Reduction and Elimination in Philosophy and the Sciences

The Metaphysical Relevance of Metric and Hybrid Logic
Martin Pleitz, Münster, Germany
1. Temporal Reasoning
To most of our temporal statements, a quantitative element
is essential. The statement that there will be rain says
more than that, in the infinitely long stretch of future time,
there exists a rainy moment. Rather, it conveys that rain
will come a reasonable temporal interval hence. And in
reasoning, our temporal reference often is much more
precise. When recording events, we often reason in the
following way:
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P1: Presently, it is windy.
P2: Presently, it is 4 p.m.
C1: It is* windy at 4 p.m.
In planning for the future, we often reason like this:
P3: The talk starts* at 5 p.m.
P4: Presently, it is 4 p.m.
C2: The talk will start in one hour.
(An asterisk (*) indicates that the verb is used tenselessly.)
There are four kinds of temporal statements involved here.
Present statements are made by temporally indefinite sentences
in the present tense which do not refer to a date
(P1). Clock statements are made by tenseless sentences
of the form “Presently, it is t” (P2, P4). Diary statements
are made by tensed sentences that give a date (C1, P3).
Metric tense statements are made by temporally indefinite
sentences which give the duration from the present moment
to a past or future event (C2).
A temporal logic that captures our temporal
reasoning must have the resources to express all four
kinds of temporal statements. Standard logic (propositional
and predicate logic) cannot formalize any of these
statements in a way which preserves their temporal
characteristics. Date logic states relations between events
and dates and nothing else. Thus it can express diary
statements (“@tp” for “At t, it is* the case that p”), but
neither present, clock nor metric tense statements.
Standard tense logic allows temporally indefinite
statements and therefore can formalize present
statements, but none of the others, because its operators
F and P cannot cope with quantities.
2. Metric Tense Logic and the Reduction of
Dates
Metric tense logic with its operators F(t) for “It will be the
case t time-units hence that” and P(t) for “It was the case t
time-units ago that” can express metric tense statements
and present statements. But it allows neither clock nor
diary statements because it cannot deal with dates. Dates
can only be dealt with by a metric tense logic that is
grounded, i.e. that is supplemented by a unique proposition,
i.e. a proposition c guaranteed to be true at exactly
one moment. The clock statement “Presently, it is t” then
can be translated as “P(t)c” and the diary statement “At t, it
is* the case that p” as “Sometimes: p ∧ P(t)c”.
For a realistic example of a grounding unique
proposition c, we only need “some standard event which is
presumed to be unique” (Prior 1957, 19). For the current
system of time-keeping, c is the proposition that presently,
Christ is born. “Presently, it is 2008” can be translated as
“It was the case 2008 years ago that, presently, Christ is
born.” Note that in metric tense logic, one unique proposition
helps to reduce the whole system of dates (cf. section 4).
If temporal instants are nothing more than dates,
then grounded metric tense logic and Ockham’s Razor
lead naturally to a tensed metaphysics of time, albeit one
where past and future are graded.
3. Hybrid Tense Logic and the Reduction of
Instants
Although Arthur Prior described metric tense logic in detail
(Prior 1957, 18-28; 1967, 95-112; 2003, 159-171) and
used it to translate date statements (Prior 1957, 19; 1967,
103ff.), he took another way to reach the metaphysics free
of temporal instants that his tense-theoretical intuitions
made him look for: He invented hybrid modal logic, that
provides, for each point of the frame, a unique proposition,
which Prior called “world-proposition” (Prior 1967, 89) and
nowadays is known as a “nominal”, as in a sense it names
an instant (Blackburn 2006, 343ff.). Nominals (i, j …) allow
the translation of date logic (Prior’s “logic of earlier and
later”) into hybrid tense logic. “@ip” becomes “Sometimes:
i ∧ p”, “i is earlier than j” becomes “Always: i → Fj”, etc.
(Prior 1967, 88ff. and 187ff.; 2003, 124ff.).
Prior took this logical result to be of metaphysical
importance: “A world-state proposition in the tense-logical
sense is simply an index of an instant; indeed, I would like
to say that it is an instant, in the only sense in which
‘instants’ are not highly fictitious entities.” (Prior 1967,
188f.)
4. Metric or Hybrid Tense Logic?
For those sharing Prior’s metaphysical tense-theoretical
convictions, there are reasons to prefer metric tense logic
to its hybrid alternative. (I) Hybrid tense logic does not
capture the quantitative side of many temporal statements
(section 1). (II) Nominals, though formally innocent (e.g.
Blackburn 2006, 343ff.; Øhrstrøm et al. 1995, 221ff.), are
suspect from a natural language stance. As only some
times can be characterized by unique events that are publicly
known, the only natural candidates we have for nominals
are (a) complete descriptions of instants and (b) clock
statements. But (a) leads to modal problems, because it
makes impossible that something else could have happened
at a certain time than what actually did. And (b)
translates hybrid tense logic into metric tense logic. (III)
The reduction of dates to grounded metric tense logic is
better suited than hybrid tense logic to capture the epistemic
side of time-keeping. Not only can a person forget
the date (cf. Müller 2002, 193ff.), but we can also imagine
a whole time-keeping community losing track of their
grounding event, but still knowing the truth of many metric
tense statements.
A fourth reason to prefer metric to hybrid logic
concerns its generalizability from time to other dimensions
of logical space (sections 5-8).