Basics of Second-Order Predicate Logic




logic, symbolic logic, predicate calculus, second-order predicate logic


The article presents the basics of second-order predicate logic (SOL). The need for a symbolic representation of the general quantifier is pointed out. A distinction is made between the first-order predicate logic (FOL) and the second-order predicate logic (predicates of predicates, relations of relations). The syntax and semantics of the second-order predicate logic are introduced. Logical and non-logical designators and operators, terms, rules for forming logical formulas, status of variables, and rules for variable substitution are introduced. Reference is made to Henkin's semantics of controlled predicates, and an axiomatic system of second-order predicate logic. Russell's analogy for the axiom of choice and methods of proving the validity of the deduction for second-order predicate logic are given.


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Cantor, G. ( 1883)."Die Grundlagen eine allgemeine Manningfaltigkeitslehre" in:Cantor, George: (1932) Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Hildesheim. Georg Olms.

Carnap, R. (1958).Introduction to Symolic Logic and its Applications. New York: Dover Publications, Inc.

Fisk, M (1964, p.63) A Modern Formal Logic. Prentice-Hall Inc., Englewood Cliffs, New Jersey.

Fitting, Melvin (2002): Types, Tableaus, and Gödel’s God. Kluwer Academic Publishers DOI:

Frege, G. (1879). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle, 1879 (Concept script, a formula language, modeled upon that of arithmetic, for pure thought).

Hilbert, D., Ackermann (1967). Grundzüge der tehoretischen Logik. DOI:

Hilbert, D., Ackermann, W., Luce, R. E (1950): Principles of Mathematical Logic. American mathematical Society Chelsea, R.I.

Jeffrey, R. C., Burgess , J. P. (2006): Formal Logic. Its Scope and Limits. Hackett Publishing Company, Inc., pp.125 –

Manzano, M., Sain, I., Alonso, E. (eds.) (2014): The Life and Work of Leon Henkin. Essays on His Contributions. Birkhäuser Basel DOI:

Moore, G. H. (1982): Zermelo’s Axiom of Choice. Its Origins, Development, and Influence.Springer

Quine, W.V.O. (1970). Philosophy of Logic . Englewood Cliffs, N.J. : Prentice-Hall, Inc.

Rossberg, M. (2004). ''First-Order Logic, Second-Order Logic, and Completeness.'' Hendricks et al. (eds.): First-Order Logic Revisited. Berlin: Logos Verlag , 303–321.

Russell, B. (1919). Introduction to mathematical Logic (London, 1919) in chapter XII (Selections and the Multiplicative Axiom),

Shapiro, S. (1991): Foundations without Foundationalism. A Case for Second Order Logic. Clarendon Press

Shapiro, S. (2002): ''Classical Logic II: Higher-Order Logic''. In: Goble, Lou ed. (2001): The Blackwell Guide to Philosophical Logic. Blackwell Publishers. pp. 33-55. DOI:

Väänänen J. (2007): Dependence logic. New approach to independence friendly logic. Cambridge University Press DOI:




How to Cite

Ibrulj, N. (2023). Basics of Second-Order Predicate Logic . THE LOGICAL FORESIGHT - Journal for Logic and Science, 3(1), 1–14.

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