Logic 13: Predicate Logic

Question 1

Predicate Logic (QL)

Answer 1

A more complex system,
want to translate sentences and be able to note when they have stuff in common,
QL- quantifier logic
ex.
1) Obama is human.
2) Hillary is human.
In PL, they would be two different sentences. But being human is a common element here, and we want to note that.
3) Obama is male
4) Hillary is female
- no common elements in 3 and 4
But same people talked about in all sentences

Question 2

Propositional Logic

Answer 2

The smallest unit is a complete sentence

Question 3

QL

Answer 3

has names and predicates as basic units, lets us build up complete sentences from them

Question 4

H(…)

Answer 4

is a predicate, for is human

Question 5

o

Answer 5

is a name, for Obama

Question 6

predicates

Answer 6

have various adicities- the number of slots that need to be filled by a name in order to get a complete sentence

Question 7

predicates with adicity greater than one

Answer 7

more than one name is required to make a complete sentence

these predicates signify relations- ways in which things can stand to one another

Question 8

dyadic predicates

Answer 8

two place predicates, signify dyadic relations
ex. is taller than, or is to the north of
can take 2 names to make a complete sentence
ex. can take “Obama” and “Hillary” and say “Obama is taller than Hillary”

things can stand in relations to themselves, we can use the one name twice to make a complete sentence with a dyadic predicate

ex. “is the same height as”
- “Nicole is the same height as Brad”
- “Nicole is the same height as Nicole”– everything is the same height as itself

Question 9

monadic predicates

Answer 9

monadic predicates– have adicity one– only one name is required to make a complete sentence

signify properties

Question 10

triadic predicates

Answer 10

three place predicates, signify triadic relations, will take 3 names and make a complete sentence
“…is between…and…”
“Kansas is between Colorado and Missouri”

can also have 4-place… 100-place predicates

Question 11

dyadic example

Answer 11

"Obama and Hillary are both Democrats, and Obama is Hillary's boss"
o: Obama
h: Hillary
D(x): x is a Democrat
B(x,y): x is y's boss

(D(o)^D(h))^B(o,h)

Question 12

What are Quantifiers?

Answer 12

When we want to say that something is a certain way without saying what thing in particular it is that is that way
– use quantifiers

ex. something is the president without saying that that thing is Obama
P(x), this says that x is the president
x is not a name for some thing, it is a VARIABLE– it is holding place for a name
- can bind that variable with a quantifier

Question 13

QL Quantifiers

Answer 13

‘∃xF(x)’ says ‘something is F’
- existential

‘∀xF(x)’ says ‘everything is F’
- absolute

∃xF(x) = ¬∀x¬F(x)
(If something is F, it’s not the case that everything is not F.)

∀xF(x) = ¬∃x¬F(x)
(If everything is F, it’s not the case that something is not F.)

Question 14

Quantifier examples

Answer 14

P(x): x is the President
∃xP(x): something is the president
∀xP(x): everything is the president
∃x¬P(x): something is not the president
∀x¬P(x): everything is not the president
¬∃xP(x): not something is the president = nothing is the president
¬∀xP(x): not everything is the president

Question 15

PL vs. QL example 1

Answer 15

1) all humans are mortal
2) socrates is human
c) socrates is mortal

in PL:

1) P
2) Q
c) R

in QL:
s: Socrates
H(x): x is human
M(x): x is mortal
1) ∀x(H(x)→M(x))
all, if human then mortal
2) H(s)
C) M(s)

Question 16

QL example 2

Answer 16

“There is a logician who loves a philosopher”
L(x): x is a logician
P(x): x is a philosopher
R(x,y): x loves y

∃x∃y(L(x)∧P(y)∧R(x,y))
some person x, some person y, some person x is a logician, some person y is a philosopher, some person x who is a logician loves person y who is a philosopher

Question 17

QL example 3- deciphering ambiguity

Answer 17

“Everyone loves some celebrity”

C(x): x is a celebrity
L(x,y): x loves y

Case 1
- everybody has their own favorite celebrity that they love?
∀x∃y(C(y)∧L(x,y))

OR

Case 2
- there’s a special celebrity that everybody loves
∃x(C(x)∧∀yL(y,x))