Vagueness and Indiscriminability by Failure

Ariel CohenBen-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. Contradiction!

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:1. the solutions to the sorites;2. our intuitions. Specifically: There is a cut-off point,

but this does not lead to contradiction.

1. ReflexivityEvery individual is indiscriminable

from itself.

2. SymmetryIf a is indiscriminable from b , then b

is indiscriminable from a.

3. Weak Transitivity Indisc. is not transitive: otherwise, a1 would

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 Indisc. is not substitutive: otherwise,

if P(a1), it would follow that P(an). Instead—“weak” substitutivity:

If P(a) but P(b), then a is 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 SymmetricBut: Not weakly transitive Not weakly substitutiveHence, 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 Axioms1. Reflexivity: x x=x2. Symmetry: x y( x=yy=x)3. Transitivity: x y z(( x=yy=z)x=z)4. (functional substitutivity)5. Predicate substitutivity: For every k-ary

predicate symbol P (apart from ‘=‘) and every 1ik:

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 AxiomsChange every = to DISC(,),

and apply contraposition to simplify:

DS1. AX1: x DISC(x,x)2. AX2: x y( DISC(y,x)DISC(x,y))3. AX3: x yz (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 1ik: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 a is indiscriminable by failure

from b with respect to iff DS| DISC(a,b)

Indiscriminability of IndiscerniblesDS|= DISC(a,b) iff there is some

property s.t DS|= (a)andDS|= (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) A contradiction 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 RestrictionFollows 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)

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

Conclusion 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.