Reduction and Elimination in Philosophy and the Sciences

5. Prior’s Problem
As Patrick Blackburn has pointed out, Prior ran into deep
problems because his tool for the reduction of temporal
instants, hybrid logic, worked rather too well and allowed to
reduce objects of other categories, as well – especially
troublesome in the case of persons (Prior 2003, 213ff.; cf.
Blackburn 2006, 362ff.). If we interpret the points of the
frame of our modal logic as persons and personally indefinite
propositions to express predicates, we can translate
statements of standard predicate logic to a “personal” analogue
of date logic: “Socrates is wise” becomes “@Socrates
be-wise”. Nominals used to name persons then dissolve
the points of personal space; we go over to “Somepersonally:
be-Socrates ∧ be-wise” and have lost Socrates
(cf. Prior 2003, 215-219).
Prior was aware of this problem. Yet he found no
solution but to simply proclaim that “persons just are
genuine individuals […] whereas […] instants are not
genuine individuals” (Prior 2003, 219f.; cf. Blackburn 2006,
364). Does metric tense logic, as a tool of metaphysical
reduction, run into similar difficulties? I will argue in
sections 6-8 that it does not.
6. Putting Time and Space into Perspective:
Standpoint Logic
Thomas Müller has shown how to generalize metric tense
logic to cover physical space and relativistic frames of
reference (Müller 2002, 268-279): Atomic propositions are
indefinite concerning time, space and frame, and there are
operators changing the temporal or spatial perspective or
which lead from one frame to another. All operators are
metrical, corresponding to temporal intervals, spatial vectors
and Lorentz transformations. Müller points out that
each set of operators constitutes a group in the algebraic
sense.
Now, the reduction of dates (section 3) generalizes
to the reduction of all context-free coordinates (Müller
2002, 276ff.). We only need to add to Müller’s standpoint
logic one proposition uniquely characterizing a place
(“Here is Greenwich”) and another proposition uniquely
characterizing a frame (“At-this-velocity is Earth”). The
metaphysical upshot is that places and frames are no
more genuine objects than instants; our spatiotemporal
world ultimately is perspectival.
7. General Perspectival Identification
But why is there no metric logic for other dimensions of
logical space, like persons and possible worlds? To answer
this question, we have to isolate those features of
metric logic and standpoint logic that allow the reduction of
instants, places and frames. When is a modal logic suited
for a similar reduction of the points of its frame?
It is not the numbers: The metrical operators of a
distance logic do not allow the reduction of spatial
coordinates. E.g., “Two meters from here it is the case
that” takes us to a sphere of two meters radius and thus is
hopelessly ambiguous. We need each operator to transfer
us to exactly one point. This condition of identification must
be general in the sense that it is satisfied at every point of
the model. But general identification is only a necessary
condition, because it is met by the satisfaction operator @t
of date logic, which of course does not allow a reduction of
dates. For genuine reduction, we need our operators to be
perspectival, i.e. the point the operator takes us to must
The Metaphysical Relevance of Metric and Hybrid Logic — Martin Pleitz
depend on the point where it is employed. I argue that
every general perspectival identificatory modal logic can
reduce the points of its frame.
To be honest to the metaphysical project of reducing
the objects these points purport to be, we have to move
from a model-theoretical to a syntactical (i.e. prooftheoretical)
characterization of general perspectival
identification. Otherwise, the objects we want to get rid of
will recur on the meta-level. I will discuss the following
meta-theorems: (M1) A modal logic is generally
identificatory just in case for all its operators V(n),
V(n)¬p ↔ ¬V(n)p is a theorem, i.e. iff all operators are
self-dual. (M2) A modal logic is perspectival just in case its
operators form a group. The group axioms rule out
satisfaction operators, because none of them has an
inverse element. In sum, I suggest that a modal logic can
reduce the points of its frame iff its operators are self-dual
and form a group.
8. Solving Prior’s Problem
To solve Prior’s problem, we thus have to find reasons
why, for persons and possible worlds, we cannot construct
modal logics that allow general perspectival identification,
i.e. whose operators are self-dual and form a group. For
possible worlds, self-duality clearly fails, because it is essential
to alethic modality that the necessary and the possible
do not fall together. For persons, the obvious candidate
for a relation of accessibility allowing perspectival
identification is the system of family ties (everyone can
perspectivally identify her mother). Here generality fails: To
construct a modal logic that would reduce persons, we
would need a relation that allows everyone to perspectivally
identify everyone. But there is no relation that holds
among all persons that is systematic and static enough for
this purpose.
So, where hybrid logic is too powerful a tool of
metaphysical reduction, generalized metric logic is just
right. It can translate context-free temporal and spatial
statements to perspectival statements, but not statements
about persons and possible worlds. This fits Prior’s
metaphysical intuition that while times and places are
logical fictions, persons are genuine objects.
9. Worlds and Selves: What is the Role for
Hybrid Logic?
But what about possible worlds? If we replace hybrid logic
by metric logic tout court, we are left with possible worlds
seeming to be genuine objects just as much as persons.
But while the reasons counting against hybrid tense logic
(section 4) generalize easily to space and frame, this is not
so for possible worlds. There is no quantitative element to
modality (reason I), and (as shown in section 8) we cannot
identify possible worlds by their relation to a unique world
(reason III). What we actually do to identify a possible
world is to describe it. Therefore, in the case of alethic
modality, we do a have a natural interpretation for nominals,
namely for each world, the maximally compatible
proposition that uniquely describes it (reason II). In sum,
what counts against the employment of hybrid logic concerning
time and space, does not count against Prior’s
project of reducing possible worlds to world-propositions.
So, from a natural language stance, hybrid logic is
well-suited to deal with possible worlds. But not with
persons (or other enduring things): Here, the only natural
candidates for nominals contain proper names. “I am
263