NDLI: Does the deduction theorem fail for modal logic?

Content Provider	Springer Nature Link
Author	Hakli, Raul Negri, Sara
Copyright Year	2011
Abstract	Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction- and cut-free sequent calculus equivalent to the Hilbert system for basic modal logic shows the standard failure argument untenable by proving the underivability of $${\square\,A}$$ from A.
Starting Page	849
Ending Page	867
Page Count	19
File Format	PDF
ISSN	00397857
Journal	Synthese
Volume Number	187
Issue Number	3
e-ISSN	15730964
Language	English
Publisher	Springer Netherlands
Publisher Date	2011-03-29
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Deduction theorem Modal logic Sequent calculus Metaphysics Philosophy of Language Logic Epistemology Philosophy of Science
Content Type	Text
Resource Type	Article
Subject	Philosophy Social Sciences

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