1. 1. A VAGUE CONCEPT: BALDNESS
Borderline
(Clearly)
not bald
(Clearly)
bald
“Almost
bald”

2. 2. BORDERLINE ITEMS: CLASH OF INTUITIONS
Borderline cases
are “real” and
violate
classical logic
For a border case:
neither P nor not P
Classical logic
is correct
E.g.
Principle of
Excluded Middle –
either P or not P
One solution is to deny that borderline cases are real
(all persons are in reality clear-cut bald or not bald).
“Borderline-ness” is just an epistemic phenomenon

3. 3. THE SORITES ARGUMENT FOR SHARP CUT-OFFS
A. Peter with zero hairs is bald
B: One hair does not change the justice with which we call someone bald. Or explicitly:
B0: If Peter with zero hairs is bald, then he is bald with 1 hair AND
B1: If Peter with 1 hair is bald, then he is bald with 2 hairs AND
B2: If Peter with 2 hairs is bald, then he is bald with 3 hairs AND …
B : If Peter with hairs is bald, then he is bald with
hairs.
C. Hence: Peter with 1 Mio hairs is bald.
One of the Bis must be false.
This means there is a sharp cut-off!
But that is counterintuitive…

4. 4. THREE FORMS OF IGNORANCE
Example
Characterization
My ignorance of the number of people in this room
I can practically know
if I make the necessary
epistemic effort
Weak
A (theoretical)
epistemic position
exists from which
I could know
A randomly printed and destroyed word that nobody has seen
Strong
Sharp borders for vague predicates (according to the epistemic account)
Necessarily and
completely
unknowable
Deep

5. 5. EPISTEMIC VIEW: T. WILLIAMSON
There are sharp cut-offs !
We cannot know sharp borders because such “knowledge” were unstable
This is so because a “Principle of margin for error” holds in the border area:
Had the meaning of “bald” been slightly different (and it could have easily been) a contrary judgment would have resulted. But we must judge in a stable way !
Problem: we cannot belief in any sharp cut-off value,
we cannot belief that there are sharp cut-offs
and want to know (ideally) what draws the sharp border

6. 6. SUBJECTIVE EXTENSIONS
extension of “bald” = the set of all people to which “bald” applies.
PROPOSAL:
Extensions are determined by the disposition to assert (or not) of a person
From this it follows that extensions are:
subjective
Each individual has its “own extensions”
adaptive and hence change over time
E.g. concept acquisition of children, accept “expertise”, etc.
context-sensitive
The surroundings, mood, etc. may bias my judgment
sharply bordered
“ Button test”: Do you press the “decision (assertion) button” or not when confronted with a person and the question “Is this person bald?”

7. 7. BORDERLINE ITEMS DO NOT VIOLATE THE “EXCLUDED MIDDLE”
Principle of
Excluded
Middle (PEM)
BALD
NOT BALD
Three
possible
outcomes
of a subjective
judgment
BALD*
BORDER-BALD*
NOT BALD*
Actively considered and judged
Determined “by elimination”

8. 8. CONCLUSIONS What we have gained: Sorites paradox solved
Conservation of classical logic
Borderline cases exist (not an epistemic illusion)
Understanding that there are sharp borders
We still ignore the exact cut-offs, but not “deeply” (“no conceptual blind-spot”)
We cannot practically map out our dispositions to assert the applicability of a vague predicate but we can conceive of a theoretical method to do so
Possible objections and problems:
Truth subjectivism and relativism ?
Infallibility ?
Etc.