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.
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 1-1+1−1+1−1+1−1... 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 1/2. A different way of looking at this sum is through the formula 1+1x+1x2+1x3+1x4+...=11−x . If x = −1, then the series 1−1+1−1+... becomes 11−x=12 . This suggests that, to the extent that this sum has a value, it is 12. 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:1,0,1,0,1,0,... and 12 now represents the concept of ‘tralse’ . In the case of your sequence, it would be the set of partial products of an infinite product(−1)(−1)(−1)(−1)(−1)(−1)(−1)... . 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 e to the power of iπ , which suggests that the infinite product is given by eiπ(1+1+1+1+1+1+...)=eiπ+iπ+iπ+iπ+iπ+iπ+iπ...=eiπ1−11
. In complex analysis, there is a well defined infinity which is the reciprocal of 0, therefore eiπ+iπ+iπ+iπ+iπ+iπ+iπ...=eiπ1−11=e∞=∞. Though ∞ is not 0, it is the single other number on the Riemann sphere which is its own negative, and is therefore also a solution to the equation x=−x !
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.
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 1-1+1−1+1−1+1−1... 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 1/2. A different way of looking at this sum is through the formula 1+1x+1x2+1x3+1x4+...=11−x . If x = −1, then the series 1−1+1−1+... becomes 11−x=12 . This suggests that, to the extent that this sum has a value, it is 12. 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:1,0,1,0,1,0,... and 12 now represents the concept of ‘tralse’ . In the case of your sequence, it would be the set of partial products of an infinite product(−1)(−1)(−1)(−1)(−1)(−1)(−1)... . 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 e to the power of iπ , which suggests that the infinite product is given by eiπ(1+1+1+1+1+1+...)=eiπ+iπ+iπ+iπ+iπ+iπ+iπ...=eiπ1−11
. In complex analysis, there is a well defined infinity which is the reciprocal of 0, therefore eiπ+iπ+iπ+iπ+iπ+iπ+iπ...=eiπ1−11=e∞=∞. Though ∞ is not 0, it is the single other number on the Riemann sphere which is its own negative, and is therefore also a solution to the equation x=−x !
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.