way will avoid running into the quicksand of higher order vagueness. Once this is taken for granted, there are, however, a number of further options available.
Vagueness-Adaptive Logic: A Pragmatical Approach to Sorites Paradoxes Bart Van Kerkhove Guido Vanackere ∗
Abstract The paper defends a pragmatical approach to vagueness. The vagueness-adaptive logic VAL is a good reconstruction of and an excellent instrument for human reasoning processes, in which vague predicates are involved. Apart from its proof-theory and semantics, we present a Soritestreating model based on it. The paper opens perspectives w.r.t. the construction of theories by means of vague predicates.
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Introduction
Sorites paradoxes have been haunting generations of logicians, ever since their original formulation in early Greece. In recent history, attempts at their solution clearly intensified twice. A first time, in the former half of the twentieth century, following the birth of modern logic. And again, more considerably, in the latter half of that century, with the rise of alternative logic. As is explained in general terms in section 2, the present paper inscribes itself in a particular segment of the second movement: the pragmatical one. Particularly, this is done by means of the vagueness-adaptive logic VAL, presented in section 4. The major advantage VAL has over other, more classical, contenders in the logical debate surrounding vagueness, is that the procedure of its proof theory is very close to that of human reasoning processes. More specifically: the proofs are dynamic, in that new information or further analysis of premises may result in a revision of the set of derived formulas. Moreover, the semantics of VAL are very intuitive. Some elaborated examples (section 5) evidence that VAL can be successfully applied, and turns out to suit excellently within a contextual and pragmatical approach to vagueness. Before turning to the presentation of VAL, however, we believe it is instructive also to give a brief introduction to adaptive logics in ∗ The authors are research assistants in logic and philosophy of science at the universities of Brussels (VUB) and Ghent (RUG) respectively. The former author is indebted to the Fund for Scientific Research - Flanders (Belgium). The research for this paper was also supported by the Flemish Minister responsible for Science and Technology (contract BIL01/80).
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general, and to ambiguity-adaptive logics, of which VAL is one, in particular. This will be done in section 3. The philosophy behind a pragmatical approach to vagueness and the paradoxes it engenders, is summarized in section 2. This will place the present discussion in a larger perspective on the use of language. In section 6, we comment on the results of the logic VAL with respect to the problems touched upon there, and on a more general result, related to the field of the construction of scientific theories. We believe the present paper establishes that there is no reason to banish vague predicates from scientific languages. In other words, the lack of an exact, domain-specific language should not be an obstacle when wanting to construct theories about the things that matter to us.
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Pragmatical considerations
In the course of the past thirty years, quite a number of tools have been proposed claiming to handle the vagueness phenomenon adequately, some fairly simple, others sophisticated. The vast majority, however, have remained entirely ‘semantical’ in nature, i.e. taking the linguistic issue to be (just) one of objective reference, and not (also) one of subjective use. These semantical accounts can be subdivided in two large categories, one coherentist, the other incoherentist.1 To the former group belong popular answers such as the epistemic, supervaluational or various multi-valued ones. The latter group comprises of those who consider the paradox, as it is formulated within the borders of classical logic (henceforth CL), to be ‘genuine’, in that it is insoluble, and inevitably leads into inconsistency. As is briefly explained below, only biting the bullet in this way will avoid running into the quicksand of higher order vagueness. Once this is taken for granted, there are, however, a number of further options available. One can either despair, and lapse into a form of nihilism, or try to make the best of this delicate situation, and opt for paraconsistent models or pragmatical qualifications to the existing ones. The strength of the present proposal, and thus its advantage over rival ‘alternative’ solutions such as those put forward in Manor [15] or Hyde [10], is that it flexibly combines the latter two features. In our opinion, indeed, you should not have the one without the other. We shall come back to this point in section 6. Nevertheless, the present approach clearly shares the philosophical and technical inspiration of the ones just mentioned. So we will have a brief look at them here,2 and in section 6 we shall weigh them against the present proposal, i.e. VAL. Higher order vagueness. An instructive way of seeing the Sorites paradox is as a problem of characterization of truth-functionality, i.e. of determining the proper number of values plus their mutual relations, and attributing them to statements. This has indeed been the insurmountable challenge: linking the 1 For a concise overview and appreciation, see, e.g., [24] (sections 3 and 4). More elaborate recent surveys are Keefe [13] and Williamson [26]. 2 More details may be found in [25] (§4.3 and §4.4).
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vague object-language with a (any) precise meta-language. In a multi-valued logic, there are at least three options for validating a formula: either it is true, or false, or something in between, where the latter gets filled in depending on the number of intermediate values available (from one to an infinite amount, i.e. a continuum, of them). Unfortunately, the objections raised against twovalued CL by the original paradox can easily be generalized. In the words of Williamson: “As grain is piled on grain, we cannot identify a precise point at which ‘That is a heap’ switches from false to true. We are equally unable to identify two precise points, one for a switch from false to neutral, the other for a switch from neutral to true. If two values are not enough, three are not enough” ([26], p.111). And so on: for exactly the same reason, four, five, six or more truth-values will not suffice. At least to a large extent, fuzzy logics might very well have been able to ease out obstacles met in real-life applications, reaching from washing machines to expert systems providing with medical diagnosis. Through these uncontested practical successes, however, the philosophical pressing question of why one should switch from one value to another at this or that particular point, has not been met at all, but instead has got duplicated at every threshold. Philosophically, as long as more truth-values are added to the analysis, one is dragged ever deeper into trouble, up to the point that one is left with an infinity of these ‘arbitrary’ thresholds to justify. The road to pragmatics. The account currently presented remains unimpressed by the strength of the latter phenomenon as much as by the original paradox, for it is not allowed to pop up in the first place. Together with other pragmatical proposals, it is indebted to Ludwig Wittgenstein’s ‘family resemblances’ alternative to the Oxford received view in the philosophy of language.3 There, the principle of rule-boundedness in linguistics gets challenged, in favour of a non-propositional approach, which says that speakers indeed implicitly (have to) follow rules, no matter what, but that, first, the bulk of them are absolutely unexpressible (elusive as well as permanently changing), and that, second, nothing whatsoever is gained by having them formulated anyway. On the contrary, as is shown in the exemplary cases of vague predicates, “utility and point of the classifications expressed [. . . ] would be frustrated if we supplied them with sharp boundaries. [. . . ] It is not generally a matter simply of lacking an instruction where to draw the line; rather the instructions we already have determine that the line is not to be drawn” (Wright [28], p.330). The very idea of incoherently codified practice is in itself considered to be incoherent. For how would it otherwise be that, in daily life, we are hardly ever bothered by its consequences? In the proposed alternative accounts, the notion of meaning will 3 “I can give the concept [number] rigid limits [in this way], that is, use the word [‘number’] for a rigidly limited concept, but I can also use it so that the extension of the concept is not closed by a frontier. And this is how we do use the word ‘game’. For how is the concept of a game bounded? What still counts as a game and what no longer does? Can you give the boundary? No. You can draw one; for none has so far been drawn. But that never troubled you before when you used the word ‘game’” (from Philosophical Investigations, §68; see [27], p.32e -33e ).
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not become pointless, however, but will largely concern non-cognitive knowledge, comparable to that constituting practical skills. The latter, we just seem to be able to perform, without ever being describable in full detail. Because we learn them by doing, in the course of an ever-lasting ostensive training, their meaning always remains under construction. In pragmatical approaches, theorists are concerned with circumstances in which language is used rather than with establishing a rigid mapping between word and object. Whenever lines are drawn, they are drawn in practice, not in principle. Two proposals are now briefly presented, prior to our own. Manor’s approach. Ruth Manor ’s basic idea is that vague predicates may denote so-called ‘foggy’ objects, i.e. objects whose parts cannot unequivocally be determined. In formal terms, objects d are represented by sets of (alternative) sets of atoms belonging to the domain A : d ∈ PP(A). Being a subset of the powerset of A, any object d is a set of possible delineations. It is a distinct object dd , if it has only one such a delineation (dd = {s}, with s ⊆ A). It is a foggy object df , if it has at least two (df = {s1 , s2 , . . . }, with s1 , s2 , . . . ⊆ A). The opposite ]d of an object d relative to A is defined as ]d = {A\s for all s ∈ d}, with A\s reading: the complement of s relative to A. An object d1 is then considered a boundary case of an object d2 , iff every member of the former is a subset of some member(s) of the latter as well as a subset of some member(s) of its complement ]d2 . So the boundary case d1 for an object d2 is partly enclosed in that object d2 (more exactly, to remain faithful to the terminology: some of the delineations of d1 are enclosed in delineations of d2 ), and partly enclosed in its complement ]d2 . This leads to the following possibility to ‘treat’ vagueness. Consider the domain of people all over the world, and assume that in it, for every n, there is at least one person with n hairs on his scull. Hn is the non-empty extension of ‘has n hairs’. Now consider an object p = {H0 , H0 ∪ H1 , . . . , H0 ∪ H1 ∪ H2 . . . ∪ Hn }, the elements of which are the sets of people with a number of hairs less than or equal to n. Together with its complement ]p, we assume it denotes the bald and non-bald people respectively. Both are vague (or foggy) in nature. Yet, every set of people with a particular number of hairs, e.g. {Hn }, neither belongs to p nor to ]p. This means that the corresponding object {{Hn }} can never belong to the extension of ‘is bald’ or ‘is non-bald’, or, put differently, that it is always a boundary case of both. This contextual account thus proves to be fruitful, for it appears to depend on how the domain gets divided, i.e. by determining n (from occasion to occasion), whether the group of n-haired will count as bald (together with the rest of the bald) or non-bald (together with the rest of the non-bald), while taken by itself, this group cannot be but borderline. Hyde’s approach. Accommodating the intuition that the daily use of vague vocabulary more often has an overdetermined than an underdetermined feel about, Dominic Hyde transfers the idea of supervaluational semantics to paraconsistent logic, and dubs the result subvaluation (henceforth SV ). In it, a
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sentence is true (resp. false) if it is true (resp. false) for at least one of its disambiguations. Since indeterminate sentences in effect receive contradictory disambiguations, it is clear that, in SV , “indeterminate sentences take on both truth-values” ([10], p.647). The exclusivity of truth-values, i.e. the principle that for all sentences A, either A is true or A is false, no longer holds. Nevertheless, CL’s validity conditions remain intact. An inference is SV -valid whenever if the premises are true in a precisification, then the conclusion is valid in a precisification. Non-contradiction, viz. for all A : ∼(A&∼A), holds in SV , as boundary cases, where A&∼A, clearly does not render invalid |=SV ∼(A&∼A). Disarming the Sorites paradox, then, is done by means of the validity clause expressing non-contradiction without exclusivity of truth-values, for this clause gives rise to a non-standard conjunction.4 More specifically, the fact that Modus Ponens is not unrestrictedly SV -valid is exploited. A boundary case Bxn (e.g. ‘a man with n hairs is bald’) is true ´and false. Because it is false, the material conditional Bxn ⊃ Bxn+1 is always true. Because it is true, we are together with this true conditional able to infer Bxn+1 . However, it is in no way to be excluded that Bxn+1 can be clearly (‘definitely’) false. As a consequence: Bxn , Bxn ⊃ Bxn+1 6|=SbV Bxn+1 (for an n). The one exception to this is when both conditional and antecedent are the case within one and the same precisification of the premises. At this point, an obvious connection becomes clear with the notions of ‘local’ vs. ‘global’ validity.5 Also, it recalls Manor’s above strategy of properly marking out the domain under consideration in order to dissolve any apparent paradoxes. Pragmatically spoken, the major merit of paraconsistent logic in general has been to take incoherence or overdetermination in our daily language at face value, and trying to get it formally under control rather than theorizing it away.6 For in the case of vagueness, the latter attitude has been typical for the bulk of semantically coherentist solutions, which in spite have all proven unable to escape higher order vagueness conclusively.7
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Adaptive logics
In view of the philosophical position explained above, the idea to develop an adaptive logic for vagueness is as plausible as can be. Starting from the pragmatical maxim that specific problems demand specific solutions,8 the Ghent group of Diderik Batens has developed a wide range of such logics. When applying some rich logic to some set of premises, one may find out —whether 4 This is the clause: A , . . . , A |= n 1 CL B iff A1 & . . . &An |=SbV B. For a proof, see [10], p.648n. 5 Surprisingly, ‘dialetheist’ Priest has been looking for a translation of these notions in fuzzy terms rather than in paraconsistent ones (see [16] and [17]). 6 As far as the latter point is concerned, though, note that question marks can be put next to Hyde’s ‘solution’. We maintain that, in this respect, the array of adaptive logics VAL is part of, do a far better job. 7 Recently, we have seen this point confirmed by JC Beall and Mark Colyvan (see [7]). 8 Compare William James’ words: “The concepts we talk with are made for purposes of practice and not for purposes of insight” ([11], p.11).
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right from the beginning or only in the long run— that the premises contradict the norms or presuppositions of the logic. There are two bad solutions to this problem. (i) Maintaining the rich logic, resulting in the derivation of triviality. (ii) Replacing the rich logic by a weak logic the norms and presuppositions of which are not contradicted by the premises, usually resulting in the loss of the rich consequences one wanted to derive. Adaptive logics avoid the troubles related to the first horn of this dilemma, and at the same time they allow the rich consequences to be derived, viz. conditionally. Whether the condition is overruled or not, depends solely on (unconditional) consequences of the given set of premises. As far as the present purposes are concerned, it should be stressed most adaptive logics allow for meaning change. People unfamiliar with adaptive logics, may be surprised to hear that the language of a well-defined formal logic does not need to meet the standard of “one term, one meaning”. There have in fact been developed several adaptive logics which allow for meaning change with respect to both logical and non-logical constants by the Ghent group.9 The formal part of this paper relies heavily on their results, and especially on the results with respect to ambiguity-adaptive logics. Some general ideas will now briefly be explained. The upper limit logic. Every adaptive logic has an upper limit logic (henceforth ULL), i.e. a rich logic that one would like to apply as long as no problems are met in the domain of application. In many situations CL turns out to be the most appropriate ULL, although some situations might call for another ULL; its choice should be motivated by the user’s purposes. Presuppositions of the ULL. Let us focus for a while on CL as upper limit logic. When one applies the rules and axioms of CL, one implicitly agrees with the fact that these axioms should not be contradicted by the premises which one wants to apply CL to. One accepts that whenever we have A&B, we also have A and B separately. One accepts that whenever we have (∀x)A(x), we also have A(a) for every individual constant a of the language. An almost dogmatic assumption is that, for any formula A, A and ∼A cannot be simultaneously true. A non-logical assumption of CL, is the following: one has to assume that every non-logical constant C has a strictly unique meaning, so that when C occurs twice in the premises, one can identify the two occurrences with one another; all occurrences of one and the same non-logical constant (predicative, individual or propositional constant) are supposed to have one and the same meaning. The latter presupposition is hard to maintain in the presence of ambiguity and vagueness. We believe that there are many situations in which one would like to apply CL, whereas some presuppositions of CL are not met. In other words, we believe that there are many situations in which we should allow for specific 9 For meaning change with respect to logical-constants see, e.g, [1], [3], [5], [6]. For meaning change with respect to non-logical terms see, e.g, [9], [19], [21], [22], [23].
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abnormalities with respect to CL. If for instance someone wants to apply CL to an inconsistent set of premises, one is in big trouble if one does not tolerate any inconsistency. We think it is very rational to allow for inconsistencies (that do not cause triviality) in such a situation, although it would not be very clever to decide that, e.g., disjunctive syllogism becomes invalid in general. The lower limit logic. The lower limit logic (henceforth LLL) is obtained from the ULL by dropping the specific axiom or rule that causes the problems. A well known example: if CL is the ULL, and some set of premises might be inconsistent, the rule that causes the problem is ex contradictione quodlibet. If ruled out, the paraconsistent logic CLuN is obtained.10 On the syntactical level, CLuN is obtained from the positive part of CL by adding just one negation-axiom, viz. (A ⊃ ∼A) ⊃ ∼A. On the semantical level, a model verifies ∼A if it falsifies A, but not vice versa. It is easily seen that the LLL avoids the problems, but generally will be much too weak (with respect to the user’s purposes). The adaptive logic. Intuitively, an adaptive logic (henceforth AL) can be characterized as oscillating between an ULL and a LLL. In any case all LLL-consequences of a set of premises are AL-consequences. Moreover, we assume that not a single abnormality is true unless and until it is derived from the premises. Thus we obtain conditionally derived formulas. E.g., the inconsistency-adaptive logic ACLuN2 allows for all rules valid in CLuN, and moreover, it assumes that we can derive q from p ∨ q and ∼p unless and until p&∼p is derivable from the premises.11 The procedure of adaptive logics is such that the derivation of abnormalities blocks the application of the rule of the ULL to these specific formulas, whereas all rules of the ULL can be applied to all other formulas occurring in, or derivable from, the set of premises. The choice of an appropriate adaptive logic. All abnormalities of CL surface as inconsistencies. In view of this, one might be inclined to use some inconsistency-adaptive logic whenever some presupposition of CL is not fulfilled. Given the fact that, for a consistent set of premises, the ACLuN2-consequent set equals the CL-consequence set, whereas, for an inconsistent set of premises, ACLuN2 avoids triviality and gives a richer consequence set than CLuN, one may be inclined to consider ACLuN2 as the best logic in most situations. Nevertheless, in many situations we are able to indicate the abnormalities more accurately. If we work with rules that might have exceptions, it is more appropriate to allow for exceptions on the instantiation rule than to allow for inconsistencies in general. In this paragraph we introduce the kind of abnormalities we expect to meet in the field of the Sorites paradoxes. We start with 10 CLuN stands for the logic obtained from CL by allowing for gluts with respect to Negation (see [3]). Analogously CLaN stands for the logic obtained from CL by allowing for gaps with respect to Negation. 11 For the accurate definition of the ACLuN2-consequence relation, see [3].
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an example. Suppose John is rich, and John is just a little bit richer than Mary. From this premises, we want to conclude that Mary is rich too. Please notice that the expression “rich too” does not have exactly the same meaning as “rich”; exactly this fact is the cause of the paradox: in almost any situation it is alright to identify “rich too” with “rich”. Let us continue the example. Suppose Mary is just a little bit richer than Paul; again we want to conclude that Paul is rich too. When we formalize these sentences, we can indicate very accurately what is going on. First of all, we work with the vague predicates of rank 1 R (“is rich”) and R0 (“is rich too”), with predicate of rank 2 “
