Not Much Higher-Order Vagueness in Williamson's 'logic of clarity' Nasim Mahoozi , Thomas Mormann June 27, 2016 Abstract This paper deals with higher-order vagueness in Williamson's 'logic of clarity'. Its aim is to prove that for 'fixed margin models' (W,d, α, [ ]) the notion of higher-order vagueness collapses to second-order vagueness. First, it is shown that fixed margin models can be reformulated in terms of similarity structures (W,∼). The relation ∼ is assumed to be reflexive and symmetric, but not necessarily transitive. Then, it is shown that the structures (W,∼) come along with naturally defined maps h and s that define a Galois connection on the power set PW of W . These maps can be used to define two distinct boundary operators bd and BD on W . The main theorem of the paper states that higher-order vagueness with respect to bd collapses to second-order vagueness. This does not hold for BD, the iterations of which behave in quite an erratic way. In contrast, the operator bd defines a variety of 1 tolerance principles that do not fall prey to the sorites paradox and, moreover, do not always satisfy the principles of positive and negative introspection. Keywords: Higher-order vagueness, logic of clarity, fixed margin models, similarity structures, graphs, boundary operators, tolerance principles, Galois connection, topology. 1 Introduction This paper deals with the problem of higher-order vagueness in Williamson's 'logic of clarity'. Its aim is to show that for 'fixed margin models' (W,d, α, [ ]) (Williamson 1994), the notion of higher-order vagueness completely collapses to second-order vagueness. We give a new equivalent formulation of these models in terms of similarity structures (W,∼). The similarity relation ∼ is reflexive and symmetric but not necessarily transitive. It is related to d by x ∼ y ⇔ d(x, y) ≤ α. More precisely, let W be a set of worlds endowed with a binary similarity relation ∼. If two worlds w and v are similar to each other, this is denoted by w ∼ v . According to Williamson, vagueness is due to our lack of knowledge or our ignorance. He appeals to the margin for error principle to explain this ignorance. When the knowledge is inexact there are some margins for error principles. He formulates clearly operator in a fixed margin model as: [CA] = {w ∈ W : ∀x ∈ W d(w, x) ≤ α→ x ∈ [A]}. In this formulation, d is a metric that measures the similarity, α a margin for error and [A] a 2 set of worlds in which A is true. The 'fixed margin for error' model, satisfies KTB system. A fixed margin model (W,d, α, [ ]) is essentially equivalent to a similarity structure (W,∼). Define the similarity neighborhood co(w) of w ∈ W by co(w) := {v;w ∼ v}. Formulas and operators of propositional logic are interpreted in the familiar way as subsets and Boolean operators on the set W . Following Breysse & De Glas(2007) we define an operator h by hA := {w; co(w) ⊆ A}. Then we will show that hA is just Williamson's [CA], to be interpreted as 'Clearly A' (Williamson 1994, 279). Let PW denote the power set of W . Then the operator h is interpreted as a map PW PWh , taking A to hA. To define the concept of boundary we also need another operator which is in fact the dual of h. We call it s and define it as sA := ChC(A) where CA is a set theoretic complement of A. The operators h and s are interdefinable, namely hA := CsC(A). Now we can define another boundary by bdA := hsA ∩ hs(CA). In section 3, we will prove the following theorem: bd3A = bd2A. In Williamson (1994, 1999) he demonstrates that if there is second-order vagueness there will be higher-order vagueness as well. We will show that Williamson's operator 'C' can be redefined as h and with a slightly different approach we will see that higher-order vagueness collapses to the second-order one. Furthermore, we will show that (W,∼, h, s) satisfies the axioms of KTB. The crucial idea is that h and s are Galois-related; i.e., A ⊆ hB ⇔ sA ⊆ B for all A,B ∈ PW. The fourth section is devoted to the concept of boundary. We distinguish our concept of boundary, defined in section 3, from two other existing concepts of boundary in the 3 literature; namely, BDA, defined as the intersection of clearly A and clearly not A and the topological one, bdtA, defined as the intersection of closure of A and the closure of the complement of A. Moreover, for bd and bdt we make a difference between two concepts: thick and thin boundaries, to explain when the second-order vagueness collapses to the first order vagueness. In the last section we shall discuss that the model, explored in sections 3 and 4, in comparison with other accounts (Égré and Bonnay 2010; Bobzien 2012, 2015) can be considered as minimal and rather general in the sense that the axiom of positive introspection hA → hhA or of negative introspection ¬hA → h¬hA need not hold for h. Moreover, (W,∼, h, s) satisfies a weak version of tolerance principle slightly different from Williamson′s and Shapiro′s definitions. The outline of the paper is as follows: we first recall the 'logic of clarity', mentioned in the appendix of Williamson(1994). Then in the next step we shall introduce the basic notions and definitions in detail and will prove the central theorem. Section four is devoted to specific cases of boundaries and finally, we discuss the important philosophical role of using Galois connection operators in the higher-order vagueness disputes. 2 Williamson's fixed margin models and Similarity Structures To set the stage, let us recall Williamson's fixed margin model. Williamson (1994) described vagueness in terms of 'logic of clarity'. In the appendix he introduced a logic for 'clearly' to 4 define the phenomena of vagueness. He introduced different models based on fixed and variable margins for error. The latter contains a family of accessibility relations while the former just uses a single accessibility relation. In this paper, we will focus on the fixed one. Moreover, the discussion would be at a formal level and would remain neutral with regards to the sources of vagueness. Definition 2.1. (Williamson, 1994: 270-271) A fixed margin model is a quadruple (W, d, α, [ ]). In this model, W is a set of possible worlds, d a metric on W that measures their similarity, α a margin for error and [A] is the set of worlds at which A is true. In fact, [A] maps formulas to subsets of W such that for all formulas A,B: [¬A] = W − [A] (1) [A ∧B] = [A] ∩ [B] (2) [CA] = {w ∈ W : ∀x ∈ W d(w, x) ≤ α→ x ∈ [A]} (3) The formulas of a fixed margin model (W, d, 1, [ ]) with C operator satisfy the following modal axioms: (Taut) ` A if A is a truth-functional tautology (K) ` C(A ⊃ B) ⊃ (CA ⊃ CB) (T) ` CA ⊃ A (B) ` ¬A ⊃ C¬CA (MP) If ` A ⊃ B and A then ` B (RN) If ` A then ` CA 5 Now let us consider the metric d of a fixed margin model (W, d, α, [ ]) in more detail. Williamson observes that d defines a reflexive and symmetric relation R on W : ∀x, y ∈ W (xRy iff d(x, y) 6 α). The modal logic of the fixed margin model is based on the reflexive and symmetric relation R. On the other hand, any reflexive and symmetric accessibility relation R is defined by a metric d and a fixed margin α and there is a metric for any reflexive and symmetric relation. He gives an example of such a metric (Williamson 1994, 270-271): d(x, y) =  0 if x = y α if xRy but x 6= y α + 1 if ¬(xRy) We will introduce an equivalent formulation of this model. This new formulation is based on similarity structures. We will show that Williamson's reference to metrics is not necessary, rather, similarity structures do suffice; what is important for fixed margin models is the relation R. The upshot of these considerations is that what is logically significant about fixed margin model (W, d, α, [ ]) is encapsulated in the similarity structure (W,∼) = (W,R) where R = R(d, α). In the rest of the paper we write (W,∼) instead of (W,R(d, α)). Finally, let us ellucidate the relation between fixed margin models and similarity structures: 6 Lemma 2.2. Let (W,d, α) be a fixed margin model. We define maps (W,d, α) (W,∼, α) φ ψ (W,d, α) (W,∼, α)φ φ(W,d, α) := (W,∼, α) with x ∼ y := d(x, y) ≤ α (W,∼, α) (W,d, α)ψ ψ(W,∼, α) := (W,d, α) with d(x, y) :=  0 if x = y α if x ∼ y but x 6= y α + 1 if otherwise Often it is useful to represent similarity structures by undirected graphs: Definition 2.3. Similarity structures (W,∼) can be represented as simple undirected graphs (G(W ), E(G)) where the vertices are members of W and there is an edge between each two different members of W that are similar. G(W ) is defined in the following way: ∀x∀y((x, y) ∈ E(G)↔ (x ∼ y)). The next definition will be a key definition in the paper. Definition 2.4. Let (W,∼) be a similarity structure and w ∈ W . Define the similarity neighborhood co(w) of w: 7 co(w) := {v;w ∼ v} The following example is to familiarize the reader with the definitions introduced thus far. Example 2.5. Let (W,∼) be a similarity structure, W = {1, 2, 3, 4, 5, 6, 7, 8} and 1 ∼ 2, 1 ∼ 3, 2 ∼ 3, 3 ∼ 4, 4 ∼ 5, 5 ∼ 6, 6 ∼ 7, 7 ∼ 8, 8 ∼ 4. Then (W,∼) is represented by the graph: Figure 1. 12 3 45 6 7 8 By definition, the similarity neighborhoods co(w) of w ∈ W are given by: co(1) = {1, 2, 3} co(2) = {1, 2, 7} co(3) = {1, 3, 4} co(4) = {3, 4, 5, 8} co(5) = {4, 5, 6} co(6) = {5, 6, 7} co(7) = {2, 6, 7, 8} co(8) = {4, 7, 8} In this paper graphs are used to provide a rich source of examples and counterexamples (See section 4). 8 3 Similarity Structures and Galois connections Following Breysse & De Glas(2007) we define hA based on neighborhood co(w) and claim that it is Williamson's clarity operator C for a fixed margin model. Definition 3.1. Let (W,∼) be a similarity structure. Then for A ⊆ W , hA is defined as: hA := {w; co(w) ⊆ A} The operator h is interpreted as a map PW PWh , taking A to hA, where PW denotes the power set of W . It can easily be shown that the map h is distributive with respect to arbitrary intersections, i.e. h( ⋂ Ai) = ⋂ h(Ai), Ai ∈ PW . The operator h is identical to Williamson′s C operator. The complement of a set A ⊆ W is denoted as CA. Define s := ChC. s is called the dual of h. The concatenations of operators h and s, and s and h are denoted as hs and sh respectively. The main novelty of our approach is based on the observation that h is a component of a Galois connection (h, s) (cf. Giertz et al. 2003). Definition 3.2. Let PW PWh be an order-preserving map with respect to settheoretical inclusion ⊆. An order-preserving map PW PWs is said to be Galois related to h, denoted by (h, s) iff ∀A,B ∈ PW,A ⊆ hB ⇔ sA ⊆ B. 9 hA shA A hsA sA Figure 2. Multi-layer model Proposition 3.3. b)s i) Given a ⋂ -distributive map h, the map s := ChC, is Galois-related to h. ii) If h and s are Galois related, then the following series of inclusions will be obtained: ...hhA ⊆ shhA ⊆ hA ⊆ shA ⊆ A ⊆ hsA ⊆ sA ⊆ hssA ⊆ ssA... For the details of the proof see Lemmas 3.8 and 3.10. Also (cf. Giertz et al. 2003, 25). Definition 3.4. Let (W,∼, h, s) be a similarity structure and (h,s) Galois connection and s0A = h0A = A. A layer system, Lk, k ∈ Z for A ⊆ W is defined as: Definition 3.5. Define a binary relation as: 10 Table 1. Layers Lk k k > 0 k ≤ 0 |k| = 2i siA− hsiA hiA− shi+1A |k| = 2i− 1 hsiA− si−1A shiA− hiA x y iff ∃i, j ∈ Z; for j − i ≥ 2, x ∈ Li and y ∈ Lj. x is said to be distinguishible from y with respect to the predicate A iff x y or y x. Definition 3.6. Let (W,∼, h, s) be a similarity structure. Define a binary relation ∼A as: x ∼A y iff ¬ (x y) and ¬ (y x). x ∼A y is true iff ∃i, j ∈ Z, |i− j| ≤ 1, x ∈ Li, y ∈ Lj (cf. Rooij 2007, 208). It can easily be seen that ∼A is reflexive and symmetric but not transitive. Proposition 3.7. Let and ∼A be two binary relations on a set W. Then (,∼A) is a semiorder of W (cf. Luce 1956; Rooij 2007). In the following easy lemma some crucial properties of h, needed for the proof of the pivotal Theorem 3.14, are collected: Lemma 3.8. Let (W,∼) be a similarity structure, A,B ⊆ W . Then the operator h has the following properties: (i) hA ⊆ A. 11 (ii) For Ai ∈ W, h( ⋂ Ai) = ⋂ h(Ai). (iii) ChChA ⊆ A. (iv) if A ⊆ B then hA ⊆ hB. Proposition 3.9. For the systems (W,∼, h, s) the Brouwersche scheme B is equivalent to the Galois axiom: (A ⊆ hB ↔ sA ⊆ B)←→ shA ⊆ A. Proof. −→: Suppose A ⊆ hB ↔ sA ⊆ B. We prove that shA ⊆ A. By assumption, h and s are Galois-related.Thus, by Proposition 3.3, shA ⊆ A. ←−: Assume shA ⊆ A and A ⊆ hB. A ⊆ hB, since s is monotone sA ⊆ shB. By the supposition shB ⊆ B. So, sA ⊆ shB ⊆ B. Now assume shA ⊆ A and sA ⊆ B. By monotonicity of h, hsA ⊆ hB. By the assumption, shA ⊆ A. On the other hand, by the definition of h and s, A ⊆ hsA. So, A ⊆ hB. The next lemma gives the most important properties of the operator hs, based on which the boundary is defined. 12 Lemma 3.10. Let (W,∼) be a similarity structure, (h,s) the corresponding Galois connection and A,B ⊆ W. Then hs is a closure operator with the following properties: (i) hs∅ = ∅, hsW = W. (ii) hshsA = hsA. (iii) A ⊆ hsA. (iv) If A ⊆ B then hsA ⊆ hsB. For the proof see Breysse & De Glas(2007). It is worth mentioning that in general, hs is neither a topological closure operator, i.e., hs(A ∪ B) 6= hsA ∪ hsB, nor a nucleus, i.e., hs(A ∩B) 6= hsA ∩ hsB. Example 3.11. Consider example 2.4 and let A be a subset of W . We will calculate the operators h, s, hs and sh for it: Let A = {4, 6, 7, 8}. Then hA = {8}, sA = {2, 3, 4, 5, 6, 7, 8}, hsA = {4, 5, 6, 7, 8} and sh(A) = {4, 7, 8}. Now we are in a position to give the exact definition of the boundary bd: Definition 3.12. Let (W,∼) be a similarity structure. The boundary bdA of A ⊆ W is defined as: bdA := hsA ∩ hs(CA). (cf. Breysse & De Glas 2007). 13 Remark 3.13. Let (W, ∼, h, s ) be a similarity structure and A ⊆ W. Then bdA is closed; i.e., hs(bdA) = bdA, since the the intersection of two closed sets is closed. Now the following theorem can be proved: Theorem 3.14. Let (W,∼, h, s) be a similarity structure and A ⊆ W. Then the following holds: (i) bdbdA ⊆ bdA. (ii) bdbdbdA = bdbdA. Proof. (i): bdbdA = hs(bdA) ∩ hs(CbdA) ( By Definition 3.12) = bdA ∩ hs(CbdA) ⊆ bdA (By Remark 3.13.) (ii): By definition, bd(A) = hs(A) ∩ hs(CA). So, bdbdbdA = hs(bdbdA) ∩ hs(CbdbdA)). The boundaries bdA and bdbdA are closed so hs(bdbdA)=bdbdA. Therefore, it is enough to show that hs(CbdbdA) = 1. hs(CbdbdA) = 1 ⇐⇒ hs(C(hs(bdA) ∩ hs(CbdA))) = 1 ⇐⇒ Csh(hs(bdA) ∩ hs(CbdA)) = 1 (shA = ChsCA) ⇐⇒ sh((bdA) ∩ hs(CbdA)) = 0 ⇐⇒ h((bdA) ∩ hs(CbdA)) = 0 (By proposition 3.3. h ⊆ sh.) 14 ⇐⇒ h(bdA ∩ hs(CbdA)) = 0 (By Remark 3.13 hs(bdA)=bdA.) ⇐⇒ hbdA ∩ hhs(CbdA) = 0 By (3.3. ii) hhs(CbdA) ⊆ s(CbdA). So, hbdA ∩ hhs(CbdA) ⊆ hbdA ∩ s(CbdA) = hbdA ∩ Chbd(A) = 0. Remark 3.15. The following counterexample shows that in general bdbdA $ bdA: Take (N,∼) with N = {0, 1, 2, ...}, and n ∼ m := |n − m| ≤ 1 Define A := 2N = {0, 2, 4, ..., 2n, ...}. Then bdA = N and bdbdA = bdN = ∅. Corollary 3.16. Let (W,∼, h, s) be a similarity structure corresponding to a fixed margin model (W,d, α, [ ]). Denote the nth iteration of the boundary operator by bdn. Then bdn(A) = bd2(A), n ≥ 2. Proof. By Theorem 3.14, bd3A = bd2A. By induction on n, one can prove that for any n greater than 3, bdnA = bdn−1. bd3A = bd2A, entails that bdnA = bd2A for n > 2. 4 Boundaries It is expedient to compare the concept of boundary as defined in 3.12., with some other ones that appear in the literature. Perhaps, the most common one is the following (Bobzien 2012, 2015; Égré 2010 ; Williamson 1994): 15 Definition 4.1. BDA := ¬CA ∩ ¬C¬A. Remark 4.2. Since h is just the clearly operator C, we can formulate (4.1.) as: BDA = ChA ∩ ChCA = sCA ∩ sA. The iterations of the boundary BD generally blow up: Example 4.3. Let (N,∼) be the similarity structure defined in Remark 3.15. For A = {0}, one obtains: BDA = {0, 1} BD7A = {0, 1, 2, 4, 5, 6, 7} BD2A = {1, 2} BD8A = {2, 3, 4, 7, 8} BD3A = {1, 2, 3} BD9A = {1, 2, 4, 5, 6, 7, 8, 9} BD4A = {0, 1, 3, 4} BD10A = {0, 1, 2, 3, 4, 9, 10} BD5A = {1, 2, 3, 4, 5} BD11A = {4, 5, 8, 10, 11} BD6A = {0, 1, 5, 6} BD12A = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12} One observes that in some of the iterations of BDA there are some gaps. For example in BD4A, {2} is a gap. Larger gaps occur in BD6A, {2, 3, 4} and in BD8A, {5, 6}. The number of gaps increases in BD11A where there are two gaps, {6, 7} and {9}. These gaps in the iterations of BD look quite irregular and that makes the concept of BD implausible. In contrast to BD which has an erratic behavior, bd is quite well-behaved since by Theorem 3.14., the iterations of the boundary operator bd do not grow so that for n greater than 2, bdn = bd2. 16 The relation between BD and bd, defined in 3.14., is succinctly expressed by the following proposition: Proposition 4.4. bdA = h(BDA). Proof. Since h is distributive with respect to ⋂ , bdA = hsA∩hsCA = h(sA∩sCA) = h(BDA) (see Lemma 3.8.). In topology, the concept of boundary is also central . Recall that a topological space is given by (X, cl) where cl : PX PX is a closure operator (Kuratowski 1968 ). Definition 4.5. The topological boundary operator bdt is defined based on the Kuratowski closure operator. bdtA := clA ∩ clCA. The dual operator of cl is called the interior kernel operator, denoted by int. It is defined as: int := CclC. The boundaries bd and bdt are somewhat analogous, i.e., h and s are almost analogous to int and cl respectively. Recall that topological closure not only satisfies all the conditions mentioned in Lemma 3.13., but also it is distributive with respect to the union: cl(A∪B) = clA∪ clB. The difference between these operators is discussed in more detail in Breysse & De Glas (2007). The boundary operators bd and bdt allow us to distinguish between two kinds of concepts: 17 Definition 4.6. Let W be a set for which (W,∼) is a similarity structure and (W, cl) is a topological space and A ⊆ W. Then i. A has a thin boundary bd if shbdA = ∅. Otherwise, A has a thick boundary. An analogous distiction can be made for the topological boundary bdt : it. A has a thin topological boundary bdt if int(bdtA) = ∅. Otherwise, A has a thick boundary. This distinction helps determine under what conditions the equation bdbdA = bdA holds. The following lemma shows that the second-order vagueness collapses to the first one if the boundary is thin. Lemma 4.7. Let (W,∼) be a similarity structure and (W, cl) be a topological space and A ⊆ W. Then i. bdbdA = bdA iff shbdA = ∅. ii. bdtbdtA = bdtA iff intbdtA = ∅. Proof. (i): Suppose bdbdA = bdA. Then by definition bdbdA = bdA ⇐⇒ hs(bdA) ∩ hs(CbdA) = bdA. ⇐⇒ bdA ∩ Csh(bdA) = bdA. (By Remark 3.15.) ⇐⇒ bdA ⊆ Csh(bdA). ⇐⇒ sh(bdA) ⊆ CbdA. sh(bdA) ⊆ bd(A). (By Proposition 3.3.) 18 Since sh(bdA) ⊆ CbdA we obtain sh(bdA) ⊆ bd(A) . So, shbdA = ∅. Now suppose shbdA = ∅. We prove that bdbdA = bdA. bdbdA = hs(bdA) ∩ hs(CbdA). (By Definition 3.12.) bdbdA = Csh(CbdA) ∩ Csh(bdA). (By interdefinability of h and s.) bdbdA = C(sh(bdA) ∪ sh(CbdA)). bdbdA = C(sh(CbdA)). (By assumption shbdA = ∅) bdbdA = hs(bdA) = bdA. (By Remark 3.13.) (ii) : −→: Suppose intbdtA = ∅. By Definition 4.5., bdtbdtA = clbdtA ∩ clCbdt. And intA = CclCA. Also, bdtA as an itersection of two closed sets is closed; i.e., clbdtA = bdtA. So, bdtbdtA = bdtA ∩ CintbdtA. By assumption, intbdtA = ∅. Therefore, bdtbdtA = bdtA. ←−: Now suppose bdtbdtA = bdtA. By Definition 4.5., bdtbdtA = clbdtA ∩ clCbdtA = bdtA. Since int and cl are dual and bdt is closed, bdtbdtA = bdtA ∩ CintbdtA = bdtA. Therefore, bdtA ⊆ Cintbdt. This is equivalent to intbdtA ⊆ CbdtA. But for the interior kernel operator, intbdtA ⊆ bdtA. So, intbdtA = ∅. For the topological boundary operator bdtA the class {A|intbdtA = ∅} is a non-complete Boolean algebra with respect to ∩,∪ and C (Stone 1936 ; McKinsey& Tarski 1944). Hence, 19 the natural question arises whether the analogous result holds for the boundaries bd and BD. The following example shows that for bd, neither the union and intersection of two concepts with thin boundaries nor the negation of a thin boundary concept need have a thin boundary. Example 4.8. For A and B with thin boundaries, bd(A ∪B) 6= bdA ∪ bdB. Let W = {1, 2, 3, 4, 5}, A = {2}, B = {4}. 1 A 2 3 B 4 5 A and B have thin boundaries. However, bd(A ∪B) is not equal to bd(A) ∪ bd(B). hs{2} = {2};hs{1, 3, 4, 5} ={1,2,3,4,5} so bdA = hs{2} ∩ hs{1, 3, 4, 5} = {2} and we can easily see that bdB = {4}. So bdA ∪ bdB = {2, 4} bd(A ∪B) = bd({2, 4}) = hs({2, 4}) ∩ hs({1, 3, 5}) = {1, 2, 3, 4, 5} ∩ {1, 2, 3, 4, 5} bd(A ∪B) = W. Now it is enough to show that A and B have thin boundaries. According to the definition, A has a thin boundary if shbdA = ∅ hbdA = hbdB = ∅. Therefore, shbdA = shbdB = ∅ . The result does not hold for BD either. The proof of it is left to the reader. 20 5 Vagueness and Galois connections There are many debates in the literature on what the prominent features of vagueness are. However, two main features are identified for vague predicates: they admit borderline cases and they give rise to the sorites paradox(Keefe 2000). Many accounts of vagueness deny the existence of sharp boundaries. In contrast, epistemists like Williamson claim that there is a fact of the matter where the boundaries are, we are just ignorant of it. More precisely, he sticks to the classical logic and contends that vague predicates have well-defined extensions of which we are ignorant. This ignorance is due to the fact that our knowledge is inexact. (Williamson 1994, 216). Inexact knowledge is rooted in the fact that indiscriminability is non-transitive (Williamson, 1990, 1994). When there is inexact knowledge, it seems reasonable to assume some margin for error principle. (5.1) Margin for error principle (MEP) : A is true in all cases similar to cases in which 'it is known that A' is true (Williamson 1994, 227). In contrast, the following principles do not hold: (5.2) Positive introspection (PI): For any proposition P, If I know that P, I know that I know that P. (5.3) Negative introspection(NI): if I do not know that P, I know that I do not know P. When it comes to vagueness, the operator I know that is to be replaced by It is clear that 21 and likewise, I do not know to it is not clear that. For systems (W,∼, h, s) the principles (MEP), (PI) and (NI) are formulated as: (MEP) ∀x, y (hAx ∧ x ∼ y → Ay) (PI) hA ⊆ hhA. (NI) ¬hA ⊆ h¬hA. (W,d, α, [ ]) and (W,∼, h, s) satisfy the principle (5.1). Égré (2010), finds PI and NI desirable. In contrast, it is easy to see from the proposition 3.5. that in (W,∼, h, s) models PI and NI principles need not hold. Nevertheless, there is higher-order vagueness but by Theorem 3.14., only of first and second order. One of the most intriguing features of vagueness is that it gives rise to the sorites paradox. (cf. Zardini 2008,2013). Vague predicates are sorites suceptible since they are not sensitive to the small changes. In other words, vague predicates are tolerant to tiny changes so that if a1 is P and a2 differs indiscriminably from a1 then a2 also is P. Wright(1974) dubbed it the "tolerance principle" . However, this leads to a paradox. Starting from a1 and applying the tolerance principle, little by little we come to a point an that is P but seems clearly not P. The paradox arises since the premises are true and also the rule of inference is valid but the conclusion appears to be false. Williamson(1994, 2000) claims that the failure of PI resolves the paradox. This failure also explains the higher-order vagueness phenomenon, commonly related to borderline cases. Recently, Pagin(2015) focused on this feature of vagueness with regard to higher-order vagueness 22 and, in particular, to the tolerance principle which is tightly related to the notion of margin for error(Égré 2015). Williamson rejects the tolerance principle and gives a weak version of it based on the margin for error principle. For him, vagueness is due to epistemic indescriminability that is a non-transitive relation and leads to the refutation of PI principle. The failure of PI not only avoids the Sorites paradox but also is associated with higher-order vagueness since it can be the case that it is clear that P but not clear that it is clear that P. In the rest of this section we will go through some versions of the tolerance principle based on formulations, proposed in Rooij(2012). Rooij(2012) starts with the following principle of tolerance as P: (P ) ∀x, y ∈ D : (Px ∧ x ∼p y)→ Py. Here D is the domain of objects and ∼p is the semiorder indistinguishability relation with respect to P as defined in Definition 3.6. (Rooij 2012, 207). Rooij mentions different ways of weakening (P ) . One of these weak versions is Williamson's. Since his clearly operator is just h, we will get: (PW ) ∀x, y ∈ D : (hPx ∧ x ∼p y)→ Py. In words, if x is clearly P and y is similar to x, then y is P . For models (W,∼, h, s) : (Psh) ∀x, y ∈ D : (hPx ∧ x ∼p y)→ shPy. 23 If the Brouwersche scheme holds, then (PW ) and (Psh) are equivalent. Shapiro(2006) introduced another weaker version of P. It relies on the concept of antonyms. P is the antonym of P if P ∩ P = ∅ and P ∪P 6= D. (PS) ∀x, y ∈ D : (Px ∧ x ∼p y)→ ¬Py. In words, if x is P and y is similar to x, then one cannot competently judge y in any other manner than x [within the same context](Shapiro 2008 , 6). Shapiro's contextual approach, using the sP as ♦P would be: (PS) ∀x, y ∈ D : (Px ∧ x ∼p y)→ sPy. For models (W,∼, h, s), (Phs) ,we propose a stronger principle than Ps: (Phs) ∀x, y ∈ D : (Px ∧ x ∼p y)→ hsPy. In general, in the multi-layer model, if x is in a layer and y is similar to x then y at most can belong to the next layer. If the B schema holds, this will be compatible with both Williamson's and Shapiro's views . Figure 2, shows the relation between the discussed principles of tolerance: 24 P Phs PS P Psh PW Figure 2. Comparison between principles of tolerance 6 Conclusion According to Williamson the validity of the margin for error principle MEP and the failure of PI and NI principles explain vagueness, higher-order vagueness, and also avoids the sorites paradox. We focused on the logic he gave for margin for error principle, 'logic of clarity'. One aim of our paper was to show that having a similarity structure is enough. There is no need for a metric. (W,∼, h, s) is equivalent to Williamson's model (W,d, α, [ ]), h being just Williamson's clearly operator C. We have shown that PI and NI principles need not hold. Furthermore, a similarity structure (W,∼) comes along with a Galois connection (h, s), and the corresponding modal system satisfies the axioms of KTB. Moreover, this has some interesting consequences: First, it explains why the hierarchy of boundaries stops at the second level and although there are higher-order vagueness, they collapse to the second one and under specific conditions where the boundary is thin they would collapse to the first-order vagueness. Secondly, it suggests a weaker version of tolerance, stronger than both Williamson's and Shapiro's versions. 25 In summary, it is well worth the effort to have a closer look at the Galois connections and their possible applications to vagueness. 7 References Bobzien, Susanne. (2015). Columnar higher-order vagueness, or vagueness is higher-order vagueness. 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