Neighborhood Contingency Logic - Logic and Its Applications - page 89

the logic of ignorance (the epistemic counterpart of contingency), in which non-
contingency operator is interpreted essentially as

˕ . His topological
models correspond to S4 Kripke models. He did not refer to the tradition on
contingency logic in his work. In this paper, we present a neighborhood semantics
for contingency logic on a much wider range of model classes.
In Section 2, we define contingency logic and a neighborhood semantics for it.
Sections 3, 4, and 5 contain our main contributions. In Section 3 we compare the
relative expressivity of contingency logic and modal logic over various classes of
neighborhood models, and investigate the frame definability of contingency logic.
Section 4 completely axiomatizes a decidable contingency logic over the class of
all neighborhood frames. This logic is called classical contingency logic ,which
is also characterized by another class of neighborhood frames. Section 5 deals
with the relationship between neighborhood semantics and Kripke semantics for
contingency logic. We conclude with some future work in Section 6.

˕ ∨ ¬

2 Syntax and Neighborhood Semantics

Let us first recall the language of contingency logic, which is a fragment of
the following logical language with both the necessity operator and the non-
contingency operator as primitive modalities.

Definition 1 (Languages CML, ML and CL). Given a set P of proposi-

tional variables, the logical language CML is defined recursively as:

˕ ::= | p |¬ ˕ | ( ˕ ∧ ˕ ) | ʔ ˕ | ˕

where p .

Without the construct ʔ ˕, we obtain the language ML of modal logic ; without
the construct

∈ P

˕, we obtain the language CL of contingency logic .

We always omit the parentheses whenever convenient. Formula ʔ ˕ is read as
“it is non-contingent that ˕ ”, and ˕ is read as “it is necessary that ˕ ”. Other
operators are defined as usual; in particular, ∇˕ is defined as ¬ ʔ ˕ , for which we
read “it is contingent that ˕ .” Note that ∇ is defined as the negation, rather than
the dual, of ʔ, although we will see from the neighborhood semantics below that
¬ ˕ . In this paper, we will mainly focus on the language

CL which has ʔ as the only primitive modality.

ʔ ˕ is equivalent to ¬ ʔ ¬

Definition 2 (Neighborhood Structure). A neighborhood model is a tuple
M ,whereS is a nonempty set of possible worlds called the domain ,

N is a neighborhood function from S to

= S, N, V

P ( P ( S )) , V is a valuation function

assigning a set of worlds V ( p ) ↆ S to each p ∈ P . Given a world s ∈ S,

the pair ( ,s ) is a pointed model ; we will omit these parentheses whenever

convenient. We also write s

M

S. A neighborhood frame is
a neighborhood model without valuation. Sometimes we write model and frame
without 'neighborhood'.

∈M to denote s ∈