Abstract
The most common logical system taught, used, and studied today is Elementary predicate logic, otherwise known as first-order logic (see Hodges’ chapter in this Volume). First-order logic has a well-studied proof theory and model theory, and it enjoys a number of interesting properties. There is a recursively-enumerable deductive system D2 such that any first-order sentence Φ is a consequence of a set Γ of first-order sentences if and only if Φ is deducible from Γ in D1. Thus, first-order logic is (strongly) complete. It follows that first-order logic is compact in the sense that if every finite subset of a set Γ of first-order sentences is satisfiable then Γ itself is satisfiable. The downward Löwenheim-Skolem theorem is that if a set Γ of first-order sentences is satisfiable, then it has a model whose domain is countable (or the cardinality of G, whichever is larger). The upward Löwenheim-Skolem theorem is that if a set Γ of first-order sentences has, for each natural number n, a model whose domain has at least n elements, then for any infinite cardinal κ, Γ has a model whose domain is of size at least κ (see Hodges’ chapter, and virtually any textbook in mathematical logic, such as Boolos and Jeffrey [1989] or Mendelson [1987]).
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Bibliography
J. Barwise. On branching quantifiers in English, Journal of Philosophical Logic, 8, 47–80, 1979.
J. Barwise. Model-theoretic logics: background and aims. In Model-Theoretic Logics, J. Barwise and S. Feferman, eds. pp. 3–23. Springer Verlag, New York, 1985.
J. Barwise and S. Feferman, eds. Model-Theoretic Logics, Springer-Verlag, New York, 1985.
J. Bell and A. Slomson. Models and Ultraproducts: An Introduction. North Holland Publishing Company Amsterdam, 1971.
G. Boolos. To be is to be a value of a variable (or to be some values of some variables). Journal of Philosophy, 81, 430–449, 1984.
G. Boolos. Nominalist platonism. The Philosophical Review, 94, 327–344, 1985.
G. Boolos. Reading the Begriffsschrift. Mind, 94, 331–344, 1985.
G. Boolos and R. Jeffrey. Computability and Logic, third edition. Cambridge University Press, Cambridge, 1989.
C. Chang. A note on the two cardinal problem. Proceedings of the American Mathematical Society, 16, 1148–1155, 1965.
C. Chang and H. J. Keisler. Model Theory. North Holland Publishing Company, Amsterdam, 1973.
A. Church. Introduction to Mathematical Logic. Princeton University Press, Princeton, 1973.
J. Corcoran. Categoricity. History and Philosophy of Logic, 1, 187–207, 1980.
J. Cowles. The relative expressive power of some logics extending first-order logic. Journal of Symbolic Logic, 44, 129–146, 1979.
R. Dedekind. Was sind und was sollen die Zahlen?, Vieweg, Brunswick, 1888; tr. as The nature and meaning of numbers. In Essays on the Theory of Numbers, W. W. Beman, ed. pp. 31–115, Dover Press, New York, 1963.
M. A. Dickmann. Larger infinitary languages. In Model-Theoretic Logics, J. Barwise and S. Feferman, eds. pp. 317–363. Springer Verlag, New York, 1985.
B. Dreben and W. Goldfarb. The Decision Problem: Solvable Classes of Quantificational Formulas. Addison-Wesley Publishing Company, Inc., London, 1979.
J. M. Dunn and N. Belnap. The sugstitution interpretation of the quantifier. Nous, 2, 177–185, 1968.
H. D. Ebbinghaus. Extended logics: The general framework. In Model-Theoretic Logics, J. Barwise and S. Feferman, eds. pp. 25–76. Springer Verlag, New York, 1985.
R. Fagin. Generalized first-order spectra and polynomial-time recognizable sets. SIAM-AMS Proceedings, 7, 43–73, 1974.
S. Feferman. Theories of finite type related to mathematical practice. In Handbook of Mathematical Logic, J. Barwise, ed. pp. 913–971. North Holland, Amsterdam, 1977.
H. Field. Deflationist views of meaning and content. Mind, 103, 249–285, 1994.
J. Flum. Characterizing logics. In Model-Theoretic Logics, J. Barwise and S. Feferman, eds. pp. 77–120. Springer Verlag, New York, 1985.
G. Frege. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Louis Nebert, Halle, 1879. In From Frege to Gödel, J. van Heijenoort, ed. pp. 1–82. Harvard University Press, Cambridge, Massachusetts, 1967.
D. Gabbay and J. Moravcsik. Branching quantifiers, English, and Montague grammar. Theoretical Linguistics, 1, 141–157, 1974.
R. Gandy. The confluence of ideas in 1936. In The Universal Turing Machine, R. Herken ed. pp. 55–111. Oxford University Press, New York, 1988.
D. Gottlieb. Ontological Economy: Substitutional Quantification and Mathematics. Oxford University Press Oxford, 1980.
Y. Gurevich. Monadic second-order theories. In Model-Theoretic Logics, J. Barwise and S. Feferman, eds. pp. 479–506. Springer Verlag, New York, 1985.
Y. Gurevich. Logic and the challenge of computer science. In Trends in Theoretical Computer Science, Egon Börger, ed. pp. 1–57, Computer Science Press, Maryland, 1988.
Y. Gurevich and S. Shelah. Interpreting second-order logic in the monadic theory of order. Journal of Symbolic Logic, 48, pp. 816–828, 1983.
L. Henkin. Banishing the rule of substitution for functional variables. Journal of Symbolic Logic, 18, 201–208, 1953.
D. Hilbert. Über ber das Unendliche. Mathematische Annalen, 95, 16–190, 1925. tr. as “On the infinite”, in From Frege to Gödel, J. van Heijenoort, ed. pp. 369–392. Harvard University Press, Cambridge, Massachusetts, 1967.
J. Hintikka. Partially ordered quantifiers vs. partially ordered ideas. Diabetica. 30, 89–99, 1976.
N. Immerman. Languages that capture complexity classes. SIAM Journal of Computing, 16, 760–778, 1987.
I. Jané. A critical appraisal of second-order logic. History and Philosophy of Logic, 14, 67–86, 1993.
R. B. Jensen. The fine structure of the constructible hierarchy. Annals of Mathematical Logic, 4, 229–308, 1972.
M. Kaufmann. The quantifier ‘there exist uncountably many’ and some of its relatives. In Model-Theoretic Logics, J. Barwise and S. Feferman, eds. pp. 123–176. Springer Verlag, New York, 1985.
P. Kolaitis. Game quantification. In Model-Theoretic Logics, J. Barwise and S. Feferman, eds. pp. 365–421. Springer Verlag, New York, 1985.
M. Krynicki and M. Mostowski. Henkin quantifiers. In Quantifiers: Logics, Models and Computation 1, M. Krynicki, M. Mostowski and L. Szczerba, eds. Kluwer Academic Publishers, Dordrecht, Holland, 1995.
F. Landman. Groups. Linguistics and Philosophy, 12, 559–605, 723–744, 1989.
S. Lavine. Understanding the Infinite. Harvard University Press, Cambridge, Massachusetts, 1994.
H. Leblanc. Truth-value Semantics, North Holland Publishing Company, Amsterdam, 1976.
D. Leivant. Descriptive characterizations of computational complexity. Journal of Computer and System Sciences, 39, 51–83, 1989.
D. Lewis. Parts of Classes. Blackwell, Oxford, 1991.
P. Lindström. On extensions of elementary logic. Theoria, 35, 1–11, 1969.
L. Löwenheim. Über Möglichkeiten im Relativkalkül. Mathematische Annalen, 76, 447–479, 1915. tr. in From Frege to Gödel, J. van Heijenoort, ed. pp. 228–251. Harvard University Press, Cambridge, Massachusetts, 1967.
E. Mendelson. Introduction to Mathematical Logic, third edition. van Nostrand, Princeton, 1987.
D. Mundici. Other quantifiers: an overview. In Model-Theoretic Logics, J. Barwise and S. Feferman, eds. pp. 211–233. Springer Verlag, New York, 1985.
M. Nadel. ℒω 1 ω and admissible fragments. In Model-Theoretic Logics, J. Barwise and S. Feferman, eds. pp. 271–316. Springer Verlag, New York, 1985.
W. V. O. Quine. From a Logical Point of View. Harper and Row, New York, 1953.
W. V. O. Quine. Philosophy of Logic, second edition. Prentice-Hall, Engle-wood Cliffs, New Jersey, 1986.
M. Rabin. Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society, 141, 1–35, 1969.
M. Resnik. Second-order logic still wild. Journal of Philosophy, 85, 75–87, 1988.
J. H. Schmerl. Transfer theorems and their applications to logics. In Model-Theoretic Logics, J. Barwise and S. Feferman, eds. pp. 177–209. Springer Verlag, New York, 1985.
S. Shapiro. Foundations Without Foundationalism: A Case for Second-order Logic. Oxford University Press, Oxford, 1991.
S. Shelah. The monadic theory of order. Annals of Mathematics, 102, 379–419, 1975.
G. Sher. The Bounds of Logic. The MIT Press, Cambridge, Massachusetts, 1991.
S. Simpson. Friedman’s research on subsystems of second order arithmetic. In Harvey Friedman’s Research on the Foundations of Mathematics, L. A. Harrington et al. (eds.). North Holland Publishing Company, Amsterdam, 1985.
T. Skolem. Begrundung der elementaren Arithmetik durch die rekurrierende Denkweise. Videnskapsselskapets skrifter I. Matematisk-naturvidenskabelig klasse, no. 6. tr. as ‘The foundations of arithmetic established by the recursive mode of thought’ in In From Frege to Gödel, J. van Heijenoort, ed. pp. 303–333. Harvard University Press, Cambridge, Massachusetts, 1967.
A. Tarski. On the concept of logical consequence. In Logic, Semantics and Metamathematics, A. Tarski, pp. 417–429. Clarendon Press, Oxford, 1956.
A. Tarski. What are logical notions? (ed by John Corcoran). History and Philosophy of Logic, 7, 143–154, 1986.
J. Väänänen. Set-theoretic definability of logics. In Model-Theoretic Logics, J. Barwise and S. Feferman, eds. pp. 599–643. Springer Verlag, New York, 1985.
J. Van Heijenoort, ed. From Frege to Gödel. Harvard University Press, Cambridge, Massachusetts, 1967.
S. Wagner. The rationalist conception of logic. Notre Dame Journal of Formal Logic, 28, 3–35, 1987.
E. Zermelo. Über stufen der Qualifikation und die Logik des Unendlichen. Jahresbericht Deutsche Mathematische Verein, 31, 85–88, 1931.
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Shapiro, S. (2001). Systems between First-Order and Second-Order Logics. In: Gabbay, D.M., Guenthner, F. (eds) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9833-0_2
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