In the dark old days of early logic, there was only syntax. People
introduced formal languages and laid down axioms and inference rules,
but there was nothing to justify these except a claim to
"self-evidence". Of course, the languages were assumed to be meaningful,
but there was no systematic theory of meaning, so the axioms and rules
could not be justified by the meaning of the logical constants.
All this changed with the development of model theory. Now one could
give a precise semantics for logical languages. The intuitive idea of
entailment as necessary truth-preservation could be formalized. One
could check that some proposed system of axioms and rules was sound, and
one could confirm – what had been impossible before – that it was
complete, so that any further, non-redundant axiom or rule would break
the system's soundness.
The standard dynamic norm of Bayesianism,
conditionalization, is clearly inadequate if credences are
defined over self-locating propositions. How should it be adjusted?
This question was popular at around 2005-2015. Chris Meacham and I
came up with the same answer, which we published in (Meacham 2010),
(Schwarz
2012), and (Schwarz 2015). I showed that the
replacement norm that we proposed has all the traditional virtues of
conditionalization. For example, (under the usual idealized conditions)
following the norm uniquely maximizes expected accuracy, and an agent is
invulnerable to diachronic Dutch books iff they follow the norm.
Assume that for any proposition A there is a proposition \(\Box A\) saying that A ought to be the
case. One can imagine an agent – call him Frederic – whose only basic
desire is that whatever ought to be the case is the case. As a result,
he desires any proposition A in proportion to his belief that it ought
to be the case:
\[\begin{equation*}
(1)\qquad V(A) = Cr(\Box A).
\end{equation*}
\]
Let w be a maximally specific proposition. Such a "world" settles all
descriptive and all normative matters. In particular, w entails either
\(\Box w\) or \(\neg \Box w\). Suppose w entails \(\Box w\). Does Frederick desire to live in
such a world? Yes. On the assumption that w is actual, the entire world
is as it ought to be. That's what Frederick wants. So he desires w.