I really like this. I think there's two different vibes here I like: (1) evaluation of arguments should compare to other possible arguments within some reference class rather than being based solely on the properties of the argument, and (2) more tangentially, the truth value of a system of statements need not just be "true" or "false", "consistent" or "inconsistent".
I don't really have much more to say about (1) other than that I liked your explanations of the loopy blue universe and tessellating beetles. Both are absurd, but have explanatory power. I would like to explore more about the relationship of arguments to their reference classes and the structure of possible argument reference classes.
I like thinking about (2) as digraphs where each node is a statement and arrows are references to other statements, then each statement has a set of possible truth values which either makes the graph inconsistent or not. Some graphs may have a single, or multiple consistent sets of truth values, but there is more to be said about inconsistent ones than merely that they are inconsistent. If at each iteration you "updated" the truth value for the node corresponding to each inconsistent arrow, you would get patterns of propagation based on the starting state. Further, an obvious way to force an inconsistent graph is to have cycles in the graph with an odd number of "bit flip" arrows, which assert some reference is false, and so must be false if the referent is true. Then you can ask about the other possible structures which introduce forced inconsistency.
( I'm sure people have focused on this, but I'm not sure what it would be called... )
One application for this might be that if you could throw out all inconsistency forcing structures you could (perhaps) find the subset of programs for which the halting problem doesn't apply.
Your comment is much appreciated and touched on an idea I find interesting but didn't intend to reference directly in the post. Thanks for your kind words about this post. You make a very interesting point about networks of interconnected statements and arguments having more than simply a true or false value. I think the simplest such network which exhibits the kind of interesting structure you mention is that of the statement: "This statement is false." . If you represent truth values of statements as numbers between 1 and -1, with 1 representing truth and -1 falsehood, then that statement can be written down as the equation x = -x, where x is the truth of the statement. The fact that this statement is neither true or false is a consequence of its actual solution being 0, which I think of as representing 'tralsehood'. More generally, the process you describe sounds a lot like the google page rank algorithm, which is a process by which a monstrously complicated family of simultaneous equations can be approximately satisfied because they have basins of attraction around the solutions, which would be consistent sets of truth values in this context. Clearly, not all of these networks have integer solutions as you point out, but there will always be real solutions, and I would interpret these as representing the statements having varying levels of 'futhhood' or 'tralsity'.
I like your "tralsity" terminology. I think I was thinking about this differently from you. I was thinking from a computer science or graph theory perspective, where the "solution" to an argument would b the set of families of state transitions for statements when the truth values are updated iteratively.
This is more related to arguments, systems of statements, being valid or invalid, tautological, or contradictory... although I think the ideas would need to be extended in some way to apply to arguments with cyclic form, as opposed to the more normal form of having premises and conclusions.
It seems like tralsity, which you are talking about, is more focused on assigning valid tralse values to statements regardless of them having logical contradictions. I think this is an interesting thing to be trying to do. I think I prefer the argument structure and state transition approach because it would point out which states are valid and what is causing the state to be invalid when it is invalid.
So rather than "this statement is false" going to "x=-x" and being solved as 0, it would go to "x<--x" and the solution would be the sequence "...-1,1,-1,1,-1,1..." which is unrolled from a state transition that might look something like {( x=1 --> x=-1 ), ( x=-1 --> x=1)}, or something similar. That creates a single connected transition graph with two nodes, one representing the state "x=1" and one representing the state "x=-1" with each pointing at the other. And that is the only system of transitions this argument can take. But other arguments might have disconnected transition graphs. Only states in transition graphs that only point to themselves would be valid, but the different kinds of subgraphs with more than one node would tell you about the way they are invalid.
Haha... sorry, this isn't fully thought through so I apologize if it isn't clear or easy to follow.
I think there's probably a relationship between this idea of assigning tralsity to states vs looking at transition graphs for arguments, but I'm not sure exactly what that relationship would be.
It is reminding me of the book "Topoi: the categorical analysis of logic" which I have not gotten into far enough to know if it is related at all or not.
Alas, so much math, so little time.
Very interesting, I am in a similar position with respect to learning the relevant mathematics as you know from my first comment. One thing that your sequence resembles to me is the divergent infinite sum 1- This sequence does not get closer and closer to any particular value, so from the most standard perspective, its sum is undefined. However, the partial sums alternate from 1, to 0, back to 1 again and continue to do so ad infinitum, which means that their average is . A different way of looking at this sum is through the formula . If x = -1, then the series becomes . This suggests that, to the extent that this sum has a value, it is . Although this series is not the same as your sequence, if we take 1 to represent true, 0 to represent false and subtraction from 1 to represent logical negation, then the equivalent sequence is the sequence of partial sums: and now represents the concept of 'tralse' . In the case of your sequence, it would be the set of partial products of an infinite product . How to obtain a generalization of the value of an infinite product when it does not converge, I am not sure. One possibility is to operate on a logarithmic scale, on which multiplication is equivalent to addition. Multiplication by -1 is equivalent to raising to the power of , which suggests that the infinite product is given by
. In complex analysis, there is a well defined infinity which is the reciprocal of 0, therefore . Though is not , it is the single other number on the Riemann sphere which is its own negative, and is therefore also a solution to the equation !
I have found another way to shoehorn into your interpretation the notion of 'tralse' . I'm not sure if this meaningful or not, but I didn't know that calculation would produce a result compatible with the idea of 'tralsity' until I carried it out, and it did produce such a result in an unexpected way .
I will mirror your disclaimer about the idea being newly encountered and not clearly explained.
In my other comment I forgot to ask you what you mean by "One application for this might be that if you could throw out all inconsistency forcing structures you could (perhaps) find the subset of programs for which the halting problem doesn't apply." Why might this be? Is there a mapping between all programs and the truth-value - propagation system?
The halting problem is the problem of looking at the structure of a program and using it to determine whether or not the program would halt if you ran it. There is a proof by contradiction that you can't have such a program (from the wikipedia page for the halting problem):
def g() -> None: if halts(g): loop_forever()
This is a contradiction because if halts returns true then it shouldn't have, because g will loop forever based on halts returning true, but if halts returns false then g will return None and halt, and so halts should have returned true.
But learning about this annoyed me, because it's obvious what's going on, halts is being forced into self reference, and a contradiction is only forced if halts is forced to return true or false. If it instead returned "I am embedded in a program with a 2 state logical contradiction" then that would be a better answer.
I think this is also Russell's paradox in set theory, and might be related to Godel's incompleteness. Basically the thing seems to be "self referential statements break our languages" and my feeling is just "create your languages so they talk about the nature of self reference! You're creating a problem where there does not need to be a problem"... but obviously I haven't fully understood the ideas or thought through all the implications, so it would be rash for me to make such a statement, but it is a feeling that I get.
I see, this is something I've wondered about... If the set is allowed to 'half contain' itself, or contain half of itself, then the paradox is kind of resolved. On your point about languages, I remember reading about the 'Santa claus sentence' which states: " If this sentence is true, then Santa Claus exists". Clearly, if the sentence were true, then Santa Claus would exist. But this is what the statement says, so Santa Claus must exist. Of course, if we instead ask what would happen if the statement is false, then we are not drawn to conclude that Santa Claus exists, merely that he wouldn't even if the sentence were true...
When thinking about this I realized that the problem might just be that the way language is structured is inherently linear, so that to expand the statement fully would require it to be an infinitely long sentence of the form: "If " If " If "If " ....infinitely deep ...." is true, then Santa Claus exists." is true, then Santa Claus exists. " is true, then Santa Claus exists. " is true, then Santa Claus exists." . This is the only way to parse the sentence using linear language, and clearly it depends on what the text 'infinitely deep' that I've used to signify the 'rock bottom' actually means. This is indeterminate, similar to how the equation can be expanded out into an infinite expression which converges to two possible values when you iterate upon it starting from a particular number. One of them is what's commonly called the golden ratio, the other is the negative reciprocal of the (what's commonly called) golden ratio.
I might think of more to say about this but don't currently have the time to do so, or to reply to your other comment. Hopefully I will soon.
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Circular arguments are often claimed not to be of explanatory value. I argue that this is true primarily as a result of the existence of inverted, 'parody arguments', which take the same circular form, but with every step inverted to yield the opposite conclusion of the opposite premises.
Example:
A demonstration that a cyclical universe with a duration of 2 hours will be filled with blue goo: a blue substance consists of tiny entities which live for at least 1+1/2 hours, and duplicate themselves every hour. If at one instance of time, space is filled with these blue entities, then it will be 1 hour later, because they will have continued to exist for the last hour and will now duplicate themselves. These duplicates will still exist another hour later, as they themselves will live for another half hour. But in this universe, the future 2 hours later is the present, which explains why the universe was filled with the blue substance at all.
Parody argument: If at one instance of time, there are no blue creatures in the universe, there won't be any after one hour has passed, because the nonexistence of these blue creatures will propagate itself into the future by one hour. No duplicate blue creatures will come into existence because there were none to begin with. These duplicates will still not exist another hour later, but in this universe, the future 2 hours later is the present, which explains why the universe didn't contain any blue substance at all.
As both the parody argument and the original seem equally logical, there is no way to differentiate them and decide which to believe, other than with a criterion like simplicity.
Circular arguments can also be logically circular even if they refer to something chronologically linear. For example, suppose there was a kind of beetle which lived in a tessellated 'honeycomb' structure along with others of its kind. The following argument suggests that its structure is likely to be cubic:
1) Having a cubic shape is evolutionarily advantageous because it allows each cubic beetle to interlock with the others, which protects it from the cold and predators.
2) This selective pressure will gradually increase the proportion and number of cubic beetles over time.
3) The abundance of cubic beetles explains why it is evolutionarily advantageous to fit in with them as opposed to beetles of some other shape.
This argument also has a parody, but not one of the kind above which is a simple inverse of the original argument:
Suppose there was a kind of animal which lived in a tessellated 'honeycomb' structure along with others of its kind. The following argument suggests that its structure is likely to be tetrahedral: 1) Having a tetrahedral shape is evolutionarily advantageous because it allows each tetrahedral beetle to interlock with the others, which protects it from the cold and predators. 2) This selective pressure will gradually increase the proportion and number of tetrahedral beetles over time. 3) The abundance of tetrahedral beetles explains why it is evolutionarily advantageous to fit in with them as opposed to beetles of some other shape.
Again, the existence of two, apparently equally compelling, arguments for two different conclusions prevents us from being able to determine which one to believe, at least on the basis of these arguments alone. However, unlike the first example, these arguments do suggest something. If the beetles were spherical, then the corresponding argument would cease to be sound, because spherical beetles would not be able to tessellate in the same way. If we modify the original argument to refer to any polyhedra capable of tiling 3-dimensional space with no gaps, it becomes impossible to negate this part of the argument in a way which doesn't weaken it, and therefore much harder to find a symmetric, equally persuasive parody argument.
For this reason, such an argument would be evidence for the existence of tessellating polyhedral beetles of some kind.