Vagueness and multivalued logic
UC Berkeley, Philosophy 142, Spring 2016
1
John MacFarlane
What is vagueness?
Three features standardly associated with vagueness:
1.1
Borderline cases
In standard semantics, we assume that each predicate has a determinate extension, a set of
objects in the domain to which it applies. The remaining objects are objects to which it
does not apply. But for vague predicates, there seem to be borderline cases. Is a pile of 20
grains of sand a heap? is someone with 100 short hairs bald? is Cool Whip food?
Note that we don’t think of our hesitation to answer here as merely epistemic (though
Williamson [11] will argue that it is!). Rather, we think there’s “no fact of the matter”
about whether the borderline cases fall into the extension of the vague term.
1.2
No sharp boundaries
In theory there might be borderline cases, but sharp boundaries between them and the
clear cases. Imagine an artificial predicate, ‘yuve’, defined as follows:
x is yuve if x is less than 16 years old, and x is not yuve if x is greater than 21
years old.
Arguably, ‘yuve’ is not vague. For vagueness, we need not just indeterminate cases, but
“no sharp boundaries” between the definite cases and the borderline cases.
1.3
Sorites susceptible
Vague terms are susceptible to sorites arguments:
(1)
If someone with n cents is rich, someone with n − 1 cents is also rich.
Someone with 1,000,000,000 cents is rich.
So, someone with 999,999,999 cents is rich.
So, . . .
So, someone with 1 cent is rich.
This version uses a universally quantified premise. We can also give a sorites that just has
a bunch of conditional premises:
(2)
If someone with 1,000,000,000 cents is rich, someone with 999,999,999 cents
is rich.
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2. Three-valued logics
If someone with 999,999,999 cents is rich, someone with 999,999,998 cents is
rich.
...
If someone with 2 cents is rich, someone with 1 cent is rich.
Someone with 1,000,000,000 cents is rich.
So, someone with 999,999,999 cents is rich.
So, . . .
So, someone with 1 cent is rich.
The only rule of inference used in this argument is modus ponens.
These arguments are problematic because we accept the premises but not the conclusion. If we don’t want to accept the conclusion, we must either reject one of the premises
or reject one of the modes of inference used. Let’s assume that the premise that someone
with 1,000,000,000 cents is rich should not be rejected. Then rejecting a premise means
rejecting a quantified statement of the form
∀n(R(n) ⊃ R(n − 1))
In classical logic, rejecting this premise means accepting its negation, which is
∃n(R(n) ∧ ¬R(n − 1)),
which posits a sharp one-penny boundary between the rich and the non-rich, and that
is difficult to swallow. (The same is true if we use the form of the argument with just
conditional premises, at least if we understand these as material conditionals.)
Rejecting the validity of the inference principles is also impossible, given classical logic.
So, it is tempting to think that an adequate account of the sorites argument will require
a nonclassical logic and semantics.
2
Three-valued logics
One attractive move is to say that we can reject the sorites premise without accepting its
negation. As long as we have bivalence, that’s not possible. Given bivalence, if the premise
is not true, it’s false, and its negation is true. But suppose we drop bivalence and allow that,
in the borderline area, the conditionals are neither true nor false?
2.1
Semantics for connectives
The simplest way to develop this idea is to use a three-valued logic: with values T (true),
F (false), and N (neither). A model is an assignment of one of these values to each propositional constant. This can be extended to compound formulas using multivalued truth
vables.
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2.1
Semantics for connectives
What should our truth vables look like? Everyone agrees with these constraints (considering just negation and conjunction):
1.
2.
3.
4.
The tables should agree with classical tables on classical “inputs” (T or F)
Negation should take T to F, F to T, and N to N.
A ∧ A should have the same value as A.
A ∧ B should have the same value as B ∧ A.
If we fill in truth tables with these constraints, we can see in which four squares where is
room for disagreement:
∧
T
N
F
T
T
2
F
N
1
N
4
F
F
3
F
1=2
3=4
There are two main strategies for filling in these squares:
Weak Kleene: N is “infectious,” so that any N component makse the whole thing N.
This makes sense if you read N as “meaningless.”
Strong Kleene: If both conjuncts are T, the conjunction is T, and if either conjunct is
F, the conjunction is F; otherwise N.
The situation is parallel with disjunction.
With the cnoditional, the Kleene tables understand it as the material conditional, so
that A ⊃ B is equivalent to ¬A ∨ B. Łukasiewicz makes a different choice: he wants a
conditional with an N antecedent and an N consequent to be T, rather than N, so that
A ⊃ A can be a tautology.
Bochvar = Weak Kleene = Halldén
¬
T
N
F
F
N
T
∧
T
N
F
T
T
N
F
N
N
N
N
F
F
N
F
∨
T
N
F
T
T
N
T
N
N
N
N
F
T
N
F
⊃
T
N
F
T
T
N
T
N
N
N
N
F
F
N
T
≡
T
N
F
T
T
N
F
N
N
N
N
F
F
N
T
∨
T
N
F
T
T
T
T
N
T
N
N
F
T
N
F
⊃
T
N
F
T
T
T
T
N
N
N
T
F
F
N
T
≡
T
N
F
T
T
N
F
N
N
N
N
F
F
N
T
⊃
T
N
F
T
T
T
T
N
N
T
T
F
F
N
T
≡
T
N
F
T
T
N
F
N
N
T
N
F
F
N
T
Strong Kleene = Körner
∧
T
N
F
T
T
N
F
N
N
N
F
F
F
F
F
Łukasiewicz
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2.2
Three ways to define validity
‘Determinately true’ operator
When Joe is borderline bald, we might want to say that it’s false that Joe is determinately
bald. We can think of ‘d’ (“determinately”) as a one-place sentential operator that works
d
T
like this:
N
F
2.2
T
F
F
Three ways to define validity
In classical logic, there is no difference between three validity definitions:
• preservation of truth
• preservation of non-falsity
• preservation truth and non-falsity (or of degree in the ordering T > F)
With three-valued logics, these come apart, and we have to choose. (The first and
third are the usual choices.) Here we can rely on philosophical considerations (what do
we really care about preserving when we use valid arguments?) and consideration of cases
(particular inferences).
Preservation of truth. Γ |= B iff for every model v, if every formula in Γ is T on v,
then B is T on v.
Only T is treated as a designated value (Łukasiewicz, Bochvar, Kleene, Tye).
Preservation of non-falsity. Γ |= B iff for every model v, if every formula in Γ is T
or N on v, then B is T or N on v.
Both T and N are treated as designated values (Halldén).
Preservation of truth and non-falsity. Γ |= B iff for every model v, if every formula
in Γ is T on v, then B is T on v, and for every model v, if every formula in Γ is T or
N on v, then B is T or N on v.
We preserve the degree in the ordering T > N > F.
Consider Disjunctive Syllogism or Modus ponens or the inference from A to B ⊃
(A ∧ B). Ask:
•
•
•
•
Valid with weak Kleene tables and preservation of truth?
Valid with weak Kleene tables and preservation of truth and nonfalsity?
Valid with strong Kleene tables and preservation of truth?
Valid with strong Kleene tables and preservation of truth and nonfalsity?
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designated
2.3
Application to sorites
2.3
Application to sorites
What kind of diagnosis of the sorites paradox do we get out of this?
Assume the Strong Kleene truth tables. Consider the form of the sorites that just uses
conditionals (??, above). The conditionals at the beginning, like
If someone with 1,000,000,000 cents is rich, someone with 999,999,999 cents is rich.
will have T antecedents and T consequents, and will therefore be T. The conditionals at
the end, like
If someone with 3 cents is rich, someone with 2 cents is rich.
will have F antecedents and F consequentsn, and will therefore be F. But in the middle,
at some point, we’ll have a conditional with a T antecedent and an N consequent (which
will be N), followed by a number of conditionals with N antecedents and N consequents
(which will also be N), and a conditional with an N antecedent and a F consequent (which
will be N too).1
So, some of our conditional premises will be T, and some will be N, but none will be
F. We now have a choice, depending on how we define validity:
• If we define validity as the preservation of truth, then the Modus Ponens inferences
are all valid. We end up with a false conclusion because, on this understanding, valid
arguments need not preserve non-falsity.
• If we define validity as the preservation of truth and non-falsity, then the Modus
Ponens inferences are not valid (because you can go from N premises to a F conclusion). So the sorites is invalid.
Either way, we’ve solved our original problem: we’ve provided a way to reject some
of the premises of the sorites argument without accepting a “sharp boundaries” claim of
the form: “Someone with n cents is rich, but someone with n − 1 cents is not rich.”
We do, however, still have “sharp boundaries” between the T’s and N’s, and between
the N’s and F’s. And, if we have a “definitely” or “determinately” operator that works like
d above, we can reproduce the sorites at a higher level, using premises like
If someone with 1,000,000,000 cents is definitely rich, someone with 999,999,999
cents is definitely rich.
3
Continuum-valued logics
We might hope to solve the problem by saying that, as we get to the borderline area, the
conditionals become less than fully true, and that modus ponens does not quite preserve
1
If we consider the form of the sorites argument that uses a universally quantified premise, then the diagnosis is much the same: the universal premise will be N. But we haven’t given multivalued semantics for the
quantifiers, so we’ll stick to the simpler version here.
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3.1
Semantics of connectives
degree of truth, so that you can go from lots of pretty-true premises to a false conclusion.
A semantics based on the idea that truth comes in degrees yields what is sometimes called
“fuzzy logic.”
In this approach the values are real numbers in the closed internal [0, 1]; that is, the
real numbers from 0 to 1 inclusive. A model is an assignment of these values to propositional constants. These assignments can be extended to compound formulas using the
rules below.
3.1
Semantics of connectives
Łukasiewicz
|¬A| = 1 − |A|
|A ∨ B| = max(|A|, |B|)
|A ∧ B| = min|A|, |B|)
|A ⊃ B| = 1 if |A| ≤ |B|
= 1 − (|A| − |B|) if |A| > |B|
|A ≡ B| = 1 − (max(|A|, |B|) − min(|A|, |B|))
Goguen
|A ∧ B| = |A| × |B|
|A ⊃ B| = 1 if |A| ≤ |B|
=
|B|
if |A| > |B|
|A|
|A ≡ B| = 1 − (max(|A|, |B|) − min(|A|, |B|))
Edgington
Non-degree-functional:
|¬A| = 1 − |A|
|A ∧ B| = |A| × |B given A|
|A ∨ B| = |A| + |B| − |A ∧ B|
3.2
Three ways to define validity
Preservation of perfect truth Γ |= B iff for every model v, if every formula in Γ is 1 on
v, then B is 1 on v.
Only 1 is treated as a designated value (Williamson, Edgington).
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3.3
Application to sorites
Preservation of degree of truth Γ |= B iff for every model v, the value of B on v ≥ the
lowest value of a member of Γ on v.
This corresponds to preservation of degree; a valid argument can’t take you to a conclusion with a lower degree of truth than any of the premises (Machina).
Conclusion can have no more falsity than the premises combined Γ |= B iff for every
model v, 1 − |B|v ≤ ΣA∈Γ (1 − |A|v ). (Edgington p. 307; she proves that every argument
that is valid in the sense of preserving perfect truth has this property too.)
3.3
Application to sorites
The premises of our sorites argument have the form
If some with n cents is rich, then someone with n − 1 cents is rich,
with different values of n. In each case, the degree of truth of the consequent will be the
same as, or only very slightly less than, the degree of truth of the antecedent. So, given
the Łukasiewicz semantics, the conditional will have degree 1 or a degree very close to 1.
The upshot is that, on this approach, all of the conditionals are either perfectly true
or nearly perfectly true.
What about the argument? Is it valid? (That is, is Modus Ponens valid?) Again, it
depends on how we define validity.
• If validity is preservation of perfect truth (1), then Modus Ponens is valid. But,
since there is no guarantee that nearly perfect truth will be preserved, we can go
from a bunch of premises that are either perfectly true or nearly so to a perfectly
false conclusion.
• If validity is preservation of degree, then Modus Ponens is not valid. The degree of
its conclusion can be less than the degrees of any of the premises. Example: consider
the inference from A and A ⊃ B to B, when A has degree 0.8 and B has degree 0.6.
In this case A ⊃ B will have degree 0.8, so we’ll go from two premises with degree
0.8 to a conclusion with degree 0.6.
Either way, we have an explanation of where the argument goes wrong that does not
require us to embrace a “sharp boundaries” claim.
4
Troubles with degree-functionality
To say that a connective is degree-functional is to say that the degree of truth of a compound
in which it is the main connective is a function of the degrees of truth of its constituent
sentences. Williamson notes some counterintuitive consequences of degree-functionality.
(Similar problems arise for the three-valued logics we looked at earlier.)
Suppose Tek is taller than Tim:
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degree-functional
5. Can we make sense of degrees of truth?
(3)
[Tek is tall] = 0.5
(4)
[Tim is tall] = 0.4
Then
(5)
[Tim is tall ⊃ Tek is tall] = 1
(6)
[Tim is tall ⊃ ¬Tek is tall] = 1
But intuitively (??) should be perfectly true (since Tek is taller than Tim), while (??) should
be perfectly false.
Similarly,
(7)
[Tim is tall ∧ ¬Tek is tall] = [Tim is tall ∧ Tek is tall] = 0.4
But intuitively ‘Tim is tall and Tek is not tall’ should get the degree 0: Tim is shorter than
Tek, so it can’t be that Tim is tall and Tek isn’t.
How bad are these results?
We’ve already accepted that an outright contradiction can have value 0.5. So “impossibility” can’t be reason for rejecting a nonzero assignment of degree, given what we’ve
already accepted.
Fine’s example: take a blob whose color is between red and pink.
(8)
[Blob is red] = 0.5
(9)
[Blob is pink] = 0.5
(10)
[Blob is red ∨ Blob is pink] = 0.5
Fine argues that intuitively the disjunction (??) should be perfectly true. But here’s an
argument that the disjunction shouldn’t be perfectly true: Blob can only be red or pink
it it’s true that it’s red or true that it’s pink. But it’s only borderline red and it’s only
borderline pink. So it’s not perfectly true that it’s red or pink.
5
Can we make sense of degrees of truth?
We can make sense of degrees of truth if we can make sense of
(11) A is truer than B.
What might be meant by (??)? That depends on what you say about truth.
As Williamson notes, some degree theorists attempt to explain degrees of truth via
comparatives. The general idea is to move from
(12)
X is balder than Y
to
(13)
‘X is bald’ is truer than ‘Y is bald’.
Forbes [4] argues more or less as follows [6, p. 92]:
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5. Can we make sense of degrees of truth?
1.
2.
3.
4.
X is balder than Y .
X is bald to a greater degree than Y .
X satisfies ‘is bald’ to a greater degree than Y does.
‘X is bald’ is truer than ‘Y is bald’.
The step from 1 to 2 is fine, but the step from 2 to 3 introduces semantic vocabulary. This
is where something funny might be going on.
Problems:
1. It works better for ‘bald’ (which has a lower bound of complete baldness) than for
‘tall’. Michael Jordan and Bill Clinton might both be clear cases of tall people. So
‘Michael Jordan is tall’ and ‘Bill Clinton is tall’ should both have degree 1. But since
Jordan is taller than Clinton, by the Forbes argument, ‘Michael Jordan is tall’ should
be truer than ‘Bill Clinton is tall’ [11].
This objection could perhaps be met by saying that degree of truth asymptotically
approaches 1 but never gets there.
2. Williamson [11]: ‘acute’ is not a vague predicate; all angles less than 90 degrees are
clear cases of "acute" angles. So ‘a 25 degree angle is acute’ and ‘a 45 degree angle
is acute’ should both have degree 1. But a 25 degree angle is more acute than a 45
degree angle.
Are we perhaps moving back and forth between a precise geometrical notion of
acuteness (being less than 90 degrees) and an ordinary sense of ‘acute’ in which it
means something like ‘pointy’? The comparative claim may only work for the latter.
3. Keefe [6]: Any approach that assigns a real number degree to every sentence is committed to connectedness: For every sentence S and T , either S is truer than T or T
is truer than S or they have the same degree of truth.
connectedness
But this seems implausible:
• Consider multidimensional predicates like ‘nice’ and ‘large’. X is 50 1100 and
weighs 240 pounds, while Y is 60 100 and weighs 220 pounds. Which is larger?
It seems indeterminate whether ‘X is large’ or ‘Y is large’ has higher degree of
truth.
• Is ‘Clinton is tall’ truer than ‘Clinton is charismatic’? Which is truer, ‘Quine
was smart’ or ‘Nixon was deceitful’? Comparatives are no help at all here.
One solution ([5], rediscovered by [10]) is to use partially ordered values instead of
real-number degrees. This allows us to reject connectedness. Define conjunction
and disjunction as greatest lower bound and least upper bound on the lattice of
values. Define implication in the natural way: the degree of the conclusion must be
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multidimensional
REFERENCES
REFERENCES
greater or equal to the least upper bound of degrees of the premises. (Negation is
harder; we can set up constraints, but they don’t determine a unique value.)
A problem with this strategy: [A∧ B] is likely to be the minimal element when [A]
and [B] are not connected! Thus, ‘Gore is charismatic and Clinton is honest’ may
end up with same value as ‘Clinton is charismatic and Gore is honest’.
Can we make sense of degrees of truth without going through comparatives? Sainsbury appeals to our understanding of truth as the aim of belief:
Truth is what we seek in belief. It is that than which we cannot do better.
So where partial confidence is the best that is even theoretically available,
we need a corresponding concept of partial truth or degree of truth. Where
vagueness is at issue, we must aim at a degree of belief that matches the degree
of truth, just as, where there is no vagueness, we must aim to believe just what
is true. [9, p. 44]
That is: to say that a proposition is true is to say that we would meet our aim in cognition
if we believed it fully. To say that a proposition is true to degree d is to say that it we
would meet our aim in cognition if we believed it to degree d .
References
[1] Roy Cook. “Vagueness and Mathematical Precision”. In: Mind 111 (2002), pp. 225–
47.
[2] Dorothy Edgington. “Vagueness by Degrees”. In: Vagueness: A Reader. Ed. by Rosanna
Keefe and Peter Smith. Cambridge, MA: MIT, 1997, pp. 294–316.
[3] Kit Fine. “Vagueness, Truth and Logic”. In: Vagueness: A Reader. Ed. by Rosanna
Keefe and Peter Smith. Cambridge, MA: MIT, 1997, pp. 119–150.
[4] Graham Forbes. The Metaphysics of Modality. Oxford: Oxford University Press,
1985.
[5] J. A. Goguen. “The Logic of Inexact Concepts”. In: Synthese 19 (1969), pp. 325–73.
[6] Rosanna Keefe. Theories of Vagueness. Cambridge: Cambridge University Press, 2000.
[7] John MacFarlane. “Fuzzy Epistemicism”. In: Cuts and Clouds. Ed. by Richard Dietz
and Sebastiano Moruzzi. Oxford: Oxford University Press, 2010, pp. 438–463.
[8] Kenton F. Machina. “Truth, Belief and Vagueness”. In: Vagueness: A Reader. Ed. by
Rosanna Keefe and Peter Smith. Cambridge, MA: MIT, 1997, pp. 174–203.
[9] R. M. Sainsbury. Paradoxes. second. Cambridge: Cambridge University Press, 1995.
[10] Brian Weatherson. “True, Truer, Truest”. In: Philosophical Studies 123 (2005), pp. 47–
70.
[11] Timothy Williamson. Vagueness. London: Routledge, 1994.
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