The Liars Paradox is typically stated as “This sentence is false”. Trying to assign a truth value to this statement seems to result in a logical contradiction, thus the proposition doesn't appear to be true or false. Although there is no way to externally verify whether the sentence is true or false, we can still analyze the two possibilities. In doing this we find, per Wikipedia, that “If it is true, then the sentence is false, but if the sentence states that it is false, and it is false, then it must be true, and so on.” Since both cases result in contradictions we have paradox.
When discussing this sentence I have found the sentence's self-reference is often a point of contention/confusion. Many have claimed the sentence lacks content or that the self-reference isn't clear. To avoid the self-reference debate, you can construct the Liars Paradox as follows:
(1) Statement (2) is false.
(2) Statement (1) is true.
Similar to the Liars Paradox, it seems like if (1) is true, then (2) must be both true and false. While if (1) is false, then (2) is true, making (1) true. In either case, we again have an apparent contradiction, and thus the propositions don't appear to be true or false.
I believe this paradox can be resolved by accounting for all claims of a proposition, both implicit and explicit. This is simply my way of explaining the Arthur Prior resolution, which I found to be completely satisfactory and I think it should be the accepted resolution.
The idea is simply that for statement (1) to be true, it require not just that statement (2) is false but also that statement (1) is in fact true. This seems obvious, but it changes everything. The claim is that propositions, by their very nature, are implicitly claiming to be true. For example the statement “The sky is blue.” is logically equivalent as “It is true that the sky is blue” or “This statement is true, the sky is blue.” I view this as self-evident and uncontroversial, but I may just be ignorant of some of the implications. Anyways, to make this implicit claim explicit, we can reconstruct the propositions as follows:
(1) Statement (2) is false (and statement (1) is true).
(2) Statement (1) is true (and statement (2) is true).
These 'modified' propositions, but logically equivalent, can be resolved without contradictions. Lets consider the two possibilities for statement (1).
If statement (1) is true, then we know two things. Statement (2) is false, and statement (1) is true.
So what exactly is false about statement (2)? It is the fact that it is claiming that statement (2) is true, when we know that statement (2) is false. In other words, statement (2) isn't false because it's claiming that statement (1) is true - it got that part right. It's false because it's implicating claiming that statement (2) is true, when it is in fact false.
Now lets think about the case where statement (1) is false. If statement (1) is false, then it follows that it must be incorrect about one of the two things it is claiming (for example, consider the proposition “Two plus two is four and dogs are not animals.” That proposition is false, but that clearly doesn't mean every aspect of the proposition is false. It is false because dogs are in fact animals, not because two plus two isn't four). So in this case, (1) being false doesn't make (2) true. The reason (1) is false is rather because it is implicitly claiming to be true, when it is in fact false. Since (1) is false, (2) is obviously false as well, since one of its claims is that (1) is true.
This resolution maps onto the more common construction of the Paradox, “This sentence is false.” by modifying it to “It (this sentence) is true that this sentence is false.” or "This sentence is true and false." For the sentence to be true, it would need to be both true and false at the same time. This is a logical contradiction, thus the sentence is false. It would be like claiming that “X is 1 and X is 2.” You don't need to have any idea about what X is in order to know that the proposition is false since it is claiming something contradictory. Therefore, I no longer see this as a real paradox. Thoughts?
When discussing this sentence I have found the sentence's self-reference is often a point of contention/confusion. Many have claimed the sentence lacks content or that the self-reference isn't clear. To avoid the self-reference debate, you can construct the Liars Paradox as follows:
(1) Statement (2) is false.
(2) Statement (1) is true.
Similar to the Liars Paradox, it seems like if (1) is true, then (2) must be both true and false. While if (1) is false, then (2) is true, making (1) true. In either case, we again have an apparent contradiction, and thus the propositions don't appear to be true or false.
I believe this paradox can be resolved by accounting for all claims of a proposition, both implicit and explicit. This is simply my way of explaining the Arthur Prior resolution, which I found to be completely satisfactory and I think it should be the accepted resolution.
The idea is simply that for statement (1) to be true, it require not just that statement (2) is false but also that statement (1) is in fact true. This seems obvious, but it changes everything. The claim is that propositions, by their very nature, are implicitly claiming to be true. For example the statement “The sky is blue.” is logically equivalent as “It is true that the sky is blue” or “This statement is true, the sky is blue.” I view this as self-evident and uncontroversial, but I may just be ignorant of some of the implications. Anyways, to make this implicit claim explicit, we can reconstruct the propositions as follows:
(1) Statement (2) is false (and statement (1) is true).
(2) Statement (1) is true (and statement (2) is true).
These 'modified' propositions, but logically equivalent, can be resolved without contradictions. Lets consider the two possibilities for statement (1).
If statement (1) is true, then we know two things. Statement (2) is false, and statement (1) is true.
So what exactly is false about statement (2)? It is the fact that it is claiming that statement (2) is true, when we know that statement (2) is false. In other words, statement (2) isn't false because it's claiming that statement (1) is true - it got that part right. It's false because it's implicating claiming that statement (2) is true, when it is in fact false.
Now lets think about the case where statement (1) is false. If statement (1) is false, then it follows that it must be incorrect about one of the two things it is claiming (for example, consider the proposition “Two plus two is four and dogs are not animals.” That proposition is false, but that clearly doesn't mean every aspect of the proposition is false. It is false because dogs are in fact animals, not because two plus two isn't four). So in this case, (1) being false doesn't make (2) true. The reason (1) is false is rather because it is implicitly claiming to be true, when it is in fact false. Since (1) is false, (2) is obviously false as well, since one of its claims is that (1) is true.
This resolution maps onto the more common construction of the Paradox, “This sentence is false.” by modifying it to “It (this sentence) is true that this sentence is false.” or "This sentence is true and false." For the sentence to be true, it would need to be both true and false at the same time. This is a logical contradiction, thus the sentence is false. It would be like claiming that “X is 1 and X is 2.” You don't need to have any idea about what X is in order to know that the proposition is false since it is claiming something contradictory. Therefore, I no longer see this as a real paradox. Thoughts?
