Abstract
First-order logic augments the expressive power of propositional logic as it links the logical assertions to properties of objects of some non-empty universe: the domain of discourse. This is achieved by allowing the propositional symbols to take arguments that range over elements of the domain of discourse. These are now called predicate symbols and are interpreted as relations on the domain. Elements of the domain of discourse are denoted by terms built up from variables, constants, and functions applied to other terms. First-order logic also expands the lexicon of propositional logic with the quantifiers “for all” and “there exists” that are interpreted consistently with their natural language meaning.
This chapter is devoted to classical first-order logic. Our presentation is similar to the one conducted for propositional logic. We first define the syntax of first-order logic, followed by its semantics. Next we define a proof system for it and present the fundamental theoretical results of soundness and completeness. We also discuss the decision problems related to this logic. The remaining sections of the chapter cover variations and extensions of first-order logic, as well as first-order theories.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This argument assumes a fixed interpretation of equality, but can easily be rephrased without that assumption.
- 2.
This is an immediate corollary of Gödel’s incompleteness theorem, see Sect. 4.7.2. Note however that there exists an alternative semantics of second-order logic, Henkin models, that support complete proof systems and results such as compactness. Thisgeneral semantics, as opposed to thestandard semantics that we have been considering, essentially reduce second-order logic to many-sorted first-order logic [9].
- 3.
Note that in fact reflexivity is redundant as it follows from symmetry and transitivity.
- 4.
- 5.
- 6.
- 7.
References
Barendregt, H.P.: Lambda calculi with types. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, vol. 2, pp. 117–310. Oxford University Press, New York (1992)
Barrett, C., Tinelli, C.: CVC3. In: Damm, W., Hermanns, H. (eds.) Proceedings of the 19th International Conference on Computer Aided Verification (CAV ’07). Lecture Notes in Computer Science, vol. 4590, pp. 298–302. Springer, Berlin (2007)
Ben-Ari, M.: Mathematical Logic for Computer Science, 2nd edn. Springer, Berlin (2001)
Bertot, Y., Castran, P.: Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science. Springer, Berlin (2004)
Bradley, A.R., Manna, Z.: Calculus of Computation: Decision Procedures with Applications to Verification. Springer, Berlin (2007)
Conchon, S., Contejean, E., Kanig, J.: Ergo: A theorem prover for polymorphic first-order logic modulo theories (2006)
De Moura, L., Bjørner, N.: Z3: An efficient smt solver. In: Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS), 2008
Dutertre, B., De Moura, L.: The Yices SMT solver. Technical report, SRI (2006)
Enderton, H.B.: A Mathematical Introduction to Logic, 2nd edn. Academic Press, New York (2001)
Gallier, J.: Logic for Computer Science: Foundations of Automatic Theorem Proving. Wiley, New York (1986). Out of print. Corrected online version available from the authors webpagehttp://www.cis.upenn.edu/~jean/gbooks/logic.html
Goubault-Larrecq, J., Mackie, I.: Proof Theory and Automated Deduction. Applied Logic Series, vol. 6. Kluwer Academic, Dordrecht (1997)
Harrison, J.: Handbook of Practical Logic and Automated Reasoning. Cambridge University Press, Cambridge (2009)
Hedman, S.: A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity. Oxford Texts in Logic, vol. 1. Oxford University Press, Oxford (2004)
Huth, M., Ryan, M.: Logic in Computer Science: Modelling and Reasoning About Systems, 2nd edn. Cambridge University Press, Cambridge (2004)
Kroening, D., Strichman, O.: Decision Procedures: An Algorithmic Point of View. Springer, Berlin (2008)
Mendonça de Moura, L., Bjørner, N.: Satisfiability modulo theories: An appetizer. In: Vinicius Medeiros Oliveira, M., Woodcock, J. (eds.) SBMF. Lecture Notes in Computer Science, vol. 5902, pp. 23–36. Springer, Berlin (2009)
Paulson, L.: Isabelle: A Generic Theorem Prover. Lecture Notes in Computer Science, vol. 828. Springer, Berlin (1994)
Robinson, J.A., Voronkov, A. (eds.): Handbook of Automated Reasoning (in 2 volumes). Elsevier and MIT Press (2001)
Shankar, N.: Automated deduction for verification. ACM Comput. Surv.41(4), 1–56 (2009)
Smullyan, R.M.: First-Order Logic. Dover, New York (1995)
van Dalen, D.: Logic and Structure, 4th edn. Universitext. Springer, Berlin (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag London Limited
About this chapter
Cite this chapter
Almeida, J.B., Frade, M.J., Pinto, J.S., Melo de Sousa, S. (2011). First-Order Logic. In: Rigorous Software Development. Undergraduate Topics in Computer Science. Springer, London. https://doi.org/10.1007/978-0-85729-018-2_4
Download citation
DOI: https://doi.org/10.1007/978-0-85729-018-2_4
Publisher Name: Springer, London
Print ISBN: 978-0-85729-017-5
Online ISBN: 978-0-85729-018-2
eBook Packages: Computer ScienceComputer Science (R0)