Truth for Graphs
April 26, 2014
Barry Smith
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Ontology of graphs
What is a graph made of?
What is a graph about?
Begin with time-series graphs
3
Ringo’s temperature: June 4, 1952
Parts of the graph
•
•
•
•
a label indicating the target of measurement, here: Ringo’s temperature
a label indicating the quality that is being measured, here: temperature
a time axis with associated actual time values and a measurement unit
a second, orthogonal axis with associated measurement values, here
degrees Celsius
• a set of graph entries (points on the graph)
• a rectangular graph background (the two-dimensional shaded area
demarcated by the two axes)
• a continuous line connecting the graph entries, which we refer to as the
graph line, within the graph background
4
Ringo’s temperature: June 4, 1952
What makes the graph true? What would
the world have to be like in order for the
graph to be true?
5
Aristotle’s Ontological Square (Lowe’s 4CO)
Particular
Universal
Substantial
Accidental
Second substance Second accident
man
headache
cat
sun-tan
ox
dread
First substance
First accident
this man
this headache
this cat
this sun-tan
this ox
this dread
6
Standard Predicate Logic – F(a), R(a,b) ...
Particular
Universal
Objects
Properties
Attributes
F, G, R
Individual things
a, b, c
this, that
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Bicategorial Nominalism
Accidental
Particular
Universal
Substantial
First substance
this man
this cat
this ox
First accident
this headache
this sun-tan
this dread
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Process Metaphysics (Heraclitus)
Accidental
Particular
Universal
Substantial
Events
Processes
“Everything is flux”
9
Davidson: Events as truthmakers
Objects
Events
Attributes
is a man
is sharp
Universal
Particular
Properties
Individuals
this man
that knife
Events
this buttering
that flying
10
BFO (6CO)
Processes
Universal
Quality entities
SubstanceQualityuniversals universals
Processuniversals
Particular
Substances
QualityIndividual
instances
Substances
(Tropes…)
Processinstances
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depends_on
Continuant
Independent
Continuant
Dependent
Continuant
thing
quality
Occurrent
process, event
quality depends
on bearer
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is_about
IAO:measurement
datum
quality: John’s
blood glucose level
has_specified
_output
inheres_in
portion of blood
concretized_by
participates_in
OBI process:
this specific assay
quality: ‘120
mg/dL’-shaped
pattern
inheres_in
derived_from
device
John
participates_in
part_of
screen
Numerical Value Example
13
Quality of portion of blood
elements of an ontological analysis:
1. the portion of blood (material entity)
2. the blood sugar level (quality)
referred to by means of
1. an expression (information artifact, thus
a BFO:generically dependent continuant)
‘100 mg/dL’.
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15
is_about
IAO:measurement
datum
process: John’s
heart beating
has_specified
_output
quality: ‘120
bpm’-shaped
pattern
has_participant
has_participant
device
OBI process:
this specific assay
has_part
concretized_by
inheres_in
screen
has_participant
John
Beat Measurement
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Heart beating at constant rate
Elements of an ontological assay:
1. the heart (object)
2. the process of beating
3. the temporal region occupied by this process
4. the spatiotemporal region that is occupied by
this process (trajectory of the beating process)
5. the rate, referred to by means of
6. an expression (information artifact, thus a
BFO:generically dependent continuant) such as
‘63 beats/minute’.
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What is a rate?
• One popular solution
– a rate is a quality of a process
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8CO (à la [old] Schneider)?
Processes
Process
qualities
Universal
Qualities
Process
Substance- QualityProcessquality
universals universals universals
universals
Particular
Substances
QualityProcessIndividual
instances
Substances
instances
(Tropes…)
Process
quality
instances
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Objects have qualities which can be
accepted as first class entities
• Why?
Because objects can gain and lose qualities – as
you can gain and lose a suntan
Processes do not have qualities which can be
accepted as first class entities
• Why not?
Because processes cannot change.
a beating process p goes from 63 to 65bpm= p
(unchangingly) has two temporal parts, the first of
which is a 63bpm segment, the second a 65bpm
segment
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Intuition:
if you have 3 apples in a box, then you do
not have 4 first-class entities: the 3 apples,
plus the number 3.
Analogously:
if you have a process that is a 63bpmprocess, then you do not have 2 first-class
entities: the process, plus the 63bpm.
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Note the difference between
• this moving m has speed v at time instant t
• this person j has temperature 63°C at time t
j might have had a different temperature at t
m could not have had a different speed at t
m could not have been different in any way
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process p has rate r is analogous not to:
rabbit r has weight w
but rather to:
rabbit r instance_of universal: rabbit
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Solution
• not ‘thick’ processes, such as runnings or
heart beats, instantiate quantitative universals
• but ‘thin’ structural parts of processes
–called ‘process profiles’
• (event patterns, …)
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Cardiac Cycle, Left Ventricle
WIGGERS DIAGRAM
25
26
Process profile = that which a correct
device output would represent
• that which a correct time-series graph would
represent
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Definition
• a is a process profile means: a is a process of the
sort that can be represented by a chart plotting
measurement results on a single dimension
against a time axis.
• Since BFO will be used to annotate process
measurement data, we always have such
measurement results
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is_about
IAO:measurement
datum
concretized_by
process: John’s heart beating
has_specified
_output
beat profile
has_participant
has_participant
device
quality: ‘120
bpm’-shaped
pattern
OBI process:
this specific assay
has_part
inheres_in
screen
has_participant
John
Beat Measurement
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Cardiac Cycle, Left Ventricle
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Process_profile_of
ELUCIDATION: a process_profile_of b holds when
• a is a process profile
• & b is a process
• & a proper occurrent_part_of b
• & a and b occupy the same temporal region
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Davidson
Early: If two processes occupy the same
spatiotemporal regions they are identical
Later: the top is spinning, the top is warming up;
same spatiotemporal region, distinct processes
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BFO:Histories vs. process profiles
• A history is a full process
• x is a history --> for some spatiotemporal
region r, x occupies r and every process taking
place in r is part of x
• A process profile is a partial process
• x is a process profile --> x is not a history
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the spinning top is at the same time
warming up (two distinct truthmakers)
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Minimal Extensional Mereology
Simons, Parts, pp. 26-31
• Antisymmetry: If x part of y, then if y part of x, then x = y.
• Transitivity: If x part of y, and y part_of z, then x part_of z.
• Unique Product: If x and y have a part in common, then
there is some unique z such that for all w (w is part of z if
and only if (w is part of x and w is part of y)).
• Weak Supplementation: If x part_of y & not x = y, then
there is some z such that (z part_of y and z has no part in
common with x).
• NOT STRONG SUPPLEMENTATION: z is not necessarily
unique.
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But what about this case: A top is spinning
and simultaneously getting warmer
same spatiotemporal region, distinct processes
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the spinning top is moving and at the same
time warming up (what’s the difference?)
38
Measurements always have a level
of precision
39
x component of the angular velocity vector
for a spinning top
(solid line = corrected for wobble)
40
Measurements always have a level
of precision
The process is continuous
The graph-representation is built out of
points plus interpolation
Graphs are affected by different sorts of
artifacts of the measurement instrument,
scale, …
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Different granularities of
measurement
...
...
-20-10 -10  0
normal
0  10 10  20 ...
massively
increased increased
chronic
...
All measurement instruments are to some degree
imprecise. Thus coarse-grained partitions are the
best that we can achieve
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Definition
• a is a process profile means: a is a process of
the sort that can be represented by a correct
chart plotting measurement results on a single
dimension against a time axis.
43
This
represents Ringo’s temperature from 9am to
4pm on this particular day.
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But it does not tell us what Ringo’s
temperature was doing at 10.30am
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Ringo’s temperature: June 4, 1952
What makes this graph true? What would
the world have to be like in order for the
graph to be true?
46
What did your temperature do over the last
month, Ringo?
Ringo’s temperature process profile, the
target of a certain sort of cognitive
selection, or cognitive profiling
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When we look at Ringo’s history
through the temperature partition
• we focus not on the full process which is the
entire history of Ringo, but rather on a partial
process
–called a ‘temperature quality process
profile’
48
Process profile  that which the output
of a correct device would represent
= that which a correct time-series graph would
represent
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The graph picks out just one dimension of
qualitative change within a much larger
conglomerate of processes
Hence ‘quality process profile’
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What did your temperature do over the
last month, Ringo?
a process profile is the target of a
certain sort of cognitive selection
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Thesis
• Almost all of the most important things in life
are a matter of process profiles
• Examples:
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Life
Cardiac Cycle, Left Ventricle
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Biological processes typically incorporate
multiple quality process profiles
Cardiac Cycle, Left Ventricle
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…corresponding to the multiple
different sorts of partition of the same
reality involved during measurement
Cardiac Cycle, Left Ventricle
55
Music
• Cognitive selection of the cello part when you
listen to a string quartet
• Picking out a certain sonic partial process
within a larger body of vibrations
• Ignoring sneezes, coughs, …
• (or sometimes focusing on sneezes and
coughs for diagnostic purposes)
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Sex
A physiologist may be interested in
• processes in the reproductive system of a single
organism,
• processes in the reproductive systems in a pair of
mating organisms
• processes in this pair of mating organisms as part
of a larger whole which includes an entire
population of organisms of a relevantly similar
type (all adult humans, all adult babies of a given
body weight, all adult human athletes, …).
Normal processes are defined for these larger
populations, and deviations from this norm are
defined for the single organism relative thereto.
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Football
• Every process is embedded within a series of
larger process wholes, each nested within yet
larger process wholes
• When a football is kicked into the goal, we can
focus
– on the ball’s motion relative to the field
– on the larger process which is the motion of the field
system relative to the motion of the two teams,
– on the motion of the team-ball-field system relative to
the entirety of processes taking place within the
league championship in this season …
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War
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Many streamiform structures in reality
• thinking
• speaking
• money
60
graph of prices of all
components of the Dow
Jones Industrial Average
61
identifying streams in
hyperfrequency trading
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identifying streams in hyperfrequency
trading
63
Eleven seconds of flash crash
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Dow Jones Industrial Average (Flash
Crash)
1399
1398
1397
1396
1395
1394
1393
1392
1391
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What did the Dow do this morning, Ringo?
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