The above formalisation of Aquinas is inaccurate. The possibility of X does not necessitate its instantiation, ever, at least not without a seperate proof to that effect. Aquinas’ argument: “[I]f everything is possible not to be, then at one time there was nothing in reality.” This may be formalised as follows: ‘for all x, either x or not-x, which implies that for all x there is a t, such that not-x and t’. ∀x(x ∨ ¬x) → ∀x∃t(¬x.t)
This can be formally disproven by showing that the negation of the conclusion does not imply contradiction with the premise(s), or else, by showing the conclusion as it is implies contradiction with the premises (which would take a few steps). This is nevertheless evident by inspection, on the grounds of relevance, since the premise(s) does not include any reference to t.
Yes, agreed. It's a while since I wrote this piece, and so I don't remember all the reading I did for it, but my memory is that the formalization I gave is roughly what scholars think Aquinas is getting at. I think the reasoning is something like charitable interpretation. If he really means possible in the modal sense, rather than in a temporal sense, the argument is even more clearly invalid. I guess this interpretation suggests a way of reading him on which the invalidity is a little less clear!! I think the paper I mention by John Anders gives the reading I favour here.
Then Anders could be accused of a strawman argument, for introducing a spurious inference as a premise. Aquinas didn’t write ‘everything is not at some time’, but “everything is possible not to be”. If language means anything, then ‘possible’ is distinct from ‘is’. In fact, it must be inferred (charitably) that Acquinas did not mean the interpretation proposed by Anders, because he would be thereby implying that there is a time when God is not, and this would amount to renunciation of his religious dogma.
An example on black ravens suggests we can do a lot without logical notation. Suppose there are just two things in the world, a raven and swan, and two possible colours black and white, assumed equally likely for simplicity. Observing that the raven is black confirms, with certainty, "all ravens are black" and also, "if anything is white, it's a swan". Now, the maintained hypothesis "the colours of the birds are independent", is equivalent to "observing that the swan is white (or black) tells us nothing about the raven", which is the answer we intuitively want.
Having started with the (1,1) case, the extension to any (n, m) is trivial. So the problem seems to be with the quantifier "all".
Without independence things get tricky. But it seems more plausible to say that observing a black swan supports the hypothesis "all birds are black". To get the opposite you need some belief like "there are both black and white birds"
Yeah, I think something like this was Hosiasson-Lindenbaum's point. The fact we don't think non-black non-ravens confirm all ravens are black really just reflects something about how many items we think there are in each cell of the fourfold partition. When we set those numbers in a plausible way, we see that the confirmation does hold, but it's tiny.
I came up with an example that supports the confirmation idea. Suppose that the objects in the box are two cards drawn from the same deck. Too distinguish them, we can mark one on the back with a star, the other with a circle. Initially, both are equally likely to be black or red. But if we turn over the circle card and it’s red, there are only 51 possibilities left for the star card - 26 black and 25 red. So, we do indeed increase our belief that the star card is black.
But I can't see how one example is more plausible than the other. And it's easy to construct examples where a black non-raven supports a general hypothesis that all-black object classes are common. You need to know something about the structure of the world to say anything about this kind of inference.
It seems as if what Hempel wants here is a method of strictly model-free inference from data. That's a fools errand, as economists have repeatedly discovered.
The above formalisation of Aquinas is inaccurate. The possibility of X does not necessitate its instantiation, ever, at least not without a seperate proof to that effect. Aquinas’ argument: “[I]f everything is possible not to be, then at one time there was nothing in reality.” This may be formalised as follows: ‘for all x, either x or not-x, which implies that for all x there is a t, such that not-x and t’. ∀x(x ∨ ¬x) → ∀x∃t(¬x.t)
This can be formally disproven by showing that the negation of the conclusion does not imply contradiction with the premise(s), or else, by showing the conclusion as it is implies contradiction with the premises (which would take a few steps). This is nevertheless evident by inspection, on the grounds of relevance, since the premise(s) does not include any reference to t.
Yes, agreed. It's a while since I wrote this piece, and so I don't remember all the reading I did for it, but my memory is that the formalization I gave is roughly what scholars think Aquinas is getting at. I think the reasoning is something like charitable interpretation. If he really means possible in the modal sense, rather than in a temporal sense, the argument is even more clearly invalid. I guess this interpretation suggests a way of reading him on which the invalidity is a little less clear!! I think the paper I mention by John Anders gives the reading I favour here.
Then Anders could be accused of a strawman argument, for introducing a spurious inference as a premise. Aquinas didn’t write ‘everything is not at some time’, but “everything is possible not to be”. If language means anything, then ‘possible’ is distinct from ‘is’. In fact, it must be inferred (charitably) that Acquinas did not mean the interpretation proposed by Anders, because he would be thereby implying that there is a time when God is not, and this would amount to renunciation of his religious dogma.
An example on black ravens suggests we can do a lot without logical notation. Suppose there are just two things in the world, a raven and swan, and two possible colours black and white, assumed equally likely for simplicity. Observing that the raven is black confirms, with certainty, "all ravens are black" and also, "if anything is white, it's a swan". Now, the maintained hypothesis "the colours of the birds are independent", is equivalent to "observing that the swan is white (or black) tells us nothing about the raven", which is the answer we intuitively want.
Having started with the (1,1) case, the extension to any (n, m) is trivial. So the problem seems to be with the quantifier "all".
Without independence things get tricky. But it seems more plausible to say that observing a black swan supports the hypothesis "all birds are black". To get the opposite you need some belief like "there are both black and white birds"
Yeah, I think something like this was Hosiasson-Lindenbaum's point. The fact we don't think non-black non-ravens confirm all ravens are black really just reflects something about how many items we think there are in each cell of the fourfold partition. When we set those numbers in a plausible way, we see that the confirmation does hold, but it's tiny.
I came up with an example that supports the confirmation idea. Suppose that the objects in the box are two cards drawn from the same deck. Too distinguish them, we can mark one on the back with a star, the other with a circle. Initially, both are equally likely to be black or red. But if we turn over the circle card and it’s red, there are only 51 possibilities left for the star card - 26 black and 25 red. So, we do indeed increase our belief that the star card is black.
But I can't see how one example is more plausible than the other. And it's easy to construct examples where a black non-raven supports a general hypothesis that all-black object classes are common. You need to know something about the structure of the world to say anything about this kind of inference.
It seems as if what Hempel wants here is a method of strictly model-free inference from data. That's a fools errand, as economists have repeatedly discovered.