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Vagueness and Indiscriminability by Failure

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Vagueness and Indiscriminability by Failure

Ariel Cohen

Ben-Gurion University

Israel

- 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!

- There is a cut-off point i.
- But i is unknown/unknowable.
- Hence, for all 0<i<n, ai is indiscriminable from ai+1.

- 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.

- 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.

Every individual is indiscriminable from itself.

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

- 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.

- 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.

- 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.)

- Reflexive
- Symmetric
But:

- Not weakly transitive
- Not weakly substitutive
Hence, not a good definition

- 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.

1.Reflexivity: x x=x

2.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)))

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 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)}

- Let be a consistent set of propositions.
- Then ais indiscriminable by failure from b with respect to iff
DS| DISC(a,b)

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

DS|= (a)

and

DS|= (b)

- By AX1, for every a,
DS|=DISC(a,a)

- By the consistency of :
DS|DISC(a,a)

- 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

- Indisc. by failure is not transitive
- For example:
- ={P(a),P(c)}
- DS|DISC(a,b) andDS|DISC(b,c)
- yet DS|=DISC(a,c)

- But weak transitivity follows directly from AX3

- 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.

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

- 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.

- “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)

- DS|= P(a1)
- DS|= P(an)
- By weak substitutivity:
- DS|= DISC(a1,an)

- 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
- 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.