In logic, sentences should be assigned circuits* instead of truth values.
Firstly, processing circuits, which execute the action described by the sentence.
“Proof” that there exists a suitable circuit for sentences such as “2+2=4“ “2+2=5” and “9^9=387420489”, exists in the form of calculators.
Secondly, circuits for checking if a truth value assignment is consistent. They receive “True” or “False” as an input, hereafter referred to as “I”, and output “True” if the assignment is consistent. This can be done by having both the processing circuit and “I” as inputs to a gate which returns “True” if they are both equal. (An equals gate.)
This is like turning the problem of evaluating the truth value of “2+2=4” into constructing a circuit that “represents” “2+2==4”.
These are necessitated by the existence of sentences which may be assigned multiple truth values. “This sentence is true.” Assigning ‘true’ would be consistent, as would ‘false’. (This sentence is unusual in that the consistent assignment circuit has the input, “I”, go into both the processing circuit, and the equals gate.)
*Several things work here. In addition to being a good place to hold a lot of technical/logical questions, circuits can be implemented fairly easily.
I came up with this when thinking through something I saw someone write on the resolving the LP (on their blog, they also wrote a book). Now I’m trying to find it again.*
What they said, paraphrased (from memory): You attempt to assign a truth value as follows: you suppose it is true. If it were true, then it would be false. So you suppose it is false. But then it would be true. At this point, they rejected this line of thinking on the grounds that, what it means for something to be true, is for there to exist something in reality that corresponds to it, and there is no operator that satisfies this criteria. So the LP is false. (This is similar to the answer that it’s “not true or false, but nonsense” (from xkcd forums results on google search, when I tried to find their blog). The author took the additional step of combining the notions of “falsehood” and “nonsense” under the label of “false”.)
When I was thinking through that, while I got their point, it sounded like a NOT-gate. That is, I figured you could assign to a sentence a circuit which takes the truth value you would assign to it, and returns what it would be if that were so. This made sense for both of the self-referential sentences I considered (LP and “This sentence is true.”), and valid assignments were fixed points. What makes LP “paradoxical” is that trying to assign it a truth value is a process that corresponds to trying to find the fixed point of a function which doesn’t have any fixed points. (It’s opposite behaves the opposite way: it has all the fixed points.) When I thought about other sentences that weren’t self-referential, this didn’t make as much sense, and that was when I came up with the other two types of circuits/(ways of thinking about this).
Idea for resolving Liar’s Paradox:
In logic, sentences should be assigned circuits* instead of truth values.
Firstly, processing circuits, which execute the action described by the sentence.
“Proof” that there exists a suitable circuit for sentences such as “2+2=4“ “2+2=5” and “9^9=387420489”, exists in the form of calculators.
Secondly, circuits for checking if a truth value assignment is consistent. They receive “True” or “False” as an input, hereafter referred to as “I”, and output “True” if the assignment is consistent. This can be done by having both the processing circuit and “I” as inputs to a gate which returns “True” if they are both equal. (An equals gate.)
This is like turning the problem of evaluating the truth value of “2+2=4” into constructing a circuit that “represents” “2+2==4”.
These are necessitated by the existence of sentences which may be assigned multiple truth values. “This sentence is true.” Assigning ‘true’ would be consistent, as would ‘false’. (This sentence is unusual in that the consistent assignment circuit has the input, “I”, go into both the processing circuit, and the equals gate.)
*Several things work here. In addition to being a good place to hold a lot of technical/logical questions, circuits can be implemented fairly easily.
I came up with this when thinking through something I saw someone write on the resolving the LP (on their blog, they also wrote a book). Now I’m trying to find it again.*
What they said, paraphrased (from memory): You attempt to assign a truth value as follows: you suppose it is true. If it were true, then it would be false. So you suppose it is false. But then it would be true. At this point, they rejected this line of thinking on the grounds that, what it means for something to be true, is for there to exist something in reality that corresponds to it, and there is no operator that satisfies this criteria. So the LP is false. (This is similar to the answer that it’s “not true or false, but nonsense” (from xkcd forums results on google search, when I tried to find their blog). The author took the additional step of combining the notions of “falsehood” and “nonsense” under the label of “false”.)
When I was thinking through that, while I got their point, it sounded like a NOT-gate. That is, I figured you could assign to a sentence a circuit which takes the truth value you would assign to it, and returns what it would be if that were so. This made sense for both of the self-referential sentences I considered (LP and “This sentence is true.”), and valid assignments were fixed points. What makes LP “paradoxical” is that trying to assign it a truth value is a process that corresponds to trying to find the fixed point of a function which doesn’t have any fixed points. (It’s opposite behaves the opposite way: it has all the fixed points.) When I thought about other sentences that weren’t self-referential, this didn’t make as much sense, and that was when I came up with the other two types of circuits/(ways of thinking about this).
*EDIT: It’s fakenous.net.