Theories that require additional premises are less likely to be true, according to the eternal laws of probability … Unfortunately, this answer seems to be based on a concept of “truth” granted from above—but what do differing degrees of truth actually mean, when two theories make exactly the same predictions?
Reading this and going back to my post to work out what I was thinking, I have a sort-of clarification for the issue in the quote. The original argument was that, before experiencing the universe, all premises are a priori equally likely, so we can generate as many hypotheses as we like out of them. Then, after experience (a posteriori) some premises are extremely likely (shorthand: true) and some are extremely unlikely (shorthand: false). Now, in our advanced position, there are a large number of false premises, a small number of true premises, and an unknown number of premises we haven’t had experiences about. We can now generate as many “prediction-equivalent” theories as we like by combining the unknown premises with the true ones. As long as we avoid the false premises, all our hypotheses will be based on true premises, and premises which we have not yet checked. To refine that argument, it is the conjunction of specifically these unknown premises that might weaken the hypothesis. Therefore, we ought to include as few of these as-yet-untested premises in our hypothesis, in order to reduce the chances of it being wrong.
Now, in the case of two theories making the same prediction: I suggest that it is possible to look at an unknown premise and decide whether we can check it. In this sense, if it is checkable, we can view it as a prediction: the hypothesis includes premise C such that if C is true then the hypothesis is true and if C is false the hypothesis is false. In other words, the hypothesis makes a prediction that C is true. If it is uncheckable, though, we don’t use the word prediction. This is the discussion that Matt_Simpson and cousin_it are having down below. If both theories make the same predictions because one says A B C and D, and the other says A B C E F G and H, (A and B are true, C is unknown-testable, D through H are unknown-untestable), then the theories are still distinguishable, still different, but they make the same predictions. In this specific case, we should in principle prefer the first theory, because it has one one fault-line and the second has four: even though we don’t think we can test any of these fault-lines, the laws of probability still apply. So this is what degrees of truth mean when all theories make the same predictions.
Now, what about two different theories that say A B C and D? I’m not familiar with the physics, but it appears that Hamiltonian and Lagrangian systems are in this scenario: both say A B C and D, in different but equivalent ways. I haven’t had enough time to think to bake up an answer for this, but I suspect it is similar to how you can express the same truth-table with different combinations of premises and operators. The question that stumps me is: in logic, we don’t care about the operators except for how they twist the premises. In physics, we actually do care about the operators, to an extent: we give them names like “the mechanism of x” and “the underlying reality”. So it seems to me that saying Lagrangian and Hamiltonian are different but equivalent is saying that two different logical formulations with the same truth-table are different but equivalent, except in physics we feel the difference actually matters.
Reading this and going back to my post to work out what I was thinking, I have a sort-of clarification for the issue in the quote. The original argument was that, before experiencing the universe, all premises are a priori equally likely, so we can generate as many hypotheses as we like out of them. Then, after experience (a posteriori) some premises are extremely likely (shorthand: true) and some are extremely unlikely (shorthand: false). Now, in our advanced position, there are a large number of false premises, a small number of true premises, and an unknown number of premises we haven’t had experiences about. We can now generate as many “prediction-equivalent” theories as we like by combining the unknown premises with the true ones. As long as we avoid the false premises, all our hypotheses will be based on true premises, and premises which we have not yet checked. To refine that argument, it is the conjunction of specifically these unknown premises that might weaken the hypothesis. Therefore, we ought to include as few of these as-yet-untested premises in our hypothesis, in order to reduce the chances of it being wrong.
Now, in the case of two theories making the same prediction: I suggest that it is possible to look at an unknown premise and decide whether we can check it. In this sense, if it is checkable, we can view it as a prediction: the hypothesis includes premise C such that if C is true then the hypothesis is true and if C is false the hypothesis is false. In other words, the hypothesis makes a prediction that C is true. If it is uncheckable, though, we don’t use the word prediction. This is the discussion that Matt_Simpson and cousin_it are having down below. If both theories make the same predictions because one says A B C and D, and the other says A B C E F G and H, (A and B are true, C is unknown-testable, D through H are unknown-untestable), then the theories are still distinguishable, still different, but they make the same predictions. In this specific case, we should in principle prefer the first theory, because it has one one fault-line and the second has four: even though we don’t think we can test any of these fault-lines, the laws of probability still apply. So this is what degrees of truth mean when all theories make the same predictions.
Now, what about two different theories that say A B C and D? I’m not familiar with the physics, but it appears that Hamiltonian and Lagrangian systems are in this scenario: both say A B C and D, in different but equivalent ways. I haven’t had enough time to think to bake up an answer for this, but I suspect it is similar to how you can express the same truth-table with different combinations of premises and operators. The question that stumps me is: in logic, we don’t care about the operators except for how they twist the premises. In physics, we actually do care about the operators, to an extent: we give them names like “the mechanism of x” and “the underlying reality”. So it seems to me that saying Lagrangian and Hamiltonian are different but equivalent is saying that two different logical formulations with the same truth-table are different but equivalent, except in physics we feel the difference actually matters.