Is the liars paradox actually a paradox at all? I don't think it is.

Question

Note - I am not versed in logic notation or whatever the particular notation is for this kind of thing. Sorry in advance.

Edit - This is to address the argument about taking the statement as a whole rather than in parts. If you take the statement as a whole, then there is actually three hypothetical possibilities. The statement is true which creates a paradox, the statement is false which creates a paradox, or that the statement is undefined with regards to the truth value. Undefined only means that the function value could not be evaluated. So, in order to prove that this is a paradox then you have to prove that the truth value of the given statement is definitively true or false. If you can't do that then the truth value is undefined which doesn't create a paradox. And I have no idea how to go about evaluating a statement whose only information relies on that very statements truth value. If you have answer not based on hypotheticals than I'd love to hear it.

The version of the liar's paradox that I am using is "this sentence is false". Supposedly this statement is a paradox because they say if the statement is true then it is false and it is false if it is true. However, this is an absolute statement and is the same way that x = 1. Moreover, it is an absolute statement with an assigned value. Think of it like this, x = 1 is just assigning a value to x so will always equal 1 so long as the value of x is assigned to one. So, if we turn this statement into a mathematical equation we assign the variable "this sentence" = x, give the numerical value of true = 0 and false = 1 this sentence mathematically becomes x = 1. In the same way "this sentence" has been assigned the truth value of false. The core of this paradox seems to be the assumption that we can evaluate the given statement in the same way that we can evaluate a conditional truth statement like "apples can be red. The last sentence is false." However, "this sentence is false" has no conditions that we can evaluate it as true or false by comparing it to known truths and falsehoods. Because the truth variable of this statement is independent of reality or conditions and is an absolute assigned value it cannot be evaluated. The truth value of the sentence has been assigned as false and therefore cannot be changed by an outside valuation of true or false assuming that we are viewing true and false as absolute mutually exclusive values. True and false without any conditions, conditional statements or context just become arbitrary mutually exclusive values.

On the more common variant of the liars paradox "this sentence is a lie" if a lie is defined as something that is false or lie = something false then in the context of the previous statement the new statement becomes "this sentence is false".

You may as well just simplify it to just "false" because "this sentence" really doesn't add anything in the context of the statement besides an extra dependent truth variable which is is equal to the assigned truth value. Also, false is treated as a negative condition where as false = -1 and true = 1 and a falsehood is false is treat as -1*-1=1 however, this makes no sense. A false falsehood has always been true so its values have never been negative in the first place. If something is true or false (treating true or false as absolutes whose probabilities add up to 1) then it only has ever been one or the other.

I'm probably missing something and you're more than welcome to correct me.

Answer 1

Let's compare two English sentences, each talking only about themselves:

• $F$: "This sentence is false."
• $T$: "This sentence is true."

Neither statement can have a unique truth value out of the usual two determined for it. But while insufficient information for evaluation may be a point of similarity, there's a crucial difference. We can assign either value to $T$ without paradox, but can assign neither to $F$ without paradox. So being unable to uniquely determine the truth value doesn't get to the heart of this paradox.

We cannot simply avoid $F$'s paradox by saying its truth value "can't be evaluated", because this doesn't change the unfortunate fact that $F$ can't be true without also being false and vice versa, and therefore we can't maintain the popular position that all propositions are exactly one of these, unless $F$ doesn't really state a proposition.

Does it state one? Maybe it only looks like it does. But if so, what makes it illegal? Simply banning self-reference has two obvious drawbacks:

• "This document contains instructions on..." doesn't in general raise our eyebrows.
• One can get the same kind of paradox from $n\ge2$ sentences in a circle, e.g. with each statement claiming the next one is true, except for the last one claiming the first is false. So one might have to ban sentences talking about each other's truth-values altogether, which would cost us even more dearly.

Nonetheless, one can use an idea like this in formal mathematical languages, in which statements can only apply a truth predicate to others lower in a hierarchy. But that's just one way to address the paradox. You may feel your ideas are similar to some others on the same page.

Answer 2

Suppose you have a list of sentences each of which are classed as either true, false or somehow indeterminate (a logical trichotomy). Suppose sentence L in the list is in the True category if and only if it is in the False category (like "This sentence is false"). Using ordinary logic, we can infer by contradiction that L cannot be in either the True category or in the False category, but must be in the third, Somehow Indeterminate category.