Mathematics
E-Mail
Comments to: opencyc-doc@cyc.com
Last
Update: 4/5/02
Copyright© 1996-2002. All rights reserved. See Terms of Usage.
Return to Documentation Contents
Return to Vocabulary Contents
Main
#$MathematicalOrComputationalThing mathematical
or computational objects
The collection of #$Intangible things
that are intrinsically mathematical (see #$MathematicalThing)
or computational (see #$ComputationalObject).
Instances of #$MathematicalOrComputationalThing
are abstract in the very strong sense of being nonspatial,
atemporal, and massless. Examples include numbers, sets,
collections, relations, algorithms, and abstract character strings.
guid: bd58e31f-9c29-11b1-9dad-c379636f7270
direct instance of: #$VariableOrderCollection
direct specialization of:
#$Intangible
direct generalization of:
#$MathematicalThing
#$Algorithm #$ComputationalObject
#$ComputationalObject computational objects
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of both #$MathematicalOrComputationalThing
and #$IntangibleIndividual.
Each instance of #$ComputationalObject
is a syntactically structured form, such as a Cyc system
expression, a Lisp string, a C variable name, or an equation
in a particular canonical form format.
guid: bd58803b-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$MathematicalOrComputationalThing #$IntangibleIndividual
direct generalization of: #$JustificationTruth
#$RuleTemplate #$CycLAssertionDirection #$CycHLTruthValue
#$ELSentenceTemplate
Numbers
#$IntervalOnNumberLine intervals (quantities)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of #$ScalarInterval.
Each instance of #$IntervalOnNumberLine
is an interval on the real number line; for example, the
interval described by `numbers greater than zero and less
than or equal to 10'. A common special case of such
intervals is that of a single point on that line, viz., a
number such as five or 125. Note that such an interval need
not be contiguous; e.g., `even numbers between Pi and the
square root of 1000' describes a legitimate instance of #$IntervalOnNumberLine.
The collection #$RealNumber is
itself a specialization of #$IntervalOnNumberLine.
guid: bd58f32f-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of:
#$ScalarInterval
direct generalization of: #$IntegerExtent #$RealNumber
#$RealNumber real numbers
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all the minimal intervals (i.e.,
points) on the number line; a subcollection of #$IntervalOnNumberLine.
Each instance of #$RealNumber is a
single point on the real number line, which has no upper or
lower bounds. Subcollections of #$RealNumber include
#$Integer, #$RationalNumber,
#$NegativeNumber,
#$PrimeNumber, and others. Note: Real numbers, like other
instances of #$IntervalOnNumberLine,
are measured along a single number `line'; but complex
numbers, quaternions, etc., are n-tuples of numbers, and
therefore are instances of #$NTupleInterval.
For example, #$ComplexNumber
is a subcollection of #$NTupleInterval
guid: bd588098-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of:
#$ScalarPointValue #$IntervalOnNumberLine #$ComplexNumber
direct generalization of: #$NonPositiveNumber #$NonNegativeNumber
#$RationalNumber
#$PositiveNumber
#$NegativeNumber
#$RationalNumber rational numbers
A specialization of #$RealNumber. A
number NUM is an instance of #$RationalNumber
just in case NUM can be expressed as the quotient of two
integers. For example, 3/4, 2 1/8, 0.3333333..., 11/5.
guid: bd58f2aa-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of:
#$RealNumber
direct generalization of:
#$SubLRealNumber
#$Integer
#$Integer integers
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all whole numbers; a
subcollection of #$RationalNumber.
Each instance of #$Integer is a whole
number, resolvable into units with no fractional remainder.
An integer may be positive (e.g., 42), zero, or negative
(e.g., -42). Note that 42.0 is a floating-point real number
which is close to the integer 42 within the tolerance of the
floating-point representation but is not necessarily equal
to the integer 42. Therefore, 42.0 is not an instance of #$Integer.
guid: bd589e12-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of:
#$IntegerExtent #$RationalNumber
direct generalization of: #$NonPositiveInteger #$OddNumber #$EvenNumber #$NegativeInteger
#$SubLInteger #$PositiveInteger
#$NonNegativeInteger
#$EvenNumber even numbers
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The set of all even numbers (integers) including
positive and negative even numbers and zero, but not
including any infinite 'numbers'.
guid: bff46760-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of:
#$Integer
#$OddNumber odd numbers
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The set of all odd numbers (integers) including
positive and negative odd numbers, but not including any
infinite 'numbers'.
guid: becb081f-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of:
#$Integer
#$PositiveNumber positive (number)
A specialization of #$RealNumber. An
instance NUMBER of #$RealNumber is also
an instance of #$PositiveNumber
just in case NUMBER is greater than zero. Instances of #$PositiveNumber
include 42 and 0.17, but not 0 or -5.
guid: bf6afb32-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of:
#$PositiveScalarInterval #$NonNegativeNumber
direct generalization of: #$Real1-Infinity #$PositiveInteger
#$NegativeNumber negative numbers
**GAFs NOT REVIEWED**
A specialization of #$RealNumber. An
instance REAL of #$RealNumber is also
an instance of #$NegativeNumber
just in case REAL is less than 0.
guid: c0fe03e3-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of: #$NegativeScalarInterval #$NonPositiveNumber
direct generalization of:
#$NegativeInteger
#$PositiveInteger positive integers
A specialization of #$Integer. An instance
INT of #$Integer is
an instance of #$PositiveInteger
just in case INT is an integer greater than 0.
guid: bd589ed9-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of:
#$PositiveIntegerExtent #$PositiveNumber #$NonNegativeInteger
#$NegativeInteger negative integers
**GAFs NOT REVIEWED**
A specialization of #$Integer. An instance
INT of #$Integer is
also an instance of #$NegativeInteger
if and only if INT is less than zero.
guid: c0fe031d-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of:
#$NegativeIntegerExtent #$NonPositiveInteger #$NegativeNumber
#$NonNegativeNumber non-negative numbers
#$NonNegativeNumber
is the sub-collection of #$RealNumber that
excludes the negative reals. Each instance of #$NonNegativeNumber
is a number greater than or equal to zero -- for example, 0,
0.173, Pi, 4, and 101. Quantities measured in units -- for
example, (#$SecondsDuration
4) and (#$Mile 42) -- are not instances of #$NonNegativeNumber,
but rather are instances of its super-collection #$NonNegativeScalarInterval.
guid: bd58f22b-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of:
#$NonNegativeScalarInterval #$RealNumber
direct generalization of:
#$PositiveNumber
#$NonNegativeInteger
#$NonNegativeInteger non-negative integers
#$NonNegativeInteger
is the sub-collection of #$Integer that excludes
the negative integers. Each instance of #$NonNegativeInteger
is a whole number greater than or equal to zero -- for
example, 0, 1, 2, 3, ....
guid: bd58bf69-9c29-11b1-9dad-c379636f7270
direct instance of:
#$MeasurableScalarIntervalType
direct specialization of:
#$NonNegativeIntegerExtent #$Cardinal-Mathematical #$NonNegativeNumber #$Integer
direct generalization of: #$CycUniversalSecond #$PositiveInteger
Predicates
#$decodingDeviceType decoding
device type
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
This predicate is used to specify the type of
device needed for decoding information contained in a
particular kind of encoding scheme. (#$decodingDeviceType
SCHEME DEV-TYP) means that DEV-TYP is the type of device
required to decode the information encoded with encoding
scheme SCHEME. For example, a telephone is the kind of
device needed to decode audio information sent over
telephone lines. A cable TV box is the device type needed to
decode cable TV signals. And so on. See also #$EncodingSchemeType.
guid: bd58cef2-9c29-11b1-9dad-c379636f7270
direct instance of: #$BinaryPredicate
#$deviceControlledBy device
controlled by
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
(#$deviceControlledBy
DEV CONTROL) means that CONTROL is a #$ControlDevice
that controls #$PhysicalDevice
DEV. Most of the time, the controls will be #$physicalParts
of the device. However there are some notable exceptions --
one's TV remote control, for example -- which control a
device but are not #$physicalParts
of that device.
guid: bd5cbb65-9c29-11b1-9dad-c379636f7270
direct instance of: #$AsymmetricBinaryPredicate
#$instrument-Generic instrument
(actor slot)
The predicate #$instrument-Generic
is used to link a particular event to any of the objects
which play an instrumental role in it. (#$instrument-Generic
EVENT OBJECT) means that OBJECT plays an intermediate causal
role in EVENT, facilitating the occurrence of EVENT, and
serving some purpose of some #$Agent. This can happen
in at least two ways: either the `doer' of EVENT acts on
OBJECT, which in turn acts on something else (as when
someone uses a hammer to pound in a nail) or the `doer' of
EVENT acts on something, making it possible for OBJECT to
act on that thing (as when someone puts wet clothes out in
the sun to dry). Typically, an #$instrument-Generic
is not significantly altered by playing that role in an
event. #$deviceUsed
is an important specialization of #$instrument-Generic.
guid: bd5d4eb3-9c29-11b1-9dad-c379636f7270
direct instance of: #$IndividualLevelPredicate #$ActorSlot
direct specialization of:
#$actors
#$instrumentalRole
#$deviceUsed device
used (actor slot)
The predicate #$deviceUsed
relates an event to a device used in that event. (#$deviceUsed
EVENT OBJECT) means that the #$PhysicalDevice
OBJECT plays an instrumental role in the #$Event EVENT (see the
more generalized predicate #$instrument-Generic),
OBJECT is intentionally used in EVENT, and standardly (for
example, in the #$HumanActivitiesMt) OBJECT's role in EVENT
is consistent with the object's #$primaryFunction.
guid: bd5d0531-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ActorSlot
direct specialization of:
#$instrument-Generic
#$hasInterfaceDevices has
interface devices
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
This predicate identifies a particular interface
device that is linked to a particular computer. (#$hasInterfaceDevices
COMPUTER INTERFACE) means that INTERFACE is a #$ComputerInterfaceDevice
for the #$Computer COMPUTER.
E.g., this predicate holds between my desktop PC and the
mouse that's connected to it, the monitor that's connected
to it, the keyboard that's connected to it, etc.
guid: bd58bb61-9c29-11b1-9dad-c379636f7270
direct instance of: #$AntiSymmetricBinaryPredicate
#$PhysicalPartPredicate
#$energySource energy
source (binary role predicate)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
This predicate is used to identify a particular
source of energy used in a particular event. (#$energySource
EVENT ENERGYSOURCE) means that ENERGYSOURCE provides (some
of) the energy used in EVENT. ENERGYSOURCE might be a
battery, the sun, a person, etc.
guid: bd58ea4e-9c29-11b1-9dad-c379636f7270
direct instance of: #$IndividualLevelPredicate
#$BinaryRolePredicate #$AsymmetricBinaryPredicate
direct specialization of:
#$temporallyIntersects
#$objectControlled controlled object
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
This predicate is used to indicate that a
particular object is being controlled in a particular event.
(#$objectControlled
EVENT OBJ) means that the object OBJ is being controlled in
the #$Event EVENT.
Note: #$objectControlled
does not assume or require physical contact between
controller and object controlled.
guid: bd58bdb0-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ActorSlot
direct specialization of:
#$objectActedOn
#$objectActedOn affected
object (actor slot)
The predicate #$objectActedOn
is used to relate an event to an entity or entities
significantly affected in that event. The entity or entities
in question must exist before the event, but may be either
destroyed in the event (see the more specific predicate #$inputsDestroyed),
or merely affected by it (for example, see the more specific
predicates #$damages and #$objectOfStateChange).
(#$objectActedOn
EVENT OBJECT) means that OBJECT is altered or affected in
EVENT, and the change that OBJECT undergoes is central or
focal to understanding EVENT. Thus, scissors are _not_ an #$objectActedOn
in a #$HairCuttingEvent. The focal change in a haircut is
hair getting shorter; thus, hair is a legitimate #$objectActedOn
in a #$HairCuttingEvent. The almost microscopic dulling that
scissors undergo in a single haircut is a comparatively
insignificant change with respect to a single haircut,
considered as a #$HairCuttingEvent.
guid: bd58bae6-9c29-11b1-9dad-c379636f7270
direct instance of: #$IndividualLevelPredicate #$ActorSlot
direct specialization of:
#$preActors
#$vehicle vehicle
(actor slot)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
(#$vehicle EVENT
VEHICLE) means that VEHICLE is a #$TransportationDevice-Vehicle
which is both the #$providerOfMotiveForce
and the #$transporter
in EVENT. If an object is a #$TransportationDevice-Vehicle
and plays the role of #$transporter
in some moving event, then it generally will play the role
of #$vehicle
in that event. Examples: a car plays the role of #$vehicle in
driving. Note, however, that a bicycle does not play the
role of #$vehicle in
bike riding since it is not a provider of motive force. A
borderline non-example is someone sitting in their car while
it's being pulled by a towtruck; their car is not playing
the role of #$vehicle in
that event.
guid: bd58a883-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ActorSlot
direct specialization of:
#$deviceUsed
#$transporter
#$providerOfMotiveForce
#$stuffUsed stuff used
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The predicate #$stuffUsed
relates an event to some tangible substance which
facilitates that event. (#$stuffUsed
EVENT STUFF) means that STUFF is a portion of an instance of
#$ExistingStuffType
which plays an instrumental role in EVENT. STUFF may or may
not be consumed in the course of EVENT. Examples: portions
of #$Water are #$stuffUsed in
instances of #$WashingDishes, #$WashingHair,
#$WashingClothesInAMachine, etc.; portions of #$EdibleOil
are #$stuffUsed in
some instances of #$Frying food and #$BakingBread.
guid: bd58cadc-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ActorSlot
direct specialization of:
#$instrument-Generic
#$transporter transporter
(actor slot)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
(#$transporter
MOVE OBJ) means that OBJ enables or facilitates the
conveyance of the #$transportees
in the #$TransportationEvent
MOVE. OBJ is an #$objectMoving in
MOVE that moves along with the #$transportees.
OBJ will generally hold, support, contain, pull, or push the
#$transportees
throughout the MOVE #$Event. OBJ may or may
not be the #$providerOfMotiveForce
in the event MOVE. If OBJ stays with each
#$primaryObjectMoving from the #$fromLocation to
the #$toLocation,
moving along the same trajectory, then it is also a
#$primaryObjectMoving. If OBJ facilitates the motion of the
#$primaryObjectMoving but does not itself engage in
translational motion it is merely the conveyor of the
action, and the role #$conveyor-Stationary
should be asserted. If it is unclear whether the conveyor is
stationary or not, the role #$conveyor-Generic is used.
Specializations of the role #$transporter
should be used when possible: although automobiles are #$transporters
in many events, they should normally have the more specific
role designation of #$vehicle
because they are also #$SelfPoweredDevices.
When a car is being towed by a towtruck, the car is just a
#$transporter
of any of its contents while the towtruck is the #$vehicle of
that event. An additional role designation for some #$transporters
-- those which are #$PhysicalDevices
-- is #$deviceUsed;
e.g., the use of crutches in hobbling or ice skates in
skating. See the #$comment on #$TransportationEvent.
Note that an organism may be a #$transportees
in a #$Bicycle
riding or #$Skating event as well as being the #$providerOfMotiveForce
in such cases.
guid: bd58c1d8-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ActorSlot
direct specialization of: #$conveyor-Generic #$objectMoving
#$transportees transportee
(actor slot)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The predicate #$transportees
relates a translational motion event to the object(s)
transported by a separate object, i.e. a distinctly separate
other participant in the event. (#$transportees
MOVE OBJ) means that some #$conveyor-Generic facilitates the
conveyance of OBJ in MOVE. For example, in a dumptruck
driving event, the dirt in the back of the truck is a #$transportees.
Any humans in the truck cab (or truck bed) during the trip
are also #$transportees;
however, a more precise role designation for humans riding
in the truck would be either #$passengers
or (for the driver) #$driverActor.
Borderline positive example #$transportees
include the clothes worn by a person walking, or a horseshoe
worn by a horse walking. A negative exemplar of a #$transportees
is the ear of the person walking. This is because #$transporters
do not transport their parts when they move. In other words,
#$transporters
only transport separate objects. #$translatesFromTo on the
other hand, does apply to parts of #$transportees.
Note also that parts of #$transportees
are not necessarily #$transportees
themselves. See also the comments on #$TransportationEvent
and #$transporter.
guid: bd58ae94-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ActorSlot
direct specialization of: #$primaryObjectMoving #$objectActedOn
#$driverActor driver
(actor slot)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
(#$driverActor
DRIVE DRIVER) means that DRIVER controls (see #$ControllingATransporter)
the #$transporter
in DRIVE. DRIVER steers the wheel, grasps the tiller,
controls the throttle, the reins, the brakes, etc., of the
#$transporter,
e.g., a boat, train, windsurfer, mule, plane, horse and
carriage, spaceship, sled, etc. DRIVER is not a #$passengers
in DRIVE. Because #$transporter
and #$transportees
are disjoint and #$driverActor
has #$transportees
as a #$genlPreds, DRIVER is distinct from the value on #$transporter.
Thus a person walking while carrying a watermelon would not
be a #$driverActor
in their own walking. DRIVER is usually in #$SittingPosture
during DRIVE. For any given instant of DRIVE there is
exactly one DRIVER. Until we have more extensive vocabulary,
#$transportees
is the most specific we can be about events in which
multiple drivers share driving responsibility throughout the
event or if there is a pilot/co-pilot combination.
guid: bd58bf45-9c29-11b1-9dad-c379636f7270
direct instance of: #$AgentiveRole
direct specialization of: #$crewMember #$performedBy #$transportees
Vectors
#$VectorInterval relative location
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of #$NTupleInterval.
Each instance of #$VectorInterval
is an n-tuple of intervals (where n > 1), one of which is a
direction. Like the instances of #$ScalarInterval,
the intervals in an instance of #$VectorInterval
may be point-valued or cover a range of values. The minimal
interval (i.e., point-valued) type of vector interval is
exemplified by a vector such as `10 meters due east'.
Vectors may also cover a range of values; e.g., `at least 10
feet away and in a horizontal direction'; `between ten to
twelve miles NNW'.
guid: bd58ec12-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$NTupleInterval
direct generalization of:
#$UnitVectorInterval
#$Vector-Precise
#$Vector-Precise precise vectors
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of #$VectorInterval.
Each instance of #$Vector-Precise
is an exactly indicated (i.e., point) vector, such as `5
feet due West'. Both direction and distance are precise.
Thus, #$Vector-Precise
is to #$VectorInterval
what #$ScalarPointValue
is to #$ScalarInterval.
#$Vector-Precise
includes all the instances of #$UnitVector-Precise.
guid: bd58dd86-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$VectorInterval
direct generalization of:
#$UnitVector-Precise
#$UnitVectorInterval unit
vector intervals
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection #$UnitVectorInterval
is a subcollection of #$VectorInterval.
Each instance of #$UnitVectorInterval
is a vector interval with a magnitude of 1. The range of the
endpoints of all the unit vectors [#$UnitVector-Precise]
form a contiguous curve (in 2 space), surface (in 3 space),
volume (in 4 space), etc. depending upon the dimentionality
of the vector. One #$UnitVectorInterval
differs from another only in range of direction, since the
magnitude of every #$UnitVectorInterval
is the same. An instance of #$UnitVectorInterval
may specify either a precise unit vector
[#$UnitVector-Precise] or a generalized range of directions
such as `in front of'. The range of directions in 3-space
may be quite irregular, such as the direction interval from
#$Chile to #$Russia.
guid: bd58eb0c-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$VectorInterval
direct generalization of:
#$TerrestrialDirection
#$UnitVector-Precise
#$DirectionExpression
#$UnitVector-Precise unit
vector - precise
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of both #$Vector-Precise
and #$UnitVectorInterval.
Each instance of #$UnitVector-Precise
is a vector interval with a magnitude of 1 and a precisely
specified direction (e.g., due North, straight down). Thus,
one precise unit vector differs from another only in
direction, since each vector consists of a magnitude and a
direction (in a space of n > 1 dimensions). So instances of
#$UnitVector-Precise
in effect indicate different directions such as `straight
up' or `due East'.
guid: bd58ec98-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$UnitVectorInterval #$Vector-Precise
direct generalization of:
#$GeographicalDirection-Direct
Subsets Of Mathematical Thing
#$MathematicalThing mathematical concepts
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of #$MathematicalOrComputationalThing.
Each instance of #$MathematicalThing
is an atemporal, nonspatial, purely mathematical thing. #$MathematicalThing
is partitioned into two main specializations, #$MathematicalObject
and #$SetOrCollection (qq.v).
guid: bd58e5b6-9c29-11b1-9dad-c379636f7270
direct instance of: #$VariableOrderCollection
direct specialization of:
#$MathematicalOrComputationalThing
direct generalization of:
#$MathematicalObject
#$SetOrCollection
#$MathematicalObject mathematical object
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of both #$MathematicalThing
and #$IntangibleIndividual.
Each instance of #$MathematicalObject
is a purely abstract mathematical thing which is also an
individual (see #$Individual).
Specializations of #$MathematicalObject
include #$Quantifier, #$RealNumber,
#$Triangle, and #$TruthValue. Note
that instances of #$SetOrCollection
are not instances of #$MathematicalObject,
since they are not instances of #$Individual.
guid: bf461f37-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$MathematicalThing #$IntangibleIndividual
direct generalization of:
#$Relation-MathematicalObject #$RelationalStructure
#$Number-General #$GeometricThing-Abstract
#$FrameOfReference
#$Relation #$Tuple #$TruthValue
#$TruthValue truths
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
#$TruthValue is a
collection of mathematical objects; it contains the
abstract, logical objects #$True and #$False.
guid: bd58e3ba-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$MathematicalObject
#$True true (mathematical concept)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
An instance of #$TruthValue. #$True is logical
truth in Cyc; this is the abstract logical notion--not to be
confused with Lisp's T, nor with the English word `true'.
guid: bd5880d9-9c29-11b1-9dad-c379636f7270
direct instance of:
#$TruthValue #$Individual
#$False false
(mathematical concept)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
An instance of #$TruthValue. #$False is logical
falsehood in Cyc; this is the abstract logical notion--not
to be confused with Lisp's NIL, nor with the English word `false'.
guid: bd5880d8-9c29-11b1-9dad-c379636f7270
direct instance of:
#$TruthValue #$Individual
#$Relation relationships
(mathematical concepts)
The collection of relations whose CycL
representations can appear in the 0th (or arg0 ) argument
position of a #$CycLFormula, i.e. as the term immediately
following the formula's opening parenthesis. An important
subcollection of #$Relation is
#$TruthFunction
(q.v.), whose instances are intimately related to
truth-values, as reflected in the fact that the CycL
expressions that denote truth-functions can appear in the
arg0 position of a #$CycLSentence; and a sentence (if
quantificationally closed; see #$CycLClosedSentence), will
generally be either true or false (with respect to a given
context or interpretation). The major subcollections of #$TruthFunction
are #$Predicate,
#$LogicalConnective,
and #$Quantifier
(qq.v.). Another important subcollection of #$Relation is
#$Function-Denotational
(q.v.), whose instances are functions the CycL expressions
for which can appear in the arg0 position of a
#$CycLNonAtomicTerm; and such terms (if closed) generally
denote things.
guid: bd5880cd-9c29-11b1-9dad-c379636f7270
direct instance of:
#$RelationshipType
direct specialization of:
#$MathematicalObject
direct generalization of:
#$MicrotheoryDesignatingRelation
#$PartiallyCommutativeRelation #$AssociativeRelation #$ScopingRelation
#$EvaluatableRelation
#$CommutativeRelation
#$FunctionalRelation
#$MacroRelation #$FixedArityRelation #$VariableArityRelation
#$Function-Denotational
#$TruthFunction
#$Quantifier quantifier (relationship)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A collection of mathematical objects. Each instance
of #$Quantifier
represents a relationship between a variable and a formula.
In Cyc, a quantifier binds the variable found in its first
argument within the formula that appears as its second
argument. Instances of #$Quantifier in CycL
include #$forAll, #$thereExists,
#$thereExistExactly,
#$thereExistAtLeast,
#$thereExistAtMost.
guid: bd58c271-9c29-11b1-9dad-c379636f7270
direct instance of:
#$RelationshipType
direct specialization of:
#$ScopingRelation #$TruthFunction
direct generalization of: #$ExistentialQuantifier
#$LogicalConnective logical connective
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A collection of mathematical objects, including the
basic logical connectives. Each instance of #$LogicalConnective
is a #$Relation
which takes one or more truth-valued expressions (sentences)
as arguments and returns a truth-valued sentence. The
instances of #$LogicalConnective
include #$and, #$or, #$not, and #$implies.
guid: bd58b9f9-9c29-11b1-9dad-c379636f7270
direct instance of:
#$RelationshipType
direct specialization of:
#$TruthFunction
#$VariableArityRelation variable
arity relation
A specialization of #$Relation.
Each instance of #$VariableArityRelation
is a relation that can take a variable number of arguments.
Examples of #$VariableArityRelation
include the predicate #$different
and the function #$PlusFn.
guid: bd5d0b56-9c29-11b1-9dad-c379636f7270
direct instance of: #$RelationshipTypeByArity
direct specialization of:
#$Relation
direct generalization of:
#$SocialOrEconomicAttributeFunction
#$VariableArityFunction #$VariableAritySkolemFunction
#$Tuple tuple
**GAFs NOT REVIEWED**
A collection of mathematical objects. Each instance
of #$Tuple is a
complex consisting of one or more ordered (or otherwise
indexed) components; it might be a pair, or a triple, or so
on; and the components might be things of any sort
whatsoever. For example, #$BloodPressureReading is a
specialization #$Tuple, each instance of
which is an ordered or column-indexed pair of numbers, where
the first number is the systolic reading and the second the
diastolic reading. Another specialization of #$Tuple is #$NTupleInterval
(q.v.), whose instances are tuples consisting exclusively of
#$ScalarIntervals
(q.v.); e.g. complex numbers and physical vectors are
n-tuple intervals. If the index set for a given #$Tuple is the counting
numbers (or an initial segment of them), then the numbers'
usual ordering serves to order the tuple's components, and
the tuple is in fact a #$List (q.v.). But in
general any set (e.g. the column names in a relational
database) may be used to index the components of a tuple.
guid: bd58f8d8-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$MathematicalObject
direct generalization of:
#$NTupleInterval
#$List
#$Fractal fractal
(mathematical concept)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A collection of functions. Each element of #$Fractal is a function
which, when applied to data, can be displayed visually as an
instance of #$FractalRepresentation.
guid: c10c1ae4-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$Function-MathematicalObject
#$GeometricallyDescribableThing geometric forms
**GAFs NOT REVIEWED**
A subcollection of #$SpatialThing.
Each instance of #$GeometricallyDescribableThing
is a spatially-connected spatial thing (of 0, 1, 2, or 3
dimensions) that either (i) has or approximates a simple
geometric shape (e.g. it is a #$Line of a #$Hemisphere)
or (ii) consists of a number of (connected) parts in a
relatively stable geometric configuration, where each such
part has or approximates a simple geometric shape (e.g. a
table consisting of a 3-D-disc-shaped top and four
cylindrical legs). A #$GeometricallyDescribableThing
might be tangible (see #$PartiallyTangible)
or intangible (see
#$GeometricallyDescribableThing-Intangible). Note that what
counts as approximating a given simple geometric shape --
and thus what spatial things count as #$GeometricallyDescribableThings
-- varies with context. In a context that was so
fine-grained shape-wise that even the shapes of the
individual molecules on the surface of an object were
considered relevant to the object's shape, perhaps nearly
every (connected, solid) tangible object would be
geometrically-describable. In more everyday contexts, on the
other hand, an unopened can of soup would be
geometrically-describable (as a cylinder), while a telephone
or an animal's body would probably not.
guid: bd58c42e-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$SpatialThing
direct generalization of:
#$GeometricallyDescribableThing-Intangible
#$GeometricThing-Localized #$LinePortion
#$HalfLine-Geometrical #$RoundShape #$Angle #$Line #$TwoDimensionalGeometricThing
#$ThreeDimensionalGeometricThing
#$GeometricThing-Abstract abstract shapes
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of #$GeometricallyDescribableThing
each of whose instances is abstract in the sense of being
intangible (see #$Intangible) as well
as lacking spatial and temporal location. Each instance of
#$GeometricThing-Abstract
is an abstract region of an abstract space (the latter
having two or more dimensions). Geometric figures that are
located in this (or another) universe are not instances of
this collection, but of #$GeometricThing-Localized.
guid: bd5885bc-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$GeometricallyDescribableThing-Intangible #$MathematicalObject
#$ThreeDimensionalGeometricThing three
dimensional shapes
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of #$GeometricallyDescribableThing.
Each instance of #$ThreeDimensionalGeometricThing
is a three-dimensional object. Examples include spatially
localized objects, such as the Pentagon, as well as abstract
three-dimensional shapes.
guid: c0fbbe61-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ThreeDimensionalShapeType
direct specialization of:
#$GeometricallyDescribableThing #$TwoOrHigherDimensionalThing
direct generalization of: #$Cruciform #$SpiralShaped
#$Screw-GenericShape #$Rectangular3DShape #$Hemisphere
#$CylinderShape #$Polygon3DShaped #$Ellipsoid
#$SphereShape #$ConeShape #$CubeShape #$PyramidShape
#$KnotShape
#$FrameOfReference frames
of reference
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of #$MathematicalObject.
Each instance of #$FrameOfReference
is a mathematical (and hence intangible) representation of
the context in which certain data are to be interpreted.
Such contexts are typically physical (i.e., spatiotemporal),
but contexts may also be purely mathematical. A Cartesian
coordinate system represents a frame of reference.
guid: bd58d4a0-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$MathematicalObject
#$Line lines
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of #$GeometricallyDescribableThing.
Each instance of #$Line
is a one-dimensional path, either curved or straight,
through one-, two-, or higher-dimensional space. Examples
include spatially localized objects, such as the equator, as
well as abstract lines.
guid: bd5906cb-9c29-11b1-9dad-c379636f7270
direct instance of: #$GenericShapeType
direct specialization of: #$Linear #$GeometricallyDescribableThing
direct generalization of: #$Arc #$Line-Straight
#$TwoDimensionalGeometricThing two
dimensional shapes
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of #$GeometricallyDescribableThings
(q.v.) that are two-dimensional. Each instance of #$TwoDimensionalGeometricThing
is a two-dimensional object whose shape is describable in
geometric terms. Examples include tangible objects, such as
the flat (two-dimensional) tangible surface of an oval
tabletop, as well as abstract two-dimensional objects.
guid: bd58c2a5-9c29-11b1-9dad-c379636f7270
direct instance of: #$GenericShapeType #$TwoDimensionalShapeType
direct specialization of:
#$GeometricallyDescribableThing #$TwoOrHigherDimensionalThing
direct generalization of: #$SectorShaped #$Oval
#$PlaneFigure-Geometrical #$HalfPlane #$Quadrilateral
#$Ellipse #$Polygon #$Circle
#$Angle angles
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of #$GeometricallyDescribableThing.
Each instance of #$Angle is formed by
two lines diverging from the same point, or two surfaces
diverging from the same line. Examples include spatially
localized objects, such as the angle formed by the
intersection of two walls, and abstract objects, such as the
angle formed by the intersection of two (abstract) lines.
guid: bd61bd87-9c29-11b1-9dad-c379636f7270
direct instance of: #$GenericShapeType
direct specialization of:
#$GeometricallyDescribableThing
#$NTupleInterval n
tuple interval
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of #$Tuple. Each instance of
#$NTupleInterval
is a tuple whose items are all intervals. Notable
specializations of #$NTupleInterval
include #$ScalarInterval,
#$VectorInterval,
and #$ComplexNumber.
guid: bd58ec55-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$Tuple
direct generalization of:
#$VectorInterval
#$ScalarInterval
#$ComplexNumber
#$ScalarInterval quantities
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of #$NTupleInterval.
Each proper subcollection SCALAR of #$ScalarInterval
(with some exceptions, such as #$ScalarPointValue)
has the following two properties: 1) the collection of point
instances of SCALAR (i.e., those instances of SCALAR that
are _not_ proper intervals - see the collection #$ScalarPointValue)
is ordered by some `natural' linear ordering (i.e., some
`natural' relation that is reflexive, antisymmetric, and
transitive on the collection of point instances of SCALAR),
2) SCALAR is closed under addition (#$PlusFn), so
that the sum of any two instances of SCALAR will also be an
instance of SCALAR. Notable specializations of #$ScalarInterval
include #$Time-Quantity,
#$Integer, and
#$Distance.
guid: bd5880a7-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$NTupleInterval
direct generalization of: #$PhysicalQuantity
#$NegativeScalarInterval #$JustificationStrength
#$TemperamentAttribute #$Interval-UnboundedAbove
#$Interval-BoundedBelow #$Interval-UnboundedBelow
#$Interval-BoundedAbove #$NonNegativeScalarInterval #$LevelOfDiscomfort
#$LevelOfPain
#$OrderOfMagnitudeInterval
#$ScalarPointValue
#$IntervalOnNumberLine
#$NonPositiveScalarInterval #$Dirtiness #$Money
#$PolitenessOfSpeech #$FormalityOfSpeech #$FeelingAttribute
#$Alertness
#$ScalarPointValue scalar values
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of minimal scalar intervals. That
is, each instance of #$ScalarPointValue
is a scalar interval which has no `width'. Scalar intervals
include both pure numbers, such as 3.14, and linear
measurements such as 3.14 meters, which in CycL would be
written `(#$Meter 3.14)'.
guid: bd58825b-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of: #$Interval-Bounded
direct generalization of:
#$RealNumber
#$Cardinal-Mathematical
#$ComplexNumber complex number
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A specialization of both #$Number-General and #$NTupleInterval.
Each instance of #$ComplexNumber
can be thought of as a vector of two numbers, which are
usually called the real part and the imaginary part of the
complex number. Complex numbers may also be considered as
corresponding to points in the real plane, where the x axis
determines the real component of a complex number and the y
axis the imaginary component. The unit value on the real
number line is 1, the unit value on the imaginary number
line is the square root of -1, generally written `i' in
mathematics and `j' in engineering.
guid: bd58b80a-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of: #$Number-General
direct generalization of:
#$RealNumber
#$MathematicalPoint mathematical point
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A collection of mathematical objects. Each element
of #$MathematicalPoint
is an atomic (i.e., structureless) abstract object.
guid: bd5903d9-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$MathematicalObject
#$Field-Mathematical fields
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A collection of spatial things. Each element of #$Field-Mathematical
is an abstract region through which forces (or other vector,
scalar, or tensor functions of position) are exerted; often,
relative to one or two objects.
guid: bd58eea6-9c29-11b1-9dad-c379636f7270
direct instance of:
#$Collection
direct specialization of:
#$SpatialThing #$IntangibleIndividual
#$SetOrCollection intensional
or extensional sets
A specialization of #$MathematicalThing.
Something is an instance of #$SetOrCollection
just in case it is a collection (i.e. an instance of #$Collection)
or a mathematical set (i.e. an instance of #$Set-Mathematical).
Instances of #$Set-Mathematical
and instances of #$Collection
(and thus instances of #$SetOrCollection)
share some basic common features. All instances of #$Collection
and all instances of #$Set-Mathematical
(and thus all instances of #$SetOrCollection)
are abstract entities, lacking spatial and temporal
properties. Nearly all instances of #$Collection
(except empty collections) and nearly all instances of #$Set-Mathematical
(except the empty set; see #$TheEmptySet) have
elements (i.e. instances or members; see #$elementOf); hence
set-or-collections may stand to one another in generalized
set-theoretic relations such as #$subsetOf and #$disjointWith
(qq.v.). (It is this shared feature of having elements that
provides the basic rationale for reifying the collection #$SetOrCollection.)
Nevertheless, sets and collections differ in two important
ways. First, each collection is intrinsically associated
with an intensional criterion for membership -- a more or
less natural property (or group of properties) possessed by
all of (and only) its elements. Collections are thus akin to
kinds. In contrast, the elements of a set are not required
to be homogeneous in any respect: any things whatsoever may
together constitute the elements of a set. The second major
difference between sets and collections is that no two
distinct sets can be coextensional (i.e. have exactly the
same elements; see #$coExtensional). Sets can thus be
identified purely on the basis of their extensions (see
#$extent). Collections, on the other hand, are individuated
by their intensional criteria for membership. So collections
that have exactly the same elements might nevertheless be
distinct, differing in their respective membership criteria.
(Note that the general relationship between collections and
their intensional criteria for membership in the above sense
is not something that is currently represented explicitly in
the Knowledge Base (though this seems a worthwhile area for
future work); still the #$comment and other
definitional assertions on a given collection should ideally
convey a reasonably clear and precise idea of its associated
membership criterion.)
guid: bd58e5fd-9c29-11b1-9dad-c379636f7270
direct instance of: #$VariableOrderCollection
#$SetOrCollectionType
direct specialization of:
#$MathematicalThing
direct generalization of: #$SetOrCollectionType #$Set-Mathematical
#$Collection
#$Set-Mathematical sets
(mathematical concepts)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of mathematical sets. An element of
#$Set-Mathematical
can be any arbitrary set, including sets whose members have
nothing in common. In contrast, the members of an instance
of #$Set-Mathematical's sibling #$Collection
(q.v.) all have some important, natural properties in
common. Sets and collections also differ in that there
cannot exist two distinct sets that have exactly the same
elements. A third point of contrast between sets and
collections is that rarely will it be desirable to create a
new constant to refer to a set. Instead, a set will either
be intensionally specified by a defining property, using #$TheSetOf, as in (#$TheSetOf ?X (#$and (#$isa ?X
#$PositiveInteger)(#$greaterThan ?X 42))), or extensionally
specified by listing its elements, using #$TheSet, as in (#$TheSet 3 4 5). (In
certain cases, a set will be extensionally specified by
means of one of the more specialized functions #$ThePartition
or #$TheCovering.
See #$partitionedInto
and #$covering.)
guid: be5d9e9f-9c29-11b1-9dad-c379636f7270
direct instance of:
#$Collection
direct specialization of:
#$SetOrCollection
#$DisjointSetOrCollection disjoint
set or collections
A collection of mathematical sets and collections
whose elements are themselves mathematical sets or
collections. A set or collection, SETORCOL, of sets or
collections is an instance of #$DisjointSetOrCollection
just in case the elements of SETORCOL are mutually disjoint
-- that is, no two elements of SETORCOL have any elements in common.
guid: be13fa12-9c29-11b1-9dad-c379636f7270
direct instance of: #$VariableOrderCollection
#$SetOrCollectionType
direct specialization of: #$SiblingDisjointSetOrCollection
direct generalization of: #$DisjointCollectionType
#$subsetOf subset
(taxonomic slot)
This predicate relates a set or collection SUB to a
set or collection SUPER whenever the extent (see #$extent)
of SUB is a subset of the extent of SUPER. That is, (#$subsetOf SUB SUPER)
means that every element of (see #$elementOf) SUB is
an element of SUPER. #$subsetOf is thus a
generalization both of the subset relation in set theory and
of #$genls
(q.v.); and (unlike either of those other relations) #$subsetOf can hold
between a set and a collection, or between a collection and
a set.
guid: bd903ed3-9c29-11b1-9dad-c379636f7270
direct instance of:
#$TaxonomicSlot
#$ReflexiveBinaryPredicate #$TransitiveBinaryPredicate
direct specialization of: #$most-GenQuant
#$generalizations
#$elementOf element of
(#$elementOf THING
SETORCOL) means that THING is an element of the mathematical
set or collection SETORCOL. #$elementOf is a more
general relation than #$isa, since #$isa is used
exclusively to talk about membership in instances of #$Collection.
#$elementOf,
unlike #$isa, can
also be used to talk about membership in arbitrarily-defined
mathematical sets (instances of #$Set-Mathematical),
such as those denoted by #$TheSet expressions.
guid: c0659a2b-9c29-11b1-9dad-c379636f7270
direct instance of:
#$TaxonomicSlotForAnyThing
#$TheSetOf the set of
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A function that results in a #$Set-Mathematical
of instances that satisfy some #$ELSentence-Assertible.
The first argument is an #$ELVariable
that appears as a free variable in the second argument. For
example, (#$TheSetOf ?X (#$and (#$isa ?X #$Dog) (#$objectHasColor
?X #$BlackColor) denotes the set of black dogs.
guid: bd58d7f6-9c29-11b1-9dad-c379636f7270
direct instance of: #$UnreifiableFunction
#$BinaryFunction #$ScopingRelation
#$SetDenotingFunction #$Individual
#$TheSet the set (relationship)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
(#$TheSet E1
E2 ... En) denotes the #$Set-Mathematical
consisting of the elements E1 through En. #$TheSet is a variable
arity relation, taking one or more arguments. All the
arguments must be ground terms; variables are not allowed.
See also #$TheSetOf
to define sets in the {x: ---x---} format.
guid: bd58e476-9c29-11b1-9dad-c379636f7270
direct instance of: #$VariableArityFunction
#$UnreifiableFunction #$CommutativeRelation
#$SetDenotingFunction #$Individual
#$TermSet term set
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
A collection of sets. Each element of #$TermSet is a set of
Cyc terms.
guid: bd597688-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$Set-Mathematical
Relational Structures
#$RelationalStructure relational structure
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all mathematical structures each
being a composite individual structure consisting of a #$baseSet with
structuring relations or operations on that set (and,
optionally, one or more selected other sets, relations,
functions or individuals). Examples include #$PartialOrderings,
#$Multigraphs,
etc. Sometimes mathematicians specify these using a #$Tuple of sets,
relations, functions, and/or individuals, as specifications.
Note that RelationalStructure is not the same as its #$baseSet since a pure
#$SetOrCollection
necessarily lacks the associated 'structure'.
guid: bf48bfa0-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$MathematicalObject
direct generalization of:
#$MathematicalOrdering
#$Multigraph
#$baseSet base set
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
(#$baseSet
STRUCTURE SET) means that SET is the base set of #$RelationalStructure
STRUCTURE. That is, SET is the domain from which the
individual elements of the structure are drawn. Although
there may be several relations involved in the relational
structure, they all relate members of the base set. As each
relational structure has a unique base set, this predicate
is functional.
guid: be51a217-9c29-11b1-9dad-c379636f7270
direct instance of:
#$FunctionalSlot
#$IntangibleObjectRelatingPredicate
#$orderingRelations ordering
relations (asymmetric binary predicate)
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
(#$orderingRelations
ORDER PRED) means that, in a #$MathematicalOrdering
ORDER, there is an order-predicate PRED that forms an
ordering relation on the #$baseSet of ORDER. The
predicate PRED, when restricted to the #$baseSet of the #$MathematicalOrdering
ORDER, is transitive on the #$baseSet of ORDER.
Note that the predicate #$orderingRelations
is not a functional predicate because we may have both (#$orderingRelations
ORDER PRED1) and (#$orderingRelations
ORDER PRED2). (This is true even if, extensionally, the two
predicates PRED1 and PRED2 denote only one set-theoretic
binary relation associated with ORDER.) Such an ordering
relation is not necessarily a full order or even a partial
order; it depends on the kind of #$MathematicalOrdering.
guid: c0416aa0-9c29-11b1-9dad-c379636f7270
direct instance of: #$AsymmetricBinaryPredicate
#$MathematicalOrdering mathematical ordering
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all those #$RelationalStructures
that are called 'orderings'. A #$MathematicalOrdering
is usually described as an ordered pair <S, R> where S is a
set and R is a binary relation on S that is transitive,
i.e., for each X, Y and Z in S, R(X Y) and R(Y Z) imply R(X
Z). We do not have to define a #$MathematicalOrdering
in Cyc as an ordered pair, but it is essential that each
such #$MathematicalOrdering
has a unique #$baseSet and a unique
ordering relation. We use, for each #$MathematicalOrdering
ORDER, (#$orderingRelations
ORDER PRED) to specify a binary predicate PRED, the
restriction of which to the #$baseSet S of ORDER
indicates the ordering relation R on S, and we require the
collections that are used to specify the arguments to PRED
to be supersets of S. In principle, there could be different
predicates PRED1 and PRED2 such that when restricted to the
same #$baseSet S of
a #$MathematicalOrdering
ORDER, they order the elements of S exactly the same way.
When we said that there is a unique ordering relation R on
S, we mean to ignore the difference between PRED1 and PRED2
when they are restricted to S, and treat the results of such
restrictions the same, as far as they are used to talk about
ORDER. Subcollections of #$MathematicalOrdering
include #$PartialOrdering,
#$PartialOrdering-Strict,
#$TreeOrdering, #$TreeOrdering-Strict,
#$TotalOrdering,
#$TotalOrdering-Strict,
#$WellOrdering and
#$Lattice-LatticeTheoretic, etc.
guid: bf3b5382-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$RelationalStructure
direct generalization of:
#$QuasiOrdering
#$PartialOrdering-Strict
#$QuasiOrdering quasi-orderings
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all those #$MathematicalOrderings
ORDER in which the ordering relation R is a reflexive and
transitive relation on the #$baseSet S of ORDER,
i.e., for each X in S, R(X X) holds, and for each X, Y and Z
in S, if R(X Y) and R(Y Z) then R(X Z). For example, if you
take the set of all people in the states today, and take the
relation '__ is at least as tall as ...' (i.e., either __ is
as tall as ... or __ is taller than ...) on this set, you
get a #$QuasiOrdering
because this relation is reflexive and transitive on the set
of all people in the states today. Note that the #$QuasiOrdering
in this example is neither a #$PartialOrdering
nor a #$PartialOrdering-Strict.
Subcollections of #$QuasiOrdering
include #$PartialOrdering,
#$TreeOrdering, #$TotalOrdering
and #$Lattice-LatticeTheoretic.
guid: c14247c0-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$MathematicalOrdering
direct generalization of:
#$PartialOrdering
#$PartialOrdering partial ordering
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all those #$MathematicalOrderings
ORDER in which the ordering relation R is a reflexive,
transitive and antisymmetric relation on the #$baseSet S of ORDER. R
is reflexive on S if and only if for each X in S, R(X X). R
is transitive on S if and only if for each X, Y and Z in S,
R(X Y) and R(Y Z) imply R(X Z). R is antisymmetric on S if
and only if for each X and Y in S, R(X Y) and R(Y X) imply X
= Y. For example, if you take a set of #$Lists and take the
#$subLists relation restricted to this set, then you have a
#$PartialOrdering
because #$subLists relation is reflexive, transitive and
antisymmetric. Since the ordering relation in each #$PartialOrderings
reflexive and transitive, the collection #$PartialOrdering
is a subcollection of #$QuasiOrdering.
Subcollections of #$PartialOrdering
include #$TreeOrdering, #$TotalOrdering
and #$Lattice-LatticeTheoretic.
If you want a #$MathematicalOrdering
in which the ordering relation is irreflexive, see #$PartialOrdering-Strict.
guid: c135aad5-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$QuasiOrdering
direct generalization of:
#$TreeOrdering
#$Lattice-LatticeTheoretic
#$Lattice-LatticeTheoretic lattice order
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all mathmetical structures called
'lattices' in Lattice Theory (this is not the same concept
as the crystalline or grid lattices studied in
Crystallography and Group Theory). A #$Lattice-LatticeTheoretic
is often defined in different but equivalent ways. To define
a lattice using ordering relation, it is a #$PartialOrdering
ORDER in which each pair of elements of the #$baseSet S of ORDER
has an R-smallest upper bound and an R-greatest lower bound,
where R is the ordering relation on S. To define a lattice
using operations, it is a mathematical structure with two
operations MEET and JOIN on the #$baseSet S of the
structure that satisfy the commutative laws, the associative
laws, the idempotent laws and the absorption laws. Note that
the correspondence of these two different ways of defining
lattices is characterized by the following: for all X, Y in
S, Y = (JOIN X Y) <=> R(X Y) <=> X = (MEET X Y). See
#$meetFunctionOnLattice and #$joinFunctionOnLattice.
guid: bdfb75ea-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$PartialOrdering
direct generalization of:
#$ModularLattice
#$ModularLattice modular lattices
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The subcollection of #$Lattice-LatticeTheoretic
that contains all those lattices called 'modular lattices'
by the lattice theorists. A #$Lattice-LatticeTheoretic
LATTICE (with the #$baseSet S, ordering
relation R, meet operation MEET and join operation JOIN) is
a #$ModularLattice
if the following condition holds: for each X, Y and Z in S,
if R(Z X) then (MEET X (JOIN Y Z)) = (JOIN (MEET X Y) (MEET
X Z)). Note that there are many conditions equivalent to the
one above, one of which is the condition that for each X, Y
and Z in S, if R(Z X) then (JOIN X (MEET Y Z)) = (MEET (JOIN
X Y) (MEET X Z)). (Other equivalent conditions may be found
by browsing the rules for #$ModularLattice.)
Note also that a #$DistributiveLattice
satisfies this condition automatically, and therefore is a
#$ModularLattice.
guid: c0caf649-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$Lattice-LatticeTheoretic
direct generalization of:
#$DistributiveLattice
#$DistributiveLattice distributive lattices
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The subcollection of #$Lattice-LatticeTheoretic
that contains all those lattices called 'distributive
lattices' by lattice theorists. A #$Lattice-LatticeTheoretic
LATICE (with the #$baseSet S, meet
operation MEET and join operation JOIN) is distributive if
the distribution laws hold, i.e., for each X, Y and Z in S,
(MEET X (JOIN Y Z)) = (JOIN (MEET X Y) (MEET X Z)). Note
that this condition is equivalent to the condition that for
each X, Y and Z in S, (JOIN X (MEET Y Z)) = (MEET (JOIN X Y)
(JOIN X Z)).
guid: bdbc5d4c-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$ModularLattice
direct generalization of:
#$TotalOrdering
#$PartialOrdering-Strict strict
partial order
The collection of all #$MathematicalOrderings
ORDER in which the ordering relation R is an irreflexive and
transitive relation on the #$baseSet S of ORDER,
i.e., for each X in S, R(X X) does not hold, and for each X,
Y and Z in S, R(X Y) and R(Y Z) imply R(X Z). For example,
if one takes the set of all people, and the relation '__ is
older than ...' on this set, one gets a #$PartialOrdering-Strict,
since the relation '__ is older than ...' is irreflexive and
transitive on this set. Note that the important difference
between a #$PartialOrdering
(q.v.) and a #$PartialOrdering-Strict
is that the ordering relation of the former is reflexive,
while that of the latter is irreflexive. Note also that the
ordering relation R of a #$PartialOrdering-Strict
ORDER is in fact antisymmetric (i.e., for each X and Y in S,
R(X Y) and R(Y X) imply X = Y) and asymmetric (i.e., for
each X and Y in S, R(X Y) and R(Y X) can never be both true)
on the #$baseSet S.
This is because both antisymmetry and asymmetry follow from
transitivity and irreflexivity.
guid: be73e9fb-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$MathematicalOrdering
direct generalization of:
#$TreeOrdering-Strict
#$TreeOrdering tree ordering
A specialization of #$PartialOrdering.
An instance ORDER of #$PartialOrdering
is also an instance of #$TreeOrdering
just in case the ordering relation R of ORDER orders
elements of the #$baseSet S of ORDER
into a tree-like structure, so that each pair of elements of
S has a common 'R-lower-bound' in S (i.e., for each X, Y in
S, there is a Z in S such that R(Z X) and R(Z Y)), and the
set of 'R-lower-bounds' of each X in S is ordered in a chain
by R (i.e., {Y: Y is in S and R(Y X)} is a chain). Note that
an instance of #$TreeOrdering can
itself be a chain, i.e., an instance of #$TotalOrdering.
guid: beb293c3-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$PartialOrdering
direct generalization of:
#$TotalOrdering
#$TreeOrdering-Strict tree
ordering strict
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all those #$MathematicalOrderings
ORDER in which the ordering relation R is irreflexive and
transitive on the #$baseSet S of ORDER,
and in which every pair of different elements of S has a
common 'R-lower-bound', and the set of 'R-lower-bounds' of
each X in S is ordered in a (possibly empty) chain by R
(i.e., {Y: Y is in S and R(Y X)} is a (possibly empty)
chain). Note that the only difference between a #$TreeOrdering and
a #$TreeOrdering-Strict
is that the ordering relation of the former is reflexive
while that of the latter is irreflexive.
guid: c136761b-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$PartialOrdering-Strict
direct generalization of:
#$TotalOrdering-Strict
#$TotalOrdering total ordering
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all those #$PartialOrdering
ORDER in which the ordering relation R orders elements of
the #$baseSet S of
ORDER into a single line. A #$TotalOrdering
is sometimes called a 'linear ordering' or a 'chain'. A #$PartialOrdering
ORDER is a #$TotalOrdering
if all elements of the #$baseSet S of ORDER
are comparable by the ordering relation R of ORDER, i.e.,
for any X and Y in S, either R(X Y) or R(Y X). For example,
if you take a set of real numbers and the usual 'greater
than or equal to' relation among these numbers, you have a
#$TotalOrdering.
Note that if you want a 'strict line', i.e., if you want the
ordering relation in a total ordering to be irreflexive, see
#$TotalOrdering-Strict.
guid: bf4734b2-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$DistributiveLattice #$TreeOrdering
#$TotalOrdering-Strict total
ordering - strict
A specialization of #$TreeOrdering-Strict.
Each instance of #$TotalOrdering-Strict
is an ordering in which the ordering relation R orders
elements of the #$baseSet S into a
strict line. An instance ORDER of #$PartialOrdering-Strict
is an instance of #$TotalOrdering-Strict
just in case all elements of the #$baseSet S of ORDER
are comparable by the ordering relation R of ORDER; i.e., if
X and Y are elements of S, then either R(X Y) or X = Y or
R(Y X). One example of a #$TotalOrdering-Strict
is the set of real numbers with the usual 'smaller than'
relation on those numbers.
guid: be4967b0-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$TreeOrdering-Strict
direct generalization of:
#$WellOrdering
#$WellOrdering well-orderings
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all those linear #$MathematicalOrderings
that are called 'well-orderings' because they (and their
subsets) have 'first' or 'smallest' members. Let ORDER be a
#$TotalOrdering-Strict,
and let S be the #$baseSet of ORDER and
R the ordering relation. ORDER is a #$WellOrdering if
every nonempty subset SUBSET of S has its 'R-smallest'
member, i.e., there is an X in SUBSET such that R(X Y) for
all Y in SUBSET except X. For example, if you take the set
of all natural numbers and the usual less-than relation
among these numbers then you have a #$WellOrdering
because there is a lowest natural number, while if you take
the set of all integers (including negative ones) and the
usual less-than relation among these numbers then you have a
#$TotalOrdering
that is not a #$WellOrdering
because there is no lowest negative number.
guid: bfa6adcd-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$TotalOrdering-Strict
direct generalization of:
#$ListWithoutRepetition
#$ListWithoutRepetition lists
without repetition
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all those #$Lists, for each of
which, no element appears more than once in the list. A #$ListWithoutRepetition
is sometimes called an 'OSET'. Note that a #$ListWithoutRepetition
amounts to a finite #$TotalOrdering-Strict.
(In general a #$List
and #$Series may
have repeated elements.) See also #$SeriesWithoutRepetition.
guid: bdb9081e-9c29-11b1-9dad-c379636f7270
direct instance of:
#$ObjectType
direct specialization of:
#$WellOrdering #$List
#$Multigraph multigraph
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
An instance of #$PathSystemType-Structural
and a subcollection of #$PathSystem.
Each instance of #$Multigraph
is an instance of #$PathSystem
in which the only points are nodes in the system and all
paths are made of links (i.e., no intermediate points along
links). Sometime such a system is called a graph or
multi-graph in graph theory. A #$Multigraph
consists of nodes interconnected by links, with loops on
single nodes allowed, and with multiple links between the
same two nodes also allowed. (For a graph with no parallel
links and no loops, see #$SimpleGraph-GraphTheoretic.
guid: bde212ef-9c29-11b1-9dad-c379636f7270
direct instance of:
#$PathSystemType-Structural
direct specialization of:
#$PointFinitePathSystem #$RelationalStructure
direct generalization of:
#$DirectedMultigraph
#$SimpleGraph-GraphTheoretic
#$SimpleGraph-GraphTheoretic simple graphs
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all #$PathSystems
that are instances of both #$SimplePathSystem
and #$Multigraph.
Each instance of #$SimpleGraph-GraphTheoretic
is a 'graph', as studied in graph theory, in which there are
neither loops nor multiple links between the same pair of nodes.
guid: be269b3d-9c29-11b1-9dad-c379636f7270
direct instance of:
#$PathSystemType-Structural
direct specialization of:
#$SimplePathSystem #$Multigraph
direct generalization of:
#$DirectedGraph
#$DirectedMultigraph directed multigraph
A specialization of both #$DirectedPathSystem
and #$Multigraph.
Each instance of #$DirectedMultigraph
is a multigraph in which every link has one direction. Note
that there can be loops and multiple links between a pair of
nodes in a given instance of #$DirectedMultigraph.
guid: c0ba0c32-9c29-11b1-9dad-c379636f7270
direct instance of:
#$PathSystemType-Structural
direct specialization of:
#$DirectedPathSystem #$Multigraph
direct generalization of:
#$DirectedGraph
#$DirectedGraph directed graph
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all directed simple graphs, i.e.,
node-and-link structure in which every link has one
direction and no multiple links (between a pair of nodes) or
loops are allowed, as studied in graph theory. This is the
intersection of #$SimpleGraph-GraphTheoretic
and #$DirectedMultigraph,
which is the same as the intersection of #$SimpleGraph-GraphTheoretic
and #$DirectedPathSystem.
guid: beb3df26-9c29-11b1-9dad-c379636f7270
direct instance of:
#$PathSystemType-Structural
direct specialization of:
#$DirectedMultigraph #$SimpleGraph-GraphTheoretic
direct generalization of:
#$DirectedAcyclicGraph
#$DirectedAcyclicGraph Directed
Acyclic Graph
**COMMENT NOT REVIEWED**
**GAFs NOT REVIEWED**
The collection of all those #$DirectedGraphs
(node-and-link structures in which each link has one
direction) each of which has no directed cycle in it. This
is the intersection of #$DirectedGraph
and #$DirectedAcyclicPathSystem
(which is the same as the intersection of #$SimpleGraph-GraphTheoretic
and #$DirectedAcyclicPathSystem).
A #$DirectedAcyclicGraph
is often used as a representation of a #$PartialOrdering.
guid: bed5ca59-9c29-11b1-9dad-c379636f7270
direct instance of:
#$PathSystemType-Structural
direct specialization of:
#$DirectedGraph #$DirectedAcyclicPathSystem
Copyright© 1996-2002. All rights reserved. See Terms of Usage.