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.1 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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
Bibliography
E. W. Adams. Transmissible improbabilities and marginal essentialness of premises in inferences involving indicative conditionals. Journal of Philosophical Logic, 10: 149–178, 1981.
R. F. Barnes, Jr. and R. D. Gumb. The completeness of presupposition-free tense logics. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 25: 192–208, 1979.
H. Behmann. Beiträge zur Algebra der Logik, in besondere zum Entscheidungsproblem. Math Annalen, 86: 163–229, 1922.
K. Bendall. Belief-theoretic formal semantics for first-order logic and probability. Journal of Philosophical Logic, 8: 375–394, 1979.
M. Bergmann, J. Moore, and J. C. Nelson. The Logic Book. Random House, New York, 1980.
P. Bernays. Review of [Behmann, 1922]. Jahrbuch über die Fortschritte der Mathematik, 48: 11–19, 1922.
E. W. Beth. The Foundations of Mathematics. North-Holland, Amsterdam, 1959.
R. Carnap and R. C. Jeffrey. Studies in Inductive Logic and Probability, volume I. University of California Press, Berkeley and Los Angeles, CA, 1971.
R. Carnap. Introduction to Semantics. Harvard University Press, Cambridge, MA, 1942.
R. Carnap. Logical Foundations of Probability. University of Chicago Press, Chicago, IL, 1950.
R. Carnap. The Continuum of Inductive Methods. University of Chicago Press, Chicago, IL, 1952.
B. De Finetti. La prévision: Ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré, 7: 1–68, 1937.
J. M. Dunn and Jr. N. D. Belnap. The substitution interpretation of the quantifiers. Noûs, 2: 177–185, 1968.
B. Ellis. Rational Belief Systems. APQ Library of Philosophy, Rowmans and Littlefield, Totowa, NJ, 1979.
H. H. Field. Logic, meaning and conceptual role. Journal of Philosoph, 74: 379–409, 1977.
F. B. Fitch. Intuitionistic modal logic with quantifiers. Portugaliae Mathematica, 7: 113–118, 1948.
G. Frege. Begriffschrift. Halle, 1879.
G. Frege. Grundgesetze der Arithmetik. Jena, 1893–1903.
H. Gaifman. Concerning measures on first-order calculi. Israel Journal of Mathematics, 2: 1–18, 1964.
J. W. Carson. The substitution interpretation and the expressive power of intensional logic. Notre Dame Journal of Formal Logic, 20: 858–864, 1979.
G. Gentzen. Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39: 176–210, 405–431.
W. D. Goldfarb. Logic in the twenties: the nature of the quantifier. Journal of Symbolic Logic, 44: 351–68, 1979.
D. Gottlieb and T. McCarthy. Substitutional quantification and set theory. Journal of Philosophical Logic, 8: 315–331, 1979.
R. D. Gumb. Metaphor theory. Reports on Mathematical Logic, 10: 51–60, 1978.
R. D. Gumb. Evolving Theories. Haven Publishing, NY, 1979.
R. D. Gumb. Comments on probabilistic semantics. In H. Leblanc et al., editor, Essays in Epistemology and Semantics. Haven Publishing, NY, 1983.
W. L. Harper, H. Leblanc, and B. C. Van Fraassen. On characterising popper and carnap probability functions. In H. Leblanc et al.,editor, Essays in Epistemology and Semantics. Haven Publishing, NY, 1983.
W. L. Harper. Counterfactuals and representations of rational belief. PhD thesis, University of Rochester, NY, 1974.
G. Hasenjaeger. Eine bemerkung zu henkin’s beweis für die vollständigkeit des prädikatenkalküls der ersten stufe. Journal of Symbolic Logic, 18: 42–48, 1953.
L. Henkin. The completeness of the first-order functional calculus. Journal of Symbolic Logic, 14: 159–166, 1949.
J. Hintikka. Two papers on symbolic logic. Acta Philosophica Fennica, 8, 1955.
E. V. Huntington. New sets of independent postulates for the algebra of logic. Transactions of the American Mathematical Society, 35: 274–304, 1933.
R. C. Jeffrey. Formal Logic: Its Scope and Limits, 3rd edition. McGraw-Hill, NY, (1st edition, 1967), 1990.
H. Jeffreys. Theory of Probability. Oxford University Press, Oxford, 1939.
J. T. Kearns: Three substitution-instance interpretations. Notre Dame Journal of Formal Logic, 19: 331–354, 1978.
J. M. Keynes. A Treatise on Probability. Macmillan, London, 1921.
A. N. Kolmogorov. Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin, 1933.
S. Kripke. Is there a problem about substitutional quantification? In G. Evans and J. McDowell, editors, Truth and Meaning, pages 325–419. Clarendon Press, Oxford, 1976.
H. Leblanc and R. D. Gumb. Soundness and completeness proofs for three brands of intuitionistic logic. In H. Leblanc, R. D. Gumb, and R. Stern, editors, Essays in Epistemology and Semantics. Haven Publishing, NY, 1983.
H. Leblanc and C. G. Morgan. Probability functions and their assumption sets: The binary case. Synthèse, 60, 91–106, 1984.
H. Leblanc and W. A. Wisdom. Deductive Logic, 3rd edition. Allyn and Bacon, Boston, MA, (1st edition, 1972), Prentice-Hall, 1993.
H. Leblanc, R. D. Gumb, and R. Stern, editors. Essays in Epistemology and Semantics. Haven Publishing, NY, 1983.
H. Leblanc. On requirements for conditional probability functions. Journal of Symbolic Logic, 25: 171–175, 1960.
H. Leblanc. Techniques of Deductive Inference. Prentice-Hall, Englewood Cliffs, NJ, 1966.
H. Leblanc. A simplified account of validity and implication for quantificational logic. Journal of Symbolic Logic, 33: 231–235, 1968.
H. Leblanc. Truth, Syntax and Modality. North-Holland, Amsterdam, 1973.
H. Leblanc. Truth-Value Semantics. North-Holland, Amsterdam, 1976.
H. Leblanc. Generalization in first-order logic. Notre Dame Journal of Formal Logic, 20: 835–857, 1979.
H. Leblanc. Probabilistic semantics for first-order logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 25: 497–509, 1979.
H. Leblanc. What price substitutivity? A note on probability theory. Philosophy of Science, 48: 317–322, 1981.
H. Leblanc. Existence, Truth and Provability. SUNY Press, Albany, NY, 1982.
H. Leblanc. Free intuitionistic logic: A formal sketch. In J. Agassi and R. Cohen, editors, Scientific Philosophy Today: Essays in Honor of Mario Bunge, pages 133–145. D. Reidel, Dordrecht, 1982.
H. Leblanc. Popper’s 1955 axiomatization of absolute probability. Pacific Philosophical Quarterly, 63: 133–145, 1982.
H. Leblanc. Probability functions and their assumption sets: The singulary case. Journal of Philosophical Logic, 12, 1983.
L. Löwenheim. Über möglichkeiten im relativkalkul. Mathematischen Annalen, 76: 447–470, 1915.
R. B. Marcus. Modallogics I: Modalities and international languages. In M. W. Wartofsky, editor, Proceedings of the Boston Colloquium for the Philosophy of Science, 1961–1962. D. Reidel, Dordrecht, 1963.
R. P. McArthur and H. Leblanc. A completeness result for quantificational tense logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 22: 89–96, 1976.
R. P. McArthur. Tense Logic. D. Reidel, Dordrecht, 1976.
G. E. Moore. Philosophical Papers. Allen and Unwin, London, 1959.
C. G. Morgan and H. Leblanc. Probabilistic semantics for intuitionistic logic. Notre Dame Journal of Formal Logic, 23: 161–180, 1983.
C. G. Morgan and H. Leblanc. Probability theory, intuitionism, semantics and the Dutch Book argument. Notre Dame Journal of Formal Logic, 24: 289–304, 1983.
C. G. Morgan and H. Leblanc. Satisfiability in probabilistic semantics. In H. Leblanc, R. D. Gumb, and R. Stern, editors, Essays in Epistemology and Semantics. Haven Publishing, NY, 1983.
C. G. Morgan. Simple probabilistic semantics for propositional K, T, B, S4 and S5. Journal of Philosophical Logic, 11: 442–458, 1982.
C. G. Morgan. There is a probabilistic semantics for every extension of classical sentence logic. Journal of Philosophical Logic, 11: 431–442, 1982.
C. G. Morgan. Probabilistic semantics for propositional modal logics. In H. Leblanc, R. D. Gumb, and R. Stern, editors, Essays in Epistemology and Semantics. Haven Publishing, NY, 1983.
A. Orenstein. Existence and the Particular Quantifier. Temple University Press, Philadelphia, PA, 1979.
C. Parsons. A plea for substitutional quantification. Journal of Philosophy, 68: 231–237, 1971.
K. R. Popper. Two autonomous axiom systems for the calculus of probabilities. British Journal of the Philosophy of Science, 6: 51–57, 1955.
K. R. Popper. Philosophy of science: a personal report. In A. C. Mace, editor, British Philosophy in Mid-Century, pages 155–191. Allen and Unwin, London, 1957.
K. R. Popper. The Logic of Scientific Discovery. Basic Books, New York, 1959.
W. V. Quine. Mathematical Logic. Norton, NY, 1940.
W. V. Quine. Ontological Relativity and Other Essays. Columbia University Press, New York and London, 1969.
F. P. Ramsey. The foundations of mathematics. Proceedings of the London Mathematical Society, Series 2, 25: 338–384, 1926.
F. P. Ramsey. Mathematical logic. The Mathematical Gazette, 13: 185–194, 1926.
R. Reichenbach. Wahrscheinlichkeitslehre. Leiden, 1935.
A. Rényi. On a new axiomatic theory of probability. Acta Mathematica Aca. Scient. Hungaricae, 6: 285–335, 1935.
A. Robinson. On the Mathematics of Algebra. North-Holland, Amsterdam, 1951.
J. B. Rosser. Logic for Mathematicians. McGraw Hill, NY, 1953.
P. K. Schotch and R. E. Jennings. Probabilistic considerations on modal semantics. Notre Dame Journal of Formal Logic, 22: 227–238, 1981.
K. Schütte. Syntactical and semantical properties of simple type theory. Journal of Symbolic Logic, 25: 305–326, 1960.
K. Schütte. Lecture Notes in Mathematical Logic, volume 1. Pennsylvania State University, 1962.
W. Seager. Probabilistic semantics, identity and belief. Canadian Journal of Philosophy, 12, 1983.
J. R. Shoenfield. Mathematical Logic. Addison-Wesley, Reading, MA, 1967.
T. A. Skolem. Logisch-Kombinatorische Untersuchungen über die Erfüllbarkeit und Beweisbarkeit mathematischer Sätze nebst einem Theorem über dichte Mengen. Skrifter utgit av Videnskapsselkapet i Kristiania, I. Mathematisknaturvidenskabelig klasse 1920, 4: 1–36, 1920.
R. M. Smullyan. First-Order Logic. Springer-Verlag, NY, 1968.
R. Stalnaker. Probability and conditionals. Philosophy of Science, 37: 64–80, 1970.
L. Stevenson. Frege’s two definitions of quantification. Philosophical Quarterly, 23: 207–223, 1973.
A. Tarski. Fondamentale Begriffe der Methodologie der deductiven Wissenschaften. I. Monatshefte für Mathematik und Physik, 37: 361–404, 1930.
A. Tarski. Grundzüge des systemkalkül. Fundamenta Math, 25, 26: 503–526, 1936.
A. Tarski. Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica, 1: 261–405, 1936.
R. H. Thomason. Studies in the formal logic of quantification. PhD thesis, Yale University, 1965.
B. C. Van Fraassen. Probabilistic semantics objectified. Journal of Philosophical Logic, 10: 371–394, 1981.
B. C. Van Fraassen. Quantification as an act of mind. Journal of Philosophical Logic, 11: 343–369, 1982.
G. R. Von Wright. The Logical Problem of Induction, volume Second, revised edition. MacMillan, NY, 1957.
A. N. Whitehead and B. Russell. Principia Mathematica. Cambridge University Press, Cambridge, 1910–1913.
L. Wittgenstein. Tratatus logico-philosphicus (Logischphilosophische Abhandlung). Annalen der Natruphilosophie, 14: 185–262, 1921.
E. W. Adams and B. Skyrms. A Primer of Probability Logic. CSLI Publications, 1998.
T. Hailperin. Sentential Probability Logic: Origins, Development, Current Status and Technical Applications. Lehigh University Press, 1996.
P. Roeper and H. Leblanc. Probability Theory and Probability Logic.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Leblanc, H. (2001). Alternatives to Standard First-Order Semantics. In: Gabbay, D.M., Guenthner, F. (eds) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0452-6_2
Download citation
DOI: https://doi.org/10.1007/978-94-017-0452-6_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5753-2
Online ISBN: 978-94-017-0452-6
eBook Packages: Springer Book Archive