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We provide a small glimpse of the foundations of our approach to constructing the
music calculus
M
; these foundations give at least a provisional account of these
three dimensions. Why do we need such a formalism? As we begin to examine the
act of musical conducting in a bit more detail, we begin to see why:
Consider a simple situation in which there is a composer
c
, a performer
p
, a listener
listener
,
and a conductor
h
. The composition, or score, in question is
score
. The performance of
score
by
p
is
performance
. Composer
c
creates
score
with the intention of inducing a single
emotional effect
effect
1
in the listener of the piece,
listener
. Performer
p
has a belief that the
composer intends the music to draw out
effect
1
in
listener
, but performer
p
might want his
performance to have effect
effect
2
on
listener
. The conductor
h
might in turn have beliefs
regarding what the composer and the performer intend, and
c
might have their own intentions
for the performance. Each participant in such a scenario can have further iterative beliefs:
for example, the conductor believing what the performer believes the composer intended
the performance should be. The conductor should also have an understanding of emotional
effects and their inter-relations. For example,
h
should know that a melancholic effect is
incompatible with a joyous effect. Such knowledge of effects should allow the conductor to
dynamically alter a performance to elicit compatible effects.
...
Obviously, even this simple, informal analysis reveals that cognitive, social, and
doxastic factors are quite real, and quite central. Our music calculus, designed to
allow formal capture of such factors, is based on the
cognitive event calculus
(
CEC
),
which we review briefly now.
The
CEC
CEC
is a first-order modal logic. The formal syntax of the
is shown in
Fig.
14.1
. The syntax specifies sorts
S
signature of the function and predicate symbols
f
, syntax of terms
t
, and the syntax of sentences
,
. We refrain from specifying
a formal semantics for the calculus as we feel that the possible-worlds approach,
though popular, falls short of the
tripartite analysis of knowledge
(Pappas [
20
]),
according to which
knowledge
is a
belief
that is true and justified. The standard
possible-worlds semantics for epistemic logics skips over the justification criterion
for knowledge.
1
Instead of giving here a full formal semantics for our calculus based
in a formalization of justification, we specify a set of inference rules that capture our
informal “justification-based” semantics (Fig.
14.2
).
We denote that agent
a
knows
φ
. The operators
B
,
P
,
D
, and
I
can be understood to align with belief, perception, desire, and intention,
respectively. The formula
S
φ
at time
t
by
K
(
a
,
t
,
φ
)
(
a
,
b
,
t
,
φ
)
captures declarative communication of
φ
from
agent
a
to agent
b
at time
t
. Common-knowledge of
φ
in the system is denoted by
C
(
t
,
φ
)
. Common-knowledge of some proposition
φ
holds when every agent knows
φ
, and so on
ad indefinitum
.The
Moment
sort is used for representing timepoints. We assume that timepoints are
isomorphic with
, and every agent knows that every agent knows
φ
N
; and function symbols (or functors)
+
,
−
, relations
>, <,
≥
,
≤
are available.
The
includes the signature of the classic
event calculus
(EC) (see Mueller's
[
18
]), and the axioms of EC are assumed to be common knowledge in the system
[
2
]. The EC is a first-order calculus that lets one reason about events that occur in
CEC
1
The possible worlds approach, at least in its standard form, also suffers from allowing logically
omniscient agents: agents which know all logically valid sentences.
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