Cobreros, Égré, Ripley and van Rooij 2012: Tolerant, classical, strict #

A T-model is a classical model whose predicates each carry an indifference relation ∼_P, reflexive and symmetric but possibly non-transitive (Definition 4), the vague predicate's "looks the same with respect to P". Besides classical truth a T-model supports two further notions defined by simultaneous induction (Definition 9): Pa is strictly true when every ∼_P-neighbour of a is classically P and tolerantly true when some neighbour is, negation swapping the two, so that the strict extension sharpens and the tolerant extension coarsens the classical one (Lemma 1) and the tolerance principle Pa ∧ a I_P b → Pb is tolerantly valid (Fact 2) up to two similarity steps (Fact 3). Borderline cases are the tolerantly-but-not-strictly P individuals (Definition 10), the ones tolerantly both P and ¬P. On the similarity-free vocabulary tolerant validity is classical validity while nothing is strictly valid (Theorems 1 and 2), so strict validity is not Fine's supervaluationist validity, which is classical; and tolerant and strict consequence are Priest's LP and Kleene's strong K3 (Theorem 3), via the model correspondences of Lemmas 4 and 5. Mixing the three notions gives nine consequence relations ⊨ᵐⁿ (Definition 17), ordered by Lemma 7, dualised by Lemma 6, collapsing to four on the restricted vocabulary (Lemmas 8 and 9) but distinct once similarity statements are expressible (§3.4), where the deduction theorem singles out the self-dual st, cc and ts (Lemma 10). A relation is reflexive iff its premise standard is at most its conclusion standard, and closed under cut when it is at least (§3.3.2); ct, sc and st fail even simple transitivity (§3.4.1). The paper's choice is st, strict premises to tolerant conclusions: it validates tolerance and modus ponens and is non-transitive, so it validates every step of a sorites while invalidating the chain (§3.6, Version 1), and it makes the tolerance-premise sorites valid but unsound, its tolerance premise not being strictly true (Version 2). The five-men model of §4.2.3 gives the paper's account of Alxatib and Pelletier's data: the median man is borderline however he is classified.

Main definitions #

TModel, TModel.Realize, Mode: T-models and their three notions of truth, the modes ordered by strength strict < classical < tolerant.
TModel.Borderline: Definition 10.
Valid, Unsat, Consequence: Definitions 11, 12 and 17, the last multiple-conclusion Consequence.MixedConsequence.
IsRestricted, TModel.identity, allBorderline, toMV, ofMV: the restricted vocabulary and the models of Lemmas 2, 3, 4 and 5.
series: the sorites series of §3.4.1 and §3.6, whose instances are the four-element model of p. 354 and the five-men model of §4.2.3.

Main results #

TModel.realize_mono (Lemma 1), isDual (Remark 1), tolerance_valid (Fact 2), st_two_step (Fact 3).
TModel.Borderline.exists_ne: a model with a borderline case has two (footnote 4); TModel.eq_tolerant_of_realize_conj_neg: a contradiction can only be tolerantly true.
valid_tolerant_iff, unsat_strict_iff (Theorem 1); not_valid_strict, exists_realize_tolerant (Theorem 2); consequence_tolerant_iff_lp, consequence_strict_iff_k3 (Theorem 3).
consequence_iff_dual (Lemma 6), Consequence.mono (Lemma 7), Consequence.restricted_iff (Lemma 8), not_consequence_tolerant_strict (Lemma 9), Consequence.iff_valid_imp, deductionTheorem_iff, Consequence.iff_valid_imp_of_restricted (Lemma 10).
not_cc_one_step, not_valid_classical_tolerance, not_sc_two_step, not_ct_two_step: the §3.4 distinctness of cc, sc, ct and st, on the four-element model.
stdRefl_consequence_iff, Consequence.cut, isTrans_consequence_iff: reflexivity and transitivity of the nine relations (§§3.3.2, 3.4.1).
not_st_version1, st_version2, not_series_realize_strict_tolerance: the sorites of §3.6.
alxatibPelletier_median_borderline: the five-men model of §4.2.3.

Implementation notes #

The language is the paper's propositional fragment over a set of constants C: negation and conjunction over predications Pa and similarity statements a I_P b, with disjunction and the conditional defined classically. The universally quantified principles appear as their instances, which the paper notes changes nothing (§3.6). Identity, which the paper adds in §2.3.1, is not represented. Consequence quantifies over T-models of every domain in the universe of C (TModels), as the paper's does; the atomic strict and tolerant clauses are ModalLogic.Box and ModalLogic.Diamond along ∼_P, so Lemma 1's atomic case is the T axiom.

Cobreros, Égré, Ripley and van Rooij 2012: Tolerant, classical, strict #

Main definitions #

Main results #

Implementation notes #

References #

T-models #

Borderline cases #

Consequence #

Tolerance #

The restricted vocabulary #

LP and K3 #

The deduction theorem #

Sorites series #