Nestes Modes, ’Qua’ and the Incarnation

A nested mode ontology allows one to make sense of apparently contradictory Christological claims such as that Christ knows everything and there are some things Christ does not know

On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Let $\Omega$ be a countable infinite product $\Omega^\N$ of copies of the same probability space $\Omega_1$, and let ${\Xi_n}$ be the sequence of the coordinate projection functions from $\Omega$ to $\Omega_1$. Let $\Psi$ be a possibly nonmeasurable function from $\Omega_1$ to $\R$, and let $X_n(\omega) = \Psi(\Xi_n(\omega))$. Then we can think of ${X_n}$ as a sequence of independent but possibly nonmeasurable random variables on $\Omega$. Let $S_n = X_1+...+X_n$. By the ordinary Strong Law of Large Numbers, we almost surely have $E_*[X_1] \le \liminf S_n/n \le \limsup S_n/n \le E^*[X_1]$, where $E_*$ and $E^*$ are the lower and upper expectations. We ask if anything more precise can be said about the limit points of $S_n/n$ in the non-trivial case where $E_*[X_1] < E^*[X_1]$, and obtain several negative answers. For instance, the set of points of $\Omega$ where $S_n/n$ converges is maximally nonmeasurable: it has inner measure zero and outer measure one

Being Sure and Being Confident That You Won’t Lose Confidence

There is an important sense in which one can be sure without being certain, i.e., without assigning unit probability. I will offer an explication of this sense of sureness, connecting it with the level of credence that a rational agent would need to have to be confident that she won’t ever lose her confidence. A simple formal result then gives us an explicit formula connecting the threshold α for credence needed for confidence with the threshold needed for being sure: one needs 1−(1−α) to be sure. I then suggest that stepping between α and 1−(1−α) gives a procedure that generates an interesting hierarchy of credential thresholds

Magnetic fields, bremsstrahlung and synchrotron emission in the flare of 24 October 1969

Magnetic fields, bremsstrahlung, and synchrotron emission in solar flare of October 24, 196