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In philosophy and logic, the liar paradoxor liar's
paradox(pseudomenonin Ancient Greek), is the statement "This
sentence is false."
Trying to assign to this statement a classical binary truth
value leads to a contradiction
(see arado!).
If "This sentence is false" is true, then the sentence is false,
hich ould in turn meanthat it is actually true, but this ould mean
that it is false, and so on ad infinitum.
#imilarly, if "This sentence is false" is false, then the
sentence is true, hich ould in
turn mean that it is actually false, but this ould mean that it
is true, and so on ad
infinitum.
History
The $pimenides parado! (circa %&& ') has been suggested
as an e!ample of the liarparado!, but they are not logically
euivalent. The fictional speaker $pimenides,
a retan, reportedly stated that "The retans are alays
liars."*citation
needed+ oever $pimenides- statement that all retans are liars
can be resolved as
false, given that he knos of at least one other retan ho does
not lie.
It is unlikely that $pimenides intended his ords to be
understood as a kind of liar
parado!, and they ere probably only understood as such much
later in history.
*citation needed+
The oldest knon version of the actual liar parado! is attributed
tothe Greek philosopher $ubulides of iletus ho lived in the /th
century '. It is
very unlikely that he kne of $pimenides-s ords, even if they ere
intended as a
parado!.*citation needed+ $ubulides reportedly asked, "A man
says that he is lying. Is
hat he says true or false0"
The parado! as once discussed by #t. 1erome in a sermon2
"I said in my alarm, -$very man is a liar3- (salm. 44%244) Is
5avid telling the truth or
is he lying0 If it is true that every man is a liar, and 5avid-s
statement, "$very man is a
liar" is true, then 5avid also is lying6 he, too, is a man. 'ut
if he, too, is lying, his
statement2 "$very man is a liar," conseuently is not true.
7hatever ay you turn theproposition, the conclusion is a
contradiction. #ince 5avid himself is a man, it follos
that he also is lying6 but if he is lying because every man is a
liar, his lying is of a
different sort."*4+
In early Islamic tradition liar parado! as discussed for at
least 8 centuries starting
from late 9th century apparently ithout being influenced by any
other
tradition. :a ;r al
8/13/2019 Liars Paradox
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The problem of the liar parado! is that it seems to sho that
common beliefs
about truth and falsity actually lead to a contradiction.
#entences can be constructed
that cannot consistently be assigned a truth value even though
they are completely in
accord ith grammar and semantic rules.
The simplest version of the parado! is the sentence2
This statement is false. (A)
If (A) is true, then "This statement is false" is true.
Therefore (A) must be false. The
hypothesis that (A) is true leads to the conclusion that (A) is
false, a contradiction.
If (A) is false, then "This statement is false" is false.
Therefore (A) must be true. The
hypothesis that (A) is false leads to the conclusion that (A) is
true, another
contradiction. $ither ay, (A) is both true and false, hich is a
parado!.
