Abstract
Recent experiments have shown that naive speakers find borderline contradictions involving vague predicates acceptable. In Cobreros et al. (Journal of Philosophical Logic, 41, 347–385, 2012a) we proposed a pragmatic explanation of the acceptability of borderline contradictions, building on a three-valued semantics. In a reply, Alxatib et al. (Journal of Philosophical Logic, 42, 619–634, 2013) show, however, that the pragmatic account predicts the wrong interpretations for some examples involving disjunction, and propose as a remedy a semantic analysis instead, based on fuzzy logic. In this paper we provide an explicit global pragmatic interpretation rule, based on a somewhat richer semantics, and show that with its help the problem can be overcome in pragmatics after all. Furthermore, we use this pragmatic interpretation rule to define a new (nonmonotonic) consequence-relation and discuss some of its properties.
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Notes
According to the strongest meaning hypothesis, if a sentence can give rise to several closely related meanings, the sentence should be interpreted in the strongest possible way. A similar principle was used by [1] as well.
This account of the conditional problem is new, and not discussed in TCS.
The notion of ‘model’ that we use here is not exactly the standard model-theoretic notion, where interpretation of non-logical constants varies between models. In contrast, the notion of ‘model’ we use in section 4 to define various notions of entailment will be the standard model-theoretic one. We hope this will never give rise to confusion.
Interestingly enough, this is exactly what Priest’s LP m predicts for this type of example as well (see [5]). We will come back to this later.
Van Fraassen uses these truth-makers to give a semantics for the notion of ‘tautological entailment’ introduced by [3], a notion of entailment that is weaker than both Kleene’s K3 and Priest’s LP. For recent work on truth-makers, see [13]. For use of the same framework for quite a different purpose, see [28].
Note that the definition of T(ϕ) parallels the construction of the disjunctive normal form of ϕ.
To give a semantics of tautological entailments, [14] looks at models that neither have to be maximal, nor consistent.
Our pragmatic interpretation rule does not account for the ‘scalar implicature’ that from ‘ (p ∧ ¬ p) ∨ q’ we conclude that only one of the disjuncts is as true as possible. It is easy to change the pragmatic interperation rule to account for this—and for embedded implicatures—as well (by changing ‘ ∃S ∈ T(ϕ)’ in the definition \(PRAG(\phi ,c) = \{\mathcal {M} |\ \exists S \in T(\phi ): \mathcal {M} \in [\![S]\!]^{t}_{c} \ \& \ \neg \exists \mathcal {N} \in [\![S]\!]^{t}_{c} : \mathcal {M}<_{S} \mathcal {N}\}\) which is equivalent to the one used in the main text, into ‘ ∃!S ∈ T(ϕ)’), but a discussion of this would go beyond the purpose of this paper.
Also APS treat conditionals in terms of material implication. Our assumption that we just use 3 truth-values instead of all the ones in [0,1] has as a consequence that the following sentences \(p \curlywedge \neg p\), \((p \curlywedge \neg p) \curlywedge p\) and \((p \curlywedge \neg p) \curlywedge \neg (p \curlywedge \neg p)\) will all have value 1 exactly if p has value \(\frac {1}{2}\). In APS, instead, it is predicted that these different sentences have value 1 in different circumstances. For instance, \((p \curlywedge \neg p) \curlywedge p\) is predicted to have value 1 iff p has value \(\frac {2}{3}\). Notice that on our analysis the three sentences are predicted to be equivalent because they are predicted to have the same exact truth-maker: {p, ¬ p}.
As it turns out, this rescaling method is virtually identical to the recalibration-method proposed by [18] to account for adjective-noun combinations.
In fact, in case the language does not have special connectives, this consequence relation is S3, the intersection of K3 and LP.
For what it is worth, we can make sense of our predictions for (9) as well as for (10), even though the predictions APS makes for (10) might seem more appropriate. Consider the reading of (10) made salient by the elaboration: ‘Adam is (surely) tall, and either he is (also) not tall or else John is rich (but I forget which)’. It seems to us that our prediction for this elaboration is correct, and that APS mispredicts here. Still, even if our prediction turns out to be incorrect, this still wouldn’t automatically mean that a semantic analysis would be preferred to a pragmatic one. It is possible to change the pragmatic interpretation rule so as to predict the same interpretation for (10) as APS does. For instance, by assuming that the definition of T(ϕ ∧ ψ) should be as given in the main text only if T(ϕ) and T(ψ) are singletons. In any other case we should only allow A∪B to be an element of T(ϕ ∧ ψ) for A ∈ T(ϕ) and B ∈ T(ψ), if A∪B does not contain a p! that was not yet in either A or B, where p! = {p, ¬ p}:
$$\begin{array}{@{}rcl@{}}T(\phi \wedge \psi) &=& \{A \cup B: A \in T(\phi), B \in T(\psi)\}, \text{if}\,\, T(\phi)\,\, \text{and}\,\, T(\psi)\,\, \text{are singletons}, \\ & =& \{A \cup B: A \in T(\phi), B \in T(\psi), \neg \exists p: p! \subseteq A \cup B \ \& \ p! \not\subseteq A\ \& \ p! \not\subseteq B\}, \text{otherwise}. \end{array} $$At this point, however, we are undecided on whether we should change the definition of T(ϕ ∧ ψ), because we feel that an assertion of a sentence like (10) would be inappropriate because it violates Grice’s maxim of Manner. Notice, though, that in case we would interpret not (10), but rather a sentence like ¬ Ta ∨ Rj in a context where Ta is known to be strictly true, we would already make the same prediction as APS. As it turns out, however, the alternative definition could be relevant for the analysis of pragmatic entailment discussed in the following section.
because we demand that x ∼ P y is true provided Px and Py have truth values not differing by more than \(\frac {1}{2}\).
Notice that the simple definition \({\Gamma } \models ^{spr}_{\sim } \phi \) iff S(Γ) ⊆ PRAG(ϕ, S(Γ)) wouldn’t work (with S(Γ) as the class of models where all the elements of Γ are strictly true) because there will be models where Tx ∧ x ∼ T y is stricly true, but where Ty is only tolerantly true.
Notice that if we had defined pragmatic consequence as follows: \( {\Gamma } \models \!^{prt}_{c}\ \psi \mbox { iff } PRAG(\bigwedge _{\phi \in {\Gamma }},c) \subseteq \{\mathcal {M}\in c: \mathcal {M}\in [\![\psi ]\!]^{t}\}\), things would have been different. In that case we could only conclude from p, ¬ p ∨ q that either p is only tolerantly true, or that q is strictly true. This has an important consequence, though: it can be that ϕ, ψ⊧prt χ although ϕ ∧ ψ ⊮prt χ, for p, ¬ p ∨ q⊧prt q but p ∧ (¬ p ∨ q) ⊮prt q. This problem could be solved, however, by adopting the alternative definition of T(ϕ ∧ ψ) as given in footnote 14.
One might think that an interesting difference with LP shows up here: can we not ensure the validity of the disjunctive syllogism ϕ, ¬ ϕ ∨ ψ⊧prt ψ by adding ¬ (ϕ ∧ ¬ ϕ) as an extra premise, even though LP cannot (because ¬ (ϕ ∧ ¬ ϕ) is an LP-tautology)? Unfortunately, adding ¬ (ϕ ∧ ¬ ϕ) as an extra premise cannot guarantee the validity of the disjunctive syllogism for ⊧prt either, as can be seen by taking ϕ to be p ∧ ¬ p.
The other direction doesn’t hold, though. Assume ⊧prt ϕ → ψ. Now it doesn’t follow that ϕ⊧prt ψ. For take ϕ: = p ∧ ¬ p and ψ:=q.
Some (e.g. [25]) have also questioned conjunction elimination.
Note though, that ϕ, ¬ ϕ ⊧prt ψ.
In fact, our consequence relation verifies (slight variants of) all the standard conditions of nonmonotonic reasoning (as stated by [20]): (i) Reflexivity: ϕ⊧prt ϕ, (ii) Left Equivalence: if PRAG(ϕ) = PRAG(ψ), then from ϕ⊧prt χ, it follows that ψ⊧prt χ (Notice that the condition PRAG(ϕ) = PRAG(ψ) cannot be replaced by ⊧prt ϕ ↔ ψ.) (iii) Right Weakening: if ⟦ψ⟧t ⊆ ⟦χ⟧t, then ϕ⊧prt χ follows from ϕ⊧prt ψ, (Of course not if ⟦ψ⟧t ⊆ ⟦χ⟧t were replaced by ψ⊧prt χ or by ⊧prt ψ → χ.), (iv) Cautious Monotonicity: if ϕ⊧prt χ and ϕ⊧prt ψ, then it follows that ϕ ∧ ψ⊧prt χ, and (v) Or: if ϕ⊧prt χ and ψ⊧prt χ, then ϕ ∨ ψ⊧prt χ. Obviously, (vi) Cautious Cut (if ϕ ∧ ψ⊧prt χ and ϕ⊧prt ψ, then ϕ⊧prt χ) does not hold, at least in case of \(\models ^{prt}_{\sim }\): a \(\models ^{prt}_{\sim }\)-conclusion indeed may have a lower truth-value than that of the premises it is based on. Notice that in these rules we have always interpreted the premises as one conjunction. For the reason behind that, see footnote 17.
Beall [5] notes that \((q \wedge \neg q) \models ^{LP^{m}} (q \wedge \neg q) \vee r\) and \((q \wedge \neg q) \vee r\models ^{LP^{m}} r\), but \((q \wedge \neg q)\not \models ^{LP^{m}} r\).
In fact, we think that ⊧prt is strictly weaker than \(\models ^{LP^{m}}\), but leave a discussion of this to another occasion.
Van Rooij [29] uses this notion in relation to the well-known knowability-paradox.
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Acknowledgments
We wish to thank the reviewers for their helpful comments. Thanks also to audiences and colleagues at the universities of Amsterdam, Tilburg, NYU, and Barcelona where part of this material was presented, for helpful feedback. Financial support for this work was provided by the Marie Curie Initial Training Network ESSENCE-project, the NWO-sponsored ‘Language in Interaction’-project, and the project ‘Logicas no-transitivas. Una nueva aproximacion a las paradojas’, funded by the Ministerio de Economa y Competitividad, Government of Spain. Thanks also to grants ANR-10-LABX-0087 IEC and ANR-10-IDEX-0001-02 PSL, as well as to the the European Research Council (FP7/2007-2013) under ERC Advanced Grant agreement number 229 441-CCC (PI: Recanati).
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Cobreros, P., Egré, P., Ripley, D. et al. Pragmatic Interpretations of Vague Expressions: Strongest Meaning and Nonmonotonic Consequence. J Philos Logic 44, 375–393 (2015). https://doi.org/10.1007/s10992-014-9325-7
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DOI: https://doi.org/10.1007/s10992-014-9325-7