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Why use logic? |
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CycL Syntax |
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Collections and Individuals (#$isa and #$genls) |
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Microtheories |
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A sampling of some constants: |
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#$Dog, #$SnowSkiing, #$PhysicalAttribute |
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#$BillClinton,#$Rover,
#$DisneyLand-TouristAttraction |
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#$likesAsFriend, #$bordersOn, #$objectHasColor, #$and,
#$not, #$implies, #$forAll |
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#$RedColor, #$Soil-Sandy |
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CycLFormula:
a relation applied to some arguments, enclosed in parentheses |
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Examples: |
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(#$isa #$GeorgeWBush #$Person) |
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(#$likesAsFriend #$GeorgeWBush #$AlGore) |
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(#$BirthFn #$JacquelineKennedyOnassis) |
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A CycL Sentence is a well-formed CycLFormula
with a Truth Function, such as a predicate in the arg0 position. Sentences have truth values. |
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A CycL Non-atomic Term is a well-formed
CycLFormula with a Function-Denotational in the arg0 position. |
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A TruthFunction: |
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is a relation that can be used to form
sentences. |
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begins with a lower-case letter. |
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Types of TruthFunctions: |
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Predicates: #$likesAsFriend, #$bordersOn,
#$objectHasColor, #$isa |
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Logical Connectives: #$and, #$or, #$not |
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Quantifiers: #$implies#$forAll, #$thereExists |
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Sample CycLSentences: |
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(#$isa #$GeorgeWBush #$Person) |
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(#$likesAsFriend #$GeorgeWBush #$AlGore) |
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CycLSentences are used to form assertions and
queries. |
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A Function-Denotational |
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is a relation that can be applied to some
arguments to pick out something new |
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usually ends in “Fn” |
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Examples of Function-Denotational: |
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#$BirthFn, #$GovernmentFn, #$BorderBetweenFn |
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Sample CycL Non-atomic Terms: |
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(#$GovernmentFn #$France) |
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(#$BorderBetweenFn #$France #$Switzerland) |
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(#$BirthFn #$JacquelineKennedyOnassis) |
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CycL Non-atomic Terms are denotational terms.
They can be used like any other, as in: |
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(#$residenceOfOrganization (#$GovernmentFn
#$France) #$CityOfParisFrance) |
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Arity constraints are represented in CycL with
the predicate #$arity: |
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(#$arity #$performedBy 2) |
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Represents the fact that #$performedBy takes two arguments, e.g.: (#$performedBy
#$AssassinationOfPresidentLincoln #$JohnWilkesBooth) |
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(#$arity #$BirthFn 1) |
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Represents the fact that #$BirthFn takes one
arguments, e.g.: |
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(#$BirthFn #$JacquelineKennedyOnassis) |
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Argument type constraints are represented in
CycL with the following 2 predicates: |
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1 #$argIsa |
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(#$argIsa #$performedBy 1 #$Action) means that
the first argument of #$performedBy must be an individual #$Action, such as
the assassination of Lincoln in: |
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(#$performedBy #$AssassinationOfPresidentLincoln
#$JohnWilkesBooth) |
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2 #$argGenl |
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(#$argGenl #$penaltyForInfraction 2 #$Event) means
that the second argument of #$penaltyForInfraction must be a type of
#$Event, such as the collection of illegal equipment use events in: |
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(#$penaltyForInfraction #$SportsEvent
#$IllegalEquipmentUse #$Disqualification) |
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CycL also includes logical terms to allow us to
stick formulas together and quantify into them |
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Logical connectives |
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truth functions |
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take sentences as their arguments |
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(#$and
(#$performedBy #$GettysburgAddress
#$Lincoln)
(#$objectHasColor #$Rover #$TanColor)) |
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(#$or
(#$objectHasColor #$Rover #$TanColor)
(#$objectHasColor #$Rover #$BlackColor)) |
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(#$implies
(#mainColorOfObject #$Rover #$TanColor)
(#$not (#$mainColorOfObject #$Rover #$RedColor))) |
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(#$not
(#$performedBy #$GettysburgAddress |
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#$BillClinton)) |
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By adding variables and quantifiers to the
logical connectives, predicates and other CycL components we’ve already
covered, we gain the ability to represent many pieces of ordinary
knowledge. |
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Sentences involving concepts like “everybody,”
“something,” and “nothing” require variables and quantifiers: |
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Everybody loves somebody. |
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Nobody likes spinach. |
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Some people like spinach and some people like
broccoli, but no one likes them both. |
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Adding variables and quantifiers, we can
represent more general knowledge. |
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Two main quantifiers: |
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1. Universal Quantifer -- #$forAll |
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Used to represent very general facts, like: |
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All dogs are mammals |
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Everyone loves dogs |
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2. Existential Quantifier -- #$thereExists |
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Used to assert that something exists, to state
facts like: |
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Someone
is bored |
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Some people like dogs |
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Universal Quantifier |
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(#$forAll ?THING |
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(#$isa ?THING #$Thing)) |
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Existential Quantifier: |
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(#$thereExists ?JOE |
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(#$isa ?JOE #$Poodle)) |
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Others: |
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(#$thereExistsExactly 12 ?ZOS (#$isa ?ZOS
#$ZodiacSign)) |
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(#$thereExistsAtLeast 9 ?PLNT (#$isa ?PLNT
#$Planet)) |
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All variables occurring “free” in a formula are
understood by Cyc to be implicitly universally quantified. |
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So, to CYC, the following two formulas represent
the same fact: |
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(#$forAll ?X |
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(#$implies |
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(#$isa ?X #$Dog) |
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(#$isa ?X #$Animal)) |
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(#$implies |
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(#$isa ?X #$Dog) |
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(#$isa ?X #$Animal)) |
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What does this formula mean? |
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(#$thereExists ?PLANET
(#$and
(#$isa ?PLANET
#$Planet)
(#$orbits ?PLANET
#$Sun))) |
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What does this formula mean? |
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(#$thereExists ?PLANET
(#$and
(#$isa ?PLANET
#$Planet)
(#$orbits ?PLANET
#$Sun))) |
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What does this formula mean? |
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(#$forAll ?PERSON1
(#$implies |
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(#$isa ?PERSON1 #$Person) |
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(#$thereExists ?PERSON2 |
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(#$and
(#$isa ?PERSON2
#$Person)
(#$loves ?PERSON1
?PERSON2))) |
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What does this formula mean? |
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(#$forAll ?PERSON1
(#$implies |
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(#$isa ?PERSON1 #$Person) |
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(#$thereExists ?PERSON2 |
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(#$and
(#$isa ?PERSON2
#$Person)
(#$loves ?PERSON1
?PERSON2))) |
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How about this one? |
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(#$implies
(#$isa ?PERSON1
#$Person)
(thereExists ?PERSON2
(#$and
(#$isa ?PERSON2 #$Person)
(#$loves ?PERSON2
?PERSON1)))) |
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How about this one? |
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(#$implies
(#$isa ?PERSON1
#$Person)
(thereExists ?PERSON2
(#$and
(#$isa ?PERSON2 #$Person)
(#$loves ?PERSON2
?PERSON1)))) |
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And this? |
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(#$implies |
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(#$isa ?PRSN #$Person) |
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(#$loves ?PRSN ?PRSN)) |
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And this? |
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(#$implies |
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(#$isa ?PRSN #$Person) |
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(#$loves ?PRSN ?PRSN)) |
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Denotational Functions can be applied to some
arguments to pick out something new.
The result of applying a denotational function is a term that
denotes something. Function names
are always capitalized, and often end in “Fn”. |
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Examples: |
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#$FruitFn |
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#$GovernmentFn |
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#$DeadFn |
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A non-atomic term (NAT) is a denoting term like
a constant. |
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NATs are formed by applying a denotational
function to a denoting term. |
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(#$FruitFn #$AppleTree) |
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(#$GovernmentFn #$France) |
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(#$DeadFn #$Cockroach) |
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NATs can be used just like atomic terms (i.e.,
constants). |
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(#$implies |
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(#$isa ?APPLE (#$FruitFn #$AppleTree)) |
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(#$colorOfObject ?APPLE #$RedColor)) |
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The denotation of a NAT is determined by the
denotations of the inputs to the denoting function. |
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Uniformity |
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All kinds of fruits, nuts, etc., are represented
in the same, compositional way: |
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(#$FruitFn
PLANT) * |
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Inferential Efficiency |
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Forward rules can automatically conclude many
useful assertions about NATs as soon as they are created, based on the
function and arguments used to create the NAT. |
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what kind of thing that NAT represents |
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how to refer to the NAT in English |
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… |
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Some functions return concepts that we want to
“reify” and keep in the KB. These
are reifiable functions, such as: |
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#$GovernmentFn |
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#$BirthFn |
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When a new NAT is created using a reifiable
function, that new term is itself reified (kept around separately in the
KB) and becomes a Non-Atomic Reified Term, or NART, such as |
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(#$GovernmentFn #$France) |
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(#$BirthFn
#$JacquelineKennedyOnassis) |
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Other functions return concepts that we don’t
want to store separately in the KB.
These are unrefiable functions, such as: |
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#$TimesFn |
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When a new NAT is created using an unrefiable
functions, it does not get created as a persistent term in the KB. Unrefiable functions result in
Non-Atomic Unreified Terms, such as (#$TimesFn 3 7). |
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CycL components |
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Constants |
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Formulas |
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Sentences (and Truth Functions) |
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Non-atomic Terms (and Denotational Functions) |
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Logical Constants |
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Variables and Quantifiers |
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Well-Formedness |
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arity |
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argument constraints |
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Notes