Here I will discuss a principle very similar to Lauwers and Vallentyne's Weak Catching-Up, which I will call Agential Betterness. I want to be able to show that if Agential Betterness is true, then there is no further principle P such that (i) P delivers a betterness verdict in any double-infinity pairs (as defined below), and (ii) P is consistent with Agential Betterness. In other words, any principle for ordering worlds must either be silent on all double infinity pairs, or it must be inconsistent with Agential Betterness.
Although Lauwers and Vallentyne formulate Weak Catching-Up in terms of locations, I will only be concerned with cases in which the relevant locations are agents (we can return to generalizations of these results to locations other than agents in a later post):
Agential Betterness (AB): If (1) U and V contain the same agents, and (2) for all possible enumerations of agents, the lower limit, as T approaches infinity, of the sum of the values of U-V of agents 1 to T is greater than zero $($i.e., $lim_{T\rightarrow\infty} inf \sum_{t=1,..,T}(U_t - V_t) > 0)$, then U is strictly better than V. And if $lim_{T\rightarrow\infty} inf \sum_{t=1,..,T}(U_t - V_t) \geq 0$ then U $\succcurlyeq$ V).
Agential Betterness is restricted to cases in which all of the agents are the same between the worlds we are comparing. I will argue that this makes denials of this principle highly implausible, even if we would wish to reject a similar principle that applied to distinct populations.
Pairs (or sets) of worlds can have fully distinct populations, fully overlapping populations, or populations that partly overlap. I will call pairs of the first kind disjoint, pairs of the second kind fully intersecting (rather than 'identical', since these worlds may have distinct structures, even if they contain identical agents), and pairs of the third kind partially intersecting.
If U and V contain the same agents, Agential Betterness will fail to deliver a verdict iff U and V form a double-infinity pair. Lauwers and Vallentyne define double infinity pairs for fully intersecting worlds. But we can try to generalize the concept of double-infinity pairs to worlds that don't fully intersect. Lauwers and Vallentyne define a world U-V as follows:
"For two worlds, U and V, with the same locations, U+V (respectively, U-V) is a world with the same locations, with the value at each location equal to value at U at that location plus (respectively, minus) the value at V at that location" (p. 15)
Building on this, we can define U+V and U-V for all worlds, and not merely fully intersecting worlds, as follows: For two worlds, U and V, U+V (respectively, U-V) is the set of all possible worlds with the value at each shared location equal to value at U at that location plus (respectively, minus) the value at V at that shared location, and the value at each non-shared location equal to the value at U at that location plus (respectively, minus) the value at V at some non-shared location.
Given this, we can define double-infinity pairs for any pairs of worlds (and not just those that fully intersect) as follows:
Double-Infinity Pairs (def): For any worlds U and V, if (1) there is some world in {U-V} such that the sum of the non-negative values of that world is infinite, and (2) there is some world in {U-V} such that the sum of the negative values of that world is infinitely negative, then U and V are a double-infinity pair.
Now let us assume the following principle, which I take to be plausible:
Agent Permutation (AP): For any property p or relation r that we use to enumerate the agents of an infinite world U, and for any arbitrary possible enumeration e of U, there exists a world U' such that (i) U and U' are fully intersecting, and (ii) p/rwill enumerate U' by e
Agent Permutation essentially says that there is no property or relation that we can use to enumerate or order agents in an infinite world that will always pick out the same enumeration (or indeed block any possible enumeration) in a world containing exactly the same agents. Agent permutation holds if worlds can retain the same agents in a world while adjusting whether they have property P (or to what degree they have property P, if P is gradable) or bear relation R to the relevant relatum or relata.
There are some properties or relations that will violate agent permutation. For example, if we can appeal to haecceitistic properties then we could simply enumerate the agents of a world by associating each agent of the infinite world with an integer. But if the betterness relation must appeal to a qualitative property of agents like their spatio-temporal location, then it seems plausible that we will be able to find a world that alters the relevant qualititative properties of agents (note that if adjusting the property changes the identity of agents, then Agential Betterness will no longer apply). I think it will be useful to discuss this principle, and some proposed properties and relations, in more detail in a future post.
If enumerating U by enumeration e results in U', then we will say that U' is a permutation of U (such that U is also a permutation of itself). Notice that since U' does not differ from U in any respect other than the enumeration of its agents, agent identity is preserved between U and U'. In other words, we suppose that merely changing the order in which evaluate agents does not alter the identity of those agents. (Notice also that we are only considering countably infinite worlds at the moment.)
Finally, I will assume that the betterness ordering is transitive, defined as follows:
Transitivity (T): For any worlds U, V, W: if U $\succcurlyeq$ V and V $\succcurlyeq$ W, then U $\succcurlyeq$ W
The claims I want to consider in the next post are as follows:
Claim 1: If Agent Permutation and Transitivity are true, then there is no principle P such that (i) P delivers a betterness verdict in fully intersecting double-infinity pairs, and (ii) P is consistent with Agential Betterness.
Claim 2: If Agent Permutation and Transitivity are true, then there is no principle P such that (i) P delivers a betterness verdict in any double-infinity pairs, and (ii) P is consistent with Agential Betterness.
Claim 3: If Agent Permutation and Transitivity are true, then there is no principle P such that (i) P delivers a betterness verdict in disjoint or partly intersecting infinite worlds (including worlds that are not double infinities), and (ii) P is consistent with Agential Betterness.
We might need further assumptions, or to adjust the ones given above, but I want to think about each of these claims in turn and make adjustments along the way.
Showing that claim 1 is true would be enough to show that if AB is necessary then it's also sufficient, because AB delivers a verdict about the betterness of world pairs in all and only those world pairs that are fully intersecting non-double infinity pairs (i.e. it leaves no 'gaps' for additional principles in fully intersecting non-double infinity pairs, and so if no other principle delivers verdicts in fully intersecting double infinity pairs, then AB is sufficient).
I will return to these three claims in a future post.
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