Abstract
Logic is concerned with reasoning and with establishing the validity of arguments. It allows conclusions to be deduced from premises according to logical rules, and a valid deductive argument establishes the truth of the conclusion provided that the premises are true. Logic plays a key role in reasoning and deduction in mathematics but is regarded as a separate discipline from mathematics. There were attempts in the early twentieth century to show that all mathematics can be derived from formal logic. However, the Austrian logician, Kurt Goedel showed that there are truths in the formal system of arithmetic that cannot be proved within the formal system (i.e. first-order arithmetic is incomplete).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Basic truth tables were first used by Frege, and developed further by Post and Wittgenstein.
- 2.
This institution is now known as University College Cork and has approximately 18,000 students.
- 3.
This is stated more formally that if H È {P} Q by a deduction containing no application of generalization to a variable that occurs free in P then H ├ P Þ Q.
- 4.
It is best to avoid undefinedness by taking care with the definitions of terms and expressions.
- 5.
The above expression would evaluate to true under Jones three-valued logic of partial functions.
- 6.
The above expression evaluates to true for Parnas logic (a two-valued logic).
- 7.
It is a little strange to assign the value false to the primitive predicate calculus expression y = 1/0.
- 8.
The approach avoids the undefined logical value (^) and preserves the two-valued logic.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2013 Springer-Verlag London
About this chapter
Cite this chapter
O’Regan, G. (2013). Logic. In: Mathematics in Computing. Springer, London. https://doi.org/10.1007/978-1-4471-4534-9_3
Download citation
DOI: https://doi.org/10.1007/978-1-4471-4534-9_3
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4533-2
Online ISBN: 978-1-4471-4534-9
eBook Packages: Computer ScienceComputer Science (R0)