So, I like GRTd, insofar as it captures both what is so plausible about physicalism, and insofar as the ‘totality fact’ expresses an important kind of empirical inference: from even a small subset of all the physico-logical facts, we can get a good general picture of how the universe works, and what kinds of things are real.
I still have questions about the GRTd as a principle however. I don’t see how the following three statements are consistant with one another:
S1) GRTd: ‘Every true statement can be deductively derived from the set of purely physical and logical truths combined with statements of the semantics of the non-physical and non-logical terms.’
S2) The totality fact is true.
S3) ‘The totality fact can’t be strictly deduced from any other fact.’
One of these three has to go, and I strongly suspect I’ve misunderstood S3. So my question is this: Given all the physical and logical facts, combined with statements of the semantics of any non-physical and non-logical terms one might care to make use of, do you think we could deduce the totality fact?
The totality fact is one of the physical/logical facts, and can be expressed in purely physical/logical terms. For instance, in a toy universe where the only properties were P (‘being a particle’) and C (‘being a spacetime point’), the totality fact would have the form ∀x(P(x) ∨ C(x)) to exclude other categories of entity. A more complete totality fact would exclude bonus particles and spacetime points too, by asserting ∀x(x=a ∨ x=b ∨ x=c...), where {a,b,c...} is the (perhaps transfinitely large) set of particles and points. You can also express the same idea using existential quantification.
S1, S2, and S3 are all correct, provided that the totality fact is purely physical and logical. (Obviously, any physical/logical fact follows trivially from the set of all physical/logical facts.) GRTd says nothing about which, if any, physical/logical facts are derivable from a proper subset of the physical/logical. (It also says nothing about whether there are non-physicological truths; it only denies that, if there are some, their truth or falsehood can fail to rest entirely on the physical/logical facts.)
A single giant totality fact would do the job, but you could also replace it (or introduce redundancy) by positing a large number of smaller totality facts. Suppose you want to define a simple classical universe in which a 2x2x2-inch cube exists. You can quantify over a specific 2x2x2-inch region of space, and assert that each of the points within the interval is occupied. But that only posits an object that’s at least that large; we also need to define the empty space around it, to give it a definite border. A totality fact (or a small army of them) could give you the requisite border, establishing ‘there’s no more cube’ in the same way that the Giant Totality Fact establishes ‘there’s no more reality.’ But if you get a kick out of parsimony or concision, you don’t need to do this again and again for each new bounded object you posit. Instead, you can stick to positive assertions until the very end, and then clean up after yourself with the Giant Totality Fact. That there’s no more reality than what you’ve described, after all, implies (among other things) that there’s no more cube.
(Obviously, any physical/logical fact follows trivially from the set of all physical/logical facts.)
Ah, I took GRTd to mean that ‘every true statement (including all physical and logical truths) can be deductively derived from the set of purely physical and logical truths (excluding the one to be derived)...’.Thus, if the totality fact is true, then it should be derivable from the set of all physico-logical facts (excluding the totality fact). Is that right, or have I misunderstood GRTd?
I may, I think, just be overestimating what it takes to plausibly posit the totality fact: i.e. you may just mean that we can have a lot of confidence in the totality fact just by having as broad and coherent a view of the universe as we actually do right now. The totality fact may be false, but its supported in general by the predictive power of our theories and an apparent lack of spooky phenomena. If we had all the physico-logical facts, we could be super duper confident in the totality fact, as confident as we are about anything. It would by no means follow deductively from the set of all physico-logical facts, but it’s not that sort of claim anyway. Is that right?
The edit is fine. Let me add that ‘the’ totality fact may be a misleading locution. Nearly every model that can be analyzed factwise contains its own totality fact, and which model we’re in will change what the ‘totality’ is, hence what the shape of the totality fact is.
We can be confident that there is at least one fact of this sort in reality, simply because trivialism is false. But GRTd does constrain what that fact will have to look like: It will have to be purely logical and physical, and/or derivable from the purely logical and physical truths. (And the only thing we could derive a Big Totality Fact from would be other, smaller totality facts like ‘there’s no more square,’ plus a second-order totality fact.)
I didn’t intend for you to read ‘(excluding the one to be derived)’ into the statement. The GRTd I had in mind is a lot more modest, and allows for totality facts and a richer variety of causal relations.
GRTd isn’t a tautology (unless GRTm is true), because if there are logically underivable nonphysical and nonlogical truths, then GRTd is false. ‘X can be derived from the conjunction of GRTd with X’ is a tautology, but an innocuous one, since it leaves open the possibility that ‘X’ on its lonesome is a garden-variety contingent fact.
So, I like GRTd, insofar as it captures both what is so plausible about physicalism, and insofar as the ‘totality fact’ expresses an important kind of empirical inference: from even a small subset of all the physico-logical facts, we can get a good general picture of how the universe works, and what kinds of things are real.
I still have questions about the GRTd as a principle however. I don’t see how the following three statements are consistant with one another:
S1) GRTd: ‘Every true statement can be deductively derived from the set of purely physical and logical truths combined with statements of the semantics of the non-physical and non-logical terms.’
S2) The totality fact is true.
S3) ‘The totality fact can’t be strictly deduced from any other fact.’
One of these three has to go, and I strongly suspect I’ve misunderstood S3. So my question is this: Given all the physical and logical facts, combined with statements of the semantics of any non-physical and non-logical terms one might care to make use of, do you think we could deduce the totality fact?
The totality fact is one of the physical/logical facts, and can be expressed in purely physical/logical terms. For instance, in a toy universe where the only properties were P (‘being a particle’) and C (‘being a spacetime point’), the totality fact would have the form ∀x(P(x) ∨ C(x)) to exclude other categories of entity. A more complete totality fact would exclude bonus particles and spacetime points too, by asserting ∀x(x=a ∨ x=b ∨ x=c...), where {a,b,c...} is the (perhaps transfinitely large) set of particles and points. You can also express the same idea using existential quantification.
S1, S2, and S3 are all correct, provided that the totality fact is purely physical and logical. (Obviously, any physical/logical fact follows trivially from the set of all physical/logical facts.) GRTd says nothing about which, if any, physical/logical facts are derivable from a proper subset of the physical/logical. (It also says nothing about whether there are non-physicological truths; it only denies that, if there are some, their truth or falsehood can fail to rest entirely on the physical/logical facts.)
A single giant totality fact would do the job, but you could also replace it (or introduce redundancy) by positing a large number of smaller totality facts. Suppose you want to define a simple classical universe in which a 2x2x2-inch cube exists. You can quantify over a specific 2x2x2-inch region of space, and assert that each of the points within the interval is occupied. But that only posits an object that’s at least that large; we also need to define the empty space around it, to give it a definite border. A totality fact (or a small army of them) could give you the requisite border, establishing ‘there’s no more cube’ in the same way that the Giant Totality Fact establishes ‘there’s no more reality.’ But if you get a kick out of parsimony or concision, you don’t need to do this again and again for each new bounded object you posit. Instead, you can stick to positive assertions until the very end, and then clean up after yourself with the Giant Totality Fact. That there’s no more reality than what you’ve described, after all, implies (among other things) that there’s no more cube.
Ah, I took GRTd to mean that ‘every true statement (including all physical and logical truths) can be deductively derived from the set of purely physical and logical truths (excluding the one to be derived)...’.Thus, if the totality fact is true, then it should be derivable from the set of all physico-logical facts (excluding the totality fact). Is that right, or have I misunderstood GRTd?
I may, I think, just be overestimating what it takes to plausibly posit the totality fact: i.e. you may just mean that we can have a lot of confidence in the totality fact just by having as broad and coherent a view of the universe as we actually do right now. The totality fact may be false, but its supported in general by the predictive power of our theories and an apparent lack of spooky phenomena. If we had all the physico-logical facts, we could be super duper confident in the totality fact, as confident as we are about anything. It would by no means follow deductively from the set of all physico-logical facts, but it’s not that sort of claim anyway. Is that right?
The edit is fine. Let me add that ‘the’ totality fact may be a misleading locution. Nearly every model that can be analyzed factwise contains its own totality fact, and which model we’re in will change what the ‘totality’ is, hence what the shape of the totality fact is.
We can be confident that there is at least one fact of this sort in reality, simply because trivialism is false. But GRTd does constrain what that fact will have to look like: It will have to be purely logical and physical, and/or derivable from the purely logical and physical truths. (And the only thing we could derive a Big Totality Fact from would be other, smaller totality facts like ‘there’s no more square,’ plus a second-order totality fact.)
Excellent, I think I understand. GRTd sounds good to me, and I think you should convince EY to adopt it as opposed to GRTt/m.
I didn’t intend for you to read ‘(excluding the one to be derived)’ into the statement. The GRTd I had in mind is a lot more modest, and allows for totality facts and a richer variety of causal relations.
GRTd isn’t a tautology (unless GRTm is true), because if there are logically underivable nonphysical and nonlogical truths, then GRTd is false. ‘X can be derived from the conjunction of GRTd with X’ is a tautology, but an innocuous one, since it leaves open the possibility that ‘X’ on its lonesome is a garden-variety contingent fact.
Sorry, I didn’t expect you to read my post so quickly, and I edited it heavily without marking my edits (a failure of etiquette, I admit).