Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Logics for vague predicates:
Multiple truth values and Supervaluationism
Peter Sutton
[email protected]
HHU Düsseldorf, Institute für Sprache und Information
12th January, 2015
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Some sorites arguments:
(P1) 1000 grains make a heap
(P2) If 1000 grains make a heap, then 999 grains make a heap.
(P3) So, 999 grains make a heap
(...) ...
(C) 0 grains make a heap
——————————————————————————(P1) 1000 grains make a heap
(P2) Removing one grain doesn’t ever make a difference between a
heap and a non-heap
(C) 0 grains make a heap
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Why is this a problem?
It looks as though lots of predicates admit of these kinds of
‘indistinguishability’ or ‘too small to make a difference’ qualities.
∀x.∀y.F(x) ∧ SmallDiff (x, y, F) ⊃ F(y)
But at the same time admit of ‘big difference’ properties:
∀x.∃y.F(x) ∧ BigDiff (x, y, F) ⊃ ¬F(y)
But this will always take us to an absurd conclusion.
Call predicates that admit of both of these properties ‘vague’.
Then it looks as though classical logic and vague predicates don’t mix
very well.
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
What is vagueness?
1 Semantic/ Metaphysical/ Epistemic
G Semantic: Something about the meaning of words is vague.
(Can this be reduced to another semantic phenomenon?)
G Metaphysical: Something about the objects in the world are
vague.
G Epistemic: What we know about the meanings of words is
vague.
1 Blurred boundaries / Borderline cases
G Blurred boundaries: -ish, no clear end point for extensions.
G Borderline cases: Something can be/seem as F as ¬F
1 Sorites Susceptibility
G Can we form a sorites argument like we did just now for
‘heap’.
Introduction
Trivalency
Supervaluationism
What is vagueness? (cont.)
1 Indeterminacy, Degrees and Degrees of Indeterminacy
G Indeterminacy/Truth-value gaps
G cf. Paraconsistency/Truth-value gluts
G A matter of degree?
1 Vagueness across categories.
G As: tall, blue, bald, fast, brave.
G Ns: heap, mud, bravery.
G Advs: quickly, punctually.
G Vs/VPs?: run, close the door
G Ds/DPs: many students, most students
G Ps: in, under
Alternatives and Rejoinders
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Theory One: trivalent and multi-value logics
1 A possibly natural thought:
G Classical logic fails for vagueness because every statement
is either true or false (principle of bivalence).
Ý nb. PoB is not the same as the Law of the Excluded
Middle (φ ∨ ¬φ, for all propositions, φ).
G If bivalence fails then not every statement is either true or
false.
1 So perhaps we require a third truth value: True, False and
Indeterminate.
1 However, even for propositional logic, implementing this is not
straight forward.
G Presumably, we still want connectives to be
truth-functional.
G But, what truth-functions should we choose?
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Trivalent Logic
1 We have three truth values: True (1), False(0), and Indeterminate
(0.5).
1 There are different ways to implement this (see [Kleene, 1952]).
1 We will try to stay close to classical logic for limit cases (1 and
0), and will not let 0.5 overly dominate (not all complexes with
indeterminate parts are themselves indeterminate).
1 We will adopt Zadeh’s [Zadeh, 1965] Fuzzy logic (which
generalizes to truth-values in the range [0, 1]).
[[φ]]
[[¬φ]]
[[φ ∧ ψ]]
[[φ ∨ ψ]]
=def
=def
=def
=def
{0, 0.5, 1}
1 − [[φ]]
min{[[φ]], [[ψ]]}
max{[[φ]], [[ψ]]}
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Trivalent Truth-Tables
This yields the following truth tables:
φ
ψ φ∧ψ
1
1
1
1 0.5
0.5
φ ¬φ
1
0
0
1
0
0.5 1
0.5
0.5 0.5
0.5 0.5
0.5
0
1
0.5 0
0
0
1
0
0 0.5
0
0
0
0
φ
1
1
1
0.5
0.5
0.5
0
0
0
ψ
1
0.5
0
1
0.5
0
1
0.5
0
φ∨ψ
1
1
1
1
0.5
0.5
1
0.5
0
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Examples
Andy is tall and bald. Bob is borderline tall and borderline bald. Colin is
neither tall nor bald.
[[tall(a)]] = [[bald(a)]] = 1
[[tall(b)]] = [[bald(b)]] = 0.5
[[tall(c)]] = [[bald(c)]] = 0
1. Andy is tall and Colin is bald:
[tall(a) ∧ bald(c)]] = min{[[tall(a)]], [bald(c)]]} = min{1, 0} = 0
2. Andy is tall and Bob is bald:
[[tall(a) ∧ bald(c)]] = min{[[tall(a)], [bald(c)]]} = 0.5
3. Andy is bald or Colin is tall:
[[tall(a) ∨ bald(c)] = max{[[tall(a)]], [bald(c)]]} = max{1, 0} = 1
4. Bob is tall or Colin is bald:
[[tall(b) ∧ bald(c)]] = max{[[tall(b)]], [[bald(c)]]]} = 0.5
5. Andy is tall and Colin is not bald:
[[tall(a)∧¬bald(c)]] = min{[[tall(a)]], [¬bald(c)]]} = min{1, 1−0} = 1
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Edgington’s Objection
“Suppose we have a collection of balls of various sizes and colours.
These are independent variables: how close a ball is to a clear case of
small (in the context) is unaffected by its colour; and how close it is to
a clear case of red is unaffected by its size. Let Ra, Rb and Rc be the
statements that balls a, b, and c are red, respectively, and Sa, Sb and
Sc be the statements that they are small. Suppose:
[Ra] = 1
[Sa] = 0.5
[Rb] = 0.5 [Sb] = 0.5
[Rc] = 0
[Sc] = 0
According to [the conjunction rule], [Ra ∧ Sa] = [Rb ∧ Sb] = 0.5. But
it is plausible that a is a better case for red and small, than b... Would
not a–perfectly red and arguably small–be a better choice than b?”
[Edgington, 1997]
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Fine’s Objection
Keep part of Edgington’s example:
[Rb] = 0.5
[Sb] = 0.5
1 According to the conjunction and negation rule,
[Rb ∧ Sb] = [Rb ∧ ¬Rb] = 0.5. But that is absurd; Rb ∧ ¬Rb is a flat
contradiction. Surely it should receive a value of 0!
1 So sometimes conjunctions of indeterminates should be indeterminate
(red and small when something is reddish and smallish), but sometimes
it should be false (red and not red when something is reddish).
1 But that means that there cannot be a truth function for ∧ if we treat
Indeterminate as a truth value! (trivalent and multi-value logics cannot
fit the semantic data and be truth/degree functional.)
Fine’s Solution: Opt for gaps–there are only two truth values, but the
truth-function is incomplete (not every statement is either true or false).
This abandons bivalence, but maintains excluded middle (as we shall see).
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Truths on a penumbra (Penumbral truths)
1 Fine calls the fact that there are logical truths/relations that hold
between statements that are neither true nor false Truths on a
Penumbra or Penumbral Truths
G He claims that multiple value logics cannot accommodate
penumbral truths
1 For example. Even if ‘x is tall’ and ‘x is not tall’ are borderline
cases, we could:
(i) sharpen the meaning of ‘tall’ to make ‘x is tall’ false
(ii) loosen the meaning of ‘tall’ to make ‘x is not tall’ true
1 but there is no loosening or sharpening that would make ‘x is tall
and x is not tall’ true.
G This is because is is a penumbral truth that φ ∧ ¬φ is a flat
contradiction and so always false
G This can be so, even though φ ∧ ¬ψ is not always a flat
contradiction.
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Theory 2: Supervaluationism
1 Suppose that in English, there are different ‘meanings’ for tall
(say, with respect to the comparison class of adult males).
Tall1 (x)
Tall2 (x)
Tall3 (x)
Tall4 (x)
=
=
=
=
x ≥ 155cm
x ≥ 165cm
x ≥ 175cm
x ≥ 185cm
1 Then, the following can be said:
(i) if x is 150cm, ‘x is tall’ isn’t true on any meanings of Tall.
(ii) if x is 170cm, ‘x is tall’ is true on Tall1 and Tall2 , but not on
Tall3 or Tall4 .
(iii) if x is 190cm, ‘x is tall’ is true on all meanings of Tall.
1 So for some heights, it must be true that x is tall. For others, it
must be false that x is tall. For some heights in between, it may
be true or false that x is tall.
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Supertruth I
1 Fine’s dictum: “Truth is secured if it doesn’t turn on exactly one one
means” [Fine, 1975, p.278]
1 Define truth of φ, not as truth on some meanings of φ, but truth on all
meanings of φ
G “Truth is supertruth, truth from above” [Fine, 1975, p. 273]
1 Mutatis mutandis for falsity
1 If φ is true on some meanings, but not all, then φ is neither true nor
false (it lies in the truth value gap).
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Supertruth II
We can interpret tall relative to a context c: [[tall(x)]]c . Each interpretation
relative to a context will relate to a total disambiguation of tall (which
means that it is totally non-vague/sharp).
Say we have four contexts, C = {c1 , c2 , c3 , c4 }, and two truth-values {0, 1}.
[[tall(x)]]c1
[[tall(x)]]c2
[[tall(x)]]c3
[[tall(x)]]c4
=
=
=
=
x ≥ 155cm
x ≥ 165cm
x ≥ 175cm
x ≥ 185cm
John is tall is (super)true if and only if (i) [[tall(john)]]c = 1 for all c ∈ C.
John is tall is (super)false if and only if (ii) [[tall(john)]]c = 0 for all c ∈ C.
John is tall is neither (super)true nor (super)false if and only if neither (i) nor
(ii) hold.
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Logic for supervaluationism
1 Two-valued classical, bivalent logic for (sharp)
meanings/disambiguations of vague expressions.
1 The Law of the Excluded middle still holds:
G [[φ ∨ ¬φ]]c = 1 for all c ∈ C because φ is non-vague relative
to every context, and ∨ is classical.
1 Similarly non-contradiction still holds:
G [[¬(φ ∧ ¬φ)]]c = 1 for all c ∈ C
1 Many have taken this to be a big improvement on multi-value
theories.
1 However, we do not have bivalence:
G It is not the case that, for all φ, either φ is (super)true or φ is
(super)false
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Sorites and Supervaluationism I
People who are 190cm in height are tall, so
(1)
∀x[height(x)=190cm ⊃ tall(x)
But one centimetre surely can’t make the difference between being
tall and being not tall, so
(2)
∀x∀y[tall(x) ∧ height(y)=height(x)−1cm ⊃ tall(y)]
But this implies that 1m (or even 0m!) tall people are tall.
(3)
∀x[height(x)=100cm ⊃ tall(x)
(1) and (2) seem to be clearly true, the argument is classically valid,
but (3) is clearly false
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Sorites and Supervaluationism II
Here’s the supervaluationist solution...
(For simplicity) We can treat (2) as a conjunction of its instances. So let’s
read
(2)
∀x∀y[tall(x) ∧ height(y)=height(x)−1cm ⊃ tall(y)]
As
(tall(a1 ) ∧ height(a2 )=190cm ⊃ tall(a2 )) ∧
(tall(a2 ) ∧ height(a3 )=189cm ⊃ tall(a3 )) ∧
(tall(a3 ) ∧ height(a4 )=188cm ⊃ tall(a4 )) ∧ ...
... ∧ (tall(a90 ) ∧ height(a91 )=100cm ⊃ tall(a91 ))
Where a1 , a2 , ...a91 can be thought of as arbitrary names for values of the
variables x, y.
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Sorites and Supervaluationism III
We can evaluate this conjunction at every context:
[[(tall(a1 ) ∧ height(a2 )=190cm ⊃ tall(a2 )) ∧
(tall(a2 ) ∧ height(a3 )=189cm ⊃ tall(a3 )) ∧
(tall(a3 ) ∧ height(a4 )=188cm ⊃ tall(a4 )) ∧ ...
... ∧ (tall(a90 ) ∧ height(a91 )=100cm ⊃ tall(a91 ))]]c
[[tall(x)]]c1
[[tall(x)]]c2
[[tall(x)]]c3
[[tall(x)]]c4
=
=
=
=
x ≥ 155cm
x ≥ 165cm
x ≥ 175cm
x ≥ 185cm
At every context, one of the conjuncts is false. e.g. at context c3, the
conjunct: (tall(a17 ) ∧ height(a18 )=174cm ⊃ tall(a18 )) is false.
This means that (2) is superfalse. Sorites arguments are therefore
valid but unsound.
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Problems for supervaluationism I
Problem 1: The wrong result.
We just saw that (2) is meant to be superfalse:
(2)
∀x∀y[tall(x) ∧ height(y)=height(x)−1cm ⊃ tall(y)]
But wasn’t the point that 1cm cannot make a difference between tall
and not-tall. Supervaluationism claims that this is superfalse. In other
words that the following is supertrue:
(4)
∃x∃y[tall(x) ∧ height(y)=height(x)−1cm ∧ ¬tall(y)]
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Problems for Supervaluationism II
Problem 2: Higher-Order Vagueness.
Supervaluationism replaces a single sharp true/false boundary with
two sharp boundaries.
(i) A boundary between (super)true and neither (super)true nor
(super)false.
(ii) A boundary between (super)false and neither (super)true nor
(super)false.
This can be exploited. If tall is vague, then clearly/definitely/truly tall
should be vague. the supertrue extension of this is not vague on
supervaluationism.
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
Rejoinders and Alternatives
The issues discussed today are not final:
1 Supervaluationism does have its defenders [Keefe, 2000]
1 As does fuzzy logic [Smith, 2008]
For alternatives, the literature is vast. To name but a few:
1 Epistemicism [Williamson, 1994]
1 Degree-semantics [Barker, 2002]
1 Probabilistic [Edgington, 1997, Lassiter, 2011]
Introduction
Trivalency
Supervaluationism
Alternatives and Rejoinders
References I
[Barker, 2002] Barker, C. (2002).
The Dynamics of Vagueness.
Linguistics and Philosophy 25, 1–36.
[Edgington, 1997] Edgington, D. (1997).
Vagueness by Degrees.
In Vagueness: A Reader, (Keefe, R. and Smith, P., eds), pp. 294–316. MIT Press Cambridge,
MA.
[Fine, 1975] Fine, K. (1975).
Vagueness, Truth and Logic.
Synthese Vol. 30, 265–300.
[Keefe, 2000] Keefe, R. (2000).
Theories of Vagueness.
Cambridge University Press.
[Kleene, 1952] Kleene, S. C. (1952).
Introduction to Metamathematics.
North Holland.
[Lassiter, 2011] Lassiter, D. (2011).
Vagueness as Probabilistic Linguistic Knowledge.
In Vagueness in Communication, (Nouwen, R., Sauerland, U., Schmitz, H. and van Rooij,
R., eds),. Springer.
Introduction
Trivalency
Supervaluationism
References II
[Smith, 2008] Smith, N. J. J. (2008).
Vagueness and Degrees of Truth.
Oxford University Press.
[Williamson, 1994] Williamson, T. (1994).
Vagueness.
Routledge, Abingdon.
[Zadeh, 1965] Zadeh, L. (1965).
Fuzzy Sets.
Information and Control 19, 328–353.
Alternatives and Rejoinders