Neighborhood Contingency Logic - Logic and Its Applications - page 88

Neighborhood Contingency Logic

Jie Fan 1 and Hans van Ditmarsch 2

1 Department of Philosophy, Peking University
[e-mail]
2 LORIA, CNRS — Universite de Lorraine
[e-mail]

Abstract. A formula is contingent, if it is possibly true and possibly
false; a formula is non-contingent, if it is not contingent, i.e., if it is
necessarily true or necessarily false. In this paper, we propose a neigh-
borhood semantics for contingency logic, in which the interpretation of
the non-contingency operator is consistent with its philosophical intu-
ition. Based on this semantics, we compare the relative expressivity of
contingency logic and modal logic on various classes of neighborhood
models, and investigate the frame definability of contingency logic. We
present a decidable axiomatization for classical contingency logic (the
obvious counterpart of classical modal logic), and demonstrate that for
contingency logic, neighborhood semantics can be seen as an extension
of Kripke semantics.

1 Introduction

Like necessity and possibility, contingency is a very important notion in philo-
sophical logic. This notion goes back to Aristotle, who develops a logic of state-
ments about contingency [2]. As first defined in [11], a formula is contingent ,if
it is possibly true and possibly false; otherwise, it is non-contingent , i.e., if it
is necessarily true or necessarily false. (Non-)Contingency also arose in the area
of epistemic logic but with different terminology: ignorance [8,14] and 'know-
ingwhether'[6];'aformula ˕ is non-contingent' there means 'the agent knows
whether ˕ ', and ' ˕ is contingent' there means 'the agent is ignorant about ˕ '.
Non-contingency can be defined with necessity, namely as ʔ ˕ = df ˕ .

But necessity is not always definable in terms of non-contingency [4, 5, 11].
Moreover, a known diculty for contingency logic is the absence of axioms
characterizing Kripke frame properties, which makes it hard to find axiomat-
izations of contingency logic over various classes of Kripke frames (refer to [5]
and the reference therein). As shown in [6], contingency logic is not normal, since
ʔ( ˕

˕ ∨ ¬

ʔ ˈ ) is invalid. This suggests that it may be interesting to
investigate neighborhood semantics for contingency logic.
Neighborhood semantics was proposed independently by Scott and Montague
in 1970 [10,13]. Since it was introduced, neighborhood semantics has become a
standard semantics tool for studying non-normal modal logics [3]. To our know-
ledge, Steinsvold can be said to have been the first to have explored neighbor-
hood semantics for contingency logic [14]. He gave a topological semantics for

→ ˈ ) → (ʔ ˕ →