I have just now submitted an attempted solution to this problem to “Geometry and Topology”. I claim that the space you are looking for is ( being the least uncountable cardinal) with the ``generalised Cantor space topology”, that is for each countable well-ordered bit-string you have a basic open set consisting of all bit-strings of length with as an initial fragment. Since this topological space has quite a large cardinality I’m somewhat unclear whether this is helpful for your proposed application and would need to think about it more. (Matthew Barnett just now directed me to this post of yours.) I sent you an early draft of my paper, which argues the point in detail, on FB Messenger, and can send the latest version to you if you wish.
Rupert
I made the following observation to Chris on Facebook which he encouraged me to post here.
My point was basically just that, in reply to the statement “If we don’t have such a model to reject, the statement will be tautological”, it is in fact true relative to the standard semantics for first-order languages with equality that there is indeed no model-combined-with-an-interpretation-of-the-free-variables for which “x=x” comes out false. That is to say, relative to the standard semantics the formula is indeed a “logical truth” in that sense, although we usually only say “tautology” for formulas that are tautologies in propositional logic (that is, true under every Boolean valuation, a truth-valuation of all subformulas starting with a quantifier and all subformulas which are atomic formulas which then gets extended to a truth-valuation of all subformulas using the standard rules for the propositional connectives). So most certainly “x=x” is universally valid, relative to the standard semantics, and in the sense just described, there is no counter-model.
I take it that Chris’ project here is in some way to articulate in what sense the Law of Identity could be taken as a statement that “has content” to it. It sounds as though the best approach to this might be to try to take a look at how you would explain the semantics of statements that involve the equality relation. It looks as though it should be in some way possible to defend the idea that the Law of Identity is in some way “true in virtue of its meaning”.
My main thought would be that you consider the risk factors of physical intimacy but possibly ought to look a bit more closely at whether there are risk factors associated with avoiding physical intimacy (are you sure that that’s not harmful as well if taken to sufficient lengths?)
My main thought would be that you consider the risk factors of physical intimacy but possibly ought to look a bit more closely at whether there are risk factors associated with avoiding physical intimacy (are you sure that that’s not harmful as well if taken to sufficient lengths?)
Point C is a particular combination of utilities. The particular combination of utilities is not attainable via re-distribution while the economy is in state a. If a change took place so that the economy was now in state c, then point C would be attainable by re-distribution.
(And there is a point common to both the curves a and c, but just from knowing that the utilities of Citizens 1 and 2 were at that particular point wouldn’t allow you to know whether the economy is in state a or c, that would be extra information, and this extra information would be necessary in order to know which other points you could get to via re-distribution from your current situation.)
A Kaldor-Hicks improvement is a change of state of the economy from A to B such that B can be converted into a Pareto improvement by re-distribution, and such that A cannot be converted into a Pareto improvement of B by re-distribution.
Every labelled point on that diagram permits re-distribution (because it lies on a curve).
Each curve corresponds to a “state of the economy”. To get to a state where you could start re-distributing by moving along a different curve to the one you were originally moving along would require a change of state. When re-distributing, you can only move along one curve corresponding to the current state of the economy.
Oh yes, and when the economy is in a given state you are only allowed to move along one curve. The curves are allowed to intersect, but you can’t change which curve you are moving along when doing re-distribution. The explanation I just added of why the curves are allowed to intersect may help.
All of the curves represent states of the economy such that a re-distribution of resources will correspond to a movement along that curve. A change in state of the economy can be explained by a change in technological knowledge, a change in climate, the discovery of a deposit of a particular resource, stuff like that. “Re-allocation of resources” can correspond to re-allocation of quite a complex bundle of goods. The projections of the points on the co-ordinate axes merely represent the utilities of each of the two citizens, where only the order relations between utilities matters.
Let me be sure I understand what you’re saying? If someone wants to argue on the internet that abortion should be prohibited by the criminal law, or that there isn’t any moral obligation to be vegan, then I shouldn’t moralize about the fact that I disagree with them? I mean, I can think of ways that you could maybe argue that point, I just want to make sure I understand you though.
Thanks for the clarification. Possibly that reduces the interest of the observations about computational complexity.
However, because topology on is finer than topology on here, this still shows how the proof of the Lawvere fixed point theorem can be applied here to give Brouwer fixed point theorem as corollary, which could conceivably be a publishable result (see what “Geometry and Topology” think about that), and this could still be sorta kinda maybe relevant to Scott’s original motivation for looking at the problem (if you’re okay with working with two different topologies on the space of agents, one finer than the other). But this is a very big space of agents you’re talking about here.
Correction: need not only that topology on is finer than topology on , but also, given arbitrary open subset of , take pre-image under evaluation map in , projection onto first factor and then pre-image of that under the continuous surjection , it needs to be shown that this set is open in both topologies. I believe that this can indeed be done for an appropriate class of spaces for the pair of topologies in question.
Let be with generalised Cantor space topology, and be with product topology, a closed disc in a finite-dimensional Euclidean space. Then there is a continuous surjection . I don’t know how to show that there is a topological space with carrier set and a continuous surjection . Thanks to Alex Mennen for pointing out the problem.
When I look at my post the LaTeX code isn’t formatting properly; if anyone can let me know how to fix that.
My main thought would be that you consider the risk factors of physical intimacy but possibly ought to look a bit more closely at whether there are risk factors associated with avoiding physical intimacy (are you sure that that’s not harmful as well if taken to sufficient lengths?)