Abstract
A deductive system is consistent iff no two contradictory formulae, A and ~ A, are derivable in the system. An equivalent formulation reads that no explicit contradiction A & ~ A is derivable. Another, seemingly weaker, condition is equivalent if the system is based on the classical propositional calculus: there exists an underivable formula. To prove the equivalence, assume that B is underivable and A is abritrary. If both A and ~ A were derivable then B would be derivable as well using modus ponens from the propositional tautology (Duns Scotus law): ~ A ⊃ (A⊃ B).
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
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1981 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Krajewski, S. (1981). Consistency. In: Marciszewski, W. (eds) Dictionary of Logic as Applied in the Study of Language. Nijhoff International Philosophy Series, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1253-8_15
Download citation
DOI: https://doi.org/10.1007/978-94-017-1253-8_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8257-2
Online ISBN: 978-94-017-1253-8
eBook Packages: Springer Book Archive