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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

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## Vagueness and Indiscriminability by Failure

Ariel Cohen

Ben-Gurion University

Israel

### 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

• 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

• 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?

• 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

Every individual is indiscriminable from itself.

### 2. Symmetry

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

### 3. Weak Transitivity

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

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

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

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

• Weak substitutivity is entailed by substitutivity.

### 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.

### Red

• 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.

### Rich

• 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

• Reflexive

• Symmetric

But:

• Not weakly transitive

• Not weakly substitutive

Hence, not a good definition

### 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

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

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

### DS

1.AX1: x DISC(x,x)

2.AX2: x y( DISC(y,x)DISC(x,y))

3.AX3: x yz

(DISC(x,z)(DISC(x,y)DISC(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:

={P(a1),P(an)}

### 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

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

DS|= (a)

and

DS|= (b)

### 1. Reflexivity

• By AX1, for every a,

DS|=DISC(a,a)

• By the consistency of :

DS|DISC(a,a)

### 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:

DS|=DISC(a,b)

• Hence, indisc. by failure is symmetric

### 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 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 Restriction

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

### 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

• “Bare bones” sorites:

={P(a1),P(an)}

• Hence, for all 1<i<n:

DS|P(ai)

DS|P(ai)

• Therefore, for all 0<i<n:

DS|DISC(ai,ai+1)

### a1 is Discriminable from an

• DS|= P(a1)

• DS|= P(an)

• By weak substitutivity:

• DS|= DISC(a1,an)

### A Cut-off Point Exists

• By repeated applications of weak transitivity:

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

• All consecutive pairs of elements are indiscriminable

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