]> 2008/06/10 OpenCyc Knowledge Base Copyright© 2001-2008 Cycorp, Inc., http://www.cyc.com/, Austin, TX, USA This file contains an OWL representation of information contained in the OpenCyc Knowledge Base. The content of this OWL file is licensed under the Creative Commons Attribution 3.0 license whose text can be found at http://creativecommons.org/licenses/by/3.0/legalcode. The content of this OWL file, including the OpenCyc content it represents, constitutes the "Work" referred to in the Creative Commons license. The terms of this license equally apply to, without limitation, renamings and other logically equivalent reformulations of the content of this OWL file (or portions thereof) in any natural or formal language, as well as to derivations of this content or inclusion of it in other ontologies. externalID A unique, language-neutral, variable-sized identifier for a concept that can be used to refer unambiguously to that concept across OWL exports or across Cyc inference engines. label A natural-language representation for a concept that is both human readable and readable by the Cyc inference engine. These terms are not guaranteed to refer to the same concept across time but are guaranteed to be consistent within a particular OWL export. Use 'cycAnnot:externalID' for unambiguously referring to a concept across OWL exports or across Cyc inference engines. SubsetsOfMathematicalThing-Math-Topic subsets of mathematical thing math topic LogicalConnective logical connective A collection of mathematical objects, including the basic logical connectives. Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVi5-ZwpEbGdrcN5Y29ycA" class="cyc_term">LogicalConnective</a> is a <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzZwpEbGdrcN5Y29ycA" class="cyc_term">Relation</a> which takes one or more truth-valued expressions (sentences) as arguments and returns a truth-valued sentence. The instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVi5-ZwpEbGdrcN5Y29ycA" class="cyc_term">LogicalConnective</a> include <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA-ZwpEbGdrcN5Y29ycA" class="cyc_term">and</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA-pwpEbGdrcN5Y29ycA" class="cyc_term">or</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA-5wpEbGdrcN5Y29ycA" class="cyc_term">not</a>, and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA-JwpEbGdrcN5Y29ycA" class="cyc_term">implies</a>. A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzZwpEbGdrcN5Y29ycA" class="cyc_term">Relation</a>. Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvV0LVpwpEbGdrcN5Y29ycA" class="cyc_term">VariableArityRelation</a> is a relation that can take a variable number of arguments. The degree of variability for a given such relation can be constrained using the predicates <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwSjE8ZwpEbGdrcN5Y29ycA" class="cyc_term">arityMin</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvWotlJwpEbGdrcN5Y29ycA" class="cyc_term">arityMax</a>. Examples of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvV0LVpwpEbGdrcN5Y29ycA" class="cyc_term">VariableArityRelation</a>s include the predicate <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvWPzQ5wpEbGdrcN5Y29ycA" class="cyc_term">different</a> and the function <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViArpwpEbGdrcN5Y29ycA" class="cyc_term">PlusFn</a>. Thus the terms '(<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViArpwpEbGdrcN5Y29ycA" class="cyc_term">PlusFn</a> 1 2)' and '(<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViArpwpEbGdrcN5Y29ycA" class="cyc_term">PlusFn</a> 1 2 3)' are both well-formed. Cf. <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv_MjepwpEbGdrcN5Y29ycA" class="cyc_term">FixedArityRelation</a>. VariableArityRelation variable-arity relation the empty set TheEmptySet <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvfjtrpwpEbGdrcN5Y29ycA" class="cyc_term">TheEmptySet</a> is the empty (or "null") set: the unique set that has no elements. Note that <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvfjtrpwpEbGdrcN5Y29ycA" class="cyc_term">TheEmptySet</a> is an instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvl2en5wpEbGdrcN5Y29ycA" class="cyc_term">Set_Mathematical</a> and thus _not_ an instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzJwpEbGdrcN5Y29ycA" class="cyc_term">Collection</a>. A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjl_ZwpEbGdrcN5Y29ycA" class="cyc_term">SetOrCollection</a> (q.v.); the collection of mathematical sets. An instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvl2en5wpEbGdrcN5Y29ycA" class="cyc_term">Set_Mathematical</a> can be any arbitrary set of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA9JwpEbGdrcN5Y29ycA" class="cyc_term">Thing</a>s. A good way to explain this notion with respect to the Cyc ontology is to contrast <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvl2en5wpEbGdrcN5Y29ycA" class="cyc_term">Set_Mathematical</a> with <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzJwpEbGdrcN5Y29ycA" class="cyc_term">Collection</a> (q.v.). First, while the instances of a given collection all have some more-or-less significant (often "natural") property or properties in common, the elements (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwGWaK5wpEbGdrcN5Y29ycA" class="cyc_term">elementOf</a>) in a given set might have nothing in common (besides membership in that set). Second, while it is in principle possible for two distinct collections to have exactly the same elements (with respect to a given context), this cannot happen in the case of sets, which are individuated strictly in terms of their extensions (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwN2YGZwpEbGdrcN5Y29ycA" class="cyc_term">extent</a>). Third (and specifically regarding their expression in the CycL language), unlike with collections, rarely will it be desirable to create a new constant to denote a particular set. Instead, a set will often be either (a) intensionally specified by a defining property via <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjX9pwpEbGdrcN5Y29ycA" class="cyc_term">TheSetOf</a> (q.v.), as in `(<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjX9pwpEbGdrcN5Y29ycA" class="cyc_term">TheSetOf</a> ?X (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA-ZwpEbGdrcN5Y29ycA" class="cyc_term">and</a> (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViBBJwpEbGdrcN5Y29ycA" class="cyc_term">isa</a> ?X <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVieEpwpEbGdrcN5Y29ycA" class="cyc_term">Integer</a>) (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAspwpEbGdrcN5Y29ycA" class="cyc_term">greaterThan</a> ?X 42)))', or (b) extensionally specified by enumerating its elements via <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjkdpwpEbGdrcN5Y29ycA" class="cyc_term">TheSet</a> (q.v.), as in `(<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjkdpwpEbGdrcN5Y29ycA" class="cyc_term">TheSet</a> 3 4 5)'; see also <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvcUeB5wpEbGdrcN5Y29ycA" class="cyc_term">ThePartition</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwMTOWpwpEbGdrcN5Y29ycA" class="cyc_term">TheCovering</a>. Set-Mathematical set A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4r0pGnmMbrEdmAAAACs0uFOQ" class="cyc_term">LinearObject_OneDimensional</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rREw8nsfnEdmAAAACs0uFOQ" class="cyc_term">LinearObject_Unbounded</a>, and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rUeq9IMb4EdmAAAACs0uFOQ" class="cyc_term">LinearObject_Infinite</a> (qq.v.). Instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVkGy5wpEbGdrcN5Y29ycA" class="cyc_term">Line</a> are one-dimensional, unbounded, infinite portions of space. They might be straight or curved. Specializations of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVkGy5wpEbGdrcN5Y29ycA" class="cyc_term">Line</a> include <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvbsAspwpEbGdrcN5Y29ycA" class="cyc_term">Line_Planar</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rRbPm7sl4EdmAAAACs0uFOQ" class="cyc_term">Line_NonPlanar</a>. And cf. <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwKUKtJwpEbGdrcN5Y29ycA" class="cyc_term">Point</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvg1ZeJwpEbGdrcN5Y29ycA" class="cyc_term">Surface</a>, and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvmaaAZwpEbGdrcN5Y29ycA" class="cyc_term">SpaceChunk</a>. Line space line A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvg1ZeJwpEbGdrcN5Y29ycA" class="cyc_term">Surface</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjELpwpEbGdrcN5Y29ycA" class="cyc_term">GeometricallyDescribableThing</a>, and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwJRrmJwpEbGdrcN5Y29ycA" class="cyc_term">ShapedObject</a> (qq.v.). Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjCpZwpEbGdrcN5Y29ycA" class="cyc_term">TwoDimensionalGeometricThing</a> is an intangible, two-dimensional, self-connected object whose shape is describable in (relatively simple) geometric terms. Instances include planes, as well as all (planar or "curved") polygons and elliptical regions. For bounded two-dimensional geometric objects that can be embedded in a <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rN1T3Nlv9QdeJ7LFO9u6POw" class="cyc_term">Plane</a>, see the specialization <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rviHxm5wpEbGdrcN5Y29ycA" class="cyc_term">PlaneFigure</a>. TwoDimensionalGeometricThing two dimensional shape MathematicalThing mathematical concept A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjjH5wpEbGdrcN5Y29ycA" class="cyc_term">MathematicalOrComputationalThing</a>. Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjltpwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalThing</a> is an atemporal, nonspatial, purely mathematical thing. <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjltpwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalThing</a> is partitioned into two main specializations, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv0YfN5wpEbGdrcN5Y29ycA" class="cyc_term">MathematicalObject</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjl_ZwpEbGdrcN5Y29ycA" class="cyc_term">SetOrCollection</a> (qq.v). A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv0YfN5wpEbGdrcN5Y29ycA" class="cyc_term">MathematicalObject</a>. Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjUoJwpEbGdrcN5Y29ycA" class="cyc_term">FrameOfReference</a> is a mathematical (and hence intangible) representation of the context in which certain data are to be interpreted. Such contexts are typically physical, but may be purely mathematical. For example, a <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwIU_XZwpEbGdrcN5Y29ycA" class="cyc_term">CartesianCoordinateSystem</a> can be used to represent the positions of things on the surface of the Eath, but can also be used to represent an abstract gepometric space. FrameOfReference frame of reference individual mathematical object A specialization of both <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjltpwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalThing</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjj45wpEbGdrcN5Y29ycA" class="cyc_term">IntangibleIndividual</a>. Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv0YfN5wpEbGdrcN5Y29ycA" class="cyc_term">MathematicalObject</a> is a purely abstract mathematical thing which is also an individual (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjaApwpEbGdrcN5Y29ycA" class="cyc_term">Individual</a>). Specializations of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv0YfN5wpEbGdrcN5Y29ycA" class="cyc_term">MathematicalObject</a> include <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjCcZwpEbGdrcN5Y29ycA" class="cyc_term">Quantifier</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAmJwpEbGdrcN5Y29ycA" class="cyc_term">RealNumber</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjrapwpEbGdrcN5Y29ycA" class="cyc_term">Triangle</a>, and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjjupwpEbGdrcN5Y29ycA" class="cyc_term">TruthValue</a>. Note that instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjl_ZwpEbGdrcN5Y29ycA" class="cyc_term">SetOrCollection</a> are not instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv0YfN5wpEbGdrcN5Y29ycA" class="cyc_term">MathematicalObject</a>, since they are not instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjaApwpEbGdrcN5Y29ycA" class="cyc_term">Individual</a>. MathematicalObject This predicate relates a set or collection <code>SUB</code> to a set or collection <code>SUPER</code> whenever the extent (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwN2YGZwpEbGdrcN5Y29ycA" class="cyc_term">extent</a>) of <code>SUB</code> is a subset of the extent of <code>SUPER</code>. That is, <code>(<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvZA-05wpEbGdrcN5Y29ycA" class="cyc_term">subsetOf</a> SUB SUPER)</code> means that every element of (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwGWaK5wpEbGdrcN5Y29ycA" class="cyc_term">elementOf</a>) <code>SUB</code> is an element of <code>SUPER</code>. <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvZA-05wpEbGdrcN5Y29ycA" class="cyc_term">subsetOf</a> is thus a generalization both of the subset relation in set theory and of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViBDpwpEbGdrcN5Y29ycA" class="cyc_term">genls</a> (q.v.); and (unlike either of those other relations) <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvZA-05wpEbGdrcN5Y29ycA" class="cyc_term">subsetOf</a> can hold between a set and a collection, or between a collection and a set. subsetOf subset 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 <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rk8dxOVcGEdaLwgACs0uFOQ" class="cyc_term">DisjointSetOrCollectionType</a> just in case the elements of SETORCOL are mutually disjoint -- that is, no two elements of SETORCOL have any elements in common. DisjointSetOrCollectionType disjoint set or collection elementOf Element Of A very general binary predicate that relates a thing to any set or collection (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjl_ZwpEbGdrcN5Y29ycA" class="cyc_term">SetOrCollection</a>) that it is a member or element of. <code>(<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwGWaK5wpEbGdrcN5Y29ycA" class="cyc_term">elementOf</a> THING SETORCOL)</code> means that <code>THING</code> is an element of <code>SETORCOL</code>. <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwGWaK5wpEbGdrcN5Y29ycA" class="cyc_term">elementOf</a> is a more general relation than <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViBBJwpEbGdrcN5Y29ycA" class="cyc_term">isa</a>. Whereas <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViBBJwpEbGdrcN5Y29ycA" class="cyc_term">isa</a> is used exclusively to talk about membership in <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzJwpEbGdrcN5Y29ycA" class="cyc_term">Collection</a>s, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwGWaK5wpEbGdrcN5Y29ycA" class="cyc_term">elementOf</a> can also be used to talk about membership in mathematical sets (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvl2en5wpEbGdrcN5Y29ycA" class="cyc_term">Set_Mathematical</a>). geometric form GeometricallyDescribableThing A subcollection of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjpUZwpEbGdrcN5Y29ycA" class="cyc_term">SpatialThing</a>. Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjELpwpEbGdrcN5Y29ycA" class="cyc_term">GeometricallyDescribableThing</a> is a spatially-connected spatial thing (of one, two, three, or four dimensions) that either (i) has or approximates a simple geometric shape (e.g. it is a <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVkGy5wpEbGdrcN5Y29ycA" class="cyc_term">Line</a> or a <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4ryStbyFITEdqAAAACs0uFOQ" class="cyc_term">Hemisphere</a>) or (ii) consists of a small number of parts in a relatively simple (and stable) simple 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 geometrically-describable thing might be tangible or intangible. Note that what counts as "approximating" a given simple geometric shape -- and thus what spatial things count as <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjELpwpEbGdrcN5Y29ycA" class="cyc_term">GeometricallyDescribableThing</a>s -- 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. False An instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjjupwpEbGdrcN5Y29ycA" class="cyc_term">TruthValue</a> (q.v.). <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA2JwpEbGdrcN5Y29ycA" class="cyc_term">False</a> is the logical notion of falsehood. That is, the term '<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA2JwpEbGdrcN5Y29ycA" class="cyc_term">False</a>' is used as a sentential constant of CycL that is false under every model theoretic interpretation. For example, (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwP1T85wpEbGdrcN5Y29ycA" class="cyc_term">booleanResult</a> T/F <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA2JwpEbGdrcN5Y29ycA" class="cyc_term">False</a>) means that the result obtained from the true-or-false test T/F is False. Cf. <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA2ZwpEbGdrcN5Y29ycA" class="cyc_term">True</a>. false Fractal A collection of functions. Each element of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwQwa5JwpEbGdrcN5Y29ycA" class="cyc_term">Fractal</a> is a function which, when applied to data, can be displayed visually as an instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwQwbk5wpEbGdrcN5Y29ycA" class="cyc_term">FractalRepresentation</a>. fractal SetOrCollection intensional or extensional set A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjltpwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalThing</a>. Something is an instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjl_ZwpEbGdrcN5Y29ycA" class="cyc_term">SetOrCollection</a> just in case it is a collection (i.e. an instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzJwpEbGdrcN5Y29ycA" class="cyc_term">Collection</a>) or a mathematical set (i.e. an instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvl2en5wpEbGdrcN5Y29ycA" class="cyc_term">Set_Mathematical</a>). Instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvl2en5wpEbGdrcN5Y29ycA" class="cyc_term">Set_Mathematical</a> and instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzJwpEbGdrcN5Y29ycA" class="cyc_term">Collection</a> (and thus instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjl_ZwpEbGdrcN5Y29ycA" class="cyc_term">SetOrCollection</a>) share some basic common features. All instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzJwpEbGdrcN5Y29ycA" class="cyc_term">Collection</a> and all instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvl2en5wpEbGdrcN5Y29ycA" class="cyc_term">Set_Mathematical</a> (and thus all instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjl_ZwpEbGdrcN5Y29ycA" class="cyc_term">SetOrCollection</a>) are abstract entities, lacking spatial and temporal properties. Nearly all instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzJwpEbGdrcN5Y29ycA" class="cyc_term">Collection</a> (except "empty" collections) and nearly all instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvl2en5wpEbGdrcN5Y29ycA" class="cyc_term">Set_Mathematical</a> (except the empty set; see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvfjtrpwpEbGdrcN5Y29ycA" class="cyc_term">TheEmptySet</a>) have "elements" (i.e. instances or members; see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwGWaK5wpEbGdrcN5Y29ycA" class="cyc_term">elementOf</a>); hence set-or-collections may stand to one another in generalized set-theoretic relations such as <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvZA-05wpEbGdrcN5Y29ycA" class="cyc_term">subsetOf</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA45wpEbGdrcN5Y29ycA" class="cyc_term">disjointWith</a> (qq.v.). (It is this shared feature of having elements that provides the basic rationale for reifying the collection <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjl_ZwpEbGdrcN5Y29ycA" class="cyc_term">SetOrCollection</a>.) 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 <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVkIOpwpEbGdrcN5Y29ycA" class="cyc_term">coExtensional</a>). Sets can thus be identified purely on the basis of their extensions (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwN2YGZwpEbGdrcN5Y29ycA" class="cyc_term">extent</a>). 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 <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViBCZwpEbGdrcN5Y29ycA" class="cyc_term">comment</a> and other "definitional" assertions on a given collection should ideally convey a reasonably clear and precise idea of its associated membership criterion.) TruthValue truth value <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjjupwpEbGdrcN5Y29ycA" class="cyc_term">TruthValue</a> is a collection of mathematical objects; it contains the abstract, logical objects <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA2ZwpEbGdrcN5Y29ycA" class="cyc_term">True</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA2JwpEbGdrcN5Y29ycA" class="cyc_term">False</a>. Relation A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv0YfN5wpEbGdrcN5Y29ycA" class="cyc_term">MathematicalObject</a> and the collection of all relations. Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzZwpEbGdrcN5Y29ycA" class="cyc_term">Relation</a> is a relation that can hold among one or more things, depending on whether the relation is unary, binary, ternary, or whatever (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzpwpEbGdrcN5Y29ycA" class="cyc_term">arity</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv3v3BpwpEbGdrcN5Y29ycA" class="cyc_term">relationalArity</a>). A unary relation (such as <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvh5RNpwpEbGdrcN5Y29ycA" class="cyc_term">unknownSentence</a>) is a sort of degenerate case that holds of certain individual things (in this case, all sentences that are unknown to Cyc). A binary relation (such as <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVi6bJwpEbGdrcN5Y29ycA" class="cyc_term">likesAsFriend</a>) relates one thing to another (in this case, it relates one sentient animal to another just in case the first likes the second). A ternary relation relates certain triples of things. And so on. There are also relations of no particular fixed arity; see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvV0LVpwpEbGdrcN5Y29ycA" class="cyc_term">VariableArityRelation</a>. Names of relations can be used to construct sentences and other formulas. More precisely, CycL terms that denote <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzZwpEbGdrcN5Y29ycA" class="cyc_term">Relation</a>s can appear in the "0th" argument (or "arg0") position of a <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwD7DupwpEbGdrcN5Y29ycA" class="cyc_term">CycLFormula</a> (q.v.), i.e. as the term immediately following the formula's opening parenthesis. An important subcollection of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzZwpEbGdrcN5Y29ycA" class="cyc_term">Relation</a> is <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvqZFUZwpEbGdrcN5Y29ycA" class="cyc_term">TruthFunction</a> (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 <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAoJwpEbGdrcN5Y29ycA" class="cyc_term">CycLSentence</a>; and a sentence (if quantificationally closed; see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvieo7JwpEbGdrcN5Y29ycA" class="cyc_term">CycLClosedSentence</a>), will generally be either true or false (with respect to a given context or interpretation). The major subcollections of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvqZFUZwpEbGdrcN5Y29ycA" class="cyc_term">TruthFunction</a> are <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA1pwpEbGdrcN5Y29ycA" class="cyc_term">Predicate</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVi5-ZwpEbGdrcN5Y29ycA" class="cyc_term">LogicalConnective</a>, and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjCcZwpEbGdrcN5Y29ycA" class="cyc_term">Quantifier</a> (qq.v.). Another important subcollection of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAzZwpEbGdrcN5Y29ycA" class="cyc_term">Relation</a> is <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVxAsJwpEbGdrcN5Y29ycA" class="cyc_term">Function_Denotational</a> (q.v.), the collection of all functions. A CycL term that denotes a function can appear in the arg0 position of a <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvx-VHZwpEbGdrcN5Y29ycA" class="cyc_term">CycLNonAtomicTerm</a> (q.v.). See also <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwP41gJwpEbGdrcN5Y29ycA" class="cyc_term">relationExtension</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvayXMZwpEbGdrcN5Y29ycA" class="cyc_term">relationHoldsAmong</a>. relationship scalar value A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAp5wpEbGdrcN5Y29ycA" class="cyc_term">ScalarInterval</a> (q.v.). Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViCW5wpEbGdrcN5Y29ycA" class="cyc_term">ScalarPointValue</a> is a specific number or quantity, as opposed to a range of numbers or quantities (cf. <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rLOwZCvkjQdea-sfI2QK3kQ" class="cyc_term">ScalarProperInterval</a>). Instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViCW5wpEbGdrcN5Y29ycA" class="cyc_term">ScalarPointValue</a> include all reals numbers and other <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvxwC-ZwpEbGdrcN5Y29ycA" class="cyc_term">Number_General</a>s (q.v.), as well as specific instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rSGNLQK45EdqAAAACs6hRXg" class="cyc_term">Quantity</a> such as (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjRp5wpEbGdrcN5Y29ycA" class="cyc_term">Meter</a> 3) and (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjrsZwpEbGdrcN5Y29ycA" class="cyc_term">SecondsDuration</a> 10). ScalarPointValue A specialization of both <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv0YfN5wpEbGdrcN5Y29ycA" class="cyc_term">MathematicalObject</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4r6iVgyruPQdefX9D0i1i7MQ" class="cyc_term">AbstractIndividual</a> (qq.v.). Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVj42JwpEbGdrcN5Y29ycA" class="cyc_term">Tuple</a> is a complex consisting of one or more indexed (and possibly ordered) components; it might be a single, a pair, a triple, or so on; and the components might be things of any sort whatsoever (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rx1wVcpg-EdaAAACQJ5qciw" class="cyc_term">memberOfTuple</a>). For example, a specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVj42JwpEbGdrcN5Y29ycA" class="cyc_term">Tuple</a> is <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjsVZwpEbGdrcN5Y29ycA" class="cyc_term">TupleOfIntervals</a> (q.v.), whose instances are tuples consisting exclusively of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAp5wpEbGdrcN5Y29ycA" class="cyc_term">ScalarInterval</a>s (q.v.); e.g. complex numbers and physical vectors are n-tuple-intervals. Another specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVj42JwpEbGdrcN5Y29ycA" class="cyc_term">Tuple</a> is <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvtUAU5wpEbGdrcN5Y29ycA" class="cyc_term">List</a> (q.v.), whose instances are ordered. Each tuple has an associated "index set": the set of things that serve (via an associated "indexing function") to index or individually represent the tuple's members (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rZ9tXZphDEdaAAACQJ5qciw" class="cyc_term">tupleIndexSet</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rCY9tbphIEdaAAACQJ5qciw" class="cyc_term">tupleMemberIndex</a>). If the index set for a given tuple is the set of positive integers (or an initial segment thereof), then the integers' usual ordering serves to order the tuple's components, and the tuple is in fact an ordered n-tuple, i.e. it is a <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvtUAU5wpEbGdrcN5Y29ycA" class="cyc_term">List</a>. But in general any set (e.g. the column names in a relational database) may be used to index the components of a tuple. tuple Tuple A <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjuppwpEbGdrcN5Y29ycA" class="cyc_term">Field_Mathematical</a> is a unary function which assigns a scalar or vector (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rRh__wMhMQdeRr7ffGIZWMw" class="cyc_term">ScalarOrVectorInterval</a>) to each <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwKUKtJwpEbGdrcN5Y29ycA" class="cyc_term">Point</a> in a region of space. Fields are often used in physics to model quantities that vary throughout space. For example, the speed and direction of a moving fluid; the strength and direction of some force, such as the magnetic or gravitational force; the distribution of temperature throughout a region; the distribution of pressure in a gaseous region. The first two examples are types of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rCkakrkynEdqAAAACs6hO_g" class="cyc_term">VectorField</a>s, while the latter two examples are types of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rC1VvTEynEdqAAAACs6hO_g" class="cyc_term">ScalarField</a>s. field Field-Mathematical quantifier A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv2jAEFlOEdaAAAACs2S-ew" class="cyc_term">SententialRelation</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwJYN4pwpEbGdrcN5Y29ycA" class="cyc_term">ScopingRelation</a> (qq.v.). Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjCcZwpEbGdrcN5Y29ycA" class="cyc_term">Quantifier</a> takes as its arguments (at least) a variable (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv3gAv5wpEbGdrcN5Y29ycA" class="cyc_term">CycLVariable</a>) and a sentence (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwRZBCJwpEbGdrcN5Y29ycA" class="cyc_term">CycLSentence_Assertible</a>), and is used to make a certain kind of generic quantitative statement regarding the things that satisfy the sentence. Typically, the variable VAR will occur free in the sentence SENT, and in the quantified sentence (QUANT VAR SENT ...) these occurrences of VAR are bound by that occurrence of QUANT. (If VAR does not occur free in SENT, then the quantified sentence is a "vacuous quantification" that is equivalent to SENT by itself. For the definitions of 'free' and 'bound' occurrences of variables, see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwJYN4pwpEbGdrcN5Y29ycA" class="cyc_term">ScopingRelation</a>.) For example, '(<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA9pwpEbGdrcN5Y29ycA" class="cyc_term">thereExists</a> ?X (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViBBJwpEbGdrcN5Y29ycA" class="cyc_term">isa</a> ?X <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjaoJwpEbGdrcN5Y29ycA" class="cyc_term">Dog</a>))' means that there exists at least one dog. Other instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjCcZwpEbGdrcN5Y29ycA" class="cyc_term">Quantifier</a> are <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA95wpEbGdrcN5Y29ycA" class="cyc_term">forAll</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwQrnuJwpEbGdrcN5Y29ycA" class="cyc_term">thereExistExactly</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwQr155wpEbGdrcN5Y29ycA" class="cyc_term">thereExistAtLeast</a>, and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwQr5MpwpEbGdrcN5Y29ycA" class="cyc_term">thereExistAtMost</a>. Quantifier sentential relation A collection of mathematical objects. Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv2jAEFlOEdaAAAACs2S-ew" class="cyc_term">SententialRelation</a> is by definition either a <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVi5-ZwpEbGdrcN5Y29ycA" class="cyc_term">LogicalConnective</a> or a <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjCcZwpEbGdrcN5Y29ycA" class="cyc_term">Quantifier</a>. SententialRelation complex number A specialization of both <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvxwC-ZwpEbGdrcN5Y29ycA" class="cyc_term">Number_General</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViCW5wpEbGdrcN5Y29ycA" class="cyc_term">ScalarPointValue</a>. Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVi4CpwpEbGdrcN5Y29ycA" class="cyc_term">ComplexNumber</a> 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. Every complex number can be represented as a + ib where i is the square root of negative one (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvx4CzZwpEbGdrcN5Y29ycA" class="cyc_term">I_TheNumber</a>), and a and b are reals (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAmJwpEbGdrcN5Y29ycA" class="cyc_term">RealNumber</a>). When b=0, the complex number has no imaginary part, and therefore is a real number. When b is not zero the complex number has an imginary part and so it is an imaginary number (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwL5mhJwpEbGdrcN5Y29ycA" class="cyc_term">ImaginaryNumber</a>). 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. ComplexNumber ScalarInterval quantity A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjsVZwpEbGdrcN5Y29ycA" class="cyc_term">TupleOfIntervals</a> (q.v.). Roughly put, this is the collection of all things that can be ranked according to some one-dimensional scale. Instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAp5wpEbGdrcN5Y29ycA" class="cyc_term">ScalarInterval</a> are numbers or quantities possessing only sign and magnitude. They are construed as one-tuples (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVj42JwpEbGdrcN5Y29ycA" class="cyc_term">Tuple</a>) of intervals. They are to be contrasted with <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjsEpwpEbGdrcN5Y29ycA" class="cyc_term">VectorInterval</a>s (q.v.), which possess a direction as well as a magnitude, and are construed as two-membered tuples of intervals. <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAp5wpEbGdrcN5Y29ycA" class="cyc_term">ScalarInterval</a> is partitioned into the two collections <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rnVZQSiCmEdaAAACgyZzFrg" class="cyc_term">NumericInterval</a> (which is the collection of numbers and number-ranges of all kinds) and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rSGNLQK45EdqAAAACs6hRXg" class="cyc_term">Quantity</a> (qq.v). A <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rSGNLQK45EdqAAAACs6hRXg" class="cyc_term">Quantity</a> is usually specified with a numeric-interval, as with (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjRp5wpEbGdrcN5Y29ycA" class="cyc_term">Meter</a> 3)), but it might also be given in a generically-ranked way, as with (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvgDGhZwpEbGdrcN5Y29ycA" class="cyc_term">HighAmountFn</a> <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVi1AJwpEbGdrcN5Y29ycA" class="cyc_term">Happiness</a>); see the specializations <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rr-2NGp7sQdidoL5VyxYMGg" class="cyc_term">MeasurableQuantity</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rkmGj_Bv1EdaAAACgyZzFrg" class="cyc_term">NonNumericScalarQuantity</a>. The magnitude of a scalar might be given by a specific number (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViCW5wpEbGdrcN5Y29ycA" class="cyc_term">ScalarPointValue</a>) or by a proper range of numbers (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rLOwZCvkjQdea-sfI2QK3kQ" class="cyc_term">ScalarProperInterval</a>). Note that the instances of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjarZwpEbGdrcN5Y29ycA" class="cyc_term">MathematicalFunctionOnScalars</a> (q.v.), which include artithmetic functions such as addition (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViArpwpEbGdrcN5Y29ycA" class="cyc_term">PlusFn</a>) and division (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAq5wpEbGdrcN5Y29ycA" class="cyc_term">QuotientFn</a>), are defined broadly so as to apply not only to numbers, but to (numerically-measured) scalar intervals generally. Thus (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViArpwpEbGdrcN5Y29ycA" class="cyc_term">PlusFn</a> (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjRp5wpEbGdrcN5Y29ycA" class="cyc_term">Meter</a> 3) (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjRp5wpEbGdrcN5Y29ycA" class="cyc_term">Meter</a> 5)) is equal to (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjRp5wpEbGdrcN5Y29ycA" class="cyc_term">Meter</a> 8). n-tuple interval TupleOfIntervals A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVj42JwpEbGdrcN5Y29ycA" class="cyc_term">Tuple</a> (q.v.). Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjsVZwpEbGdrcN5Y29ycA" class="cyc_term">TupleOfIntervals</a> is a tuple whose members (see <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rx1wVcpg-EdaAAACQJ5qciw" class="cyc_term">memberOfTuple</a>) are all <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAp5wpEbGdrcN5Y29ycA" class="cyc_term">ScalarInterval</a>s (q.v.). Notable specializations of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjsVZwpEbGdrcN5Y29ycA" class="cyc_term">TupleOfIntervals</a> include <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAp5wpEbGdrcN5Y29ycA" class="cyc_term">ScalarInterval</a> itself (whose instances are one-tuples), <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjsEpwpEbGdrcN5Y29ycA" class="cyc_term">VectorInterval</a> (whose instances are two-tuples of scalars), and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvgCtPZwpEbGdrcN5Y29ycA" class="cyc_term">BloodPressureReading</a>. A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4r1jUXeq00EdmAAAACs0uFOQ" class="cyc_term">ThreeDimensionalThing</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjELpwpEbGdrcN5Y29ycA" class="cyc_term">GeometricallyDescribableThing</a>, and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwJRrmJwpEbGdrcN5Y29ycA" class="cyc_term">ShapedObject</a> (qq.v.). Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwPu-YZwpEbGdrcN5Y29ycA" class="cyc_term">ThreeDimensionalGeometricThing</a> is a three-dimensional, self-connected object whose shape is (at least roughly) describable in relatively simple geometric terms. Examples include intangible 3-D objects (such as ideal cones and cylinders) as well as certain tangible objects (such as billard balls and dice). Specializations of this collection include <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvtjhbpwpEbGdrcN5Y29ycA" class="cyc_term">Polyhedron</a>, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwQwzSJwpEbGdrcN5Y29ycA" class="cyc_term">Cone</a> and <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4ryO43AFsjQdeAPO4LYlJlgA" class="cyc_term">Torus</a>. ThreeDimensionalGeometricThing three dimensional shape True An instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjjupwpEbGdrcN5Y29ycA" class="cyc_term">TruthValue</a> (q.v.). <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA2ZwpEbGdrcN5Y29ycA" class="cyc_term">True</a> is the logical notion of truth. That is, the term '<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA2ZwpEbGdrcN5Y29ycA" class="cyc_term">True</a>' is used as a sentential constant of CycL that is true under every model theoretic interpretation. For example, (<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwP1T85wpEbGdrcN5Y29ycA" class="cyc_term">booleanResult</a> T/F <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA2ZwpEbGdrcN5Y29ycA" class="cyc_term">True</a>) means that the result obtained from the true-or-false test T/F is True. Cf. <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA2JwpEbGdrcN5Y29ycA" class="cyc_term">False</a>. true A specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rwIE23ZwpEbGdrcN5Y29ycA" class="cyc_term">TimeDependentCollection</a> (q.v.). This is the collection of all and only those collections <code>COL</code> such that any thing's being an instance of <code>COL</code> depends on the current state of the <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvVjqoZwpEbGdrcN5Y29ycA" class="cyc_term">CycKB</a>. Such a <code>COL</code> corresponds to a property defined (at least partly) in terms of the contents, features, or implementation of the Cyc Knowledge Base itself, as opposed to a property that "exists out there" in the KB-independent world. For example, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvV281JwpEbGdrcN5Y29ycA" class="cyc_term">CycLAssertion</a> is a KB-dependent collection, membership in which requires of a given CycL sentence that it has in fact been asserted to the KB. <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViAoJwpEbGdrcN5Y29ycA" class="cyc_term">CycLSentence</a> is also KB-dependent, as being a CycL sentence requires being composed (ultimately) out of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvu6KTZwpEbGdrcN5Y29ycA" class="cyc_term">CycLReifiedDenotationalTerm</a>s (and perhaps variables) -- i.e. terms that are currently reified in the KB. Conversely, <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv5ddsJwpEbGdrcN5Y29ycA" class="cyc_term">Sentence</a> is not a KB-dependent collection. See also <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvip9N5wpEbGdrcN5Y29ycA" class="cyc_term">KBDependentRelation</a>. KB dependent collection KBDependentCollection SubsetsOfMathematicalThing-Math-Topic subsets of mathematical thing math topic CycVocabularyTopic An instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvprlOZwpEbGdrcN5Y29ycA" class="cyc_term">FacetingCollectionType</a> and a specialization of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rgQsP1F2xEdif1wACs2IMlQ" class="cyc_term">Topic</a>. Each instance of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rAmoSCGJbQdiSXZJvYiNhkQ" class="cyc_term">CycVocabularyTopic</a> is a collection of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA9JwpEbGdrcN5Y29ycA" class="cyc_term">Thing</a>s falling under a topic for which some Cyc Vocabulary exists. <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rgQsP1F2xEdif1wACs2IMlQ" class="cyc_term">Topic</a> is neither a strict [<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvcZ1FpwpEbGdrcN5Y29ycA" class="cyc_term">facets_Strict</a>] nor a covering [<a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rv86JWpwpEbGdrcN5Y29ycA" class="cyc_term">facets_Covering</a>] faceting of <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA9JwpEbGdrcN5Y29ycA" class="cyc_term">Thing</a>: many <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rvViA9JwpEbGdrcN5Y29ycA" class="cyc_term">Thing</a>s may be instances of no <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rgQsP1F2xEdif1wACs2IMlQ" class="cyc_term">Topic</a>, or of multiple <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rgQsP1F2xEdif1wACs2IMlQ" class="cyc_term">Topic</a>s. cyc vocabulary topic Math-Topic math-topic A <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rAmoSCGJbQdiSXZJvYiNhkQ" class="cyc_term">CycVocabularyTopic</a> and a <a href="http://sw.opencyc.org/2008/06/10/concept/Mx4rtGXkHpNaEdqAAAACs0uFOQ" class="cyc_term">KBDependentCollection</a>. SubsetsOfMathematicalThing-Math-Topic subsets of mathematical thing math topic