Sorry, I missed the edit.
EDIT: If you want to create a system free of Russels “Paradox,” then xϵy cannot be a primitive atomic formula unless you specifically disallow x=y, ie, . Otherwise xϵy is undefined if x=y (circular definition).
Is “I am a liar” a paradox?
If “I am a liar is true”, then “I am a liar is false.”
If “I am a liar is false” then “I am a liar is true.”
So of course it is not a paradox. It is simply a variable X which cannot be given a True or False value (they give contradictions).
For example “Air Force1 is flying” can’t be given a True or False value without further information.
Suppes gives a “modern day version:”
The only sentence on this board is false.
Since there is nothing else on the board, the only sentence being referred to has to be: “The only sentence on this board is false,” so that
The only (The only sentence on this board is false) on this board is false.
The only [The only (The only sentence on this board is false) on this board is false] on this board is false.
.
which is not a paradox, it is simply nonsense.
Wicki gives the version:
This statement is false: (A)
Call “This statement is false: (A)” X(A)
Then X(T) =F and X(F)=T, ie, X(A) = notA
If “statement” is “This statement is false: (A)” then you get the same nonsensical circular definition as above, which reminds one of “Russels Paradox,”
On this subject Suppes mentions Cantor’s paradox of the greatest Cardinal number, an “intuitive proof” of which he begins with “…the cardinal number n of the set S of all sets.” Also “….set of all Subsets of S and its cardinal p.” There is no such thing as the set of all sets or subsets.
I seem to recall seeing Logic has been reformed so as to avoid Russels Paradox. Which eliminates any rational discussion because it is impossible to follow. However, I recall statements such as xεx, which is still Russels paradox.
Sorry, I missed the edit.
EDIT: If you want to create a system free of Russels “Paradox,” then xϵy cannot be a primitive atomic formula unless you specifically disallow x=y, ie, . Otherwise xϵy is undefined if x=y (circular definition).
In general, “I am (swimming,walking,talking,jumping,lying,….)” only makes sense if I can determine the truth or falsity of the statement independent of the statement. I can not conclude that you are swimming by the fact that you say “I am swimming,” just as I can not conclude you are lying by the fact that you say “I am lying.”
So if somenone says “I am lying” you can’t assume it is True and use the statement itself to test your assumption.
Epimenides is in a public square proclaiming loudly and repeatedly “I am lying.” Out of curiousity people go up to him and ask, “what are you lying about,” but he just keeps repeating the same thing and won’t finish his sentence. What they conclude I leave to the imagination out of deference to Epimenides.
If A is false when it is true and true when it is false, it is not a proper variable, it can’t be True and False at the same time so I don’t see see how any system of logic can be applied to it, assuming a TRANSPARENT system of logic. You can bury anything in a complex system of arbitrarily defined symbols and rules of derivation and then pull something out of the middle of it and challenge me to prove you are wrong: Don’t bother- you win every time. I can only question the foundations.
EDIT: I did not edit anything. I opened it up to edit out one of (see see) but then changed my mind because I thought the ensuing "last edited by..." might be confusing or misleading. So I opened it for editing and closed it without changing anything, within a minute. I believe noting "last edited by..." in this case is misleading. Now you can say "last edited by..."
To test the truth of "I am swimming," you can't take my word for it.
To test the truth of "I am lying," you can't take my word for it, there has to be a reference to something I said.
Again the paradox has nothing to do with the truth or falsity of Epimenides' statement or its decidability. Under conventional logic is true.
Epimenides' statement is false (as its assumed truth leads to a contradiction) but also its negation is false for the same reason so is false ...
.
I am (walking, talking, swimming, lying, flying…) says I am engaged in the specified activity.
I am swimming is true says it is true that I am engaged in the activity of swimming.
I am lying is true says I am engaged in the activity of lying, it does not alter the fact that I am lying.
I am not repeating myself. I have given another interpretation and addressed both in previous discussion to show there is no paradox.
1) I am lying: This statement is a lie.
2) I am lying: I am engaged in the act of lying.
EDIT: zzephod: The second statement is clearly not a paradox, and if you think both statements are the same, then you are agreeing with my conclusion. Alas, the first is different and more subtle. Posts by "others" either repeat the paradox or re-phrase it in logical symbols, which I agree is not worth repeating, as I have addressed them throughout this thread.
Sorry, I originally thought this was just a repitition of the paradox (I still do). I didn’t catch the error in logic.
Under conventional logic, E has to have either a T or F value. If you use EVnotE to prove E can be neither True or False, then you can’t use EVnotE in the first place. It’s circular reasoning.
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.
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.
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.”
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 be the set of all people. Define by . Now, we are right back into the question of how does one define a liar?