite Consequences, Deduction Theorem for, 32, 47, 101

finite model property,190 finite strong completeness, 45, 53, 239 finite-valued semantics, 233

logics having none, 254-255 Finitistic Fully General Abstraction, 94 first degree entailment,141 Fitting, M. C., 195, 200-201, 225 'follows from', 44, 49, 82, 87,96 form vs. content, 7-8 formal language. See language, formal formal system, 1 formal semantic structure.

See semantic structure, formal formalizations are false, 320 formalized English.

See language, semi-formal frame,152, 200,332,336

complete, 188 equivalence, 169 finite,152 sound,185 See also name of property

frame of reference, 119 framework, conceptual, 117-119, 135 Frege, Gottlob, 7, 13, 39, 51, 69, 135 Fregean Assumption, 13, 14, 67, 70, 89,

90,98 for contents, 99-100, 119 for relations, 99-100, 123

full collection of wffs, 205 Fully General Abstraction

Classical, 27, 28 Dependence,122 DPC, 142 Eq, 136 Finitistic, 94 forint, 202 in general framework, 93-94,95,98 Kripke semantics, 152 Relatedness, 73 Set-Assignment Semantics for Classical

Modal Logics, 157 functional completeness of connectives,

32,101,312 forD, 125

forEq, 140 forint, 222 general problem of, 101-102 forJ3. 102 for L 3 , 102, 236 for classical modal logics, 159 forPC,32 for R, S, 77-79

INDEX 373

future, proposition about the, 230, 232, 234-235

G (modallogic),179-182, 254,295,331-334 and arithmetic, 179

G* (modal logic), 161,179,182-184, 254,295, 331-334

and arithmetic, 180, 211 G-evaluation (many-valued), 256, 345-346 G 0 , G3, Gl!t, Gl!t 0 (GOdel's many-valued

logics), 256-258, 345-346 Gentzen, Gerhard, 213 Gentzen's translation of PC into Int, 213-214,

294,310 Glivenko, V., 211 Godel,Kurt,210-211,214,231,254,256,260 Goldblatt, Rob, xxii, 61, 74, 180, 211 Goldfarb, Warren D., 52 grammatical map, 291 grammatical translation, 291

with parameters, 302 why preferable, 302

Grzegorczyk, Andrzej, 171

Haack, Susan, xxii, 106-107, 155, 230,234 Hanson, William H., 116, 131, 142 Henkin, Leon, 50, 52 Heyting, Arend, 195,198, 199, 210, 224,257,

335,338 Hilbert, David, 39 holism, 63-64, 118 homophonic translation, 213, 291 Hughes, George, 59, 130, 146,159-160, 164,

165,174,191,259 hypothetical reasoning, xix, 231. See also

agreement

!(class of models forInt), 218, 308, 338

374 INDEX

Identity,law of, 37 'if and only if', formalization of, 24 'if ... then .. .', xix-xx, 8, 14, 58-60,

74-75,83,87,96,112 formalization in PC not intuitive, 58-59 See also conditional; connectives,

English; implication iff,24 immediate constituent, 30 implication, 9, 20, 87, 96, 146-147, 224

analytic, 118-119 and content as consequences in PC,

142-143 dependent,120, 142,222 first degree, 141 material, 57, 59,74-75, 82, 124, 147,

152-153 modal semantics for (of),156-157,

158,176-177,333-334 paradoxes of, 37,59-60,74-75, 124,

152-153 strict, 37, 60,74-75,147, 152-153 vs. validity of an 'if . . . then .. .'

proposition, 20 weak, 92-93, 157 weak modal semantics for (of),

157-158, 159, 175-176, 179, 181, 333-334

See also conditional; entailment; 'if ... then .. .'; transitivity of~; use-mention confusion

Importation, 37, 76, 84, 125, 140, 141 impredicative explanation of connectives, 223 inconsistency, 42. See also consistency; contradiction inconsistent information, 224, 225, 228 independent axiom (schema), 55, 126,

139,230,258-261 independent connectives in Int, 213,261 indeterminate proposition, 236 induction, 28

on length of a wff, 30, 67 double, 67

inevitability at time t, 150 infinite-valued semantics, 233,248

any logic has, 234 for SS, 255-256

infinite totalities, 27-28,73, 196-197,200, 317 infinitistic assumptions, 27-28,44-45,73, 101,

196-197, 337. See also completeness proofs, nonconstructive vs. constructive

initial point of a Kripke tree, 200 information, 116-117, 133,201,215,220,222

inconsistent, 224, 225, 228 Int (intuitionistic logic) Chapter VIII, 335-338

axiomatization, 199, 209, 227, 335-336 basis of axiomatization of PC, 203, 215, 325 canonical model, 207 Completeness Theorem, Kripke-style

semantics, 201,202,209,309-310 Completeness Theorem, set-assignment

semantics, 217-218,223-224 Completeness Theorem, why Finite Strong,

202 connectives and classical negation, 214,222 connectives, explanation of, 198,223 connectives are independent, 213, 261 contained in PC, 211 correspondence between Kripke and set-

assignment models, 218-219, 308-309 decidability, 209 Deduction Theorem, 204-205 Double Negation fails in, 202-203 evaluation, 200, 336 Excluded Middle fails in, 202-203 finite-valued semantics, has no, 254 full theory, 205 Fully General Abstraction, 202 andGn,GN,257 inductive definition of truth in, 223 Kripke semantics, 200-210, 215, 336-337 -model, 216 negation, 216,337 other semantics, 221, 310 and PC, 210-215,222-223, 309 and provability translation into P A, 211 and S4, 210,219, 293-294, 303, 308,

313 and S4Grz, 210-211,294, 308 set-assignment semantics, 216-224,

337-338 translation into PC?, 301 translations into S4, 210, 211,219,

293-294,303,313

translations of PC into Int, 211-214, 222,291,294,303,309-311

See also intuitionism; intuitionistic logic; intuitionist notion of proof; intuitionist view of; J

Int K, 218,338 lnt 1-10, 216,338 intensional connective, 112, 273, 283 intentionality, 315 Interchange of Premisses, 37, 125 internal dualization,135 interpretation, 88 intersection set-assignments, 133 intuitionism, Chapter VIII, 195-199,313,

318 bivalence in, 219-222 rejection of Excluded Middle, 197,

202-203 See also Int; J

intuitionist minimal propositional calculus. See J

intuitionist negation, 198,216,337 intuitionist notion of proof, 198 intuitionist truth-conditions (truth-tables),

216,218-219,223-224,310,337, 341,342,343

minimal, 224, 227, 339 used for L 3 , 244 used for LN, 249-250 used for GN• 257

intuitionist view of language, 196,198,220-221 negation, 198 role of logic, 196-199

intuitionistic logic, xx, 44, 56, 59, 83, 89, 90,99,198

problem theoretical explanation, 225 See also Int; J

irrational numbers, 197 Iseminger, Gary, 61 iterated modalities, 155, 170

J (intuitionist minimal propositional calculus), 224-228, 338-339

axiomatization, 235,338 basis of axiomatization of Int,

226-227

INDEX 375

Completeness Theorem, 226, 228 Deduction Theorem, 225 -evaluation, 225,338-339 finite-valued semantics, has no, 254 and Gn• GN• 257 -model,227 negationin,224,225,227,339 a paraconsistent logic, 265 translations, 226-227,294,295, 310

JaSkowski, S., 264-265 J 3 (a paraconsistent logic), Chapter IX,

347-349 axiomatization, 275, 276, 279,347-349 complete theory, 274, 276 consistent theory, 274, 276 Deduction Theorem, 269,276,280 definability of connectives, 269-271,

347-348 -evaluation, 269, 271, 347 as extension of classical logic, 272-274,

283 and L 3 , 266-267,271 -model, 282,284 negation in, 113,263-268,271-275,285 and PC, 271-273 presentation, choice of, 271-273 Strong Completeness Theorem for many­

valued semantics, 271, 279, 282 Strong Completeness Theorem for set-

assignment semantics, 284, 285 tautology, 269, 271-272 translations, 271, 272-273, 295 truth-tables, 266-269 validity, 269

Johansson, lngebrigt, 224-225, 228, 265 J6nsson, Bjarni, 167

K (modallogic),l75-l77, 179, 184-193,254, 295,331-335

K-set-assignment, 175, 334 K3 (Kleene's 3-valued logic), 250-253,

344-345 Kalmar, Uszl6, 52 Kielkopf, Charles, 131, 141 Kleene, Stephen Cole, 55-56,214,231,

250-251 Kneale, William and Martha, 40, 55

376 INDEX

knowledge and intuitionism, 197,200-201, 215,

220 logic brings us, 77 projective, xx, 199,313-314

Kohler, P., 177 Kolmogoroff, A. N., 224,225,228 Krajewski, Stanislaw, xxii, 64, 289 Kripke, Saul A., 146, 148, 199,231, 251 Kripke tree, 200, 337 Kripke-style semantics

for classical modal logics, 103, 151, Appendix to Chapter VI, 332-333

converted to many-valued semantics for S5,255-256

correspondence with set-assignment semantics, 103, 164, 168,219, 308-309

forlnt,200-210,215,308-309, 336-337

forJ,225-226,338-339 for other logics, 103, 130

L-evaluation, 246-247 L3 (Lukasiewicz's 3-valued logic),

§VIII.C.l, 339-342 axiomatization, 239,240-241,

340-341 complete and consistent theory,

241-242 Completeness Theorem, 239 Deduction Theorem, 238,239,241 deviant?, 107 -evaluation, 235, 339-340 functional completeness of connectives,

236 and J 3 , 266-267, 271 -model, 244,341-342 andPC,236 rich model, 245-246, 342 set-assignment semantics, 244-246 Strong Completeness Theorem for

many-valued semantics, 243 Strong Completeness Theorem for set­

assignment semantics, 245-246 truth-tables, 235-238 translations, 238, 271,294, 295, 304

L8 , LN, LN0 (Lukasiewicz's n-valued, infinite­valued logics), 246-250, 342-343

Langford, C. H., 87 language

constrains view of world, 219 formal, 16, 29, 64-65, 87-88, 106, 108 formalized English.

See language, semi-formal formalizing, 8-9 intuitionists and, 196, 198,220--221,318 meta-,42 object, 16 semi-formal, 17, 40, 72, 88, 92, 142-143.

See also type I vs. type II model Western, 318 See also English

Leibniz, Gottfried Wilhelm, 116, 148 Leivant, Daniel, 226, 227 Lemmon, E. J., 146, 168 length of a wff, 30 Lewis, C. I. 59, 87, 107, 130, 146 Lewis, David K., 68, 146-147, 155, 164

liar paradox, 4, 231 strengthened, 236

Lindenbaum, Adolf, 52 logic(s), 1

brings us knowledge, 77 compatible with classical logic, 107 criteria for good, 58 deviant, 105-107 differences between, 319 founded in natural language and reasoning, xxi general notion of, xxi, 44,94-97, 231 how to give semantics for, 94-95 how to understand, 315

intuitionist view of role of, 196, 198, 199 logicist conception of, 26 presented semantically, 44 presented syntactically, 44 propositional, 8 relation of mathematics to, 26-28, 93-94

relative to the logician?, xxi same, 35, 98, 271,292,312-313 as a spectrum, xx

unity of, 318 why so many?, xix

See also metalogic

'logical', xix, 319 logical form of a proposition, 21-25 logical necessity,149,170, 237,255. See

also necessity logicist conception of logic, 26 l..os, Jerzy, 52 Lukasiewicz, Jan, 54, 55,231,230,234-

237,248,251,265-266,325,344

ML (a modal logic), 177-178, 295, 331-332

MSI (modal semantics of implication), 177-178,254,295,333-334

Ml-M15 (complete list of uses), 333-335 Maddux, Roger D., 116, 123, 142, 143,

145, 146, 165 madness, 315-316 main connective, 30 many-valued logic, 230-235

evaluation, 233 falsity-weighted table, 287 finite, 233 general problem of translating into set­

assignment semantics, 103 infinite, 233 and possibility and necessity, 146, 230.

See also necessity operator; possibility operator

standard table, 287 truth-weighted table, 287 See also name of logic

Marciszewski, Witold, xxii Marconi, Diego, 265 material implication, 57, 59, 82, 124,147

paradoxes of, 37, 74-75, 152-153 material implication form of the Deduction Theorem, 82, 130, 140, 192 Mates, Benson, 39, 58, 87, 116, 147, 148,

318 mathematics, role in logic of, 26-28,

93-94. See also logic, intuitionist view of role of

matrix, 233 McCarty, Charles, 221 McKinsey, J. C. C., 213, 261 meaning, 3, 7, 199,318

and abstract propositions, 6

INDEX 377

connection of, 59, 90, 95, 99, 159-161,318 is use, 221 same, 6, 25. See also extensionally

equivalent propositions; semantic equivalence

theory of, 221-222,228 translation preserves?, 289, 302-303, 306,

313-314 metalanguage, 42 metalogic, 82, 99, 125 metavariable, 16, 17, 20 Meyer, R. K., 131 minimal intuitionist negation, 223-224,227,

339 minimal intuitionist propositional calculus.

See J minimal intuitionist truth-conditions, 227, 339 modal algebra, 168 modallogic(s), classical, Chapter VI, 331-335

canonical model,186-187 Completeness Theorems for Kripke

semantics, 184-193 consequence relations for, 82-83, 96, 163,

191-194 and connection of meanings, 59-60, 90,

159-161 content of a proposition in, 100 Deduction Theorems, 192-194 evaluation,151-152,332 Kripke-style semantics, xx, 103, 151,

Appendix to Chapter VI , 332-333 and many-valued semantics, 255 andmodusponens,l56,158,162,177 normal,163 not deviant, 107 primitives of, 147, 155, 161 quasi-normal, 163,177 set-assignment semantics, general form,

155-161 translation into itself,178, 293

See also J 3; L3; name of logic; paradoxes of implication; use-mention confusion

modal operators. See necessity operator; possibility operator. See also connectives, finite number of only; modality; necessity; possibility; use­mention confusion

378 INDEX

modal semantics for (of) implication, 156-157,158,116-111,333-334

list of conditions for (M1-M15), 333-334

weak,157-158, 159,175-176,179, 181,333-334

See also ML, MSI modality, 146

iterated, 155, 170 model{s), 17, 31, 88, §B-D and Appendices

B, C, E of Chapter IV (108-110,112) of a collection of wffs, 18 elementarily equivalent, 304 formal relation based,109-110 formal set-assignment, 109 with intensional connectives, 112 type 1,17, 72, 92, 94, 121 type n. 26, 21, 12, 93, 97, 122 type 1 vs. type n, 26-28,72-73,93,

142-143 See also canonical model; name of

logic-model; truth model preserving map, 305-306

content variant, 305-306 model preserving up to elementary

equivalence map, 304-306 modus ponens, 40, 92, 324

in classical modal logic, 156, 158, 162, 177

See also material detachment modus to/lens, 37, 55, 125 Mob Shah-Kwei, 231,236 Monteiro, A., 238 moon, 316 'must', 319-320

n-valued semantics, 233 'nand', 34 native, the, 105-106 necessary truth, xxi, 320 necessitation, rule of, 163, 192-194 necessity, xix, 149-151, 164, 234, 320-321

classical, 149 logical, 149,170, 237, 255 physical, 150-151 time dependent, 149-150 See also agreement; 'must'; possibility

necessity operator classical,154-155,161 inL3 ,237 in J3 , 267 as provability, 178-180,210 set-assignment evaluation of,157-158 See also possibility operator; use-mention

confusion negation, 9, 89-91,108

classical definable in lnt?, 214,222,310 content of, 119-120 inD,l20 general table for, 91, 285 as intensional connective, 273 intuitionist, 198,216,337 in J, 225, 227 minimal intuitionist, 223-224, 227, 339 and paraconsistency, 263-268, 271-275, 285 in relatedness logic, 70-71 set-assignments for in lnt, 223 strong,268 truth-default, 285-287 weak,266-267,271-272,274-275 See also English connectives; name of

logic, truth-conditions (truth-tables); 'not'

'neither ... nor .. .', 23, 24, 34 neuter proposition, 235, 237 nonconstructive reasoning. See infinite

totalities; infinitistic assumptions nonconstructive vs. constructive completeness

proofi,44-45,49-50,52, 129,239 Noncontradiction, law of, 36 Nonsymmetric Relatedness Logic. See R normal form of a proposition

in classical logic, 37-38 in completeness proofs, 52, 81,239 conjunctive, 38 forD, 125, 129 disjunctive, 38, 52 forS, 79,81

normal modal logic, 163 quasi-,163,177

'not', 8, 87, 90, 112, 275, 285-287. See also English connective; negation

object language, 16

objectivity, xix, 1, 3, 6, 7, 315-316,318-321 obscene proposition, 232 obvious logical truth, 106 Ockham, 235 'or', 8, 87, 147

inclusive vs. exclusive, 13, 24 in relatedness logic, 71-72 See also disjunction; English

connectives order

strict partial, 171 weak partial, 180

ordering of connectives, binding,J6 ordering of wffs, 30

P (condition on modal semantics of implication),J68, 334

PA (Peano Arithmetic), 179-180, 211, 317-318

paraconsistent logic, 112, 113, 263-265, 275. See also J 3

paradox, liar, 4, 231, 236 paradoxes of implication, 37, 59, 60,

74-75, 124, 152-153 parameter (in translations), 291, 302 parentheses, 8, 9, 29

conventions on deleting, 16 Parry, William Tuthill, 131 partial order

strict,l7J weak,l80

PC (classicallogic),J9, 31,323-325 axiomatization

history of, 51-52 independent,55,259-260 Lukasiewicz's, 54,325 relative to Int, 203,215,325 using 1, ~. 45,54,324 using 1, A, 57, 162,324 using 1, ~.A, 57,279,324 using 1, ~.A, v, 56, 279,324

background,320 complete collection of wffs, 46 complete and consistent theory, 46,

148,185,208

Completeness Theorem, 44, 45, 49, 50, 51-52,53,56

consistent collection of wffs, 46 and D, 123-123, 141 and DPC, 142-143 decidability, 35-36 Deduction Theorem, 32, 47 and deviant logics, 106-107 andEq, 141

INDEX 379

as extension of lnt, 203, 215,325 functional completeness of connectives,

32-34 and Int, 203, 210-215, 222-223 and J, 226,294, 310 and J 3 , 271-274, 283, 295 andL2 ,247 and L 3 , 236, 294, 301 language, choice of, 34-35 in language of J 3, 279 in language of modal logic, 162 model,16-17,3J,3JJ,323 not suitable to model antecedent contains

consequent, 116-117 and possible worlds, 148 Quine on, 105-106 andS, 74-78 semantic consequence, 31, 96 simplest symbolic model of reasoning, xix Strong Completeness Theorem, 50, 56 substitution, rule of, 32,51 tautologies, list of, 36-37 tautology, 18, 31 translations into, 295,300-302, 311-313 translations into lnt, 211-214, 222, 291,

303,309-310 translations to itself, 34, 292, 307 truth-tables, 13-15 within spectrum of all logics, 86 See also completeness proofs, constructive

vs. nonconstructive; name of logic and PC; paradoxes of implication

pcD (PC closed under necessitation), 177-118

Peano Arithmetic. See PA perception, xix, 315. See also experience Perzanowski, Jerzy, 194 Philo, 147

physical necessity, 150-151 Pigozzi, Donald, xxii, 101, 103

380 INDEX

platonist, 6 on abstract propositions, 6, 28

on assertions, 220 on completed infinite totalities,

196 on logical form, 22 on possible worlds, 161 on type I vs. type II models, 28 on undefined (unknown to be true)

propositions, 250 Porte, Jean, 101,194

possible world(s), 148, 151, 155-156,

160-161,332 Abstraction, 148 cardinality of set of, 152, 187

possibility in classical modal logic, 146-149 degree of, 248 in many-valued logic, 230, 234-237,

248 in paraconsistent J 3, 266-267

possibility operator

in classical modal logic, 154, 155,

157-158 in J3 , 266,267 in L 3 , 236-237

See also necessity operator; use-mention confusion

Post, Emil L., 39, 52, 57, 230

pragmatics, xix, 87,315,318

predicate logic, xx-xxi, 64, 321 predicates, xx-xxi, 117-118,250-251 predication, 64, 118, 120 presentation of a logic, 44

simple, 98-100

principal connective, 30 Principia Mathematica, 26, 39, 51, 52, 57,

146 Prior, Arthur, 66, 117, 119-120, 146, 235,

237,255

processes, 321

projective knowledge, 199,313-314

proof, formal, 41, 83, 97

using only rules, 55-56 See also consequence relation,

syntactic

proposition(s), xix, 1-9, 3, 27,86-87,

232, 237,318-319

abstract, 5-7, 21-22, 25, 161

appears in, 31 aspect(s) of, xx-xxi, 86, 95, 106, 146,315,

319 atomic, 13, 17, 72, 89 compatible (compossible), 156 compound (complex), 17, 89,318

extensionally equivalent, 104-105, 130,282

indeterminate, 236 and facts, 87, 117 future, 230, 232, 234-235 identified with worlds in which true, 160

logical form of, 21-22 many-values, classification leads to, 232

meaningless, 86 necessary. See necessity neuter, 235, 237 nonsensical, 65 obscene, 232 paradoxical, 4, 231, 236,246,251. See also

paradoxes of implication

possible. See possibility and rule of substitution, 51

same, 4, 6, 25. See also equivalence,

semantic; propositions, extensionally

equivalent semi-formal. See language, semi-formal

timeless, 4 as types, 4 undecidable (unknown), 231 valid,18-19,31,40,96,152,233 See also content; truth-value; tautology

propositional constant, 108 propositional logic, 8

criteria for good, 58 how to give semantics for, 94-95

propositional variable(s), 16

and rule of substitution, 51

sharing criteria for tautologies, 82

See also PV provability in arithmetic, 178-180, 211

provability translation, 1 79

provability-and-truth translation, 180, 211

psychology, 1,3,315

Putnam, Hilary, 160 PV (collection of propositional variables),

27,93,108 not viewed as completed infinite

totality, 200

QT (modal logic), 176-177, 179, 188-189,254,295,331-335

quantification and simple presentations, 100

quantum mechanics, 77 quasi-normal modal logic, 163,177, 182 Quesada, F. Miro, 265 Quine, Willard Van Orman, 35, 63, 74,

105,106 quotation marks, 5 quotation names, 5, 9 quotes, scare, 5

R (Nonsymmetric Relatedness Logic), 74, 88,89,90, 100,103,254,329

axiomatization, 81, 329 and D, 124-125 not suitable for modeling containment

of contents, 124 translations, 295,298-302 variable sharing criteria for tautologies,

141 R-closed subset, 200 'raining', 264,275 Rasiowa, Helena, xxii, 222 real analysis, 316-318 reality, 319 realization of a wff, 17, 88, 179 reasoning, xix-xxi, 1, 2, 7, 59, 63, 89, 111,

319-321 recursive collection, 28 reductio ad absurdum, 37, 55 reference, 64 referential content, xx, 115-120

Referential Content Assumption, 118 referring, 316 reflexive relation, 165 related conditional, 71

Relatedness Abstraction, 64, 70 Fully General, 73

Relatedness Assumption, 64 relatedness logic, 74. See R, S Relatedness Logic. See S

INDEX 381

Relatedness Logic, Nonsymmetric. See R Relatedness Logic, Subject Matter. See S Relatedness Logic, Symmetric. See S relatedness relation, 65-69 relation

accessibility, 150, 151, 332 anti-reflexive, 180 anti-symmetric, 150, 171 dependence,122-123, 138,328 DuaiD-,329 Eq-, 136-139 euclidean,185 reflexive, 165 relatedness, 65-69 symmetric, 169 transitive, 165 universal, 74, 92, 107, 109 See also order

relation based semantics, 97-98,109-110, 326 truth-default, 113

relations governing truth-tables, 91-92, 94, 98-99,102-103,109,110-113

uniformly presented, 111 See also relation

relevance, 141 relevance fallacy, 221-222 Rescher, Nicholas, 230, 231 Rosser, J. Barkley, 2, 57,324 rule(s), 40

closure under, 42 derived, 42,43 proofs using only, 55 See also name of rule, e.g., substitution,

rule of Russell, Bertrand, 26, 39, 51, 52, 147

S (Subject Matter Relatedness Logic),

Chapter m, 73. 98, 102-103, 325-327 axiomatization, 80, 326-327 conjunction defined in, 78, 135, 140 consequence relation, 73, 83-84, 97 and D, 124-125, 141

decidability, 75

382 INDEX

S (continued) Deduction Theorem, 82, 100, 130 deviant?, 107 di~unctionin,65, 71-72,76,78,80 finite-valued semantics, has no,

254-255 functional completeness of connectives,

77-79, 101 -model, 72-73 Nonnal Fonn Theorem, 79, 81 and PC, 74-77,78, 141 -relation, 67, 65-69 and rule of substitution, 81, 104 set-assignment semantics, 68-70 Strong Completeness Theorem, 81 tautology, 73-75, 124-125, 141 transitivity of-+ fails in, 76-77 translations, 75, 292, 295, 300-301,

311-313 truth-tables, 70-72 variable sharing criteria for tautologies,

141 Sl (modal logic), 130 S4 (modal logic), 165-168, 184-194, 254,

331-334 andlnt,210,293-294,308,314 translations, 210,211,294, 295, 309,

314 S4Grz (modallogic),J71-172, 190, 254,

331-334 and arithmetic, 180 and Int, 210-211, 294, 309 translations, 180, 210-211, 294, 295, 309

S5 (modallogic},169-171, 184-194,254, 331-334

inimite-valued semantics for, 255-256 same logic, 35, 98,271,292, 312-313 same meaning. See meaning, same schema, 19

independent,55,258 vs. rule of substitution, 54-55

Scott, Theodore Kennit, 69 Searle, John R., 120,315, 316 Segerberg, Krister, 128, 146, 171, 177,226,

336,338 semantic consequence.

See consequence, semantic

Semantic Deduction Theorem. See Deduction Theorem, Semantic

semantic equivalence, 20, 102,269-270 and decidability, 35 See also extensionally equivalent

propositions; meaning, same semantic structure, fonnal, 109-114 semantically faithful translation, 306-307,

313-314. See also translation, preserves meaning?

semantics, 7, 94-95 algebraic, 103, 105, 130, 168 compact, 44, 50, 101 dependence-style, 132 dual dependence sty1e,132-133 dualization of,135 falsity-default,112-114, 286 how to give, 94-95 many-valued, 103,234-235 relation based, 97-98,109-112 set-assignment, Chapter IV passim, 88-94,

108-111 simple presentation of, 98-100, 174, 335 sound,44 truth-default, 112-114,285-287 weak dependence-style,132 See also Kripke-style semantics; modal

semantics of implication; name of logic, truth-conditions (truth-tables)

semi-fonnallanguage. See language, semi-fonnal

sentence(s) and abstract propositions, 6 declarative, 2, 3 same,6

set-assignment,109 designated subset, 98,173-175 equational, 111 intersection,133 subject matter, 68-70 union,120-123, 133,135,325-326 unifonnly presented, 111 See also name of logic, set-assignment

set-assignment semantics, 88-97, 108-111 simple presentation of, 98-100

set-theoretic condition, 98-99, 110-111 Sextus Empiricus, 39,87,116,147

Shah-Kwei, Mob, 231,236 sharing a variable

and Deduction Theorem for S, 82 criteria for tautologies,141

Sheffer stroke, 34 relatedness version, 79

Silver, Charles, 49 simple presentation of semantics, 98-100,

174,335 simplicity constraint, invoked forD, 121 Simplification for Conjunction, law of, 36 Sluga, Hans, 135 Slupecki operator, 236 Smiley, T. J., 77, 97, 232, 234 Smullyan, Raymond, 179 Solovay, R., 179 sound axiomatization, 44 sound semantics, 44 Specker, Ernst, 77 standard (many-valued) table, 287 state of information. See information strawberries, 2 strict implication,147. See also

paradoxes of implication strict partial order, 180 strong completeness, 44, 101,290

finite, 45, 53,239 See also completeness proofs,

nonconstructive vs. constructive; name of logic, Strong Completeness Theorem

strong connectives (K 3 ), 251, 344 strong definability of a connective,

312-313 strong negation (J 3 ), 268 subject matter

informal discussion of, 62-64, 90, 228 vs. referential content, 115-116, 120 as set-assignment, 68-70 ofa tautology, 66, 84

Subject Matter Relatedness Logic. See S subject matter set-assignment, 68-69

and three-way overlap, 70, 80 subjunctive conditional.

See conditional, subjunctive substitution of logical equivalents, rule of,

51

forD, 130 forL 3 ,240 for PC, 51 for S, fails, 81

INDEX 383

substitution of provable logical equivalents, 81 substitution, rule of

general problem of, 104-105 for lot, 199, 335 forJ3 ,282 for PC, 32,51 for S, fails, 81, 104

Surma, Stanislaw, 52, 54, 194 symmetric relation, 169 Symmetric Relatedness Logic. See S syncategorematic connective, 66, 118 syntactic consequence. See consequence,

syntactic Syntactic Deduction Theorem. See Deduction Theorem, Syntactic syntax, 7

of a logic, 44

T (modal logic), 99,172-174, 184--193, 254, 295,331-335

-set-assignment,l73, 334 tableaux, analytic, 80, 116, 140, 244 Tarski, Alfred, 26, 52, 54, 55, 103, 167, 230,

237,246,325 tautology,18, 96,109

content of, 120 K 3 has none, 251-252 list for PC, 36-37 subject matter, 73 subject matter of, 66, 84 universal, 113-114 See also decidability; name of logic,

tautology tertium rwn datur. See Excluded Middle 'that', 155 theorem, formal, 41

theory (formal), 42

classical (PC), 46 complete. See complete collection of wffs complete and consistent. See complete and

consistent theory consistent. See consistency full, 205

384 INDEX

theory (continued) ofr,42,46 forlnt,205 modal, 191 hiviw,264-265,275

'therefore', 20 thought, 7, 315, 320 time dependent necessity, 149-150 Tokarz,~arek,238

transitive closure, 153 transitive relation, 165 transitivity

of--+, 37, 76-77, 124 oft-, 43 of 1=, 31,77

translation(s) (general notion of), 290 and aspects of propositions, 88 composition of, 291 of connectives, 302, 306-307 content variant model preserving,

305-306 and Deduction Theorems, 291, 302 finite consequence, 292,294, 309 gnunmnaticw,291,302 no grammatic& into PC, 295 grammatic& with parameters, 302 homophonic, 213,291 model preserving, 305-307 model preserving up to elementary

equivwence, 304-307 preserves meaning?, 289, 302-303, 306,

313-314 provability, 179 provability-and-truth,180, 211

translations (specific) to arithmetic, 178-180, 211, 214 classic& mod& logic to itself, 178, 293 D to Dual D, 133, 293, 307 D to Eq, 140, 298 double negation, 211-212,226,294,310 Dual D to D, 133,293,307-308 Eq to D,136, 293, 307 Eq to Dual D, 293,307 Gentzen's, 213-214,294, 310 GOdel on PC into Int, 214 Heyting's. See translations (specific),

Int to S4

Int to J, 227, 294 Int to S4, 210-211, 219, 293-294,

308,313 Int to S4Grz, 210,294 and PA, 179-180,211,214 PC to D, 292-293 PC to Dual D, 292 PC to Int, 211-214, 222, 291, 303,

309-31 PC to Int, 211-214, 222, 291, 303,

309-310 PC to itself, 34-35, 292 PCtoJ, 226 PC to J 3 , 272-273, 295 PC to L3, 238,294 PC toR, 292 PCtoS, 292 R to PC, 298, 299-302 S to PC, 75,300-301,311-312 summary of, 292-295, 307

Transposition, 37 tree, Kripke, 200 hiviw theory, 264-265, 275 Troelstra, A. S., 199 truth,2-4, 7,17,86-87,161,315,318

absolute, 266 and abstract propositions, 6 and assertibility, 220-221 and bivwence in intuitionism, 219-222 content, not part of, 118-120, 142 contrary of 'not-true' rather than fwsity, 230 degree of, 265-266 as designated vwue in many-vwued logic, 232 and facts, 117 as a hypothesis, xix, 231 in a model, 18, 31, 72, 83, 92, 93,151 necessary, xxi, 320 and nonsensic& propositions, 65 in paraconsistent logic, 265-266 as proof, 197 and reasoning, xix and subject matters, 65-66 at world w, 151, 201 at world w, eventuw, 183 See also fwsity; necessity

truth-conditions. See name of logic, truth­conditions; name of logic, truth-tables

truth-and-content functional connective, 90, 102, 125

truth-default semantics, 112-114, 285-287

truth function, 13, 15,108 truth-functional connective, 32, 109 truth-functional evaluation, 233 truth-functionally complete connectives, 32 truth-and-relatedness function, 70 truth-and-relatedness functional connective,

78 truth-and-relation functional connective,

98,102 truth-table( s)

in antiquity, 39, 147 classical, 13-15 dependent implication, 120 dual dependent implication, 133 falsity default, 112 in general framework, 90-93,

109-110, 318 intensional connective, 112 many-valued, 233 many-valued, falsity weighted, 287 many-valued, standard, 287 many-valued, !ruth weighted, 287 minimal intuitionist negation, 198,216 necessity in classical modal logic,

157-158 possibility in classical modal logic,

157-158 related conditional, 71 relations governing, 91-92,94,98-99,

102-103,109,110-113 truth-default, 113, 285-281 weak conditional, 92-93,157 See also name of logic, truth-

conditions; name of logic, truth-tables

truth-weighted (many-valued) table, 287 truth-value(s), xix, 2-3, 86-89

absolute, 266, 269 assignment. See valuation commonly agreed on, 18 default, 285 designated, 232-233 hypothetical, xix, 3, 231

many-valued, 233 more than two, 230-232 sentence has, 3 See also falsity; truth

Turing, Alan, 317 Turquette, Atwell R., 2, 248 type, 3

proposition is a, 4 word is a, 3

type I model, 17, 12, 92, 94 type II model, 26, 27, 72, 93

INDEX 385

type I vs. type II model, 26-28,72-73,93, 142-143

unary connective, 9 undecidable proposition, 231 uniformly interpreted connective, 110-111 uniformly presented relation, 111 uniformly presented set-assignment, 111 union set-assignment, 120-123, 133, 135, 136,

325-326 Unique Readability of wffs, 29, I 08 utility, 317-318 utterance, 2, 6 universal anti-tautology, 113-114 universal relation, 74, 92, 107, 109 universal tautology, 113-114 use-mention confusion, 5, 65, 78, 96, 155, 266 use-mention distinction, 5, 20

valid argument, 20 valid deduction, 20 valid proposition, 18-19,31, 40,96

in a frame, 152 in a many-valued system, 233 See also tautology

valid schema, 19 valuation, 18, 88, 109-110 van Dalen, D., 199 van Frassen, Bas C., 171 van Heijenoort, Jean, 52 variable

meta-, 16-17, 20 propositional, 16, 29, 298-299 and rule of substitution, 51

variable sharing criteria for tautologies, 141 verification, 201,220

386 INDEX

Wajsberg, Mordechaj, 239, 340 Walton, Douglas N., 61, 74, 133 weak negation, 266-267,271-275 weak modal semantics for (of) implication,

157-159, 175-176, 179, 181, 333-334

weak partial order, 171 weak table for the conditional, 92-93, 157 well-formed-formula(s). See wff(s) wff(s), 29, 108

collection of all, 29, 93, 108,200 length, 30 ordering, 30 unique readability, 29

Wffs. See wff(s) collection of all Whitman, Walt, 264

Whitehead, Alfred North, 26, 39, 51, 52, 57, 147

wholly intensional connective. See connective, intensional

Williamson, Colwyn, 7 W6jcicki, Ryszard, 97, 230,234,238,248,

289 word, 3 world. See possible world Wrigley, Michael, 317

Y (condition on modal semantics of implication), 1 71, 334

Yablo, Stephen, 231

zero-ary connective, 108

Nijhoff International Philosophy Series

1. Rotenstreich, N.: Philosophy, History and Politics. Studies in Contemporary English Philosophy of History. 1976. ISBN 90-247-1743-4.

2. Srzednicki, J.T.J.: Elements of Social and Political Philosophy. 1976. ISBN 90-247-1744-2. 3. Tatarkiewicz, W.: Analysis of Happiness. 1976. ISBN 90-247-1807-4. 4. Twardowski, K.: On the Content and Object of Presentations. A Psychological fuvestigation.

Translated and with anfutroduction by R. Grossman. 1977. ISBN 90-247-1926-7. 5. Tatarkiewicz, W.: A History of Six Ideas. An Essay in Aesthetics. 1980. ISBN 90-247-2233-0. 6. Noonan, H.W.: Objects and Identity. An Examination of the Relative Identity Thesis and Its

Consequences. 1980. ISBN 90-247-2292-6. 7. Crocker, L.: Positive Liberty. An Essay in Normative Political Philosophy. 1980.

ISBN 90-247-2291-8. 8. Brentano, F.: The Theory of Categories. 1981. ISBN 90-247-2302-7. 9. Marciszewski, W. (ed.): Dictionary of Logic as Applied in the Study of Language. Concepts,

Methods, Theories. 1981. ISBN 90-247-2123-7. 10. Ruzsa, I.: Modal Logic with Descriptions. 1981. ISBN 90-247-2473-2. 11. Hoffman, P.: The Anatomy of Idealism. Passivity and Activity in Kant, Hegel and Marx.

1982. ISBN 90-247-2708-1. 12. Gram, M.S.: Direct Realism. A Study of Perception. 1983. ISBN 90-247-2870-3. 13. Srzednicki, J.T.J., Rickey, V.F. and Czelakowski, J. (eds.): Lesniewski's Systems. Ontology

and Mereology. 1984. ISBN 90-247-2879-7. 14. Smith, J.W.: Reductionism and Cultural Being. A Philosophical Critique of Sociobiological

Reductionism and Physicalist Scientific Unificationism. 1984. ISBN 90-247-2884-3. 15. Zumbach, C.: The Transcendent Science. Kant's Conception of Biological Methodology.

1984. ISBN 90-247-2904-1. 16. Notturno, M.A.: Objectivity, Rationality and the Third Realm. Justification and the Grounds

of Psychologism. A Study of Frege and Popper. 1985. ISBN 90-247-2956-4. 17. Dilman, I. (ed.): Philosophy and Life. Essays on John Wisdom. 1984. ISBN 90-247-2996-3. 18. Russell, J.J.: Analysis and Dialectic. Studies in the Logic of Foundation Problems. 1984.

ISBN 90-247-2990-4. 19. Currie, G. and Musgrave, A. (eds.): Popper and the Human Sciences. 1985.

ISBN 90-247-2998-X. 20. Broad, C.D.: Ethics. Edited by C. Lewy. 1985. ISBN 90-247-3088-0. 21. Seargent, D.A.J.: Plurality and Continuity. An Essay in G.F. Stout's Theory of Universals.

1985. ISBN 90-247-3185-2. 22. Atwell, I.E.: Ends and Principles in Kant's Moral Thought. 1986. ISBN 90-247-3167-4. 23. Agassi, J. and Jarvie, I.Ch. (eds.): Rationality. The Critical View. 1987. ISBN 90-247-3275-1. 24. Srzednicki, J.T.J. and Stachniak, Z. (eds.): S. Lesniewski's Lecture Notes in Logic. 1988.

ISBN 90-247-3416-9. 25. Taylor, B.M. (ed.): Michael Dummett. Contributions to Philosophy. 1987.

ISBN 90-247-3463-0. 26. Bar-On, A.Z.: The Categories and Principle of Coherence. Whitehead's Theory of Categories

in Historical Perspective. 1987. ISBN 90-247-3478-9. 27. Dziemidok, B. and McCormick, P. (eds.): On the Aesthetics of Roman Ingarden. futerpreta­

tions and Assessments. 1989. ISBN 0-7923-0071-8 28. Srzednicki, J.T.J. (ed.): Stephan Komer. Philosophical Analysis and Reconstruction. 1987.

ISBN 90-247-3543-2. 29. Brentano, F.: On the Existence of God. Lectures given at the Universities of Wiirzburg and

Vienna (1868-1891). 1987. ISBN 90-247-3538-6. 30. Augustynek, Z.: Time. Past, Present and Future. Forthcoming. 31. Pawlowski, T.: Aesthetic Values. 1989. ISBN 0-7923-0418-7. 32. Ruse, M. (ed.): What the Philosophy of Biology Is. Essays Dedicated to David Hull. 1989.

ISBN 90-247-3778-8. 33. Young, J.: Willing and Unwilling: A Study in the Philosophy of Arthur Schopenhauer. 1987.

ISBN 90-247-3556-4. 34. Lavine, T.Z. and Tejera, V. (eds.): History and Anti-History in Philosophy. 1989.

ISBN 0-7923-0455-1. 35. Epstein, R.L.: The Semantic Foundations of Logic. Volume 1: Propositional Logics. 1990.

ISBN 0-7923-0622-8.

Nijhoff International Philosophy Series

36. Geach, P. (ed.): Logic and Ethics. Forthcoming. 37. Winterbourne, A.: The Ideal and the Real. 1988. ISBN 90-247-3774-5. 38. Szaniawski, K. (ed.): The Vienna Circle and the Lvov-Warsaw School. 1989. ISBN 90-247-

3798-2. 39. Priest, G.: In Contradiction. A Study of the Transconsistent. 1987. ISBN 90-247-3630-7.