COMPLETENESS FOR COUNTER-DOXA CONDITIONALS – USING RANKING SEMANTICS

Abstract

Standard conditionals $\varphi > \psi$, by which I roughly mean variably strict conditionals à la Stalnaker and Lewis, are trivially true for impossible antecedents. This article investigates three modifications in a doxastic setting. For the neutral conditional, all impossible-antecedent conditionals are false, for the doxastic conditional they are only true if the consequent is absolutely necessary, and for the metaphysical conditional only if the consequent is ‘model-implied’ by the antecedent. I motivate these conditionals logically, and also doxastically by properties of conditional belief and belief revision. For this I show that the Lewisian hierarchy of conditional logics can be reproduced within ranking semantics, provided we slightly stretch the notion of a ranking function. Given this, acceptance of a conditional can be interpreted as a conditional belief. The epistemic and the neutral conditional deviate from Lewis’ weakest system $V$, in that ID ($\varphi > \varphi$) or even CN ($\varphi > \top$) are dropped, and new axioms appear. The logic of the metaphysical conditional is completely axiomatised by $V$ to which we add the known Kripke axioms T5 for the outer modality. Related completeness results for variations of the ranking semantics are obtained as corollaries.

Definable Conditionals

“…A backdefinition (modulo backtranslation) of into the language with > and is 18 : ϕ > ψ := (ϕ ψ) ∨ (¬(ϕ ϕ) ∧ ¬(⊥ ψ)) (doxastic) ψ := ⊥ ϕ I analysed the doxastic and metaphysic conditional in ranking semantics (Raidl 2019), where is an S5 necessity. In fact, both are disjunctive conditionals.…”

“…, by f (w, A) = {v : κ w ({v}|A) = 0} if κ(A) < ∞ and else f (w, A) = ∅. For the converse, see the proof of Theorem 3.6 in Raidl (2019). Thus R ≈ M (Theorem 13).…”

“…C is believed given A (provided A is not crazy) if and only if κ(C|A) > 0. A semantics for conditionals can now be defined (Raidl 2019):…”

“…I used similar ideas inRaidl (2019), without theorising them.8 Where rule simulation is phrased in the obvious way.…”

Defeasible normative reasoning

“…The latter condition makes conditionals with impossible antecedents vacuously true. For a full treatment of those exceptional antecedents seeRaidl (2018).…”