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Vagueness and Indiscriminability by Failure. Ariel Cohen Ben-Gurion University Israel . The Sorites. A sequence of elements: a 1 …a n For all 0<i<n , a i is indiscriminable from a i+1 . For some (vague) property P : P(a 1 ) and  P(a n ) .

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Vagueness and indiscriminability by failure

Vagueness and Indiscriminability by Failure

Ariel Cohen

Ben-Gurion University


The sorites
The Sorites

  • A sequence of elements: a1…an

  • For all 0<i<n, ai is indiscriminable from ai+1.

  • For some (vague) property P:

    P(a1) and P(an).

  • Intuitively: a1 is discriminable from an.

  • Intuitively: there is some cut-off point i dividing Ps from Ps.

  • Hence: ai is discriminable from ai+1.


Epistemic solution
Epistemic Solution

  • There is a cut-off point i.

  • But i is unknown/unknowable.

  • Hence, for all 0<i<n, ai is indiscriminable from ai+1.

Supervaluation solution
Supervaluation Solution

  • There is a cut-off point i.

  • But for no i does it follow that i is the cut-off point.

  • Hence, for all 0<i<n, ai is indiscriminable from ai+1.

What is indiscriminability
What is Indiscriminability?

  • The paradox, then, hinges on what we take indiscriminability to mean.

  • A definition is required which is compatible with:

  • the solutions to the sorites;

  • our intuitions.

  • Specifically: There is a cut-off point, but this does not lead to contradiction.

1 reflexivity
1. Reflexivity

Every individual is indiscriminable from itself.

2 symmetry
2. Symmetry

If ais indiscriminable from b , then b is indiscriminable from a.

3 weak transitivity
3. Weak Transitivity

  • Indisc. is not transitive: otherwise, a1would be indisc. from an

  • Instead—“weak” transitivity:

    If a is discriminable from b, then for all c, c is discriminable from a or from b .

  • Weak transitivity is entailed by transitivity

  • Weak transitivity can be shown to entail the existence of the cut-off point:

    If a1 and an are discriminable, there is a cut-off point i where ai is discriminable from ai+1.

4 weak substitutivity
4. Weak Substitutivity

  • Indisc. is not substitutive: otherwise, if P(a1), it would follow that P(an).

  • Instead—“weak” substitutivity:

    If P(a) but P(b), then ais discriminable from b.

  • Weak substitutivity is entailed by substitutivity.

5 contextual restriction
5. Contextual Restriction

  • Weak substitutivity only applies to a “relevant” set of discriminating properties.

  • For example, P cannot be the indiscriminability relation itself, or indisc. would become transitive.

  • In the sorites: only the vague predicate in question is relevant for discrimination.


  • Every two consecutive tiles are indiscriminable with respect to their redness.

  • They are discriminable with respect to other properties, e.g. their size or location.

  • But these properties are not contextually relevant, hence do not make the tiles discriminable with respect to their redness.


  • If a has one cent more than b, they are indiscriminable with respect to richness.

  • Although they are not indiscriminable in an absolute sense: they have different sums of money (and different names, hair colors, etc.)

Direct pairwise indiscriminability
Direct, Pairwise Indiscriminability

  • Reflexive

  • Symmetric


  • Not weakly transitive

  • Not weakly substitutive

    Hence, not a good definition

Indiscriminability by failure
Indiscriminability by Failure

  • Based on Cohen and Makowsky's (1993) Equality by Failure.

  • Two elements are equal by failure iff they cannot be proved to be different.

  • Similar to Negation by Failure: if we cannot prove that something is true, we conclude that it is false.

Equality axioms
Equality Axioms

1. Reflexivity: x x=x

2. Symmetry: x y( x=yy=x)

3. Transitivity: x y z(( x=yy=z)x=z)

4. (functional substitutivity)

5. Predicate substitutivity: For every k-ary predicate symbol P (apart from ‘=‘) and every 1ik:

x y z1...zi-1 zi+1...zk

(x=y 

(P(z1...zi-1,x,zi+1...zk) P(z1...zi-1,y,zi+1...zk)))

Indiscriminability axioms
Indiscriminability Axioms

Change every = to DISC(,), and apply contraposition to simplify:


1. AX1: x DISC(x,x)

2. AX2: x y( DISC(y,x)DISC(x,y))

3. AX3: x yz


4. (AX4)

5. AX5: For every k-ary predicate symbol P (apart from DISC) and every 1ik:

x y z1...zi-1 zi+1...zk

( (P(z1...zi-1,x,zi+1...zk) P(z1...zi-1,y,zi+1...zk)) DISC(x,y))

  • A consistent set of propositions (that do not contain DISC), representing what is known (epistemic theories) or definite (supervaluation theories).

  • In a “bare bones” sorites:


Indiscriminability by failure defined
Indiscriminability by Failure Defined

  • Let  be a consistent set of propositions.

  • Then ais indiscriminable by failure from b with respect to  iff

    DS| DISC(a,b)

Indiscriminability of indiscernibles
Indiscriminability of Indiscernibles

DS|= DISC(a,b) iff there is some property  s.t

DS|= (a)


DS|= (b)

1 reflexivity1
1. Reflexivity

  • By AX1, for every a,


  • By the consistency of :


2 symmetry1
2. Symmetry

  • Suppose indisc. by failure were not symmetric. Then, for some a, b:

    DS|DISC(a,b) yet DS|=DISC(b,a)

  • But, by AX2:


  • A contradiction

  • Hence, indisc. by failure is symmetric

3 weak transitivity1
3. Weak Transitivity

  • Indisc. by failure is not transitive

  • For example:

    • ={P(a),P(c)}

    • DS|DISC(a,b) andDS|DISC(b,c)

    • yet DS|=DISC(a,c)

  • But weak transitivity follows directly from AX3

4 weak substitutivity1
4. Weak Substitutivity

  • Indisc. by failure is not substitutive

  • For example:

    • ={P(a)}

    • DS|DISC(a,b)

    • yet DS|P(b)

  • But weak substitutivity follows directly from AX5.

5 contextual restriction1
5. Contextual Restriction

Follows from the definition: indisc. by failure is defined relative to 

Helping to solve the sorites
Helping to Solve the Sorites

  • Indisc. by failure can be superimposed on an epistemic or supervaluation theory to solve the sorites.

  • As desired, the existence of a cut-off point is entailed, yet no contradiction follows.

Two consecutive elements are indiscriminable
Two Consecutive Elements are Indiscriminable

  • “Bare bones” sorites:


  • Hence, for all 1<i<n:



  • Therefore, for all 0<i<n:


A 1 is discriminable from a n
a1 is Discriminable from an

  • DS|= P(a1)

  • DS|= P(an)

  • By weak substitutivity:

  • DS|= DISC(a1,an)

A cut off point exists
A Cut-off Point Exists

  • By repeated applications of weak transitivity:

  • DS|= DISC(a1,a2)  DISC(a2,a3)...DISC(an-1,an)

No contradiction
No Contradiction

  • All consecutive pairs of elements are indiscriminable

  • There is a cut-off point i where ai is discriminable from ai+1

  • Is this a contradiction?

  • No: The existence of a cut-of point follows from , but it does not follow for any i that ai is discriminable from ai+1


  • Indisc. by failure is an intuitively plausible definition of indiscriminability: reflexive, symmetric, weakly transitive, weakly substitutive, and contextually restricted.

  • In the sorites, it entails the existence of a cut-off point, yet does not lead to contradiction.

  • Hence, it is the right sort of indisc. to be incorporated into a solution to the sorites.