Possibility and Dyadic Contingency

Abstract

The paper aims at developing the idea that the standard operator of noncontingency, usually symbolized by Δ, is a special case of a more general operator of dyadic noncontingency Δ(−, −). Such a notion may be modally defined in different ways. The one examined in the paper is Δ(B, A) = _df ◊B ∧ (A ⥽ B ∨ A ⥽ ¬B), where ⥽ stands for strict implication. The operator of dyadic contingency ∇(B, A) is defined as the negation of Δ(B, A). Possibility (◊A) may be then defined as Δ(A, A), necessity (□A) as ∇(¬A, ¬A) and standard monadic noncontingency (ΔA) as Δ(${\textsf{T}}$, A). In the second section it is proved that the deontic system KD is translationally equivalent to an axiomatic system of dyadic noncontingency named KDΔ², and that the minimal system KΔ of monadic contingency is a fragment of KDΔ². The last section suggests lines for further inquiries.

Appendix

T1. Let L be is a normal modal logic extended with the definition Δ⁻(A, B) = _df □(A ⊃ B) ∨ □(A ⊃ ¬B). Suppose that among the frames for L there is a frame containing some worlds with no R-successors and a frame containing a world with exactly one R-successor. Then □ is not definable in terms of Δ⁻.

Proof

Let us consider the disjoint union of two such frames, since the exposed condition is equivalent to requiring that there is a frame for the logic L containing both a world with no successors and a world with exactly one successor, however many additional worlds are related. For simplicity we will assume the worlds w₀, w₁ come from a single frame, w_i having i worlds R-related to it (i = 0; 1). This means that w_i is either and end-point or there is only one world w_j s.t. w_iRw_j. Construct a model M on this frame by having w₀ verify the same propositional variables as w₁. For the present purposes a convenient choice will be to stipulate, for each p_j and each w_i, that V(p_j, w_i) = 0, i.e. that each variable p_j is false at every world of M. Then by induction on the construction of C, it turns out that all formulas C whose form is ¬A, A ∧ B, have value 1 at all the worlds of M, or value 0 at all of them. As far as Δ⁻- formulas are concerned, since Δ⁻(A, B) is equivalent to □(A ⊃ B) ∨ □(A ⊃ ¬B), it is straightforward to see that all such formulas have value 1 at w₀ and w₁ in M or in any other model on the same frame (since it takes two R-successors to falsify such a formula). To sum up, for every wff C containing the connectives ¬, ∧, Δ⁻ we are able to prove that V(C, w₀) = V(C, w₁). But this property does not hold for □-formulas. In fact, since w₀ is a dead end, we have in M that V(□p₁, w₀) = 1; but V(□p_1, w₁) = 0 since there is a w_j s.t. w₁Rw_j and p₁ has value 0 at w_j. Thus no formula constructed using the Boolean connectives and Δ⁻ can be provably equivalent in L to □p₁. ■