aa r X i v : . [ m a t h . L O ] O c t A NOTE ON THE REVERSE MATHEMATICS OF THE SORITES
DAMIR D. DZHAFAROV
Abstract.
Sorites is an ancient piece of paradoxical reasoning pertaining tosets with the following properties: (Supervenience) elements of the set aremapped into some set of “attributes”; (Tolerance) if an element has a givenattribute then so are the elements in some vicinity of this element; and (Con-nectedness) such vicinities can be arranged into pairwise overlapping finitechains connecting two elements with different attributes. Obviously, if Super-veneince is assumed, then (1) Tolerance implies lack of Connectedness, and(2) Connectedness implies lack of Tolerance. Using a very general but precisedefinition of “vicinity”, Dzhafarov and Dzhafarov (2010) offered two formal-izations of these mutual contrapositions. Mathematically, the formalizationsare equally valid, but in this paper, we offer a different basis by which tocompare them. Namely, we show that the formalizations have different proof-theoretic strengths when measured in the framework of reverse mathematics:the formalization of (1) is provable in
RCA , while the formalization of (2) isequivalent to ACA over RCA . Thus, in a certain precise sense, the approachof (1) is more constructive than that of (2). Introduction
In November of 2009, the
Reverse Mathematics: Foundations and Applications workshop at the University of Chicago asked about using mathematical logic as apossible new basis for judging and comparing alternative and competing quantita-tive approaches to problems in cognitive science. There have been several paperswritten in this direction (e.g., [5], [1]), and this note is a further such contribution.In it, we show that two ostensibly equivalent mathematical approaches for a cer-tain problem in cognitive science can be teased apart in terms of their logical (orproof-theoretic) strength.Sorites is a “paradox” attributed to Eubulites of Miletus, a philosopher of theMegarian School in the 4th century BCE. It continues to be of interest to philoso-phers of mind and cognitive scientists. The essence of the two original versions ofSorites (known as The Heap and The Bald Man) is that if natural numbers can beclassified as very large and not very large, and if N is a very large number (a heapof grains, a full head of hair), then so is N −
1; but by repeated subtractions of 1(removing grains or hairs one-by-one) one can get from N to any n that is not avery large number. There are obvious analogues of Sorites in a continuum of realnumbers (e.g., growing a very short person by a sufficiently small amount would notmake this person not very short), and spaces without linear orders (e.g., in the setof spectra of color patches, a sufficiently small change in the spectral compositionof a patch judged to be red would not change its redness). There is an opinion thatdue to the use of the notion of “small changes” (sufficiently small, or as small as The author was partially supported by NSF Grant DMS-1400267. The author is grateful toE. N. Dzhafarov for helpful discussions. possible) Sorites requires a metric space [11] or at least a full-fledged topologicalspace [10]. However, it has been shown in [3] that the most general formulation ofSorites only requires a variant of the pre-topological structure proposed by Fr´echetand dubbed by him
V-spaces (see, e.g., [7]).
Definition 1.1. A V-space is a pair ( X, {V x : x ∈ X } ), where X is a set, and foreach x ∈ X , V x is a non-empty collection of subsets of X containing x .When we have fixed a particular V-space ( X, {V x : x ∈ X } ), we call each V ∈ V x a vicinity of x . Each x ∈ X has at least one vicinity, and one can think of eachsuch vicinity as representing a “sense” in which the elements of that vicinity areclose to x . Since x belongs to each of its vicinities, it is therefore close to itself inevery sense. On the other hand, { x } need not belong to V x , and more generally,if V x ∈ V x and V ′ ⊂ V x , V need not belong to V x . Furthermore, if some y ∈ X belongs to some vicinity of x , it need not be the case that x belongs to somevicinity of y . In other words, y can be close to x in some sense, without x needingto be close to y in any sense; the notion of “being close to in some sense” is notnecessarily symmetric. The vicinities of a V-space can be used for the followingnatural definition of connectedness. Definition 1.2.
Let ( X, {V x : x ∈ X } ) be a V-space.(1) A cover of this V-space is a sequence { V x : x ∈ X } such that V x ∈ V x foreach x ∈ X .(2) Two elements a, b ∈ X are connected in this V-space if for every cover { V x : x ∈ X } there is a finite sequence x , . . . , x k of elements of X suchthat a = x , b = x k , and V x j ∩ V x j +1 = ∅ for each j < k .(3) If a, b ∈ X are not connected, then we say a cover { V x : x ∈ X } for whichthere is no finite sequence x , . . . , x k as in (2) witnesses that a and b arenot connected.Using the language of V-spaces, Dzhafarov and Dzhafarov [3] formulated thefollowing theorem central to our analysis. Theorem 1.3 ([3, Theorem 3.5]) . Let ( X, ( V x : x ∈ X )) be a V -space, Y a set,and π : X → Y a function. If a, b ∈ X are connected and π ( a ) = π ( b ) , then thereexists a x ∈ X such that π is not constant on any vicinity of x . The significance of this theorem for Sorites is as follows. The soritical reasoningis based on the assumption that the function π : X → Y can be chosen so that every x ∈ X has a vicinity V x with π ( y ) = π ( x ) for any y ∈ V x . This assumption is calledTolerance. The assumption that there are connected a, b ∈ X with π ( a ) = π ( b ) iscalled Connectedness. Using this terminology, the theorem above says that, for anyV-space and any function π , Connectedness implies lack of Tolerance.Dzhafarov and Dzhafarov [3] also considered the following alternative formula-tion, in which tolerance is assumed a priori. The theorem below can, by analogywith Theorem 1.3, be seen as a formalization of the statement that Tolerance im-plies lack of Connectedness. Definition 1.4.
Let X and Y be sets, and π : X → Y a function. The V-spaceinduced by π is ( X, {V x : x ∈ X } ), where for each x ∈ X , V x contains the singlevicinity { y ∈ X : π ( x ) = π ( y ) } . NOTE ON THE REVERSE MATHEMATICS OF THE SORITES 3
Theorem 1.5 ([3, Theorem 3.10]) . Let X and Y be sets and π : X → Y a function.If π ( a ) = π ( b ) for some a, b ∈ X , then a and b are not connected in the V-spaceinduced by π . It should be noted for completeness, that in many specific versions of Soritesthe main culprit of the ensuing contradiction is the very assumption that there is afunction mapping a specific set X , such as the set of heights, into a set of attributes Y , such as “tall, not tall.” However, this assumption (called Supervenience) oftencan be saved or at least made plausible by redefining the set X (e.g., replacingits elements by sequences in which they are listed) or the set of the attributes Y (e.g., replacing such attributes as “tall, not tall” with probability distributionsthereof). With this in mind, we can say that the two theorems above tell usthat Supervenience can only be achieved by dispensing with either Tolerance orConnectedness. Which of them it is in specific cases can be revealed by adoptingthe “behavioral” approach to Sorites [2, 3]. We do not need to discuss this approachhere: our focus in the present paper is the proof-theoretic strength of Theorems 1.3and 1.5. 2. Reverse mathematical analysis
Reverse mathematics is an area of mathematical logic devoted to classifyingmathematical theorems according to their proof theoretic strength. The goal is tocalibrate this strength according to how much comprehension is needed to establishthe existence of the sets needed to prove the theorem (i.e., according to how complexthe formulas specifying such sets must be). This is a two-step process. The firstinvolves searching for a comprehension scheme sufficient to prove the theorem, whilethe second gives sharpness by showing that the theorem is in fact equivalent to thiscomprehension scheme over some base (minimal) theory. In practice, we use forthese comprehension schemes certain subsystems of second-order arithmetic. Asour base theory, we use a weak subsystem called
RCA , which suffices to provethe existence of the computable sets, but not of any non-computable ones. Assuch, RCA corresponds to computable or constructive mathematics. A strictlystronger system is ACA , which adds to RCA comprehension for sets described byarithmetical formulas. ACA is considerably stronger than RCA , sufficing to provethe existence of, e.g., the halting set, and many other non-computable sets. Thereis, more generally, a rich and fruitful relationship between reverse mathematics onthe one hand, and computability theory on the other (see, e.g., [6] for a discussion).We refer the reader to Simpson [8] or Hirschfeldt [4] for background on reversemathematics, and to Soare [9] for background on computability.In this section, we provide the computability-theoretic and reverse mathematicalanalysis of Theorems 1.3 and 1.5. We begin by formalizing the concepts fromDefinition 1.1 in a countable setting. Definition 2.1.
Let X be a non-empty subset of ω .(1) A weak system of vicinities for x in X is a sequence W = h W n : n ∈ ω i such that x ∈ W n ⊆ X for all n .(2) A weak V-space is a pair ( X, {W x : x ∈ X } ), where for each x ∈ X , W x isa weak system of vicinities for x in X . DAMIR D. DZHAFAROV (3) A strong system of vicinities for x in X is a sequence S = h S n : n ∈ I i ,where I is a non-empty (possibly finite) initial segment of ω , x ∈ S n ⊆ X for all n ∈ I , and S n = S m for all n, m ∈ I with n = m .(4) A strong V-space is a pair ( X, {S x : x ∈ X } ), where for each x ∈ X , S x isa strong system of vicinities for x in X .Note that every strong system of vicinities for x in X computes a weak suchsystem. Namely, if h S n : n ∈ I i is a strong system of vicinities for x , define a weaksystem of vicinities h W n : n ∈ ω i for x by setting W n = S n for all n ∈ I , and W n = S for all n / ∈ I . The converse is false, because in a weak system of vicinities h W n : n ∈ ω i it could in principle be that W n = W m for some n = m , and there isno computable way to tell when this is the case. Proposition 2.2.
Let ( X, {W x : x ∈ X } ) be a computable weak V-space, Y acomputable set, and π : X → Y a computable function. If there exist a, b ∈ X with π ( a ) = π ( b ) , but every x ∈ X has a vicinity on which π is constant, then there is a ∅ ′ -computable cover witnessing that a and b are not connected.Proof. For each x ∈ X , write W x = h W x,n : n ∈ ω i . Now given x , search com-putably in ∅ ′ for the least n such that π ( y ) = π ( x ) for all y ∈ W x,n , which existsby assumption, and define V x = W x,n . Then h V x : x ∈ X i is a cover of X , and π isconstant on each V x . Since π ( a ) = π ( b ), this cover obviously witnesses that a and b are not connected. (cid:3) In the proof of the following result, we fix an computable enumeration ∅ ′ [ s ] of ∅ ′ . So for all x , we have x ∈ ∅ ′ if and only if x ∈ ∅ ′ [ s ] for some s , in which case also x ∈ ∅ ′ [ t ] for all t ≥ s . We write x ց ∅ ′ [ s ] if x ∈ ∅ ′ [ s ] and x / ∈ ∅ ′ [ t ] for any t < s . Proposition 2.3.
There exists a computable strong V-space ( X, {S x : x ∈ X } ) , acomputable function π : X → { , } , and a, b ∈ X with the following properties:(1) π ( a ) = π ( b ) ;(2) every x ∈ X has an vicinity on which π is constant;(3) every cover witnessing that a and b are not connected computes ∅ ′ .Proof. We work with X = ω . Let a < b be any two numbers not in ∅ ′ . For every x ∈ X , we uniformly construct a computable strong system of vicinities. Each of a and b will have a single vicinity, V a and V b , respectively, while every other x willhave infinitely many vicinities, V x, , V x, , . . . . Specifically, we let V a = { a } ∪ {h x, s i ∈ ω : x ց ∅ ′ [ s ] } , and V b = { b } ∪ {h x, s i ∈ ω : s ≥ ∧ x ց ∅ ′ [ s − } , and for every x / ∈ { a, b } and for every n , we let V x,n = { x } ∪ {h x, t i ∈ ω : t ≥ n } . Obviously, each of these vicinities is such that the resulting V-space is computable.Note that if x is different from a and b then V x,n = V x,m for all n = m . Also, if x isenumerated into ∅ ′ at some stage s , then h x, s i belongs to V a and h x, s + 1 i to V b ,so for any n ≤ s , V a ∩ V x,n = ∅ and V b ∩ V x,n = ∅ . It follows that, for any such n ,no cover containing V a , V b , and V x,n can witness that a and b are not connected.We shall make use of this fact below. NOTE ON THE REVERSE MATHEMATICS OF THE SORITES 5
We next define the function π : X → { , } . To begin, set π ( a ) = 0 and π ( b ) = 1. For all other numbers, we proceed by induction. Unless already defined,set π (0) = 0 and π (1) = 0. Now fix y > π ( y ) is undefined, and assumewe have defined π on all smaller numbers. Say y = h x, s i , so that in particular x < y . If x ց ∅ ′ [ s ], set π ( y ) = 0; if s ≥ x ց ∅ ′ [ s − π ( y ) = 1; andotherwise, set π ( y ) = π ( x ). This completes the definition. It is immediate that π is constant on each of V a and V b . For every x different from a and b , if x / ∈ ∅ ′ then π is also constant on each V x,n . And if x is enumerated into ∅ ′ , say at stage s , then x is constant on every V x,n for n ≥ s + 2.It follows by construction that our V-space and function π satisfy conditions (1)and (2) in the statement of the proposition. We conclude by verifying property (3).Fix any cover witnessing that a and b are not connected. Since a and b each havejust one vicinity, the cover must contain V a and V b . As noted above, if x ∈ ∅ ′ , thenthis cover cannot contain V x,n for any n ≤ s , where s is the stage at which x isenumerated into ∅ ′ . It follows that x ∈ ∅ ′ if and only if x is different from a and b ,and x ∈ ∅ ′ [ t ] for the least t such that h x, t i belongs to the vicinity of x in the cover.Thus, ∅ ′ is computable from the cover, as desired. (cid:3) Combining the above results allows us to characterize the proof-theoretic strengthof Theorem 1.3.
Theorem 2.4.
The following are equivalent over
RCA .(1) ACA .(2) Let ( X, {W x : x ∈ X } ) be a weak V-space, Y a set, and π : X → Y afunction. Suppose a, b ∈ X are connected in this V-space, and π ( a ) = π ( b ) .Then there exists an x ∈ X such that π is not constant on any W ∈ W x .(3) Let ( X, {S x : x ∈ X } ) be a strong V-space, Y a set, and π : X → Y afunction. Suppose a, b ∈ X are connected in this V-space, and π ( a ) = π ( b ) .Then there exists an x ∈ X such that π is not constant on any S ∈ S x .Proof. The implication from part 1 to part 2 follows by formalizing the proof ofProposition 2.2 in
ACA . The implication from 2 to 3 is immediate by the remarkfollowing Definition 2.1. The implication 3 to 1 follows by formalizing the proof ofProposition 2.3 in RCA . (cid:3) We obtain a very contrasting result concerning the strength of Theorem 1.5. Itis an easy observation that if X , Y , and π : X → Y are all computable, then theV-space induced by π is a computable strong V-space. Formalizing this, we havethat, given sets X and Y and a function π : X → Y , RCA can prove the existenceof the V-space (as a storng V-space) induced by π . Theorem 2.5.
RCA proves the following statement. Let X and Y be sets, and π : X → Y a function with π ( a ) = π ( b ) for some a, b ∈ X . Then a and b are notconnected in the V-space induced by π .Proof. We argue in
RCA . The only cover in the V-space induced by π is h S x : x ∈ X i , where S x is the (unique) vicinity of x , { y ∈ X : π ( x ) = π ( y ) } . Suppose h x , . . . , x k i is a finite sequence of elements of X such that a = x , b = x k , and S x j ∩ S x j +1 = ∅ for each j < k . Define h y j : j < k i such that y j is the least elementof S x j ∩ S x j +1 for all j < k , which exists by ∆ comprehension, and is well-definedby assumption. Thus, π ( y j ) = π ( y j +1 ) by assumption, so π ( y j ) = π ( a ) for all j ≤ k by Σ induction. But then π ( a ) = π ( y k − ) = π ( b ), a contradiction. (cid:3) DAMIR D. DZHAFAROV Conclusion
In the analysis of Sorites, Supervenience implies incompatibility of Connected-ness and Tolerance. We have shown that the two implications forming this incom-patibility, Connectedness → ¬
Tolerance and Tolerence → ¬
Connectedness, whenformalized using the general framework of Fr´echet spaces as in Dzhafarov and Dzha-farov [3], have different proof-theoretic strength: the formalization of the formerimplication has the strength of
RCA , while the formalization of the latter has thestrength of ACA . In this sense, the implication Connectedness → ¬ Tolerance canbe formalized by more constructive methods than its contrapositive.
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Department of Mathematics, University of Connecticut, Storrs, Connecticut U.S.A.
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