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Predicate Logic, Sets, andLambda: Tools for semantic
analysis
KELOMPOK 3
Dr. Aceng !"endi Syaef!lla",
M. #!m.
Dosen Pengam$!
AnnaK!rniani
Lee %ong#&a
Dede'atino(a
)nda"
Melisa
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Lp the language of
predicate logic
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Predicate
Logic
The basis foranalyzing thestructure of
simplepropositions.
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The simplet well-formedformula in Lp consist of :A PRE!"ATE
A number of AR#$%E&T''ymbolized by TER%'
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(here n is an integer) an n placepredicate) Pred is a predicate with
arguments numbering n
P* is a one
placepredicate+Pred,
P*y is a twoplace
predicate+Pred
P*yz is athree placepredicate+Pred/
%a*
0ogs
%a* isa
bache
lor
Theboo1
is red
2be3a3bachelor4%a*5
2be3red4the3 boo15
20og4%a*5
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a. %a* dran1 thebeer
b. The beer was
drun1 by %a*
Archie admireshimseld
Archie ga6e7eatrice the camera
%ary is +8arry9s wife
+The wife of 8arry
2drin14%a*4the3beer5
2admires4Archie4Archie5
2gi6e4Archie47eatrice4the3camera5
2be3wife3to4%ary48arry5
itransiti6eerbs
Relational"omplement
Transiti6eerbs
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A two place predicate P is ';%%ETR!" i< forany w< P*y) P*y =-> Py*
? is the parentof ; =-> ; is the childof ?
? is husband to ; =-> ; is wife to ?
? buys ; from @ =-> @ sells ; to ?
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PRE!"ATE
L#!"BuantiC
ers:
$ni6ersal
BuantiCer D2for all5
E*istensial
BuantiCer D2there is atleast one5
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E6ery 6ariable is either bound
by a BuantiCer or else it is free
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%eaning postulates
• The formal semantics of naturallanguage cannot be e*pressedwithout recourse to the nonlogical 6ocabulary of meaning
postulates which identify thesemantic properties of listemes+"arnap ,FG:-
• (hat is postulatesH – The non-logical 6ocabulary +in this
boo1) words of English used in thesemantic speciCcation of listeme
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%c"awley postulated a ͡ ͡ &P ͡ &P structure reminiscent of P*y
in Lp)
'orm Semantic s$eci*cation
I Jnow +1now that p K +pT
F 7elie6e +belie6e that p K possibly p
G Ali6e +y is ali6e M ¬ +y is deadN ead +y is dead M ¬ +y is ali6e
O Jill +* 1ill y K +* cause +ybecome¬+ali6e
/F * 1ills y Tolong lihat halaman I
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J!LL
Predicate logic, sets, and lambda: tools forsemantic analysis
Tree /F′:Predicate
raising
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Predicate raising
+a - ills y / ca!se 0E1OME 2OTAL)E y
+b * causes y to die ← ? causes y 7E"%E &TAL!E
+c * causes y to cease to beali6e
←* causes y 7E"%E &T tobe ali6e
+d HH * causes y to becomedead
← ? causes y become &TAL!E
+e H* causes y to become
not ali6e+f * causes to become y notali6e
+g * causes to become notali6e
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a ) "a(e fo!r c"ildren
b ) "a(e 4 c"ild 4 c"ild 4 c"ild 4 c"ildc ) "a(e t&o c"ildren and anot"er t&o
c"ildren
d ) "a(e one c"ild t"ree c"ildren
e ) "a(e one less t"an *(e c"ildren
f ) "a(e si-teen less t"an t&entyc"ildren
g ) "a(e one more t"an t"ree c"ildren
" ) "a(e t"ree more t"an one c"ild
i ) "a(e fo!r times one c"ild 5 ) "a(e t&o times t&o c"ildren
) "a(e t&o6s7!ared c"ildren
l ) "a(e t"e s7!are root of si-teenc"ildren
m ) "a(e t"e same n!mber of c"ildrenas t"ere are 89s in 4
There is considerablee6idence that some of
theses arithmeticalprocesses are normal
in the le*icalizing ofnumber systems.E*. Qourteen and
eighteen are 2four +ten and 2eight +
ten5
Logically eBui6alent
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The main thesis of generati6e semanticswas that syntactic and semantic
representations are labelled trees and asingle system of rules relates thesemantic representation to surface formthrough intermediate stages.
The semantic speciCcations for 1illand die cannot do without 1ill +*)yand die +y.
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'ets
and Tuples
SETS
A set is an unorderedcollection of members +or
elements
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Q +your-eye) the-atican) b) a-night-with-brigine-bardot) S
# + a,b,c +b,a,c +c,b,a
the-atican Q the-atican #
a #
a Q
*:* is one of the Crst three letters of the English
alphabet
The set of all * such that * is one of the Crst
three letters of the English alphabet
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eCnitionN.N:A is '$7'ET of 7)
AU7) i< e6erymember of A is also
a member of 7.%ore formally:
AU7 D *V*A-*7
Assumption N./:e6ery set is a subset
of it self
eCnitionN.O: A set that
consist of only onemember) e.g WbX) is a
unit set
Assumption N.I: onesubset of e6ery setis the &$LL'ET) Y WX: therefore YU#
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Qor instance +a,b,c) U # says 2the set +a,b,c) is a subsetof the set #5
The rest of the subsets of # are proper sebsetsbecause theyare included within it along with other subset
+a) Z 7 +b) Z # +c) Z # +a.b) Z # +b,c) Z #+a,c) Z #
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Any two sets are identical of they contain twoand members. !f follows that it 7 U A and also A
U 7 then A7
A +*:* is one of the Crst three letters of the Englishalphabet# +a, b, c)
A#
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E6ery member of AERLAP 7 is a
member of A andalso a member of 7
more formally
Q[# +a, b, your-eye, a-night-with-Brigitte-Bardot, √2, c, the Vatican
Any member of A%ER#E 7 is memberof A and or a
member of 7 moreformally
b Є CᴖP = b is a cat with a pinribbon
b Є C!" = b is either a cat or a dog
Predicate deCnes aset
\ac1 is the same height as himself+ \ac1
\ac1 is the same height as 8arry8arry is the same height as \ac1
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The Lambda perator) ] Also 1nown as 2the
abstraction or set operator5] can bind predicates or sets ofentities ha6ing the propertyspeciCed by the formula within ]9sscope] is not an operator within standard-Crst order predicatecalculus. !t9s a corner stone of model theoritic semanticsusing type theory
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• use of the ]-operator increases thee*planation 6alue of the
metalanguage base it allows us toe*tend the metalanguage beyond Lp.
]*^ * is free 6ariable withinformula ^ +the same functiontyped as Pred,!ts not a sentence) it9s the name offunction
A function +] from entities +*to truth 6alue +^ +pro6idedcertain conditions are met
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To become a sentence) the function must be applied to an argument
2\ac1 is tall and bald5
+T0_70
Qrom English to Lp
!s baldRepresenting the fact that \ac1 is a member of the set of tallthings and also a member of the bald things.
7ut not e*plicity represent the fact that \ac1 is a member ofthe o6erlapset` the set of things which are both tall and bald.
]*VT*_7*
2the property of being talland bald5 or 2the set of
indi6iduals that are bothtalland bald5
!ts not a sentence) and not yet a translation.
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!t9s a functional loo1ing for an argument to bacome a sentencein order to yield a truth 6alue . This reBirement can be met byapplying it to constant 0:
]*VT*_7* ]*VT*_7* +0A sentencethat yield atruth 6alue
(e con6erted it into astandard formula of Lpnamely in +T0_70 by
replacing 6ariable * withconstant 0
Qor e6ery occurrence of the 6ariable bound by ]) LA%7A"&ER'!& substitutes the term gi6en in the parenthesesimmediately following the scope of lambda. The latter in then
deleted) along with the ]-operator) is bound 6ariable) and anye*traneous brac1ets.
]* VT*_7*+0 V0* ]*V T0_70 +0 T0_70
symbolizes substitute constant 0 for 6ariable *
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Additional e6idence for the e*planatory 6alue of ]-e*pression is to be had from a discussion of acti6e-passi6epairs. 7ecause they are truth-functionally synonymous.acti6e-passi6e pairs ha6e the same translation in #p)
+a Qran1 admires 8arry
+b 8arry is admired by Qran1
$sing the ]-operator in the translation rectiCesthe situation. 2admire5 is of type Pred) two6ariables are separately bound by two ]-operators.]yV]*VA*y 2the property of * admiring by y5
]*V]*VA*y 2the property of y being admired by *5
Afh
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"on6ersion proceeds from wider scope tonarrower scope) from left to right. The lefmost ]-bound 6ariable is associated with the leftmost
constant.
]y]*VA*y+h +f Vhy
]*VA*h+f
Vf*
Afh
]*]yVA*y +f+h Vf*
]yVAfy+h
VhyAfh
The di
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]*VA*h 2the propertyof 8arry9s being admired by something+*5 or 2the set of indi6iduals that 8arry is admiredf by5.
]yVAfy 2the property of Qran19s admiring something+y5 or 2the set of indi6iduals that Qran1 admires thatare admired by Qran15
W*:* admires 8arryX) which is a Pred, type e*pression asdistinct from the sentence type: ]*VA*h +f fW*:* admires8arryX
T"e Pred4 ty$e e-$ression is ]yVAfy W y:y is admired byQran1X) and the sentence ty$e e-$ression is ]yVAfy +hh Wy:y is admired by Qran1X.
The ]-operator can occur with the
uni6ersal and the e*intensialBuantiCers. Tom ate a sausage Angeline coo1ed
*V]yV'y"ay+* Et*
*V]yV'*"a*Et*
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#losses for subformulae are:
Pred4 ty$e: ]yV'y"ay 2the property of being anAngeline-coo1ed Dsauusage5 or 2the set of indi6iduals thatare Angeline-coo1ed-sausageSentence ty$e: ]yV'y"ay +* 2* has the property ofbeing an Angeline-coo1ed Dsauusage5 or 2* Wy:y is an
Angeline-coo1ed-sasusageX
The e*istensial BuantiCer bind *) so *Vs*Et* translates2there is at least one * which has the property of beingAngeline-coo1ed-sausage and Tom ate it5
The ]-operator spells out the semantic composition of theEnglish sentence in a manner closer to the surface structureof English than is possible in the standard #p metalanguage.]-can bind predicate 6ariables as well as term
6ariables.
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