PHILOS 2 Lecture Notes - Lecture 10: Epimenides, Paraconsistent Logic, Liar Paradox
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by Prof. Bernecker
This sentence is false L1: L1 is false.
Suppose L1 is true; then it is as it says it is– false. So L1 is false.
oIts content is true, but its content states that it is false.
oSo it follows that if we assume that L1 is true, then we conclude
that L1 is false.
Suppose L1 is false. Well, false is just what it says it is, and a sentence
that tells it the way it is so it’s true. So, L1 is true.
oAssuming the sentence is false, we get the conclusion that what
it says is true because it states that it is false.
So, if L1 is true, it is false; and if it is false, it is true.
oSo it seems that L1 is neither true nor false.
This is a paradox if we assume the principle of bivalence.
oDeclarative sentences such as L1 are either true or false.
oA sentence that declares something about the world is either
true or false, but not
true nor false.
History of the Liar Paradox
Attributed to the Greek philosopher Epimenides (6th century BC), a
Cretan who reportedly stated that “All Cretans are liars.”
One version of the liar paradox is attributed to the Greek philosopher
Eublides of Miletus (4th century BC) who reportedly asked, “A man
says that he is lying. Is what he says true or false?”
The Indian philosopher Bhartrhari (late 5th century CE) was aware of a
liar paradox which he formulated as “everything I am saying is false.”
The Persian scientist Nasïr al-Dïn al-Tusi (1201-1274) could have
been the <rst to identify the liar paradox as self-referential.
A tongue-in-cheek liar-style puzzle:
A: This sentence contains seven words.
Sentence A is clearly false.
So its opposite ought to be true. Right?
B: This sentence does not contain seven words.
Sentence B is the opposite of A and it is false too.
How could this be?
oNot actually the opposite of sentence A.
The Liar Paradox involves the indexical term “this sentence.”
oIndexicals are words whose referent and meaning are
determined by such
factors as the time, location, and
intentions of the speaker.
: I, he, she, this, that
: here, now, actually, presently, today, yesterday,
: my, his, her, actual, past, present, future,
A sentence that refers to itself as a sentence
oJohn is reading this sentence
oThis sentence contains exactly threee erors.
o“Ice” has three letters.
(cf. Harold Evans,
1972, p. 182)
They are committing the very errors they tell you to avoid.
– Not using the term “truth” to cause the problem, but are simply statements
that don’t live up to the lessons they give.
Make each pronoun agree with their antecedent.
Join clauses good, like a conjunction should.
Verbs has to agree with their subjects.
Don’t write run-on sentences they are hard to read.
Don’t use commas, which aren’t necessary.
It’s important to use your apostrophe’s correctly.
Proofread your writing to see if you any words out.
The passive voice is to be avoided.
Try to not ever split in<nitives.
Don’t use no double negative.
Correct spelling is esential.
Principle of Bivalence
The Liar Paradox contradicts the principle of bivalence.
oPrinciple of Bivalence: Every declarative statement has exactly
one truth value, either true or false.
Motivation: “any non-defective representation of how things are in the
world must be either accurate or inaccurate, true or false.” (Sainsbury,
oIt has to be one of those true whether we can tell which one it is.
oNon-defective = grammatically well-formed
oAs long as a declarative statement is well-formed, it has to be
accurate or inaccurate.
Are there counterexamples to the principle of bivalence (not counting
aesthetic, theological and ethical judgments)?
oMaybe there are.
oYou have stopped beating your wife.
Analysis of Liar Paradox
L1: L1 is false.
Given the principle of bivalence, L1 is either true or false.
First, let’s assume that L1 is true:
1) “L1” is true Assumption
2) L1(1), Disquotation (means the same thing)
C) “L1” is not true (2), De<nition of L1
(1) and (C) form a contradiction.
Next, let’s assume that L1 is false:
1) “L1” is not true Assumption
2) L1(1), De<nition of L1
C) “L1” is true (2), Disquotation
(1) and (C) form a contradiction.
In either case, we get the opposite truth value of what we assumed.
Thus we can derive a contradiction from the assumption that “’L1’ is
true or ‘L1’ is not true.”
oSo we have a violation of the principle of bivalence.
Clicker Question #1
The liar paradox rests on the principle of _____.
We cannot solve the liar paradox by claiming that L1 is neither true nor
To see this, consider the following strengthened version of the paradox:
B: This statement is not true.
If (B) is neither true nor false, then it must not be true.
Since this is what (B) itself states, it means that (B) must be true.
oIf it is not true, then it is what it says it is not– true.
oSo if it is not true, then it is true?
Since initially (B) was not true and is now true, another paradox arises.
Clicker Question #2
The liar paradox assumes that a declarative sentence is ______.
A. Either true or false
Lecture 10: liar paradox by prof. bernecker (02/04/16) Suppose l1 is true; then it is as it says it is false. L1: l1 is false: its content is true, but its content states that it is false, so it follows that if we assume that l1 is true, then we conclude. Well, false is just what it says it is, and a sentence that l1 is false. that tells it the way it is so it"s true. So, l1 is true: assuming the sentence is false, we get the conclusion that what it says is true because it states that it is false. So, if l1 is true, it is false; and if it is false, it is true: so it seems that l1 is neither true nor false. Attributed to the greek philosopher epimenides (6th century bc), a. Cretan who reportedly stated that all cretans are liars.