Abstract
This chapter presents some advanced topics in logic including fuzzy logic, temporal logic, intuitionistic logic, undefined values, theorem provers and the applications of logic to AI. Fuzzy logic is an extension of classical logic that acts as a mathematical model for vagueness. Temporal logic is concerned with the expression of properties that have time dependencies, and it allows temporal properties about the past, present and future to be expressed. Intuitionism was a controversial theory on the foundations of mathematics based on a rejection of the law of the excluded middle, and an insistence on constructive existence. We discuss three approaches to deal with undefined values, including the logic of partial functions; Dijkstra’s approach with his cand and cor operators and Parnas’s approach which preserves a classical two-valued logic.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
It is best to avoid undefinedness by taking care with the definitions of terms and expressions.
- 2.
The above expression would evaluate to true under Jones’ three-valued logic of partial functions.
- 3.
The above expression evaluates to true for Parnas logic (a two-valued logic).
- 4.
It seems strange to assign the value false to the primitive predicate calculus expression y = 1/0.
- 5.
The approach avoids the undefined logical value (⊥) and preserves the two-valued logic.
- 6.
John McCarthy received the Turing Award in 1971 for his contributions to Artificial Intelligence. He also developed the programming language LISP.
- 7.
First-order logic allows quantification over objects but not functions or relations. Higher order logics allow quantification of functions and relations.
- 8.
For example, the statement ∃x such that x = \( \surd{4} \) states that there is an x such that x is the square root of 4, and the constructive existence yields that the answer is that x = 2 or x = −2, i.e. constructive existence provides more the truth of the statement of existence, and an actual object satisfying the existence criteria is explicitly produced.
References
Heyting A (1966) Intuitionist logic. An introduction. North-Holland Publishing
Jones C (1986) Systematic software development using VDM. Prentice Hall International
Martin Löf P (1984) Intuitionist type theory. Notes by Giovanni Savin of lectures given in Padua, June, 1980. Bibliopolis. Napoli
McCarthy J (1959) Programs with common sense. In: Proceedings of the Teddington conference on the mechanization of thought processes
Parnas DL (1993) Predicate calculus for software engineering. IEEE Trans Softw Eng 19(9)
Stanford Encyclopedia of Philosophy. Temporal logic. http://plato.stanford.edu/entries/logic-temporal/
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
O’Regan, G. (2020). Advanced Topics in Logic. In: Mathematics in Computing. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-030-34209-8_17
Download citation
DOI: https://doi.org/10.1007/978-3-030-34209-8_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-34208-1
Online ISBN: 978-3-030-34209-8
eBook Packages: Computer ScienceComputer Science (R0)