Sensory information is centred. Right now, for example, my visual
system conveys to me that there's a red wall about 1 metre
ahead (among much else); it does not convey that Wolfgang
Schwarz is about 1 metre away from a red wall on 22 January 2026 at
12:04 UTC.
We can quibble over what exactly is part of the sensory information.
We can also quibble over what "sensory information" is even meant to be.
But it should be uncontroversial that we gain information from our
senses. My point is that, on any plausible way of spelling this out, the
information we receive is centred: it doesn't have parameters that fix a
unique location in space and time. If I were unsure about what time it
is or who I am, looking at the wall in front of me wouldn't help. The
underlying reason, of course, is that photoreceptors are insensitive to
differences in spatiotemporal location: they don't produce different
outputs depending on where or when they are activated by photons.
I (somewhat randomly) picked up Kripke 2011 the other day. This
is Kripke's first engagement with the problem of empty names. What
struck me is the biased selection of examples. Most of the paper is
concerned with names of fictional characters like 'Sherlock Holmes', and
Kripke only seems to consider simple utterances in which they figure as
the subject, like (1).
A somewhat appealing (albeit, to me, also somewhat obscure) view of
mathematics is the pluralist doctrine that every consistent mathematical
theory is true, insofar as it accurately describes some mathematical
structure. I want to comment on a potential worry for this view,
mentioned in (Clarke-Doane 2020): that
it has implausible consequences for logic.
A famous argument, first proposed in (Lucas 1961), supposedly shows that the
human mind has capabilities that go beyond those of any Turing machine.
In its basic form, the argument goes like this.
Let S be the set of mathematical sentences that I accept as true. S
includes the axioms of Peano Arithmetic. Let S+ be the set of sentences
entailed by S. Suppose for reductio that my mind is equivalent to a
Turing machine. Then S is computably enumerable, and S+ is a computably
axiomatizable extension of Peano Arithmetic. So Gödel's First
Incompleteness Theorem applies: there is a true sentence G that is
unprovable in S+. By going through Gödel's reasoning, I can see that G
is true. So G is in S and thereby in S+. Contradiction!
I'm teaching an intermediate/advanced logic course this semester. So
I had to ask myself how to introduce the semantics of quantifiers, with
an eye on proving soundness and completeness. The standard approach,
going back to Tarski, defines a satisfaction relation between a formula,
a model, and an assignment function, and then defines truth by
supervaluating over all assignments. The main alternative, often found
in intro logic textbooks, is Mates' approach, where ∀xA(x) is defined as
true in a model M iff A(c) is true in every c-variant of M, where c is a
constant not occurring in A.
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.
Bare indicative conditionals are bewildering, but they become
surprisingly well-behaved if we add an 'else' clause.
Intuitively, 'if A then B' doesn't make an outright claim about the
world. It says that B is the case if A is the case – but what
if A isn't the case?
An 'else' clause resolves this question. 'If A then B else C' makes
an outright claim. It says that either B or C is the case, depending on
whether A is the case. That is: the world is either an A-world, in which
case it is also a B-world, or it is a ¬A-world, in which case it is a
C-world. For short: (A∧B)∨(¬A∧C).
Frege argued that number concept are, in the first place, second-order predicates. When we talk about numbers as objects, we use a logical device of "nominalization" that introduces object-level representations of higher-level properties. In Grundgesetze, he assumed that every first-order predicate can be nominalized: for every first-order predicate F, there is an associated object – the "extension" of F – such that F and G are associated with the same object iff ∀x(Fx ↔︎ Gx). The number N is then identified with the extension of 'having an extension with N elements'. Unfortunately, the assumption that every predicate has an extension turned out to be inconsistent, so the whole approach collapsed.