# Epimenides “Paradox”: I am a liar

#### Hartlw

If A right -> A wrong and A wrong -> A right the result is circular (T-F-T-F---)

So I can’t assume (I am lying) is T or F to arrive at a paradox. Circular results are not a paradox.

So what is “I am lying?” – an unverified statement.

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#### Hartlw

I am a liar = F(x), x = liar (T or F),
F(x) = not x,

#### SlipEternal

MHF Helper
What is the definition of a liar? If a liar is someone who is incapable of telling the truth, then the statement "I am a liar" is obviously false since an actual liar would be incapable of making that declaration. If a liar is someone who has at some point in time told a lie, then "I am a liar" could be true or false. If a liar is someone who is currently in the process of telling a lie, then the statement "I am a liar" depends on the definition of a lie. If a lie is intentionally making a false declaration, then "I am a liar" is false, since someone who was intentionally trying to make a false statement could not state "I am a liar", hence the falsehood they declared was unintentional. If a lie is making a false declaration whether intentional or unintentional, then we are in the case described by zzephod.

#### Hartlw

You can’t treat “I am a liar” as a variable becaues it can’t be given a T,F value. It is simply an imprecise statement (bad grammar).

“Liar” is a perfectly good variable which can be given a T,F value and “I am a liar” is a perfectly good function of the variable “liar,” and upon evaluation turns out to be the function “not liar.”

#### SlipEternal

MHF Helper
I don't see anyone treating "I am a liar" as a variable. I see it treated as a statement.

Edit: I also don't understand how "liar" can be a variable. Liar is a word with a non-Boolean definition. The statement "I am a liar" implies the existence of a speaker. The one who speaks the statement "I am a liar" may be a liar or may not be a liar. So, if you want to consider functions, then here is an example of a function:

Let $$\displaystyle P$$ be the set of all people. Define $$\displaystyle f: P \to \{T,F\}$$ by $$\displaystyle f(p) = \begin{cases}T & \mbox{if }p\mbox{ is a liar} \\ F & \mbox{otherwise}\end{cases}$$. Now, we are right back into the question of how does one define a liar?

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#### Hartlw

When you attempt to assign a T/F value to a statement you are treating it as a variable.

“John went into town” can be True or False. (If John went into town, then….)

“A liar went into town” is True or False depending on whether “liar” is True or False (assuming someone did indeed go into town).

If x = liar:
“A liar went into town” = G(x) = x
“I am a liar” = F(x) = not x

Comment
We have been confusing semantics with “logic.” Semantically, “I am a liar” is either bad grammar or a limitation of language.

liar: Someone who does not (verifiably) tell the truth.

(an interesting discussion might be treating "logic" as the mathematics of functions of variables which can only have one of 2 (undefined) values, call them T/F.)

EDIT: Compare:
"He is a liar" = G(x) = x
"I am a liar" = F(x) = not x

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#### SlipEternal

MHF Helper
When you attempt to assign a T/F value to a statement you are treating it as a variable.

“John went into town” can be True or False. (If John went into town, then….)

“A liar went into town” is True or False depending on whether “liar” is True or False (assuming someone did indeed go into town).

If x = liar:
“A liar went into town” = G(x) = x
“I am a liar” = F(x) = not x

Comment
We have been confusing semantics with “logic.” Semantically, “I am a liar” is either bad grammar or a limitation of language.

liar: Someone who does not (verifiably) tell the truth.

(an interesting discussion might be treating "logic" as the mathematics of functions of variables which can only have one of 2 (undefined) values, call them T/F.)
As you wish

#### Hartlw

You forgot to reference the edit:

EDIT: Compare:
"He is a liar" = G(x) = x
"I am a liar" = F(x) = not x

#### Hartlw

Proof by assumption (circular reasoning)

“I am a liar” as a language short-cut has already been discussed ( “I am a liar” except for this statement.). Just like (“this page intentionally left blank” except for this statement.). The short-cut is understood in commom usage. Any one who heard Epemenides statement knew exactly what he meant.

However, the logical aspect hasn’t been exhausted.

PROOF BY ASSUMPTION:

Assume “I am Napoleon” is T. “I am Napoleon” can’t be F because it violates the assumption. Therefore “I am Napoleon” is T.

Assume “I am a liar” is T. Then “I am a liar” can’t be F because it violates the assumption. Therefore “I am a liar” is T.

The above arguments are logically disallowed: the assumption alone that something is True or False is not sufficient to determine its Truth or Falsity.

Prove 2>1. Assume 2>1. Then 2>1 by assumption. (You need more than the assumption to draw a conclusion about 2>1).

SUMMARY for “I am a liar”:
As a language shortcut, it is perfectly clear.
As a logical statement, there is not enough information to arrive at a conclusion (paradox).

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