philosophy discussion 3
Valerielee
SolutionstotheLiarParadox.pdf
2/8/2018
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Phil 2: Puzzles and Paradoxes
Prof. Sven Bernecker
University of California, Irvine
Solutions to the
Liar Paradox
Attempts to Solve the Liar Paradox
• Disallow meaningless statements
• Disallow self-reference statements
• Disallow any non-hierarchical notion of truth
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Disallow Meaningless Statements
• The liar sentence is nonsense because it violates the principle of
bivalence: Every declarative statement has exactly one truth
value, either true or false.
• The liar sentence ”This sentence is false“ fails to succeed in
making any statement at all, true or false, since there is nothing
which could make it true, nothing to make it false.
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Disallow Self-Referential Statements
• Proposal: Eliminating the Liar Paradox by disallowing self-referential
statements.
• Problem: Self-reference isn’t essential to the paradox. To see this
consider the Dualized Liar Paradox:
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One one side of a blank card print: “The sentence on the other side of this
card is true“
On the other side of the same card print: “The sentence on the other side
of this card is false”
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• Neither of these pairs of statements is self-referential, since
neither refers to itself. But they are still paradoxical.
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B: The next statement by A will be false
A: B has spoken truly!
Proof that the Dualized Liar is paradoxical:
A: B is true
B: A is not true
Suppose A is true
1) A is true Assumption
2) A (1), Disquotation
3) B is true (2), Def of A
4) B (3), Disquotation
C) A is not true (4), Def of B
(1) & (C) form a contradiction
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A: B is true
B: A is not true
Suppose A is not true
1) A is not true Assumption
2) B (1), Def of B
3) B is true (2), Disquotation
4) A (3), Def of A
C) A is true (4), Disquotation
(1) & (C) form a contradiction
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Disallowing self-referential statements:
• This solution throws out the baby with the bath water. Self-
reference by itself is not the problem. There are many informative
and unproblematic self-referential statements. E.g. “This statement
is made by means of an English sentence“
• Various useful theses have to be formulated by means of self-
referential statements. E.g. “Meaningful declarative statements
have some truth-value or other.“
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Disallow Any Non-Hierarchical Notion
of Truth • The liar paradox arises because the words “true“ and “false“ are
applied to the very sentence they occur in. One solution to the liar
paradox is to disallow the application of “true“ and “false“ to the very
sentences they occur in.
• Taski‘s solution: The liar paradox does not arise if “true“ and “false“
does not apply to the very sentences these terms occur in but instead
apply to sentences at a different level.
9 Alfred Tarski
(1901-1983)
• No language can contain a word “true“ which can apply to its own
sentences. If a language contains a word “true“ it applies to a
sentence in another language, an object-language.
• The words “true“ and “false“ may not occur in the object-
language. They may only occur in the meta-language. The meta-
language includes all of the object-language plus it can talk about
the truth values of the object-language.
• Claims involving the word “true“ belong to a meta-language.
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Problems with Tarski‘s solution:
• Trows out the baby with the bathwater. Not all sentences that
predicate truth of themselves are paradoxical. E.g. “This
sentence is true“ where “this sentence“ refers to another
sentence, e.g., the sentence “Paris is the capital of France“.
• Taski‘s view misinterprets the commonsensical meaning of
sentences like “Every statement is true or false“ and ”What you
said just now is true“
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