Cobreros&al. [3] build their system on the basis of a bivalent logic. They start from a language in which predicates may combine with every constant of the language, and where they are associated with functions that return 1 (truth) or 0. Adapting their terminology4, we can speak of these functions as representing the classical meaning of a predicate. The classical meanings of connectives — the standard ones — insure that, when we speak classically, all the principles of
3 T CSstands forT olerant Classical Strict.
4 We will speak of classical (or strict or tolerant)meaning, even if strictly speaking Cobreros&al.’s definitions are in terms of classical (or strict or tolerant)truth. When we speak of the classical (or strict or tolerant)meaning of a predicateP, what we mean is a function that, given the value of a constanta, yields the classical (or strict or tolerant) value ofPa.
2 J´er´emy Zehr & Orin Percus
classical logic are preserved. In particular, the classical meaning of negation is such that a proposition’s negation is true iff the proposition itself is not true, so we never find that both a proposition and its negation are true.
Importantly, however, classical meaning is not the only type of meaning an expression can have: there are also two other types of meaning,strict meaning and tolerant meaning. Depending on how we want to assert a statement, we can associate it with its classical, strict or tolerant meaning. Usually we speak classically, but if we want to strengthen our discourse, we will associate our statements with their strict meanings; and if we want to weaken our discourse, we will associate our statements with their tolerant meanings. We give below definitions for strict and tolerant truth and falsity, following the general lines of Cobreros&al. Note that the definitions in the case of falsity correspond to definitions of negation — to say that a proposition is tolerantly, classically or strictly false is to say that its negation is tolerantly, classically or strictly true, respectively.
Definition 1 (Strict truth and falsity of a vague predicate P).
P(a)is strictly true for anyaif and only if Pis classically true of all individuals that are sufficiently likeawith respect to the measure associated with P 5. P(a)is strictly false for anyaif and only if Pis classically false of all individuals that are sufficiently likeawith respect to the measure associated with P.
Definition 2 (Duality in TCS).
A proposition is tolerantly true if and only if it is not strictly false.
A proposition is tolerantly false if and only if it is not strictly true.
Definition 3 (Connectives in TCS).
Conjunctions of the form A & Bare strictly true if and only ifais strictly true and bis strictly true.
Conjunctions of the form A & Bare tolerantly true if and only ifais tolerantly true and bis tolerantly true.
The contributions to strict and tolerant truth of the disjunction∨and the condi-tional→are defined in terms of negation and conjunction in the ways familiar from classical logic.
It is important to observe the dualities that we find in this system: to say that a predicate is notstrictly true of an individual is to say that it istolerantly falseof her; and reciprocally, to say that a predicate is nottolerantly true of an individual is to say that it is strictly false of her. Crucially, the dual of strict truth is not strict falsity nor is the dual of tolerant truth tolerant falsity.
5 In reality, Cobreros&al. imagine that vague predicates are associated with a relation of indifference between individuals. They go no further than this. But, assuming that each vague predicate is associated with a specific scale of measurement (e.g.
as Kennedy [12]), we can see this relation as holding between two individuals if the difference of their measures on the scale falls below a certain threshold of “distin-guishability”.
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TCS for presuppositions 3 Here is an example. Consider a context of only four people (let’s name them a, b, c and d) who are sorted by height: a is slightly smaller than b who is slightly smaller thanc who is slightly smaller thand. It’s worth noting that the relation slightly smaller is not transitive:a being slightly smaller thanb and b being slightly smaller thancdoes not entail thatais slightly smaller thanc. In fact we will assume thatais not slightly smaller thanc, nor isbslightly smaller than d. Assuming that the distribution of classical truth for tall is as specified in the first line of Table 1, the distribution of classical falsity would be as in the second line. Assuming that two individuals count as sufficiently alike with respect to tallness only when one is at most slightly smaller than the other, we then have the distribution of strict truth and falsity in Table 2, and accordingly the distribution of tolerant truth and falsity in Table 3.
Table 1.Classical truth and falsity oftallfora,b,candd
a b c d
not true not true true true false false not false not false
Table 2.Strict truth and falsity oftallfora,b,candd
a b c d
not true not true not true true false not false not false not false
Table 3.Tolerant truth and falsity oftallfora,b,candd
a b c d
not true true true true false false false not false
Notice that, in the case of b andc, tall is on the one hand neither strictly true nor strictly false, and on the other hand both tolerantly true and tolerantly
TCS for presuppositions 79
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false. This situation arises because of the way in which Cobreros&al. establish their dualities. In their system, for someonenot to be strictly talldoesn’t mean for himto be strictly not tall; rather, it means that he’s tolerantly not tall. At the same time, when someone (like a) is not (even) tolerantly tall, this means that he’s strictly not tall (and necessarily tolerantly not tall too). These remarks suggest that the Cobreros&al. semantics can be recast in terms of a sort of modal semantics with astrictly operator — even if on their formulation there are two special kinds of meaning, tolerant and strict, their duality allows us to formulate one in terms of the other, by playing on the relative scope of the negation and astrictly operator.
The aspect of Cobreros&al.’s system that interests us here is that it gives us a way of characterizing borderline cases for vague predicates — that is, cases that trigger an unclear truth-value judgment. Borderline cases are simply those for which a vague predicate yields a proposition that is neither strictly true nor strictly false — or, equivalently, cases for which the proposition is both tolerantly true and tolerantly false. Alternatively stated, a borderline case for a vague predicate is a case that leads to different strict and tolerant values for the proposition that results when we apply that predicate. Note that, the Cobreros&al. approach seems to predict that, to the extent that strict readings are natural, we should be able to judge predicates like neither−tall−nor− not−tall to be true of borderline cases for tall; likewise, to the extent that tolerant readings are natural, we should be able to judge predicates likeboth− tall−and−not−tallto be true of the same cases. In support of their approach, Cobreros&al. cite the results of an experiment by Alxatib&Pelletier [1], who found judgments of precisely this kind.
Of related interest is the fact that, to the extent that tolerant readings are natural, the system provides an account for the sorites paradox: the paradox arises due to the tolerant way of reading the inductive premiss. Consider the following statement, meant to exemplify the kind of statement that appears as a premiss in versions of the sorites paradox.
Example 1. If a man is slightly smaller than a tall man, then he is a tall man too.
Strictly speaking, ( 1 ) conveys that any man who is slightly smaller than a tolerantly tall man is a strictly tall man — and this is false. But tolerantly speaking, ( 1 ) conveys that any man who is slightly smaller than a strictly tall man is a tolerantly tall man — and this is true. (To see this, suppose first that ( 1 ) should be represented as a formula of the form ∀y∀z[[P z&Ryz]→ P y] -or equivalently as a conjunction of conditionals of the f-orm [[P a&Rab]→P b] , whereRabexpresses thatbis slightly smaller thana. Then it suffices to observe that a conditional of the form [p→q] is strictly true if and only if the tolerant truth ofpentails the strict truth ofq, and that it is tolerantly true if and only if the strict truth ofpentails the tolerant truth ofq.6). With this in mind, consider
6 Here is a proof of the first claim; the proof of the second is parallel.
For brevity, we abbreviate “φ is strictly true” as S[φ], “φ is not strictly true” as
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TCS for presuppositions 5 the tolerant reading. If ais a strictly tall man andbis slightly smaller than a, accepting the inductive premiss in a tolerant way does not lead you to conclude thatbis astrictlytall man; rather you have to conclude thatbis atolerantly tall man. And ifcis slightly smaller thanb, then, even if this means that he’s slightly smaller than a tolerantly tall man, he is just trivially concerned by the inductive premiss ifb happens not to be a strictly tall man — we don’t have to conclude thatcis a strictly or tolerantly tall. This is the way the TCS system handles the sorites paradox. The “trick” is that the vague predicate in the antecedent of the conditional has to be understood in the dual way of the one in the consequent.
In sum, the TCS system can be seen as a system that describes strengthening and weakening of meaning. On the one hand, there is a “classical” way of using statements on which a statement is necessarily exclusively true or false. But, on the other hand, we might wish to strengthen or weaken our way of speaking, and on these modified ways of using statements, a statement can be neither true nor false, or both true and false.