  It is often stated that the second law of thermodynamics follows from the
condition that at some given time in the past the entropy was lower than it is
now. Formally, this condition is the statement that $E[S(t)|S(t_0)]$, the
expected entropy of the universe at the current time $t$ conditioned on its
value $S(t_0)$ at a time $t_0$ in the past, is an increasing function of $t $.
We point out that in general this is incorrect. The epistemic axioms underlying
probability theory say that we should condition expectations on all that we
know, and on nothing that we do not know. Arguably, we know the value of the
universe's entropy at the present time $t$ at least as well as its value at a
time in the past, $t_0$. However, as we show here, conditioning expected
entropy on its value at two times rather than one radically changes its
dynamics, resulting in a unexpected, very rich structure. For example, the
expectation value conditioned on two times can have a maximum at an
intermediate time between $t_0$ and $t$, i.e., in our past. Moreover, it can
have a negative rather than positive time derivative at the present. In such
"Boltzmann bridge" situations, the second law would not hold at the present
time. We illustrate and investigate these phenomena for a random walk model and
an idealized gas model, and briefly discuss the role of Boltzmann bridges in
our universe.
