7/25/2019 Module02 Predicate Logic

1/22

COMP 1805Predicate Logic

Carleton University

Winter 2016

1

7/25/2019 Module02 Predicate Logic

2/22

Propositional Logic

Predicate Calculus

Set TeoryMate!atical "easoning and Proo#s

$%nctionsSe&%ences and S%!s

Co!ple'ity o# (lgorit!s

)nd%ction

*raps

2

7/25/2019 Module02 Predicate Logic

3/22

Predicates

Consider eac o# te #ollo+ing state!ents

So!eone is la%ging

,veryone is appy

- . /

/

7/25/2019 Module02 Predicate Logic

4/22

Predicates

( predicate is a property tat an oect can

ave

We can +rite a predicate as a propositional

function +ic ret%rns tr%e or #alse depending

on te val%e tat te oect as

3

7/25/2019 Module02 Predicate Logic

5/22

Predicates

)s a propositional #%nction a proposition4

5

7/25/2019 Module02 Predicate Logic

6/22

Predicates

o+ do +e say so!eting lie

So!eone lies tea

,veryone lies tea

6

7/25/2019 Module02 Predicate Logic

7/22

Universe o# 7isco%rse

Wic oects are +e taling ao%t4

Te universe of discourse or domain o#ten denoted

y UUUU is te speci#ic set o# oects +e are concerned+it in a given prole!

Ly9 is te predicate :y lies star tre;

,veryone lies star tre

L'19

L'29

L'/9

%anti#ication

Te existential quantification o# te propositional#%nction () is te proposition :tere e'ists an

ele!ent in te %niverse o# disco%rse s%c tat

() is tr%e;

()

11

7/25/2019 Module02 Predicate Logic

12/22

,'istential >%anti#ication

Wen is an existential quantification tr%e?#alse4

12

7/25/2019 Module02 Predicate Logic

13/22

,'istential >%anti#ication

So!e astrona%ts +ant to go to space

1/

7/25/2019 Module02 Predicate Logic

14/22

Ainding Bariales

( variale is bound +en

( val%e is assigned to it or

( &%anti#ier is %sed on it

Te scope o# a &%anti#ier is te part o# te

e'pression tat te &%anti#ier is applied to

13

7/25/2019 Module02 Predicate Logic

15/22

Ainding Bariales

Consider

()

15

7/25/2019 Module02 Predicate Logic

16/22

Translations

: not

: and %t also

: or : i!plies i#ten onlyi# necessary

s%##icient +enever %anti#iers

,veryone lies to s!ile

So!eone lies to dance

1=

7/25/2019 Module02 Predicate Logic

18/22

Degating >%anti#iers

= ,

18

7/25/2019 Module02 Predicate Logic

19/22

Degating >%anti#iers

()

()

1@

7/25/2019 Module02 Predicate Logic

20/22

Dested >%anti#iers

$'y9 is :' and y are #riends;

,

,

,

20

7/25/2019 Module02 Predicate Logic

21/22

,'a!ple

,very CS !aor in tis class no+s at least one CS!aor tat is taing COMP 1305

21

7/25/2019 Module02 Predicate Logic

22/22

,'a!ple

,veryone in tis class no+s e'actly one person tatgoes to %Otta+a

22