Sorites Paradox

The Sorites Paradox, or Paradox of the Heap, first proposed by Eubulides of Miletus in the 4th century BCE, asks whether removing a single grain of sand from a heap turns the heap of sand into not-a-heap. Clearly, removing a single grain of sand does not turn the heap into not-a-heap. Equally clearly however, if sand continues to be removed one grain at a time, eventually (in finitely many steps) there will no longer be a heap of sand. The paradox is that the inductive argument seems not to hold. A common variant of the Sorites Paradox, and the variant used here, is based on indistinguishablility of ‘shades of grey’. Suppose you have 100 shades of grey, from white through to black,

lined up in sequence. Due to the degree of similarity between adjacent shades, you cannot tell adjacent shades of grey apart, but can tell apart shades that are not adjacent. Distinguishability is not a transitive process.

For ease of notation, write the collection of colors as Ω = {0,1, . . . ,99}, where 0

represents white and 99represents black. This inability to tell apart adjacent nodes has a very natural interpretation in terms of Knowledge Operators. Specifically, for any collection of shadesE, you know that collection contains a particular shadeωonly when both the shadeω, and also the adjacent shades are inE. Ifω ∈E, butω+ 1 ∈/ E, then you cannot know ω ∈ E, as perhaps you are just mistakingω andω + 1. The knowledge operator is

KE ={ω∈Ω| {ω−1, ω, ω+ 1} ∩Ω⊂E} (4.10)

What is the probability that you can be sure a color paletteEcontains an arbitrarily chosen shade? Since the shade is chosen at random, letµ(ω) = 1/100for allω ∈Ω. The likelihood you are certain a paletteEcontainsωisµ(KE) =|KE|/100. Similarly, the probability you know the palette does not contain the shade isµ(K¬E). The maximum

likelihood you attach to the shade being inE is1−µ(K¬E) = µ(¬K¬E). Therefore, the imprecise probability thatE contains an arbitrary shade of grey is

P E = [µ(KE), µ(¬K¬E) ]

which is exactly the Behavioral Imprecise Probability.

For a concrete example, let E0 ={0,1,3,7,8,9,10}. Shade0∈KE0, as0and1are

inE0, shade1is not inKE0 as the adjacent shade2is not inE0, shades8and9are in

KE0 as their adjacent shades are inE0, while shades3,7, and10are not inKE0. Thus,

KE0 ={0,8,9}, andP∗E0 = 3/100. For the upper probabilityP∗E0, we need¬K¬E0.

By the construction ofK, there is a general form forK¬Efor any paletteEwhich is

¬K¬E ={ω∈Ω| {ω−1, ω, ω+ 1} ∩E 6=∅}

For ourE0 example above, P∗E0 = µ({0,1,2,3,4,6,7,8,9,10,11}) = 11/100. Overall,

P E0 = [3/100,11/100].

This knowledge structure allows for potentially quite large gaps between the lower and upper probabilities for certain events. For example, letE1 be the set of odd num-

bers inΩ, that is,E1 ={1,3,5, . . . ,99}. Then,KE1 =∅, and¬K¬E1 = Ω. The imprecise

probability is totally non-informative in thatP E1 = [0,1].

In any case, there is a natural knowledge structure associated with this problem which intuitively generates imprecise probabilities. The same cannot necessarily be said for the Multiple Prior Model. WhenK is given by Equation 4.10 andµis uniform, the imprecise probabilityP(K, µ)can be generated as a Multiple Prior Imprecise Prob- ability. This is guaranteed by Proposition 4.8. AsK(E∩F) =KE∩KF for all pairs of eventsE, F ∈2Ω_{, andKΩ = Ω, thenK is a correspondence operator. By Proposition}

4.8, this imprecise probability is a Multiple Prior Imprecise Probability. The functionγ

associated withK in the sense of Equation 4.3 is

γ(ω) ={ω−1, ω, ω+ 1}

The proof of Proposition 4.8 uses this γ function to construct a set of priors which generatesP =_P(K, µ). A simpler set of priors which generatesP is the setP such that

This set of priorsP generates the imprecise probabilityP(K, µ). However, it seems to do so in a somewhat artificial manner. Mathematically it works, but it is not clear how the description of the problem relates to the set of priors.

The Dempster-Shafer Belief Model has a greater intuitive connection to this version of the Sorites paradox than the Multiple Prior Model, but does not have as strong an intuitive connection as the Behavioral Imprecise Probability Model. To model P =

P(K, µ)using the Dempster-Shafer framework, let the Dempster-Shafer mass function be6

m(E) =

(

1/100 ; ifE ={ω−1, ω, ω+ 1}for someω∈Ω 0 ; otherwise

There is additional evidence that a stateωis inEprecisely whenE ={ω−1, ω, ω+ 1}. The lower probability of a color paletteE isP

F⊂Em(F), as usual for Dempster-Shafer

models.

Overall, while the outcome of the Sorites paradox can be modeled using the tradi- tional methods of Multiple Priors, and Dempster-Shafer models, this paradox is much more easily and intuitively modeled using the new Behavioral Imprecise Probability Model.