On truth-gaps, bipolar belief and the assertability of vague propositions

Borderlines and probabilities of borderlines: On the interconnection between vagueness and uncertainty

“…These results improve on a characterisation result for supervaluations given in [15] which required the additional axiom that ∀θ,…”

“…Hence, by induction v cbs (θ) = 1 and v cbs (ϕ) = 1 and by definition 2 v cbs (θ ∧ ϕ) = 1 as required. Also, if v cbs (θ ∧ ϕ) = 1 then by definition 2 v cbs (θ) = 1 and v cbs (ϕ) = 1 which implies by induction that Theorem 22 is related to an existing result in [15] which, while similar, only holds for sentences in SL A where either A = P or A = {¬p i : p i ∈ P} although it does hold for a slightly broader class of supervaluations.…”

“…We prove a number of results for conditional supervaluation and Kleene belief pairs under different assumptions. In some cases the work presented extends results and employs definitions which have already appeared in the literature including in [14], [15], [16] and [17]. Throughout the paper we will, where appropriate, note the nature and scope of this extension.…”

“…As noted by Keefe and Smith [9], vagueness is a multifaceted phenomenon and vague predicates exhibit blurred boundaries as well as borderline cases. We have consistently argued that the former can be understood as resulting from a type of epistemic uncertainty about what is the correct definition of predicates in language [13], [15], [17]. This semantic or linguistic uncertainty [12] naturally results from the distributed manner in which language is learned through repeated interactions between individuals [23], [13].…”

“…Belief Pairs [14], [15]: Let V be a finite set of three valued valuations and w be a probability distribution on V then we define a belief pair as a pair of lower and upper measures µ = (µ, µ) where µ, µ : SL → [0, 1] such that ∀θ ∈ SL; µ(θ) = w({v ∈ V : v(θ) = 1}) and µ(θ) = w({v ∈ V : v(θ) = 0}). 9 It can also be interesting to consider mid-point belief degrees generated from a belief pair by taking the average of the lower and upper measures as follows:…”