Vagueness and Indiscriminability by Failure

1. Vagueness and Indiscriminability by Failure Ariel Cohen Ben-Gurion University Israel

2. 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!

3. 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.

4. 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.

5. 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.

6. 1. Reflexivity Every individual is indiscriminable from itself.

7. 2. Symmetry If ais indiscriminable from b , then b is indiscriminable from a.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.)

13. Direct, Pairwise Indiscriminability • Reflexive • Symmetric But: • Not weakly transitive • Not weakly substitutive Hence, not a good definition

14. 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.

15. 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)))

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

17. 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))

18.  • 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)}

19. 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)

20. Indiscriminability of Indiscernibles DS|= DISC(a,b) iff there is some property  s.t DS|= (a) and DS|= (b)

21. 1. Reflexivity • By AX1, for every a, DS|=DISC(a,a) • By the consistency of : DS|DISC(a,a)

22. 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

23. 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

24. 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.

25. 5. Contextual Restriction Follows from the definition: indisc. by failure is defined relative to 

26. 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.

27. 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)

28. a1 is Discriminable from an • DS|= P(a1) • DS|= P(an) • By weak substitutivity: • DS|= DISC(a1,an)

29. A Cut-off Point Exists • By repeated applications of weak transitivity: • DS|= DISC(a1,a2)  DISC(a2,a3)...DISC(an-1,an)

30. 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

31. 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.