There is a general pattern that occurs wherein something is expressed as a dichotomy/binary. Switching to a continuum afterwards is an extension, but this does not necessarily include all the possibilities.
Dichotomies: True/False. Beautiful/Ugly.
True/False.
Logic handles this by looking for ‘all true’.
If ‘p’ is true, and ‘q’ is false, ‘p and q’ is false.
More generally, a sentence could be broken up into parts that can be individually rated. After this, the ratio of true (atomic) statements to false (atomic) statements could be expressed—unless all the sub-statements are true, or all false. This can be fixed be expressing the ‘score’ as a (rational) number, with two choices of score:
true(sentence) = number of true statements / number of statements
false(sentence) = number of false statements / number of statements
And since every statement is true or false:
true(s) + false(s) = 1
And if we want to express how much truth is expressed, true(s)*num(s) = # of true statements. (These functions don’t have the best relationship to each other, they’re just meant to be intuitive enough.)
Consider the assumption: every statement is true or false. (Exclusively.)
Instead of diving into paradox, consider functions. Equality(s) returns the sentence “s is true”. Negation(s) returns “s is false”. These functions don’t have a truth value, it’s dependent on the variable that’s passed in. Concat(s_1, s_2) returns “s_1 s_2”, which can be just gibberish. But why is “Equality” named Equality—it preserves truth value but there are other functions with that property? It might be better thought of as a family of functions.
Now consider the function f that, given s, returns “s is true and s is false”. And here is a function that is always false. Right?
(This next paragraph* is a framing without examples, and may be rejected or accepted. I’m treating ‘paradoxes’ in this way because, as the paragraph after it notes, truth seems to come from a system.)
But just as (self referential) sentences can be constructed that are ‘paradoxical’ - neither ‘false’ nor ‘true’, sentences may also be constructed which ‘are both’. This may be resolved by pointing out that the first are “nonsense”, and resolving that “nonsense is false”, and saying that it doesn’t matter what value is assigned to the second as there is no consequence. (For such a sentence may be false, or it may be true, but not both at once.) But these resolutions are at odds. Are not both kinds “nonsense”? Or if they are different, they seem different from both ‘statements which can only be true’ and ‘statements which can only be false’.
To get back to our ‘functions’ (which take sentences as input, and return a sentence as output), consider the sentence “1+1=2”. Is this true? In many systems yes, “base 3, base 4, base 5, …”, but not in “base 2”, where “2″ is not defined, “1+1=10”. These systems may be converted between, and we may even say that while something is expressed one way in one system, and another way in another system, they’re the same “fact” (or falsehood).
But having different systems enables much confusion, two (or more) people might disagree on what color the sky is currently, even if they both have eyes that work fine, and without any unusual atmospheric phenomena that change what the sky looks like if you take a few steps to the right, or the left, if only they disagree on what colors the words for colors mean. If you call X “red”, and I call X “blue” we may still both see X.
To get back to truth(s) which can return “2/3” (meaning s contains 3 statements, 2 of which are true, one of which is false), why return one number? Why not two: 2,1: 2 true statements, 1 false. But there could be more statements than those two kinds. And here the path splits in two.
1. A particular methods of assigning one value to a sentence may ‘fail on the paradox’, or choose to call it false.* One method, one answer—every statement is true or false, exclusively.
2. A set for each possibility: It is true, it is false, it can be true or false, it cannot be either, etc. There’s still a binary aspect to this: “is it true” receives the answer “yes” or the answer “no” exclusively. But, independently, “is it false” may also receive either answer.
Following the 2nd path, what does it mean for something to be true and false? Neither?
One way is this: “The sky is blue, and the clouds are red.” Part of it is true, and part of it false. That which holds neither truth not falsehood, is nonsense.
How does this generalize? For that another dichotomy will be required.
Beautiful/Ugly***. While this may be subjective, the quaternary** view can be seen as claiming the binary view is false, some things are both beautiful and ugly, and some things are neither. Perhaps here this view will be less controversial, after all, if a thing is judged to be beautiful by one person, and ugly by another, “subjectively”, then “objectively” might not the object be both? Perhaps something ugly and beautiful could be created by cutting something beautiful in half, and something ugly in half, and combining them? This may be trickier than combining a true statement and a false statement, but perhaps if something is both beautiful and ugly, both aspects can be seen, where something that is true and false might be swiftly proclaimed ‘all wrong’ (or all right).
Perhaps this has all just been confusing, or perhaps it will be useful. The notion of ‘logical counterfactuals/counter-logicals’ has seemed strange to me—it is not that “it could be that 2+3 = 4” but that must be a different system. What such a thing could mean in conjunction with a world, say, if you put 2 things in a container, and then three, and what results is 4, seems unclear. (Even making them creatures doesn’t make sense, for if one eats another, why won’t that happen later?) If it holds for a class of objects, then that changes the relationship between numbers and objects—an apple and an orange are together are two things, but even if all things have the property that under certain circumstances they react to produce or eliminate another of the same type, then unless this holds between classes, no more might one speak of an apple and an orange being 2, because they don’t react with each other.
*Paradoxes working this way may be avoided by system design.
**One may eliminate one of these categories, and say, that nothing is neither beautiful nor ugly. Then the category still ‘exists’ though it has no members—a broader view may include things that are not, but absent a process for creating new categories, the more expansive view may be better before examining reality. And if someday that person finds something which is neither, then the bucket will be ready for this new object unlike anything seen before.
***This is one area where things may not be fixed, in a way that we don’t see in math or logic. A view in which things don’t have properties may be more useful—but it is harder to see this for things/properties like “numbers” which ‘seem to exist’. “The tree falls in the forest” argument may also be had about beauty.
There is a general pattern that occurs wherein something is expressed as a dichotomy/binary. Switching to a continuum afterwards is an extension, but this does not necessarily include all the possibilities.
Dichotomies: True/False. Beautiful/Ugly.
True/False.
Logic handles this by looking for ‘all true’.
If ‘p’ is true, and ‘q’ is false, ‘p and q’ is false.
More generally, a sentence could be broken up into parts that can be individually rated. After this, the ratio of true (atomic) statements to false (atomic) statements could be expressed—unless all the sub-statements are true, or all false. This can be fixed be expressing the ‘score’ as a (rational) number, with two choices of score:
true(sentence) = number of true statements / number of statements
false(sentence) = number of false statements / number of statements
And since every statement is true or false:
true(s) + false(s) = 1
And if we want to express how much truth is expressed, true(s)*num(s) = # of true statements. (These functions don’t have the best relationship to each other, they’re just meant to be intuitive enough.)
Consider the assumption: every statement is true or false. (Exclusively.)
Instead of diving into paradox, consider functions. Equality(s) returns the sentence “s is true”. Negation(s) returns “s is false”. These functions don’t have a truth value, it’s dependent on the variable that’s passed in. Concat(s_1, s_2) returns “s_1 s_2”, which can be just gibberish. But why is “Equality” named Equality—it preserves truth value but there are other functions with that property? It might be better thought of as a family of functions.
Now consider the function f that, given s, returns “s is true and s is false”. And here is a function that is always false. Right?
(This next paragraph* is a framing without examples, and may be rejected or accepted. I’m treating ‘paradoxes’ in this way because, as the paragraph after it notes, truth seems to come from a system.)
But just as (self referential) sentences can be constructed that are ‘paradoxical’ - neither ‘false’ nor ‘true’, sentences may also be constructed which ‘are both’. This may be resolved by pointing out that the first are “nonsense”, and resolving that “nonsense is false”, and saying that it doesn’t matter what value is assigned to the second as there is no consequence. (For such a sentence may be false, or it may be true, but not both at once.) But these resolutions are at odds. Are not both kinds “nonsense”? Or if they are different, they seem different from both ‘statements which can only be true’ and ‘statements which can only be false’.
To get back to our ‘functions’ (which take sentences as input, and return a sentence as output), consider the sentence “1+1=2”. Is this true? In many systems yes, “base 3, base 4, base 5, …”, but not in “base 2”, where “2″ is not defined, “1+1=10”. These systems may be converted between, and we may even say that while something is expressed one way in one system, and another way in another system, they’re the same “fact” (or falsehood).
But having different systems enables much confusion, two (or more) people might disagree on what color the sky is currently, even if they both have eyes that work fine, and without any unusual atmospheric phenomena that change what the sky looks like if you take a few steps to the right, or the left, if only they disagree on what colors the words for colors mean. If you call X “red”, and I call X “blue” we may still both see X.
To get back to truth(s) which can return “2/3” (meaning s contains 3 statements, 2 of which are true, one of which is false), why return one number? Why not two: 2,1: 2 true statements, 1 false. But there could be more statements than those two kinds. And here the path splits in two.
1. A particular methods of assigning one value to a sentence may ‘fail on the paradox’, or choose to call it false.* One method, one answer—every statement is true or false, exclusively.
2. A set for each possibility: It is true, it is false, it can be true or false, it cannot be either, etc. There’s still a binary aspect to this: “is it true” receives the answer “yes” or the answer “no” exclusively. But, independently, “is it false” may also receive either answer.
Following the 2nd path, what does it mean for something to be true and false? Neither?
One way is this: “The sky is blue, and the clouds are red.” Part of it is true, and part of it false. That which holds neither truth not falsehood, is nonsense.
How does this generalize? For that another dichotomy will be required.
Beautiful/Ugly***. While this may be subjective, the quaternary** view can be seen as claiming the binary view is false, some things are both beautiful and ugly, and some things are neither. Perhaps here this view will be less controversial, after all, if a thing is judged to be beautiful by one person, and ugly by another, “subjectively”, then “objectively” might not the object be both? Perhaps something ugly and beautiful could be created by cutting something beautiful in half, and something ugly in half, and combining them? This may be trickier than combining a true statement and a false statement, but perhaps if something is both beautiful and ugly, both aspects can be seen, where something that is true and false might be swiftly proclaimed ‘all wrong’ (or all right).
Perhaps this has all just been confusing, or perhaps it will be useful. The notion of ‘logical counterfactuals/counter-logicals’ has seemed strange to me—it is not that “it could be that 2+3 = 4” but that must be a different system. What such a thing could mean in conjunction with a world, say, if you put 2 things in a container, and then three, and what results is 4, seems unclear. (Even making them creatures doesn’t make sense, for if one eats another, why won’t that happen later?) If it holds for a class of objects, then that changes the relationship between numbers and objects—an apple and an orange are together are two things, but even if all things have the property that under certain circumstances they react to produce or eliminate another of the same type, then unless this holds between classes, no more might one speak of an apple and an orange being 2, because they don’t react with each other.
*Paradoxes working this way may be avoided by system design.
**One may eliminate one of these categories, and say, that nothing is neither beautiful nor ugly. Then the category still ‘exists’ though it has no members—a broader view may include things that are not, but absent a process for creating new categories, the more expansive view may be better before examining reality. And if someday that person finds something which is neither, then the bucket will be ready for this new object unlike anything seen before.
***This is one area where things may not be fixed, in a way that we don’t see in math or logic. A view in which things don’t have properties may be more useful—but it is harder to see this for things/properties like “numbers” which ‘seem to exist’. “The tree falls in the forest” argument may also be had about beauty.