If you are guessing a binary sequence the job of a hypothesis is top say whether the next bit is going to be 1 or 0. A combination where both are fine or equally predicted fails to be a hypothesis. If you have partial predictions of X1XX0X and XX11XX you can “or” them into X1110X. But if you try to combine 01000 and 00010 the result will not be 01010 but something like 0X0X0.
Mathematically one option to handle partial predictions is to model them as groups of concrete “non-open” predictions which are compatible with them. 0X0X0 is short hand for {01010, 00010,01000,00000}. But 0X0X0 is not a hypothesis but a hypothesis schema or some other way of referring to the full set.
Statement that has room of interpretation what symbol it predicts will come next can’t serve as a hypothesis. If we have a rorcharcs ink blot and a procedure to unambigiusly turn them into symbols that might not leave any wiggle room. But if the procedure is unspecified it fails to mean or predict anything for alignment with data checking purposes. You can’t be correct or wrong if you don’t call anything.
Even if you have “inkplot1” and “inkoplot2“ as predictive hypotheses it is uncertain that “inkplo1 or inkplot2” has any procedure to turn into a prediction and even if it could have it might not be deducable for the rules for inkplots without connecting “ors”. if “inkplot1” would predict 0 and “inkplot2” would predict 1 a generic combination will yield a “anything might happen” result which fails to be a prediction at all (ie “I have no idea what the next symbol is going to be”). But having a or rule that combined 010000 and 000010 into 0100010 has to be encoding specific.
A combination where both are fine or equally predicted fails to be a hypothesis.
Why? If I have two independent actions—flipping a coin and rolling a 6-sided die (d6) - am I not able to combine “the coin lands heads 50% of the time” and “the die lands even (i.e. 2, 4, or 6) 50% of the time”?
If you have partial predictions of X1XX0X and XX11XX you can “or” them into X1110X.
This is (very close to) a binary “or”, I roughly agree with you.
But if you try to combine 01000 and 00010 the result will not be 01010 but something like 0X0X0.
This is sort of like a binary “and”. Have the rules changed? And what are they now?
In order for the predcitions to be combatible they must be silent about each other.
If the base cases are H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6
then it makes sense to say that HX (H1,H2,H3,H4,H5,H6) is 50% of XX
and that {X2,X4,X6} is 50% of XX
However if the base cases are H,T,1,2,3,4,5,6 then H would be 50% of {H,T} but not of X and {2,4,6} would be 50% of {1,2,3,4,5,6} but not X
The case where “everything works out” would like the OR to output a prediction scope of {H1,H2,H3,H4,H5,H6,T2,T4,T6}. But someone mean could argue that the OR outputs a prediction scope of {H,2,4,6}
If the claims are about separate universes then they are not predictions about the same thing. A heads claim doesn’t concern the dice world so it is not a prediction on it. Predictiontions should be alternative descriptions what happens in the world of concern. So the predictions should have the same amount of holes in them and at the root level all details shoudl be filled out ie 0 holes. An OR operation would need to produce an object that has more holes than the inputs if the inputs speak about the same universe. That is H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6 are the base level hypotheses and {H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6} and {} are prediction scopes neither which is found among the hypotheses. Applying a prediction OR would push towards the former object. But hypotheses are required to be firm in the details while scopes can be slippery. That is predction scopes {H1} and {H2} can be ored to {H1,H2} but the hypotheses H1 and H2 can’t be ored to produce a hypothesis.
If you are guessing a binary sequence the job of a hypothesis is top say whether the next bit is going to be 1 or 0. A combination where both are fine or equally predicted fails to be a hypothesis. If you have partial predictions of X1XX0X and XX11XX you can “or” them into X1110X. But if you try to combine 01000 and 00010 the result will not be 01010 but something like 0X0X0.
Mathematically one option to handle partial predictions is to model them as groups of concrete “non-open” predictions which are compatible with them. 0X0X0 is short hand for {01010, 00010,01000,00000}. But 0X0X0 is not a hypothesis but a hypothesis schema or some other way of referring to the full set.
Statement that has room of interpretation what symbol it predicts will come next can’t serve as a hypothesis. If we have a rorcharcs ink blot and a procedure to unambigiusly turn them into symbols that might not leave any wiggle room. But if the procedure is unspecified it fails to mean or predict anything for alignment with data checking purposes. You can’t be correct or wrong if you don’t call anything.
Even if you have “inkplot1” and “inkoplot2“ as predictive hypotheses it is uncertain that “inkplo1 or inkplot2” has any procedure to turn into a prediction and even if it could have it might not be deducable for the rules for inkplots without connecting “ors”. if “inkplot1” would predict 0 and “inkplot2” would predict 1 a generic combination will yield a “anything might happen” result which fails to be a prediction at all (ie “I have no idea what the next symbol is going to be”). But having a or rule that combined 010000 and 000010 into 0100010 has to be encoding specific.
Why? If I have two independent actions—flipping a coin and rolling a 6-sided die (d6) - am I not able to combine “the coin lands heads 50% of the time” and “the die lands even (i.e. 2, 4, or 6) 50% of the time”?
This is (very close to) a binary “or”, I roughly agree with you.
This is sort of like a binary “and”. Have the rules changed? And what are they now?
In order for the predcitions to be combatible they must be silent about each other.
If the base cases are H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6
then it makes sense to say that HX (H1,H2,H3,H4,H5,H6) is 50% of XX
and that {X2,X4,X6} is 50% of XX
However if the base cases are H,T,1,2,3,4,5,6 then H would be 50% of {H,T} but not of X and {2,4,6} would be 50% of {1,2,3,4,5,6} but not X
The case where “everything works out” would like the OR to output a prediction scope of {H1,H2,H3,H4,H5,H6,T2,T4,T6}. But someone mean could argue that the OR outputs a prediction scope of {H,2,4,6}
If the claims are about separate universes then they are not predictions about the same thing. A heads claim doesn’t concern the dice world so it is not a prediction on it. Predictiontions should be alternative descriptions what happens in the world of concern. So the predictions should have the same amount of holes in them and at the root level all details shoudl be filled out ie 0 holes. An OR operation would need to produce an object that has more holes than the inputs if the inputs speak about the same universe. That is H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6 are the base level hypotheses and {H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6} and {} are prediction scopes neither which is found among the hypotheses. Applying a prediction OR would push towards the former object. But hypotheses are required to be firm in the details while scopes can be slippery. That is predction scopes {H1} and {H2} can be ored to {H1,H2} but the hypotheses H1 and H2 can’t be ored to produce a hypothesis.