Predicate logic, sets, and lambda: tools for semantic analysis

8/18/2019 Predicate logic, sets, and lambda: tools for semantic analysis

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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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Predicate logic, sets, and lambda: tools for semantic analysis