Philosophy assignment 10 questions

2 2 : “ L o v e ” a n d O t h e r Tw o - P l a c e P r e d i c a t e s

P H I L 1 1 0 ; S p r i n g 2 0 2 0 ; To m D o n a l d s o n

1 : N a m e s a n d P r e d i c a t e s

Proper Names A proper name is a word that represents an individual member of the domain of quantification. For example, if our domain is philosophers, we might use the following names:

• “Socrates”

• “Mary Wollstonecraft”

• “Mozi”

Similarly, if our domain is countries, we might use the following names:

• “Canada”

• “Mexico”

• “India”

PHIL 110; Spring 2020; Lecture 22 3

Predicates

If you take a statement and remove one or more proper names from it, the result is a predicate. (If you like, a predicate is a sentence with one or more proper-name-shaped holes in it.)

• A one-place predicate has one hole.

• A two-place predicate has two holes.

• A three-place predicate has three holes.

• (And so on!)

PHIL 110; Spring 2020; Lecture 22 4

Predicates

One can make a sentence by taking a predicate, and then “filling in” the hole with a name (or filling in the holes with names):

“Ashni” + “____ likes muffins” = “Ashni likes muffins”.

PHIL 110; Spring 2020; Lecture 22 5

Predicates

• So far, we’ve considered only one-place predicates like “____ likes dancing.” and “____ is having fun.”

• Today, we’re going to look at two-place predicates.

• For simplicity, let’s restrict our attention to a single example:

x loves y. Lxy

PHIL 110; Spring 2020; Lecture 22 6

Love

Lrj Romeo loves Juliet.

Ljr Juliet loves Romeo.

Lnn Narcissus loves himself.

Lqe Quasimodo loves Esmerelda

Leq Esmerelda doesn’t love Quasimodo.

Notice that there’s a big difference between loving and being loved, so the order of the names matters.

PHIL 110; Spring 2020; Lecture 22 7

Romeo

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Juliet

Quasimodo Esmerelda

Narcissus

2 : S y m b o l i z a t i o n P r a c t i c e

An Example

In this section, let’s suppose that the domain of quantification is people at Tom’s party, and that this includes Ashni, Ben, Chiara, and nobody else.

PHIL 110; Spring 2020; Lecture 22 10

Symbolization Practice

English Sentence Symbolization

Ashni loves Ben. Lab

Ben loves Ashni. Lba

Ashni and Ben love each other. (Lab & Lba)

Ashni loves Ben, but he doesn’t love her back. (Lab & Lba)

Ben loves himself. Lbb

Ashni and Ben both love Chiara. (Lac & Lbc)

PHIL 110; Spring 2020; Lecture 22 11

Existential Quantification

• The existential quantifier works in just the same way as before!

• ‘∃x Lxa’ means someone loves Ashni, and has the same truth value as this disjunction:

((Laa  Lba)  Lca)

• ‘∃x Lax’ means Ashni loves someone, and has the same truth value as this disjunction:

((Laa  Lab)  Lac)

PHIL 110; Spring 2020; Lecture 22 12

Existential Quantification

• Now for a more complex example. This is a symbolization of “There is someone whom Ashni and Ben both love”:

∃x (Lax & Lbx)

• And here is a symbolization of “There is someone who loves both Ben and Chiara”:

∃x (Lxb & Lxc)

PHIL 110; Spring 2020; Lecture 22 13

Universal Quantification

• The universal quantifier works just as before!

• The sentence ‘∀x Lxa’ means everyone loves Ashni, and has the same truth value as this conjunction:

((Laa & Lba) & Lca)

• This is very different, of course, to ‘∀x Lax’, which means Ashni loves everyone:

((Laa & Lab) & Lac)

PHIL 110; Spring 2020; Lecture 22 14

Universal Quantification

• Now for a more complex example. Here is a symbolization of “Everyone loves either Ashni or Ben”:

∀x(Lxa  Lxb)

PHIL 110; Spring 2020; Lecture 22 15

Symbolization Practice

English Sentence Symbolization

There is someone who Ashni doesn’t love. ∃x Lax

It isn’t true that Ashni loves someone. ∃x Lax

Not everyone loves Ben.  ∀x Lxb

Nobody loves Ben. ∀x  Lxb

PHIL 110; Spring 2020; Lecture 22 16

3 : N a t u r a l D e d u c t i o n P r a c t i c e

Exercise

In each case, show that the inference is valid by constructing a natural deduction proof with the given premises and the given conclusion:

(1) Premise: Everyone loves Ashni.

Conclusion: Someone loves themself.

(2) Premise: Everyone loves Ashni.

Premise: Ashni loves Ben.

Conclusion: Someone loves both Ashni and Ben.

PHIL 110; Spring 2020; Lecture 22 18

Exercise

1. ∀x Lxa Premise (“Everyone loves Ashni”)

2. Laa 1, UI

3. ∃x Lxx 2, EG (“Someone loves themself ”)

PHIL 110; Spring 2020; Lecture 22 19

Exercise

1. ∀x Lxa Premise (“Everyone loves Ashni”)

2. Lab Premise (“Ashni loves Ben”)

3. Laa 1, UI

4. (Laa & Lab) 2, 3 Conj

5. ∃x (Lxa & Lxb) 4, EG (“Someone loves both Ashni and Ben.”)

PHIL 110; Spring 2020; Lecture 22 20

4 : S t a t e m e n t s t h a t c o n t a i n b o t h a o n e - p l a c e a n d a

t w o - p l a c e p r e d i c a t e

Love and Dancing

• Let’s continue our discussion of the party, with only Ashni, Ben and Chiara in attendance.

• Let’s use the following predicates:

Dx x is a dancer.

Lxy x loves y.

PHIL 110; Spring 2020; Lecture 22 22

Love and Dancing

English Sentence Symbolization

A Every dancer loves Ashni. ∀x(Dx → Lxa)

E No dancer loves Ashni. ∀x(Dx → Lxa)

I Some dancer loves Ashni. ∃x(Dx & Lxa)

O Some dancer doesn’t love Ashni. ∃x(Dx & Lxa)

PHIL 110; Spring 2020; Lecture 22 23

Love and Dancing

English Sentence Symbolization

Everyone who Ashni loves is dancing. ∀x(Lax → Dx)

Nobody who loves Ashni is dancing. ∀x(Lxa → Dx)

Ashni loves a dancer. ∃x(Dx & Lax)

There’s this dancer who Ashni doesn’t love. ∃x(Dx & Lxa)

PHIL 110; Spring 2020; Lecture 22 24

5 : Q u a n t i f i e r s I n s i d e Q u a n t i f i e r s

First Example: ∀x ∀y Lxy

• Consider the English sentence “Everybody loves everybody.”

• This has two universal quantifiers in it!

• The correct symbolization is this: ∀x ∀y Lxy

• This shouldn’t be confused with this: ∀x Lxx

• This latter statement means everybody loves themself.

PHIL 110; Spring 2020; Lecture 22 26

Second Example: ∀x ∀y Lxy

• This is trickier to interpret!

• Here’s one approach.

• By the QN rule, the statement is equivalent to this: ∃x∀y Lxy.

• In English: It is not the case that there is some one person who loves everybody.

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Third Example: ∃x ∃y Lxy

• This is an easy one!

• It means, someone loves someone.

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Fourth Example: (1) ∃x ∀y Lxy (2) ∀x ∃y Lxy

• Both of these statements contain “everyone”, “someone”, and “loves”. So each must mean something like Everyone loves someone, or Someone loves everyone.

• But can we get clear on the difference between them?

PHIL 110; Spring 2020; Lecture 22 29

Fourth Example: (1) ∃x ∀y Lxy (2) ∀x ∃y Lxy

• (1) is an existential generalization. Its instances are: • ∀y Lay Ashni loves everyone.

• ∀y Lby Ben loves everyone.

• ∀y Lcy Chiara loves everyone.

• So (1) amounts to: Either Ashni or Ben or Chiara loves everyone.

• In short, (1) means: There is a single (very amorous!) person who loves all.

PHIL 110; Spring 2020; Lecture 22 30

Fourth Example: (1) ∃x ∀y Lxy (2) ∀x ∃y Lxy

• (2) is a universal quantification. Its instances are: • ∃y Lay Ashni loves someone.

• ∃y Lby Ben loves someone.

• ∃y Lcy Chiara loves someone.

• So (2) means something like this: Ashni loves someone, and Ben loves someone, and Chiara loves someone.

• In short, (2) means: Every person has someone that they love.

PHIL 110; Spring 2020; Lecture 22 31

Fourth Example: (1) ∃x ∀y Lxy (2) ∀x ∃y Lxy

• In summary: • (1) means: There is a single (very amorous!) person who loves all.

• (2) means: Every person has someone that they love.

PHIL 110; Spring 2020; Lecture 22 32

Fourth Example: (1) ∃x ∀y Lxy (2) ∀x ∃y Lxy

• In summary: • (1) means: There is a single (very amorous!) person who loves all.

• (2) means: Every person has someone that they love.

(1) (2)

a

a b c

b c

PHIL 110; Spring 2020; Lecture 22 33

Fifth Example: ∃x(Dx & ∀y Lxy)

• Let’s start with this: ∀y Lxy

• This means, x loves everybody.

• So the whole statement means: There is some person x, where x is a dancer and x loves everybody.

• To put it more succinctly: There is some (amorous!) dancer who loves everybody.

PHIL 110; Spring 2020; Lecture 22 34

Mx: x is male. Pxy: x is a parent of y.

Fx: x is female. Lxy: x loves y.

Yx: x is young. Kxy: x kills y.

1. ∃x ∃y ((Yx & Mx) & (Yy & Fy) & (Lxy & Lyx) & (Kxx & Kyy))

2. ∃x ∃y ∃z ((Mx & My & Fz) & (Pyx & Pzx) & (Kxy & Lxz))

PHIL 110; Spring 2020; Lecture 22 35

Great Works of Literature, in Symbols

Continue to assume that the domain contains just three objects: a, b and c. For each of the following statements, express them in natural English, and draw an arrow diagram showing a situation in which the statement is true.

(1) ∀x Lax

(2) ∀x Lxa

(3) ∀x ∀y Lxy

(4) ∀x ∀y Lxy

(5) ∃x ∀y Lyx

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(1) ∀x Lax

Ashni loves everyone.

b a c

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(2) ∀x Lxa

Everyone loves Ashni.

b a c

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(3) ∀x ∀y Lxy

Everyone loves everyone!

a

b c

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(4) ∀x ∀y Lxy

Nobody loves anyone.

a

b c

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(5) ∃x ∀y Lyx

There is some single individual who is loved by all.

b c a

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