Abstract
Until now, we have discussed connections between particular logics and corresponding algebras, e.g., between classical logic and Boolean algebras, and also between intuitionistic logic and Heyting algebras.
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.
For example, if \(\alpha \) is \(p \rightarrow (q \vee r)\), \(h(p) = [\gamma ], h(q) = [\delta ]\) and \(h(r) = [\sigma ]\), then \(h(\alpha ) = [\gamma ] \rightarrow _\mathbf{A} ([\delta ] \vee _\mathbf{A} [\sigma ]) = [\gamma \rightarrow (\delta \vee \sigma )]\), where \(\mathbf{A}\) is \(\mathbf{F}_\mathbf{L}\).
- 2.
It should be noted that every power set Boolean algebra is isomorphic to a direct product of \(\mathbf 2\) and vice versa.
- 3.
Recall that \(f_i\) is an operation symbol of \(\mathscr {L}\), while \(f_i^\mathbf{A}\) is the corresponding operation in \(\mathbf A\) which determines an interpretation of \(f_i\). See Sect. 6.2.
- 4.
By abuse of symbols, here we use \(\vee \) for both an algebraic operation and an operation symbol.
- 5.
For a precise definition, see e.g. Burris and Sankappanavar (1981).
- 6.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Ono, H. (2019). Logics and Varieties. In: Proof Theory and Algebra in Logic. Short Textbooks in Logic. Springer, Singapore. https://doi.org/10.1007/978-981-13-7997-0_8
Download citation
DOI: https://doi.org/10.1007/978-981-13-7997-0_8
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-7996-3
Online ISBN: 978-981-13-7997-0
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)