Abstract
This chapter presents some advanced topics in logic including fuzzy logic, temporal logic, intuitionistic logic, undefined values, 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; Diijkstra’s approach; and Parnas’s approach which preserves a classical two-valued logic.
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Notes
- 1.
David Hilbert was a famous German mathematician, and Hilbert’s program is discussed in Chap. 14.
- 2.
It is best to avoid undefinedness by taking care with the definitions of terms and expressions.
- 3.
The above expression would evaluate to true under Jones three-valued logic of partial functions.
- 4.
The above expression evaluates to true for Parnas logic (a two-valued logic).
- 5.
It seems strange to assign the value false to the primitive predicate calculus expression y = 1/ 0.
- 6.
The approach avoids the undefined logical value (⊥) and preserves the two-valued logic.
- 7.
John McCarthy received the Turing Award in 1971 for his contributions to artificial intelligence. He also developed the programming language LISP.
- 8.
First-order logic allows quantification over objects but not functions or relations. Higher-order logics allow quantification of functions and relations.
- 9.
For example, the statement ∃x such that x = √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
Temporal logic. Stanford enclyopedia of philosophy. http://plato.stanford.edu/entries/logic-temporal/
Heyting A (1966) Intuitionist logic. An introduction. North-Holland Publishing
Löf PM. Intuitionist type theory. Notes by Giovanni Savin of lectures given
Parnas DL (1993) Predicate calculus for software engineering. IEEE Trans Softw Eng 19(9)
Jones C (1986) Systematic software development using VDM. Prentice Hall International
McCarthy J (1959) Programs with common sense. In: Proceedings of the Teddington conference on the mechanization of thought processes
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O’Regan, G. (2023). Advanced Topics in Logic. In: Mathematical Foundations of Software Engineering. Texts in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-031-26212-8_11
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