Advertisement

Alternatives to Standard First-Order Semantics

  • Hugues Leblanc
Part of the Synthese Library book series (SYLI, volume 164)

Abstract

Alternatives to standard semantics are legion, some even antedating standard semantics. I shall study several here, among them: substitutional semantics, truth-value semantics, and probabilistic semantics. All three interpret the quantifiers substitutionally, i.e. all three rate a universal (an existential) quantification true if, and only if, every one (at least one) of its substitution instances is true.2 As a result, the first, which retains models, retains only those which are to be called Henkin models. The other two dispense with models entirely, truth-value semantics using instead truth-value assignments (or equivalents thereof to be called truth-value functions) and probabilistic semantics using probability functions. So reference, central to standard semantics, is no concern at all of truth-value and probabilistic semantics; and truth, also central to standard semantics, is but a marginal concern of probabilistic semantics.

Keywords

Arbitrary Statement Logical Truth Model Associate Standard Semantic Probabilistic Semantic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adams, E.W.: 1981, ‘Transmissible improbabilities and marginal essentialness of premises in inferences involving indicative conditionals’, J. Philosophical Logic 10, 149–178.Google Scholar
  2. Barnes, R. F., Jr. and Gumb, R.D.: 1979, ‘The completeness of presupposition-free tense logics’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 25, 192–208.CrossRefGoogle Scholar
  3. Behmann, H.: 1922, ‘Beiträge zur Algebra der Logik, in besondere zum Entscheidungs-problem’, Math. Annalen 86, 163–229.CrossRefGoogle Scholar
  4. Bendall, K.: 1979, ‘Belief-theoretic formal semantics for first-order logic and probability’, J. Philosophical Logic 8, 375–394.Google Scholar
  5. Bendall, K.: 1982, ‘A “definitive” probabilistic semantics for first-order logic’, J. Philosophical Logic 11, 255–278.CrossRefGoogle Scholar
  6. Bergmann, M., Moore, J., and Nelson, R.: 1980, The Logic Book, Random House, New York.Google Scholar
  7. Bernays, P.: 1922, Review of Behmann [1922], Jahrbuch über die Fortschritte der Mathematik 48, 1119.Google Scholar
  8. Beth, E. W.: 1959, The Foundations of Mathematics, North-Holland, Amsterdam.Google Scholar
  9. Carnap, R.: 1942, Introduction to Semantics, Harvard University Press, Cambridge, Mass.Google Scholar
  10. Carnap, R.: 1950, Logical Foundations of Probability, University of Chicago Press, Chicago, Ill.Google Scholar
  11. Carnap, R.: 1952, The Continuum of Inductive Methods, University of Chicago Press, Chicago, Ill.Google Scholar
  12. Carnap, R. and Jeffrey, R. C.: 1971, Studies in Inductive Logic and Probability, Volume I, University of California Press, Berkeley and Los Angeles, Ca.Google Scholar
  13. De Finetti, B.: 1937, ‘La prévision: Ses lois logiques, ses sources subjectives’, Annales de l’Institut Henri Poincaré 7, 1–68.Google Scholar
  14. Dunn, J. M. and Belnap, N. D., Jr.: 1968, ‘The substitution interpretation of the quantifiers’, Noûs 2, 177–185.CrossRefGoogle Scholar
  15. Ellis, B.: 1979, Rational Belief Systems, APQ Library of Philosophy, Rowman and Littlefield, Totowa, N.J.Google Scholar
  16. Field, H. H.: 1977, ‘Logic, meaning, and conceptual role’, J. Philosophy 74, 379–409.CrossRefGoogle Scholar
  17. Fitch, F. B.: 1948, ‘Intuitionistic modal logic with quantifiers’, Portugaliae Mathematica 7, 113–118.Google Scholar
  18. Frege, G.: 1879, Begriffschrift, Halle.Google Scholar
  19. Frege, G.: 1893–1903, Grundgesetze der Arithmetik, Jena.Google Scholar
  20. Gaifman, H.: 1964, ‘Concerning measures on first-order calculi’, Israel J. Math. 2, 1–18.CrossRefGoogle Scholar
  21. Garson, J.W.: 1979, ‘The substitution interpretation and the expressive power of intensional logics’, Notre Dame J. Formal Logic 20, 858–864.CrossRefGoogle Scholar
  22. Gentzen, G.: 1934–35, ‘Untersuchungen über das logische Schliessen’, Mathematische Zeitschrift 39, 176–210,405–431.CrossRefGoogle Scholar
  23. Goldfarb, W. D.: 1979, ‘Logic in the Twenties: the nature of the quantifier’, J. Symbolic Logic 44, 351–68.CrossRefGoogle Scholar
  24. Gottlieb, D. and McCarthy, T.: 1979, ‘Substitutional quantification and set theory’, J. Philosophical Logic 8, 315–331.Google Scholar
  25. Gumb, R. D.: 1978, ‘Metaphor theory’, Reports on Math. Logic 10, 51–60.Google Scholar
  26. Gumb, R. D.: 1979, Evolving Theories, Haven Publishing, New York.Google Scholar
  27. Gumb, R. D.: 1983, ‘Comments on probabilistic semantics’, in Leblanc et al. [1983].Google Scholar
  28. Harper, W. L.: 1974, ‘Counterfactuals and representations of rational belief’, doctoral dissertation, University of Rochester.Google Scholar
  29. Harper, W. L.: 1983, ‘A conditional belief semantics for free quantificational logic with identity’, in Leblanc et al. [1983].Google Scholar
  30. Harper, W. L., Leblanc, H. and Van Fraassen, B. C.: 1983, ‘On characterizing Popper and Carnap probability functions’, in Leblanc et al. [1983],Google Scholar
  31. Hasenjaeger, G.: 1953, ‘Eine Bemerkung zu Henkin’s Beweis für die Vollständigkeit des Prädikatenkalküls der ersten Stufe’, J. Symbolic Logic 18, 42–48.CrossRefGoogle Scholar
  32. Henkin, L.: 1949, ‘The completeness of the first-order functional calculus’, J. Symbolic Logic 14, 159–166.CrossRefGoogle Scholar
  33. Hintikka, J.: 1955, Two Papers on Symbolic Logic, Acta Philosophica Fennica 8.Google Scholar
  34. Huntington, E. V.: 1933, ‘New sets of independent postulates for the algebra of logic’, Transactions of the American Mathematical Society 35, 274–304.Google Scholar
  35. Jeffrey, R. C.: 1967, Formal Logic: Its Scope and Limits, McGraw-Hill, New York.Google Scholar
  36. Jeffreys, H.: 1939, Theory of Probability, Oxford University Press, London.Google Scholar
  37. Kearns, J.T.: 1978, ‘Three substitution-instance interpretations’, Notre Dame J. Formal Logic 19, 331–354.CrossRefGoogle Scholar
  38. Keynes, J. M.: 1921, A Treatise on Probability, Macmillan, London.Google Scholar
  39. Kolmogorov, A. N.: 1933, Grundbegriffe der Wahrscheinlichkeitsrechnung, Berlin.Google Scholar
  40. Kripke, S.: 1976, ‘Is there a problem about substitutional quantification?’ in G. Evans and J. McDowell (eds.), Truth and Meaning, Clarendon Press, Oxford, 325–419.Google Scholar
  41. Leblanc, H.: 1960, ‘On requirements for conditional probability functions’, J. Symbolic Logic 25, 171–175.CrossRefGoogle Scholar
  42. Leblanc, H.: 1966, Techniques of Deductive Inference, Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
  43. Leblanc, H.: 1968, ‘A simplified account of validity and implication for quantificational logic’, J. Symbolic Logic 33, 231–235.CrossRefGoogle Scholar
  44. Leblanc, H.: 1973, Truth, Syntax and Modality, North-Holland, Amsterdam.Google Scholar
  45. Leblanc, H.: 1976, Truth-Value Semantics, North-Holland, Amsterdam.Google Scholar
  46. Leblanc, H.: 1979a, ‘Generalization in first-order logic’, Notre Dame J. of Formal Logic 20, 835–857.CrossRefGoogle Scholar
  47. Leblanc, H.: 1979b, ‘Probabilistic semantics for first-order logic’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 25, 497–509.CrossRefGoogle Scholar
  48. Leblanc, H.: 1981, ‘What price substitutivity? A note on probability theory’, Philosophy of Science 48, 317–322.CrossRefGoogle Scholar
  49. Leblanc, H.: 1982a, ‘Free intuitionistic logic: A formal sketch’, in J. Agassi and R. Cohen (eds), Scientific Philosophy Today: Essays in Honor of Mario Bunge, D. Reidel, Dordrecht, pp. 133–145.Google Scholar
  50. Leblanc, H.: 1982b, Existence, Truth, and Provability, SUNY Press, Albany, N.Y.Google Scholar
  51. Leblanc, H.: 1982c, ‘Popper’s 1955 axiomatization of absolute probability’, Pacific Philosophical Quarterly 63, 133–145.Google Scholar
  52. Leblanc, H.: 1983, ‘Probability functions and their assumption sets: The singulary case’, J. Philosophical Logic 12.Google Scholar
  53. Leblanc, H.: forthcoming, ‘Of consistency trees and model sets’.Google Scholar
  54. Leblanc, H. and Gumb, R. D.: 1983. D.: 1983, ‘Soundness and completeness proofs for three brands of intuitionistic logic’, in Leblanc et al. [1983].Google Scholar
  55. Leblanc, H., Gumb, R. D. and Stern, R. (eds.): 1983, Essays in Epistemology and Semantics, Haven Publishing, New York.Google Scholar
  56. Leblanc, H. and Morgan, C. G.: forthcoming, ‘Probability functions and their assumption sets: The binary case’, Synthèse.Google Scholar
  57. Leblanc, H. and Wisdom, W. A.: 1972, Deductive Logic, Allyn & Bacon, Boston, Mass.Google Scholar
  58. Löwenheim, L.: 1915, ‘Uber Möglichkeiten im Relativkalkul’, Mathematischen Annalen 76, 447–470.CrossRefGoogle Scholar
  59. Marcus, R. B.: 1963, ‘Modal logics I: Modalities and international languages’, in M. W. Wartofsky (ed.), Proceedings of the Boston Colloquium for the Philosophy of Science, 1961–1962, D. Reidel, Dordrecht.Google Scholar
  60. McArthur, R. P.: 1976, Tense Logic, D. Reidel, Dordrecht.Google Scholar
  61. McArthur, R. P. and Leblanc, H.: 1976, ‘A completeness result for quantificational tense logic’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 22, 89–96.CrossRefGoogle Scholar
  62. Moore, G. E.: 1959, Philosophical Papers, Allen and Unwin, London.Google Scholar
  63. Morgan, C. G.: 1982a, ‘There is a probabilistic semantics for every extension of classical sentence logic’, J. Philosophical Logic 11, 431–442.Google Scholar
  64. Morgan, C. G.: 1982b, ‘Simple probabilistic semantics for propositional K, T, B, S4, and S5’, J. Philosophical Logic 11, 443–458.Google Scholar
  65. Morgan, C. G.: 1983, ‘Probabilistic semantics for propositional modal logics’, in Leblanc et al. [1983].Google Scholar
  66. Morgan, C. G. and Leblanc, H.: 1983a, ‘Probabilistic semantics for intuitionistic logic’, Notre Dame J. Formal Logic 23, 161–180.CrossRefGoogle Scholar
  67. Morgan, C. G. and Leblanc, H.: 1983b, ‘Probability theory, intuitionism, semantics and the Dutch Book argument’, Notre Dame J. Formal Logic 24, 289–304.CrossRefGoogle Scholar
  68. Morgan, C. G. and Leblanc, H.: 1983c, ‘Satisfiability in probabilistic semantics’, in Leblanc et al. [1983].Google Scholar
  69. Orenstein, A.: 1979, Existence and the Particular Quantifier, Temple University Press, Philadelphia, Pa.Google Scholar
  70. Parsons, C.: 1971, ‘A plea for substitutional quantification’, J. Philosophy 68, 231–237.CrossRefGoogle Scholar
  71. Popper, K. R.: 1955, ‘Two autonomous axiom systems for the calculus of probabilities’, British J. Philosophy of Science 6, 51–57, 176, 351.CrossRefGoogle Scholar
  72. Popper, K. R.: 1957, ‘Philosophy of science: a personal report’, in A. C. Mace (ed.), British Philosophy in Mid-Century, Allen and Unwin, London, pp. 155–191.Google Scholar
  73. Popper, K. R.: 1959, The Logic of Scientific Discovery, Basic Books, New York.Google Scholar
  74. Quine, W. V.: 1940, Mathematical Logic, Norton, New York.Google Scholar
  75. Quine, W. V.: 1969, Ontological Relativity and Other Essays, Columbia University Press, New York and London.Google Scholar
  76. Ramsey, F. P.: 1926a, ‘The foundations of mathematics’, Proceedings of the London Mathematical Society, Series 2, 25, 338–384.CrossRefGoogle Scholar
  77. Ramsey, F. P.: 1926b, ‘Mathematical Logic’, The Mathematical Gazette 13, 185–194.CrossRefGoogle Scholar
  78. Reichenbach, R.: 1935, Wahrscheinlichkeitslehre, Leiden.Google Scholar
  79. Rényi, A.: 1955, ‘On a new axiomatic theory of probability’, Acta Mathematica Acad. Scient. Hungaricae 6, 285–335.CrossRefGoogle Scholar
  80. Robinson, A.: 1951, On the Mathematics of Algebra, North-Holland, Amsterdam.Google Scholar
  81. Rosser, J. B.: 1953, Logic for Mathematicians, McGraw Hill, New York.Google Scholar
  82. Schotch, P. K. and Jennings, R. E.: 1981, ‘Probabilistic considerations on modal semantics’, Notre Dame J. Formal Logic 22, 227–238.CrossRefGoogle Scholar
  83. Schütte, K.: 1960, ‘Syntactical and semantical properties of simple type theory’, J. Symbolic Logic 25, 305–326.CrossRefGoogle Scholar
  84. Schütte, K.: 1962, Lecture Notes in Mathematical Logic, Volume I, Pennsylvania State University.Google Scholar
  85. Seager, W.: 1983, ‘Probabilistic semantics, identity and belief’, Canadian J. Philosophy, 12.Google Scholar
  86. Shoenfield, J. R.: 1967, Mathematical Logic, Addison-Wesley, Reading, Mass.Google Scholar
  87. Skolem, T. A.: 1920, ‘Logisch-Kombinatorische Untersuchungen über die Erfüllbarkeit und Beweisbarkeit mathematischen Sätze nebst einem Theoreme über dichte Mengen’, Skrifter utgit av Videnskapsselkapet i Kristiania, I. Mathematisk-naturvidenskabelig klasse 1920, no. 4, 1–36.Google Scholar
  88. Smullyan, R. M.: 1968, First-Order Logic, Springer-Verlag, New York.Google Scholar
  89. Stalnaker, R.: 1970, ‘Probability and conditionals’, Philosophy of Science 37, 64–80.CrossRefGoogle Scholar
  90. Stevenson, L.: 1973, ‘Frege’s two definitions of quantification’, Philosophical Quarterly 23, 207–223.CrossRefGoogle Scholar
  91. Tarski, A.: 1930, ‘Fundamentale Begriffe der Methodologie der deductiven Wissenschaften. I’, Monatshefte für Mathematik und Physik 37, 361–404.CrossRefGoogle Scholar
  92. Tarski, A.: 1935–36, ‘Grundzüge des Systemkalkül’, Fundamenta Math. 25, 503–526, 26, 283–301.Google Scholar
  93. Tarski, A.: 1936, ‘Der Wahrheitsbegriff in den formalisierten Sprachen’, Studia Philosophica 1, 261–405.Google Scholar
  94. Thomason, R. H.: 1965, ‘Studies in the formal logic of quantification’, doctoral dissertation, Yale University.Google Scholar
  95. Van Fraassen, B. C.: 1981, ‘Probabilistic semantics objectified‘, J. Philosophical Logic 10, 371–394, 495–510.CrossRefGoogle Scholar
  96. Van Fraassen, B. C.: 1982, ‘Quantification as an act of mind’, J. Philosophical Logic 11, 343–369.CrossRefGoogle Scholar
  97. Von Wright, G. R.: 1957, The Logical Problem of Induction, 2nd revised edition, Macmillan, New York.Google Scholar
  98. Whitehead, A. N. and Russell, B.: 1910–13, Principia Mathematica, Cambridge University Press, Cambridge.Google Scholar
  99. Wittgenstein, L.: 1921, Tractatus Logico-Philosphicus (= Logisch-philosophische Abhandlung), Annalen der Naturphilosophie 14, 185–262.Google Scholar

Copyright information

© D. Reidel Publishing Company 1983

Authors and Affiliations

  • Hugues Leblanc
    • 1
  1. 1.Temple UniversityUSA

Personalised recommendations