Non-contingency logic and almost-definability

55 Pages·2014·0.66 MB·English

Introduction Non-contingencylogicandalmost-definability BisimulationandCharacterization Axiomatization MultimodalNon-contingencyLogic Conclusion
Almost Necessary
Jie Fan
2014.9.23
Department of Philosophy, Peking University
Introduction Non-contingencylogicandalmost-definability BisimulationandCharacterization Axiomatization MultimodalNon-contingencyLogic Conclusion
Reference
J. Fan, Y. Wang and H. van Ditmarsch. Almost Necessary.
Advances in Modal Logic, Groningen, The Netherlands,
Volume 10, pages 178-196, 2014.
J. Fan, Y. Wang and H. van Ditmarsch. Contingency and
Knowing Whether. To appear in The Review of Symbolic
Logic.
Introduction Non-contingencylogicandalmost-definability BisimulationandCharacterization Axiomatization MultimodalNon-contingencyLogic Conclusion
1 Introduction
2 Non-contingency logic and almost-definability
3 Bisimulation and Characterization
4 Axiomatization
5 Multimodal Non-contingency Logic
6 Conclusion
Introduction Non-contingencylogicandalmost-definability BisimulationandCharacterization Axiomatization MultimodalNon-contingencyLogic Conclusion
Contingency: an example
Will there be sea battles tomorrow? No!
Will there be no sea battles tomorrow? No!
Why is it the case?
The proposition “there will be sea battles tomorrow” (P) is
contingent, i.e., it is possible that P and it is possible that
not P.
Introduction Non-contingencylogicandalmost-definability BisimulationandCharacterization Axiomatization MultimodalNon-contingencyLogic Conclusion
Relating necessity and (non-)contingency
(Non-)contingency can be defined with necessity:
∆ϕ = (cid:50)ϕ∨(cid:50)¬ϕ.
df
∇ϕ = ¬∆ϕ.
df
∆: non-contingency
∇: contingency
(cid:50): necessity
Introduction Non-contingencylogicandalmost-definability BisimulationandCharacterization Axiomatization MultimodalNon-contingencyLogic Conclusion
Q1: the definability of (cid:50)
Is (cid:50) definable in terms of ∆?
1 (cid:50)ϕ=df ∆ϕ∧ϕ [Montgomery and Routley, 1966], only in the
systems containing (cid:50)ϕ→ϕ [Segerberg, 1982, page 128]
2 (cid:50) can only be defined in terms of ∆ in the Verum system, or
the systems containing (cid:50)ϕ→(cid:51)ϕ [Cresswell, 1988]
3 With Contingency Postulate, (cid:50)ϕ=df ∀p(∆(p∧ϕ)→∆p)
[Pizzi, 1999]
4 With the axiom ∇τ (τ is a propositional constant),
(cid:50)ϕ= ∆ϕ∧∆(τ →ϕ) [Pizzi, 2006, Pizzi, 2007]
df
5 (cid:2)ϕ=df (cid:86)ψ∈NCL∆(ψ →ϕ), (cid:2) behaves like, but differs from
(cid:50) [Zolin, 2001]
None of them is satisfactory!
Introduction Non-contingencylogicandalmost-definability BisimulationandCharacterization Axiomatization MultimodalNon-contingencyLogic Conclusion
Q2: fragment
∆ϕ = (cid:50)ϕ∨(cid:50)¬ϕ
df
NCL ⊆ ML
What is the exact fragment?
Introduction Non-contingencylogicandalmost-definability BisimulationandCharacterization Axiomatization MultimodalNon-contingencyLogic Conclusion
Q3: axiomatizing NCL over symmetric frames
Usual frame classes Known results
K [Humberstone, 1995, Kuhn, 1995, Zolin, 1999]
D [Humberstone, 1995, Zolin, 1999]
T [Montgomery and Routley, 1966]
4 [Kuhn, 1995, Zolin, 1999]
5 [Zolin, 1999]
B ?
Introduction Non-contingencylogicandalmost-definability BisimulationandCharacterization Axiomatization MultimodalNon-contingencyLogic Conclusion
Main results of the paper
Is (cid:50) definable in terms of ∆ in the general case?
—– almost-definability schema
How to characterize non-contingency logic within modal logic?
—– ∆-bisimulation and characterization fragement
axiomatizing NCL on B
—– axiomatization NCLB and completeness
Multimodal Non-contingency Logic: axiomatization and
completeness
Introduction Non-contingencylogicandalmost-definability BisimulationandCharacterization Axiomatization MultimodalNon-contingencyLogic Conclusion
Non-contingency logic: Language
NCL ϕ ::= (cid:62) | p | ¬ϕ | (ϕ∧ϕ) | ∆ϕ
ML ϕ ::= (cid:62) | p | ¬ϕ | (ϕ∧ϕ) | (cid:50)ϕ
∆ϕ: it is non-contingent that ϕ.
∇ϕ = ¬∆ϕ: it is contingent that ϕ.
df