Paradoxes (7): The Liar Paradox
Paradoxes
Lecture Seven
The Liar Paradox
Rob Trueman
[email protected]
University of York
Paradoxes (7): The Liar Paradox
Liars, Falsehoods and Untruths
The Liar Paradox
Liars, Falsehoods and Untruths
Self-Reference
Tarski’s Solution: Hierarchies of Languages
Objections to Tarski’s Solution
Kripke’s Solution: Truth-Value Gaps
An Objection to Kripke’s Solution
Paradoxes (7): The Liar Paradox
Liars, Falsehoods and Untruths
The Liar Paradox
• A man says that he is lying. Is what he says true or false?
Paradoxes (7): The Liar Paradox
Liars, Falsehoods and Untruths
The Liar Paradox
• If it is true, then he is lying; so it is false
Paradoxes (7): The Liar Paradox
Liars, Falsehoods and Untruths
The Liar Paradox
• If it is a lie, then he is not lying; so he is telling the truth
Paradoxes (7): The Liar Paradox
Liars, Falsehoods and Untruths
A Pointless Riddle?
Why do you bore me with that which you yourself call
the “liar fallacy”? [...] Not to know [such paradoxes]
does no harm, and mastering them does no good.
(Moral letters to Lucilius by Seneca, Letter 45: On
sophistical argumentation)
• No one agrees with Seneca anymore
• In the early 20th Century, there were huge advances in logic,
and many of them began with reflections on the Liar Paradox
• However, if we are going to get anywhere, then first we need
to get the paradox in its neatest, sharpest form
Paradoxes (7): The Liar Paradox
Liars, Falsehoods and Untruths
From Lying to Falsehoods
• Talk of ‘lying’ is an unnecessary complication, since lying
involves intention
– You are not lying if you say something false but think you are
saying something true!
• We can formulate the Liar Paradox without getting involved
with this complication
What I am saying now is false
(λ) λ is false
The 6th sentence on this slide is false
• For now we will work with λ, but we must always remember
that there are these other forms of the paradox
Paradoxes (7): The Liar Paradox
Liars, Falsehoods and Untruths
The Liar
• λ = ‘λ is false’
(T) ‘p’ is true if and only if p
– ‘Snow is white’ is true if and only if snow is white
– ‘The Paradoxes lectures have been excellent’ is true if and only
if the Paradoxes lectures have been excellent
• ‘λ is false’ is true if and only if λ is false
• λ is true if and only if λ is false
Paradoxes (7): The Liar Paradox
Liars, Falsehoods and Untruths
Truth-Value Gaps?
• λ is true if and only if λ is false
• This is paradoxical if we assume that λ must either be true or
false
– Suppose that λ is true; in that case λ is false; Contradiction!
– Suppose that λ is false; in that case λ is true; Contradiction!
– So either way, if λ is true or λ is false, then we are led to a
contradiction!
• But if we say that λ is neither true nor false, then we cannot
derive a contradiction
• Does this give us a quick response to the Liar Paradox?
– λ is a truth-value gap, it is neither true nor false
Paradoxes (7): The Liar Paradox
Liars, Falsehoods and Untruths
The Strengthened Liar
• λ = ‘λ is not true’
(T) ‘p’ is true if and only if p
• ‘λ is not true’ is true if and only if λ is not true
• λ is true if and only if λ is not true
• We cannot stop this version of the paradox just by saying that
λ is neither true nor false
– If λ is neither true nor false, then it is not true
– But that is exactly what λ says!
– So if λ is neither true nor false, then it is true!
• This is known as a revenge paradox, and they come up all
the time when we’re dealing with the Liar
Paradoxes (7): The Liar Paradox
Self-Reference
The Liar Paradox
Liars, Falsehoods and Untruths
Self-Reference
Tarski’s Solution: Hierarchies of Languages
Objections to Tarski’s Solution
Kripke’s Solution: Truth-Value Gaps
An Objection to Kripke’s Solution
Paradoxes (7): The Liar Paradox
Self-Reference
Quinean Classifications
A paradox is an apparently unacceptable conclusion
derived by apparently acceptable reasoning from
apparently acceptable premises. (Sainsbury’s definition
from Paradoxes, p. 1)
• Premise-flawed
– One of the premises turns out to be false
• Fallacious (“Falsidical”)
– The reasoning turns out to be faulty
• Veridical
– The conclusion turns out to be true
Paradoxes (7): The Liar Paradox
Self-Reference
Could the Liar Paradox be Veridical?
• Could the Liar Paradox be veridical?
• Surely not: the conclusion is a contradiction!
• Actually, the Liar Paradox has proven so tricky to deal with,
that some philosophers (most notably Graham Priest) have
been driven to say that some contradictions can be true
• This is a very radical response to the paradox, and we’ll come
back to it in Lecture 9
Paradoxes (7): The Liar Paradox
Self-Reference
The Premises of the Liar Paradox
• So we must say that the paradox is either premise-flawed, or
fallacious
• In this lecture, I am going to focus on solutions which say that
it is premise-flawed
• The Liar Paradox has just two premises:
(1) λ = ‘λ is not true’
(2) ‘λ is not true’ is true iff λ is not true
• Which of these premises are we going to reject?
Paradoxes (7): The Liar Paradox
Self-Reference
Is Self-Reference to Blame?
(1) λ = ‘λ is not true’
• (1) captures a very weird fact about λ
• λ is self-referential: it talks about itself
– λ says that λ is not true
• We might have thought that the Liar Paradox is just a proof
that this sort of self-reference is impossible
Paradoxes (7): The Liar Paradox
Self-Reference
Self-Reference and Arithmetic
• There are ways of ‘coding up’ claims about language into
arithmetic
• This took a genius (Gödel) to realise, but now the idea is very
familiar
– Think of the way that computers encode information into 1s
and 0s
• When we code things up in this way, we find that arithmetic
allows us to create a version of λ, which is self-referential in
just the way that is needed for the Liar Paradox
• So unless we want to challenge ordinary arithmetic, we cannot
complain about the kind of self-reference used in the Liar
Paradox
Paradoxes (7): The Liar Paradox
Self-Reference
In Defence of Self-Reference
• Also, remember that we have to deal with paradoxes like this:
(i) The 2nd sentence on this slide is not true
• (i) is a self-referential liar paradox, but only contingently
(ii) The 2nd sentence on the previous slide is not true
• (ii) is a perfectly meaningful, intelligible sentence
• We have no difficulty assigning it a truth-value (it is false)
• But if we are allowed to use sentences like (ii), then we
cannot eliminate contingent liar sentences like (i)
Paradoxes (7): The Liar Paradox
Self-Reference
The Premises of the Liar Paradox
(1) λ = ‘λ is not true’
(2) ‘λ is not true’ is true iff λ is not true
• So if we are going to reject any premise, it will have to be (2)
• (2) is an instance of the general schema (T):
(T) ‘p’ is true iff p
• So we need to take a closer look at (T)
Paradoxes (7): The Liar Paradox
Tarski’s Solution: Hierarchies of Languages
The Liar Paradox
Liars, Falsehoods and Untruths
Self-Reference
Tarski’s Solution: Hierarchies of Languages
Objections to Tarski’s Solution
Kripke’s Solution: Truth-Value Gaps
An Objection to Kripke’s Solution
Paradoxes (7): The Liar Paradox
Tarski’s Solution: Hierarchies of Languages
Semantically Closed Languages
• According to Tarski, the Liar Paradox comes up because we
are trying to use English to talk about truth-in-English
• In Tarski’s terminology, the Liar Paradox comes up (in
English) because English is semantically closed
• A language is semantically closed iff
(i) it contains its own truth predicate satisfying (T); and
(ii) it contains the means to form names of its own sentences
(e.g. via quotation)
Paradoxes (7): The Liar Paradox
Tarski’s Solution: Hierarchies of Languages
Avoiding Semantic Closure
• For Tarski, the lesson of the Liar Paradox was that natural
languages like English, which are semantically closed, are
irredeemably inconsistent
• Tarski abandoned natural languages, and instead studied how
we could create consistent formal languages
• His idea was that there is not one over-arching notion of
truth, but a different notion of truth for each formal language
L, truth-in-L
Paradoxes (7): The Liar Paradox
Tarski’s Solution: Hierarchies of Languages
Avoiding Semantic Closure
• No language is ever allowed to contain its own truth predicate
• If you want to talk about truth-in-L0 , then you need to move
up into a more powerful language, L1
• We can use L1 to talk about truth-in-L0 , but if we then want
to talk about truth-in-L1 , then we need to move up to an even
more powerful language, L2 , and so on
Paradoxes (7): The Liar Paradox
Tarski’s Solution: Hierarchies of Languages
A Hierarchy of Languages
Paradoxes (7): The Liar Paradox
Tarski’s Solution: Hierarchies of Languages
Back to the T-Schema
(T) ‘p’ is true iff p
• We swap this schema for:
(Tn ) ‘p’ is true-in-Ln iff p
• Since this talks about truth-in-Ln , the sentence ‘p’ must be
from Ln
• But (Tn ) itself belongs to Ln+1
– I am simplifying a bit by writing (Tn ) like this
– I am assuming that Ln is an extension of Ln−1 , so that Ln
contains all the expressions of Ln−1 (and then some)
– This does not have to be the case, but it makes things a lot
simpler
Paradoxes (7): The Liar Paradox
Tarski’s Solution: Hierarchies of Languages
Some Examples
(Tn ) ‘p’ is true-in-Ln iff p
• ‘Snow is white’ is true-in-L0 iff snow is white
– This is an acceptable instance of (Tn ), because ‘Snow is white’
is a sentence from L0
• ‘ “Snow is white”’ is true-in-L0 ’ is true-in-L1 iff ‘Snow is
white’ is true-in-L0
– This is also an acceptable instance of (Tn ), because ‘ “Snow is
white”’ is true-in-L0 ’ is a sentence in L1
• ‘ “Snow is white”’ is true-in-L0 ’ is true-in-L0 iff ‘Snow is
white’ is true-in-L0
– This is an unacceptable instance of (Tn ), because ‘ “Snow is
white”’ is true-in-L0 ’ is not a sentence in L0
Paradoxes (7): The Liar Paradox
Tarski’s Solution: Hierarchies of Languages
Solving the Lair Paradox
• How does this help with the Liar Paradox?
• The Strengthened Liar was meant to say of itself that it was
not true:
– λ = ‘λ is not true’
• But now we need to replace ‘true’ with ‘true-in-Ln ’:
– λn = ‘λn is not true-in-Ln ’
• To get a paradox going, we would then need to take the
following instance of (Tn ):
– ‘λn is not true-in-Ln ’ is true-in-Ln if and only if λn is not
true-in-Ln
• But this is actually an unacceptable instance of (Tn )
• λn cannot be a sentence of Ln : it mentions truth-in-Ln !
Paradoxes (7): The Liar Paradox
Objections to Tarski’s Solution
The Liar Paradox
Liars, Falsehoods and Untruths
Self-Reference
Tarski’s Solution: Hierarchies of Languages
Objections to Tarski’s Solution
Kripke’s Solution: Truth-Value Gaps
An Objection to Kripke’s Solution
Paradoxes (7): The Liar Paradox
Objections to Tarski’s Solution
Overkill?
• Everyone accepts that Tarski’s solution to the Liar Paradox
works in the sense that it gets us out of the paradox
• But lots of philosophers also think that the solution is too
extreme
• In effect, Tarski blocks the paradox by restricting our
expressive powers
– We can no longer talk about truth full-stop, but truth-in-L,
and we can never talk about truth-in-L in L
• This prevents us from saying lots of things that we want to be
able to say
Paradoxes (7): The Liar Paradox
Objections to Tarski’s Solution
Bivalence
• Sometimes we want to make absolutely general claims about
truth, like:
(1) Every sentence is either true or false
• This claim (bivalence) may be true or it may be false, but it
is certainly a claim that we want to be able to discuss
• But on Tarski’s picture, we can’t; all we can say is:
(2) Every sentence in Ln is either true-in-Ln or false-in-Ln
• Moreover, (2) will belong to Ln+1 , which is a language that
(2) cannot be applied to
Paradoxes (7): The Liar Paradox
Objections to Tarski’s Solution
Applying Tarski to Natural Languages
• Tarski explicitly set natural languages to one side; his theory
of truth was meant to deal with formal languages
• But what are we to say about natural languages?
• Some philosophers have tried to re-apply what Tarski said
about formal languages
• The idea is that English itself is a Tarskian hierarchy of
languages
– We have a fragment of English which talks just about the
world, and not truth, L0
– Then we have a fragment of English which talks about the
world and truth-in-L0 , L1
– Then we have a fragment of English which talks about the
world and truth-in-L0 and truth-in-L1 , L2
– ...
• The English word ‘true’ is ambiguous between ‘true-in-L0 ’,
‘true-in-L1 ’, and so on
Paradoxes (7): The Liar Paradox
Objections to Tarski’s Solution
A Problem for Applying Tarski to Natural Languages
• Simon intends his assertion to apply to everything that
Daniel says
Paradoxes (7): The Liar Paradox
Objections to Tarski’s Solution
A Problem for Applying Tarski to Natural Languages
• Daniel intends his assertion to apply to everything that
Simon says
Paradoxes (7): The Liar Paradox
Objections to Tarski’s Solution
A Problem for Applying Tarski to Natural Languages
• But there is no consistent way of assigning levels which has
this effect
Paradoxes (7): The Liar Paradox
Objections to Tarski’s Solution
A Problem for applying Tarski to Natural Languages
(1) Everything Daniel says is untrue
(2) Everything Simon says is untrue
– Suppose that Simon means true-in-L0 by ‘true’
– In that case, (1) must belong to L1
– If Daniel wants his assertion to apply to (1), he must have
meant untrue-in-L1 by ‘untrue’
– So (2) must belong to L2
– Thus (1) does not apply to (2)!
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
The Liar Paradox
Liars, Falsehoods and Untruths
Self-Reference
Tarski’s Solution: Hierarchies of Languages
Objections to Tarski’s Solution
Kripke’s Solution: Truth-Value Gaps
An Objection to Kripke’s Solution
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
Kripke’s Solution
• Kripke offered an alternative solution to the Liar Paradox,
which was not quite so restrictive as Tarski’s
• In Kripke’s system, languages are allowed to be semantically
closed
• And there is just one notion of truth
• This gets us around both of the problems we looked at for
Tarski’s system
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
Explaining Truth to the Uninitiated
• Suppose that someone was speaking a language, L, which
talks only about the world, but not about truth
• In this language you can say things like ‘Snow is white’ and
‘Grass is green’
• Suppose we wanted to explain to this person what ‘is true’
means
We may say that we are entitled to assert (or deny) of
any sentence that it is true precisely under the
circumstances when we can assert (or deny) the sentence
itself.
(Kripke 1975, p. 700)
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
Explaining Truth to the Uninitiated
• Let L+ be the language you get when you add the predicate
‘is true’ to L
• When explaining to the speaker of L how to speak L+ , we say:
– You can assert ‘s is true’ when and only when you can assert s
– You can assert ‘s is false’ when and only when you can deny s
• The speaker already knew when to assert or deny claims
about the world, like ‘Snow is white’ and ‘Grass is indigo’
• So our explanation of ‘is true’ will now allow the speaker to
understand claims like:
– ‘Snow is white’ is true
– ‘Grass is indigo’ is false
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
What about Talk about Truth?
• What about sentences like the following?
(1) ‘ “Snow is white” is true’ is true
• Well, we told the speaker that you can assert ‘s is true’
whenever you can assert s
• So the speaker knows that they can assert (1) just when they
can assert
(2) ‘Snow is white’ is true
• And they know that they can assert (2) just when they can
assert
(3) Snow is white
• (3) is a sentence from L, and so they already know when they
can assert it
• So the speaker can figure out when they can assert (1):
whenever they can assert (3)!
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
Groundedness
• Following Kripke, we can say that (1) is grounded in (3)
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
What about the Liar?
• Now suppose that the speaker tries to figure out when they
can assert λ:
(λ) λ is not true
• They know that they can assert ‘s is true’ just when they can
assert s, so they know that they can assert ‘λ is not true’ just
when they cannot assert λ
• But λ = ‘λ is not true’ !
• So all they can say is that they should assert λ when they
cannot assert λ
• This is obviously a contradictory instruction, but more
importantly for our purposes, it is a circular one
– In the explanation of when to assert λ, we assume that we
already know when not to assert λ!
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
Ungroundedness
• Following Kripke, we can say that λ is ungrounded
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
Grounded versus Ungrounded
• Kripke’s idea was that claims about truth should be grounded
in claims that do not deal with truth (i.e. sentences in L)
• Grounded claims are true or false, but ungrounded claims are
neither
• So the Liar sentence, λ is a truth-value gap: it is neither true
nor false
• To make this idea precise, Kripke had to introduce a formal
theory of groundedness
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
A Theory of Grounding
• We are used to thinking of predicates as having extensions
– ‘F ’ is true of x iff x is in the extension of ‘F ’
– ‘F ’ is false of x iff x is not in the extension of ‘F ’
• Now we need to modify that idea a bit:
– ‘is true’ comes with an extension, E , and an anti-extension, A
– A sentence is true iff it is in E
– A sentence is false iff it is in A
– A sentence is neither true nor false iff it is neither in E nor A
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
A Theory of Grounding
• Consider all of the sentences of L+ . (This picture is a bit
misleading: there are infinitely many of these sentences!)
———————
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
A Theory of Grounding
• Put all of the sentences from L that should be asserted into
the extension, and all the ones that shouldn’t be asserted into
the anti-extension
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
A Theory of Grounding
• Now we can add some sentences that mention truth into the
extension and anti-extension
—————————————————-
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
A Theory of Grounding
• Put ‘s is true’ in the extension iff s is in the extension
• Put ‘s is true’ in the anti-extension iff s is in the anti-extension
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
A Theory of Grounding
• Put ‘s is false’ in the extension iff s is in the anti-extension
• Put ‘s is false’ in the anti-extension iff s is in the extension
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
A Theory of Grounding
• Now following exactly the same rules, we can add some more
sentences into the extension and anti-extension
—————————————————-
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
A Theory of Grounding
• Kripke proved that if we carry this process on long enough, we
will eventually find that we are not adding any more sentences
to E or A
– This is called a fixed-point
• We have to carry the process on a really long time: infinitely
long!
– Mind boggling, but modern mathematics allows us to talk
sensibly about extending this process past infinity
(“transfinitely”)
• Kripke’s idea is that a sentence is true iff it is in E at the
fixed-point, and false iff it is in A at the fixed-point
• Crucially, Kripke also proved that λ is in neither E nor A at
the fixed-point
• So λ is neither true nor false!
Paradoxes (7): The Liar Paradox
Kripke’s Solution: Truth-Value Gaps
Solving the Liar Paradox
• According to Kripke, λ is neither true nor false
• What about the following instance of (T)?
(1) ‘λ is not true’ is true iff λ is not true
• It depends on how we understand the biconditional, ‘iff’ in
this context
• On Kripke’s view, ‘A iff B’ is neither true nor false whenever
A or B is neither true nor false
• So on this view, (1) is neither true nor false
• The Liar Paradox is thus premise-flawed in the sense that one
of its premises is not true (although it also is not false)
Paradoxes (7): The Liar Paradox
An Objection to Kripke’s Solution
The Liar Paradox
Liars, Falsehoods and Untruths
Self-Reference
Tarski’s Solution: Hierarchies of Languages
Objections to Tarski’s Solution
Kripke’s Solution: Truth-Value Gaps
An Objection to Kripke’s Solution
Paradoxes (7): The Liar Paradox
An Objection to Kripke’s Solution
The Strengthened Liar is not True!
• We first introduced the idea of truth-value gaps right at the
beginning of the lecture
• We raised a problem:
– If λ is neither true nor false, then it is not true
– But λ says of itself that it is not true!
– So if λ is neither true nor false, then it is true!
• How does Kripke get around this problem?
Paradoxes (7): The Liar Paradox
An Objection to Kripke’s Solution
Negation in Kripke’s System
• Kripke is forced to say that the negation in L+ , ‘not’, works
like this:
A
T
u
F
not-A
F
u
T
• From this truth-table for negation and Kripke’s definition of
truth, it follows that it is impossible to truly say that λ is
neither true nor false
• If we try, then we will just end up saying another thing which
is neither true nor false!
Paradoxes (7): The Liar Paradox
An Objection to Kripke’s Solution
A Different Kind of Negation
• Isn’t that a disaster for Kripke? His whole solution to the Liar
Paradox is to say that λ is neither true nor false!
• Kripke’s only way out of this mess is to insist that the notion
of truth he has used to describe L+ is not the same notion of
truth as the one that appears in L+
• We have now returned to something like Tarski’s hierarchy!
• The hierarchy is less restrictive — we can now say a lot about
truth without having to ascend any levels — but it is still there
Paradoxes (7): The Liar Paradox
An Objection to Kripke’s Solution
The Ghost of Tarski’s Hierarchy
Liar sentences are not true in the object language, in the
sense that the inductive process never makes them true;
but we are precluded from saying this in the object
language by our interpretation of negation and the truth
predicate [...] The necessity to ascend to a metalanguage
may be one of the weaknesses of the present theory. The
ghost of the Tarski hierarchy is still with us.
(Kripke 1975, p. 714)
Paradoxes (7): The Liar Paradox
An Objection to Kripke’s Solution
For the Seminar
• We will be looking at Mates’ ‘Two Antinomies’ (available on
the VLE)
• This paper deals with both the Liar Paradox and Russell’s
Paradox, which we will be looking at next week
• The paper is accessible, but if you are struggling with the
material on Russell’s Paradox, please do not worry: we will go
over it all later
Paradoxes (7): The Liar Paradox
An Objection to Kripke’s Solution
Next Week
• We will be looking at Russell’s Paradox
• Required reading:
– Paradoxes ch. 6 6.1 & 6.9
Paradoxes (7): The Liar Paradox
An Objection to Kripke’s Solution
References
• Kripke, S (1975) ‘Outline of a Theory of Truth’, Journal of
Philosophy 72: 690–715
• Tarski, A ‘The concept of truth in formalized languages’ in
Corcoran (ed) Woodgar (trans) Logic, Semantics,
Metamathematics (Indianapolis, IN: Hackett, 1983)
pp. 152–278