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L. Gommans

M.L. Cristea

T. DeFanti

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The Token Based Switch: Per-Packet Access Authorisation to Optical Shortcuts

Cristea, M.-L.

Gommans, L.

Xu, L.

Bos, H.J.

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The Token Based Switch: per-packet access authorisation to optical shortcuts

Cristea, M.L.

Bos, H.

International Migration, Integration and Social Cohesion online publications

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Caret, Colin Ready

Cotnoir, Aaron

St Andrews Research Repository

True, false, paranormal and ‘designated’?: areply to Jenkins∗COLIN CARET & AARON J. COTNOIR0 IntroductionJenkins [4] charges that the language advanced in Beall [2] is either expressively im-poverished, or inconsistent. We argue that Jenkins’ objections are based on unreason-ably strong constraints on formal theories of truth. Our primary concern is not todefend the ‘paranormal’ framework advanced in Beall [2], but to respond to a com-mon – and implausible – ‘revenge’-style charge directed at a certain class of formaltheories of truth and paradox. In §1, we outline Beall’s language; §2 outlines Jenk-ins’ main criticism and the resulting dilemma. In §3, we give our two-fold reply toJenkins.1 True, false and paranormalIn Beall [2] the following many-valued language (L ) is defined.The semantic values areV = {1, 0.75, 0.5, 0.25, 0}The designated values are D = {1, 0.75}The connectives and predicates are defined in the following way:ν(A&B) = min{ν(A), ν(B)}ν(A∨B) = max{ν(A), ν(B)}ν(¬A) = 1− ν(A)ν(T 〈A〉) = ν(A)ν(F 〈A〉) = ν(T 〈¬A〉)ν(piA) =0 if ν(A) ∈ {1,0}otherwise 0.75The intended interpretation is that the predicates T and F represent ‘true’ and ‘false’in the object language. These predicates satisfy the fundamental intersubstitutabilityof T 〈A〉 with A, and of F 〈A〉 with ¬A, for all sentences A in all non-opaque contexts.The connective pi represents the semantic category ‘paranormal’ which is meant as∗forthcoming in Analysis 68.3, July 2008. c©Colin Caret and Aaron Cotnoir.1a colorless term for capturing all problematic, paradoxical sentences that defy easycharacterization in terms of truth and falsity. Valid inferences for such a language arethose which are designation-preserving.Given a standard treatment of predicate extensions, the extension of T in anymodel ofL is T + = {x : 0.75≤ ν(x)}.1 As can easily be checked, this set is negation-consistent, hence it meets the goal of ‘non-triviality’ defined below as NTP. More-over, for every sentence A, in any model of L , there is a designated sentence ϕAwhere ϕ is one of T or F or pi. This demonstrates thatL meets the goal of ‘exhaus-tive characterization’ defined below as ECP.(NTP) explain how, despite having both a truth predicate (in our language, for ourlanguage) and Liar sentences, our language is non-trivial.(ECP) explain how, if at all, we can truly characterize – specify the ‘semantic statusof’ – all sentences of our language (in our language).These desiderata are achieved by employing overlapping semantic categories, an ideasometimes advanced by dialetheists as a reason to accept the existence of true contra-dictions, i.e. truths of the form A&¬A. Since the extension of T in L is negation-consistent, it follows that there are no true contradictions. Hence, L provides acounterexample to the inference from satisfaction of NTP and ECP by overlap to theconclusion of dialetheism.2 Jenkins’ argumentIn Jenkins [4] it is objected that the following predicate cannot be expressed in Beall’slanguage.ν(D〈A〉) =1 or 0.75 if ν(A) ∈D0.25 or 0 if ν(A) /∈DThe intended interpretation is that the predicate D represents the notion of designa-tion used in the semantic machinery of the meta-theory. This predicate cannot beexpressed because of the following sentence, A∗.(A∗) ¬D〈A∗〉If ν(A∗) ∈ D , then ν(D〈A∗〉) ∈ D , but then ν(¬D〈A∗〉) ∈ {0,0.25} so by the definitionof A∗ it follows that ν(A∗) /∈ D . If ν(A∗) /∈ D , then ν(D〈A∗〉) ∈ {0,0.25}, but thenν(¬D〈A∗〉) ∈ D so by the definition of A∗ it follows that ν(A∗) ∈ D . The designa-tion predicate D cannot be expressed on pain of a sentence that is designated iff it isundesignated.Jenkins calls this fact aboutA∗ a ‘model-theoretic analogue of true contradictions’([4], 81). From this, she draws the following dilemma for Beall.[E]ither no model language is fully adequate as a model of English,which can talk about its own semantic machinery, or, like the model lan-guages, English cannot express claims about its own semantic machinery.1For convenience, we write ‘T +’ to denote the extension of T .2If one takes the first option, then one has to admit that (Beall’s dis-cussion of) these models can’t help us defend exhaustive characterizationor understand the Liar-like paradoxes. [. . . ] Finally, of course, if onetakes the second option, one places a familiar, and implausible, kind oflimitation on the expressive powers of English. ([4], 83-84)Against Jenkins, we argue that this is a false dilemma. Both horns assume that theexpressibility of a predicate like D is essential to adequately model the semantics ofEnglish; that any model language which fails to do so fails to satisfy ECP. But thisis an assumption that one might reject.2 On the intended reading of ‘exhaustive’, anexhaustive characterization would simply require that for every sentence there is atleast one semantic predicate we can use to (truly) characterize it. As we previouslynoted, Beall’s language features enough predicates to achieve this goal.3 ReplyJenkins appears to be assuming a much more robust ‘exhaustive’ characterizationprinciple when she writes,[T]he ability of a language to express claims about its own semantic ma-chinery is what the exhaustive characterization project is supposed tosecure[. . . ] The kind of exhaustive characterization we want, and whichPriest thinks leads to acceptance of true contradictions, is an exhaustivecharacterization of all the sentences in the model in terms of their seman-tic values. ([4], 82)This suggests a requirement like ECP*.(ECP*) explain how, if at all, we can truly characterize all sentences of our languagein terms that uniquely specify their treatment under the ‘semantic machinery’of the proffered formal model.This robust conception of ‘exhaustive characterization’ requires that formal modelsof our language be capable of far more than the expression of semantic predicatessufficient to categorize English sentences. It requires that such models be capable ofexpressing every semantic value that plays a role in the formal mechanism. This con-ception of ‘exhaustive characterization’ explains Jenkins’ demand for the expressibil-ity of a designation predicate, but Jenkins has failed to provide reasons for preferringECP* over ECP.We believe that Jenkins is conflating purely model-dependent, instrumental no-tions like ‘semantic value’ and ‘designation’ with predicates that actually play a rolein the semantics of English. As such, the predicate D is not semantically significant,and need not be expressible. This gives the lie to ECP*: such a conception of the char-acterization project is based on a too-literal understanding of the explanatory powerof the formal tools involved in modeling our language.2Indeed, Beall [1] explicitly argues against such an assumption, as does Field [3].33.1 Semantic values?First, why think the semantic values need to be expressed individually? There arefive semantic values in L which correspond to the ways in which the three seman-tic categories ‘true’, ‘false’, and ‘paranormal’ overlap. The below diagram illustrateshow Beall conceives of the relation of these semantic categories and how the semanticvalues fall out of these relations.T P F&%'$&%'$&%'$1 .75 .5 .25 0The semantic values, then, are heuristic tools used to signal a sentence’s locationamong the overlapping semantic categories. The three categories which give rise tothese five ‘locations’ are non-trivially expressible in the model-language. Beall can, infact, predicate (truly) of each sentence its location among the semantic categories.3What one cannot do in L is to (truly) characterize each sentence in terms of thesemantic categories that don’t apply, as it were. For instance, given a sentence A suchthat ν(A) = 0.5 we might think of A as ‘not true’ and ‘not false’, but we cannot say asmuch since ν(¬T 〈A〉&¬F 〈A〉) /∈ D . If this failure of the language is what Jenkins isconcerned about, it points in the direction of a very robust conception of ‘exhaustive’characterization indeed.This suggests a project which aims to (truly) characterize all sentences both ex-haustively and exclusively, that is, under semantic predicates that express each se-mantic value to the exclusion of every other value. Why expect this? Jenkins has notargued for the desirability of ‘exclusive’ characterization, nor has Beall said anythingto commit him to such a project. In fact, the fruitfulness of overlapping semanticcategories as a tactic for understanding the non-triviality of truth seems to depend ona lack of such predicates. If Jenkins is advocating this aim, then her argument cutsequally against the semantics for Graham Priest’s LP and Anderson and Belnap’sF DE .4 Ruling out the adequacy of this entire family of semantics on the grounds ofan unargued methodological stance is unreasonable.3.2 Designation?Second, why think designation is semantically significant? Jenkins suggests it is thenotion of designation which is playing the semantic role that ‘true’ plays in English.But ‘T ’, in Beall’s model, does not express the model-theoretic analogueof the semantic property that our English predicate ‘true’ expresses. Inthis model, the role of that property is played by designation, which, as3As can be checked for all A, if ν(A) = 1, then ν(T 〈A〉) = 1; if ν(A) = 0.75, then ν(T 〈A〉&piA) = 0.75;if ν(A) = 0.5, then ν(piA) = 0.75; if ν(A) = 0.25, then ν(piA&F 〈A〉) = 0.75; if ν(A) = 0, then ν(F 〈A〉) = 1.4For example, in the case of Priest’s LP , consider the predicate Z , where ν(Z〈A〉) = 1 if ν(A) = 0, andotherwise ν(Z〈A〉) = 0. Now consider the sentence λ : Z〈λ〉. Clearly, ECP* causes triviality for LP as well.4we have seen, is not definable in the language. Hence the language is not,despite appearances, semantically closed. ( [4], 82)We argue that this is mistaken. Designation in the meta-language is meant to modelT in the object-language. We note that the designated values, D = {1,0.75}, and theextension of ‘true’, T + = {x : 0.75≤ ν(x)}, are such that for any sentence A, ν(A) ∈Diff A∈ T +. That is, A has a designated value exactly when it is true. So, designationis a meta-linguistic device which models the extension of ‘true’, and this should beenough to answer Jenkins’ stated charge.5There is, however, an implicit, stronger requirement in Jenkins’ charge. In defin-ing D as her desired truth predicate, Jenkins indicates that whatever plays the role of‘truth’ should be such that D〈A〉 is false in cases where A is undesignated. This en-ables us to truly say (by use of negation) that a sentence is not true. That is, she wantsthe behavior of ‘true’ to deliver a dual notion of ‘untrue’ that is distinct from ‘false’.She seems to hold that such a feature is essential to the role of the English predicate‘true’,6 but why would one require or think this?Jenkins has provided no argument for the requirement that ‘untruth’ be express-ible; but the requirement appears to be based on an unwarranted assumption, namelythat ‘true’ in English is a classical notion. Jenkins seems to be exploiting the fact thatour meta-language is formulated in classical set-theory, according to which, for any xand set Y either x ∈ Y or x /∈ Y . It is no wonder, then, that for any sentenceA, eitherν(A) ∈ D or ν(A) /∈ D . And it is because of this fact that ‘D’ is not expressible. Aswe know, classical logic cannot satisfy both NTP and ECP (and hence ECP*) whenliar sentences are included in the language, so it comes as no surprise that classicallybivalent notions will yield inconsistency.We conclude that Jenkins has provided no motivation for the semantic signifi-cance of designation or for ECP*; hence, her arguments in no way undermine Beall’sprincipal claim. In general, the above considerations suggest that ECP* is far too un-reasonable of a requirement. It appears doubtful, then, that ‘revenge’-style objectionsbased on such assumptions pose any problems for formal theories of truth which aimto solve the paradoxes.7University of ConnecticutStorrs, CT 06269–2054, USA5Although Jenkins never mentions validity, the notion of designation most obviously plays a role inthe definition of argument forms which are valid-in-L . On any reading, validity counts as a ‘semanticallysignificant’ notion of English. The question is how to understand the relation between validity and valid-in-L . We deny the inference from (1) designation is used to model validity, and (2) validity is semanticallysignificant, to the conclusion that (3) designation is semantically significant. For the above reasons, weclaim that purely model-theoretic notions useful for modeling semantically significant phenomena neednot themselves be semantically significant. There may be a related concern in this area about whetherBeall’s language can enjoy a validity relation. This hangs largely on whether he can construct a suitableconditional that avoids Curry paradoxes. Since this is not addressed in any length in Beall [2], we leavethe issue aside.6In fact, Jenkins has pointed out in private correspondence that she thinks the expression of the ‘un-truth’ of a sentence in the above sense is an important requirement. She maintains that any languagewhich fails to meet this requirement in expressively incomplete, although has not provided any argumentin support of it.7Thanks to JC Beall and Carrie Jenkins for helpful comments on earlier drafts of this paper.5aaron.cotnoir@uconn.educolin.caret@uconn.eduReferences[1] Beall, JC. 2007. Prolegomena to future revenge Revenge of the Liar. JC Beall(ed.) Oxford University Press.[2] Beall, JC. 2006. True, false and paranormal. Analysis. 66: 102-14.[3] Field, H. 2007. Solving the paradoxes, escaping revenge. Revenge of the Liar. JCBeall (ed.) Oxford University Press.[4] Jenkins, C. 2007. True, false, paranormal, and designated: a reply to Beall.Analysis 67: 80–83.6

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