This is a Facebook conversation a friend of mine and I had about the
Liar's Paradox.
Let's assume False is 0 and True is 1
Pat's point in the discussion is that if the sentence is true, then it must be false- and then, because it is then false, it is not true. This creates an endless logical spiral where the sentence = 0 and that situation = 1, thereby making the sentence = 1 and the situation now equaling 0, which means the sentence = 0...throwing us into a clear paradox.
My point is that because we have no actual "thing" to judge as true or false to begin with, we have no paradox at all. If you turn the phrase into an equation, we don't have a simple x = y like "sun = hot". If we assumed (this sentence) = (false) is our equation, we must ask ourselves "what is 'this sentence' "? The answer is the four words "this sentence is false". Therefore, (this sentence is false) = (false), which means
((this sentence is false) is false) = (false). This can, and will, go on indefinitely. If you used the words "this sentence" (negating what sentence exactly is being discussed) as X, then we have (X
(n+1) = 0) = 0, or
0 = 0- which is no paradox at all.
BUT, in order to have an illogical conclusion to logical reasoning, we must have a logical item of comparison; and reflexive sentences can be tricky. If I said "This sentence is English", we have points for validity; we have English (Latin) letters, English words, English syntax, English sounds, etc. But if I said to you "This sentence is true", what points do we have for validity? For "this sentence" is indeed "this sentence is true" therefore making "this sentence is true" true. We have no "thing" in the sentence, or referred to us BY the sentence, to make true or false. And because we have no basis in our quest for validity, we have no logical quest at all...
...and therefore no paradox.
But don't take my word for it. Enjoy!
Pat- must.. resist.. liar's paradox...
Me- everything i say is a lie?
Pat- something like that, yes
Me- this sentence is false?
Me- hhmm
is that even a legitimate sentence?
architecturally, not grammatically
Pat- not sure what you mean by "architecturally"
Me- it's grammatically correct
subject, verb, object
but "this sentence" = "false"
assuming you have true or false, it's a binary subject
1 or 0
so "this sentence" = 0
but "this sentence" can't be "x" by itself
no x = 0
because x (the sentence) encompasses 0
if 0 = false, and 1 = true, then the paradox basically says 0 = 1
and since "this sentence" is both a subject AND a qualifier, can you really have a sentence?
besides...there's no subject being verified
it's not like you're saying "all i say are lies"
but i suppose that's why it's paradoxical
Pat- you have successfully demonstrated that it is a paradox
I'm not sure why it wouldn't qualify as a sentence
Me- if i said "every word i say is a lie" we have something to judge
ie, the words i say
"my words" = "lies"
but with the liar's paradox we don't have that
"this sentence" = "false"
but "this sentence" is nothing. there's no substance here to test for validity
it's not like "my words".
BUT
if you take the string of words "this sentence is false" as what the sentence IS
then the paradox becomes hollow
Me- i can't really think of a good term...
um...
not real? it's not really paradoxical because there's no substance to deal with initially?
i'm seeing it as a mathematical sentence
Pat- so "this sentence is grammatically correct" is not a sentence either?
Me- if you look at the word "sentence" as a stand alone non-variable, then TECHNICALLY it is correct
because "is" qualifies as a verb so long as you have an object to compare the subject too
but because "this sentence" conceptually encompasses every word from capitalization to punctuation, it's hard to make
qualifier statements
Pat- it seems as though you are suggesting that we cannot talk about sentences
Me- this is, of course, unless you're saying "this sentence" with regards to ANOTHER sentence being examined
Pat- so is it the reflexive nature of it that is the problem?
Me- that's what i'm seeing
i see no "x"
it's more "x = 0" = "0"
but if that's the case then logically we have a 1
for if the sentence is false, then we have a truth
Pat- that is what a paradox is
Me- but we have no substance to make false to begin with
Pat- unless you are suggesting that sentences can't be true or false
Me- ...something i never said
Pat- I didn't think you had
thus "unless"
Pat- but if sentences can be true or false, then that sentence can be true or false
it certainly seems to be making a truth claim
Me- but what have we to prove true?
"this sentence is false" is the sentence itself
Me- so those four words must be false
and i can't parsel out a variable
so "this sentence is false" = "false" because "this sentence" = "this sentence is false"
Pat- yes
you're essentially demonstrating that it is a paradox. but I don't think that demonstrates your asserted solution
Me- your view of paradox lies in the 1 or 0 argument
mine lies in the fact that there is nothing concrete to be proven either true or false
because every time you try to make "this sentence" into an X, you have to take into account the entire line
Pat- are you asserting that reflexive statements never are true or false?
Me- no
hold on...
you could say "this sentence is not a latin sentence"
and that would be correct
because you would have substance for comparison (ie, the language in which it was written)
but to make an assertion about a sentence that has no real conceptual material FOR comparison makes the comparison
invalid
besides, assuming we don't use the X(n+1) for every time we use "this sentence"...
"this sentence is false" = "false"
"x = 0" = "0"
"0" = "0"
is true
and 0 = 0 being true does not invalidate either 0 or (x = 0)=0 because they've just been proven to be equal
Me- "this sentence is not latin"
"this sentence" = "not latin"
this is a valid comparison because we have something to compare
ie, english to latin
"this sentence is not true"
"this sentence"= "not true"
this is an invalid comparison because we have nothing to validate
because "this sentence" encompasses its own validity
in the latin case, the sentence encompassed its own language (so to speak)
in order to prove a truth true, you must have some item or thought or fact to prove true
"this sentence is false" has no "thing" to prove false
the only "thing" we have is false
and false = false is no paradox
Pat- would you agree that "this sentence is true" is neither true nor false?
Me- yes- for, inversely, we have nothing to validate. the problem with the paradox is that we're attempting to compare or
validate something, and our logical conclusion is an illogical one
my point is that we have nothing to compare or validate to begin with, and therefore cannot have a paradox
Pat- but sentences Are things. if "this sentence is in english" is both a sentence and true, then it is not immediately obvious to
me that "this sentence is true" is not both of those things as well.
Me- yes- sentences are things. i agree with that.
but the english example gives us something to work with- we have an agreeable example of english writing. it's something
that can be agreed upon outside of the concept of the sentence itself.
Pat- we don't have an agreeable concept of truth?
11:46pm
Me- we do, but truth is not as concrete as english
let's ignore pidgin and L337 and all that for now
if i asked you "why is the sentence english?" you'll point out that the words are in english, and the structure is english, and
the letters are latin
but if i asked you "why is the sentence false?", what would you tell me?
what points in or about or of the sentence do we have to judge as false?
we don't. we have no points.
the only point we could have, in a way, is an equation to falsehood
thus the x = 0 argument i made earlier
and if x = 0 = 0, then 0 = 0
and that is true
but 0 being 0 isn't really a paradox
Pat- so, at base, you are asserting that "this sentence is true" or "this sentence is false" do not actually have truth values
yes?
Me- yes, but because it has no values to begin with
i'm disputing existence, not validity
or rather the inability to determine validity on the basis on non-existent values for comparison
Pat- those are different things. are you making an epistemic or an ontologic claim?
Me- i don't know- let me pull out my dictionary here...
nm. give me the cliff's notes.
Pat- are you making a claim about what we can know, or about what is?
Me- what is, in that the sentence "is" not one containing material for validity
i suppose we "can know" what is false but only if we have some"thing" to prove false
"this sentence is false" gives us no"thing" except the term "false"
Pat- if we substitute the variables in your equation, does the outcome differ? where true = 0 and false = 1?
Pat- what would it look like?
Me- this sentence is false
this sentence = 1
(this sentence = 1) = 1
(X(n+1) = 1) = 1
or, not using X, 1 =1
Me- the sentence itself
every time you use the term "this sentence" you have to use all four words
Me- the words "this sentence"
so ((this sentence is false) is false) = false
it's not a big n as in X(n+1) = Xn + X
it's supposed to be a subscript n+1
to show that everytime you use the phrase "this sentence" you must muse the words "this sentence is false"
since "this sentence" is what we're declaring is false
Pat- gotcha
assuming that the sentence is true, you derive that it is false
(at least, I am pretty sure that is an accurate summary of what you've done)
Me- lol
sounds about right
been a long time since i've done math
Pat- it's akin to symbolic logic, which I haven't done in a while I think you've taken a more convoluted route than is necessary, but that's ok
if you assume that the sentence is false, what would your equation look like?
Me- what do you mean?
as in false is 0?
12:32am
Pat- no. however you set your variables, you've assumed the truth of the statement (that the statement is in fact
false), and derived not not false -> true.
