Epimenides paradox

Question

So today I am in my Theory of knowledge class at my high school and my teacher presents Epimenides paradox. At first I was excited but then I noticed something. the statement that he put up was "All cretans are liars." This statement fails to address that liars lie all the time. I feel as though if the statement read "All cretans are always lying." or even "Cretans always lie." then the statement would truly be a paradox.

If we consider the statement my teacher showed us, "All Cretans are liars" then we can say that this Cretan wasn't lying at this point in time, he could possibly be telling the truth. I feel like since we can say that the Cretan may be or may not be lying it isn't a paradox.

I got into an argument with my teacher about this and he wouldn't let me talk aside from when I said that it wasn't really a paradox. He was stuck on his ideas. I am completely okay with being wrong I just don't see any logical argument saying that the statement is truly a paradox.

Can someone please let me know?

Answer 1

You are right in the sense that Epimenides paradox is not an antinomy. It has some issues. You've detected one of them, namely that it depends on the interpretation of 'liar'. If a liar is free to say the truth some times, then the sentence does not contradict itself.

Even in the strong sense of a liar being someone who never tells the truth, the sentence isn't self contradicting at all. The point most people are missing is, that that the negation of "all Cretans are liars" isn't "all Cretans are honest", it's "at least one Cretan is honest".

To get this straight and to prevent misunderstandings (in the strong sense): Epimenides can't tell the truth, because otherwise he would call himself a liar. Therefore he's a liar and his statement is false, hence tells us that not all Creatans are liars, so there is at least one Creatan who is not a liar. No contradiction, because Epimenides wasn't the only Creatan.

Answer 2

According to this Wikipedia article, even if you take "liar" to mean a person that always lies, this isn't really a paradox.

If Epimenides is a Cretan and he utters the statement "All Cretans always lie" and we assume that the statement is false, then we assume that there is at least one Cretan, not necessarily Epimenides, that does not always lie.

But this assumption does not imply that Epimenides is not lying so there is no paradox.

Answer 3

If it is required that the statement be false for it to be a lie, then one of the conditions required for the claim "I am lying" to be a lie is that the claim be false -- that I am not lying. So: I am a lying if I am not lying (and whatever other conditions are required for a statement to be a lie):

L := ¬L [∧ ...]

This is a contradiction. The claim "I am a lying" is necessarily false. But that the claim is false is not that it is a lie; not every false statement is a lie. And that the claim is not a lie is not that it is true; not every non-lie is a true statement.

Answer 4

MichaelRushton's answer addresses the fundamental issues with your variations which is that a lie and a false case are not the same thing. "All cretans are always lying" or "Cretans always lie" are always false whereas a paradox is both true and false.

That said, "All Cretans are liars" can be true or false depending on the circumstances. As you pointed out, if you are a liar, but not lieing at that moment, then it may be true, otherwise it is false. So, your professor would have be right to call this a paradox, but not your examples.

This goes into one of the fundamental misunderstandings about paradoxes which is that people assume they are entirely true and false at the same time (often called a True Paradox). Instead, most paradoxes are the result of over-arching statements with multiple and exclusive case scenarios where sometimes they are true in one way, and sometimes in another.

For example:

• "This car moves and does not move." is a paradox, but it can be factored into "This car moves." and "This car does not move." which are are each sometimes true and sometimes false, but never true at the same time. The fact that they are not true at the same time makes the paradox false, but the fact that one case is always true also makes the paradox true because I did specify enough qualifiers like "sometimes", "always", or "either/or"

As you can see, a paradox is just a logical anomaly caused by abstracting ideas together.

The only time you see a "True Paradox" is when talking about the theoretical. Much like math can be broken by non-real numbers such as zero and infinity, logic can be broken by things that cannot exist. For example, imagine you have a box that contains everything that can exist AND everything that can not exist. By definition that which is outside of the box absolutely does not exist which means it is also absolutely inside of the box no matter how you break it down.

Here, infinity is applied to logical recursion creating a "True Paradox".

Answer 5

According to wolfram this is not a paradox.

http://mathworld.wolfram.com/EpimenidesParadox.html

But this is.

http://mathworld.wolfram.com/EubulidesParadox.html

Which is probably what your teacher wanted you to investigate.