Reply to Ondrej Majer’s Comments on Vagueness, Tolerance and Non-Transitive Entailment

In the paper I suggested that for many predicates P, the 'P-er than' relation is a strict weak order. Ondrej Majer asks why I assume that, given that I concentrate myself later in the paper on semi-orders. Well, in case the ordering relation can be made more precise by means of measurement phrases, there is no choice but to use a strict weak order. In fact, speaker's intuitions about a relation like 'taller than' very much correspond with that of strict weak orders. Having said that, I realize very well that there are many comparative relations which have only weaker properties (for instance only a partial order), 'smarter than' is arguably one of them. My point for introducing strict weak orders was not so much to claim that most comparative relations are of this type, but rather that if this is assumed, it can hardly be explained why the Sorties paradox is a problem. I believe that the Sorites paradox only arises because the way we normally observe tallness, for instance, the observably taller than-relation should be a semi-order, rather than a strict weak one. Can the use of degrees in the measurement of semi-orders not be used to formulate a degree-based analysis of vagueness?, Ondrej Majer wonders. Perhaps, I didn't try out, and I invite Majer to work out this interesting thought. But I agree with Ondrej that there exists a certain similarity between the approach I favor and fuzzy logic-based analyses of logic. In contrast to most standard logicians and semanticists I am not impressed by the standard objections that formulas like Pa ∧ ¬Pa and Pa ∨ ¬Pa are not predicted to be contradictions and tautologies, respectively, if P is a vague predicate. In fact, this is at it should be (see Atxatib and Pelletier, 2011 and Ripley, 2011), and my notions of strict and tolerant satisfaction account for this. Moreover, I believe that the most natural notion of consequence that can be formulated using using fuzzy logic is a non-transitive one. Having said that, I don't believe that we need as many values of truth that fuzzy logicians standardly assume: three or four can do a whole lot. Ondrej Majer correctly observes that given the semantics I propose, there are a number of tolerant (non-transitive) consequence relations that can be formulated. In the paper I suggest to use consequence relation ct (according to which the conclusion …