Predicate Logic Splitting the Atom What Propositional Logic can’t do It can’t explain the...

Predicate LogicPredicate LogicSplitting the AtomSplitting the Atom

What Propositional Logic can’t do• It can’t explain the validity of the Socrates Argument

• Because from the perspective of Propositional Logic

its premises and conclusion have no internal

structure.

The Socrates Argument

1. All men are mortal

2. Socrates is a man

_______________________

3. Therefore, Socrates is mortal

Translates as:

1. H

2. S

__

3. M

The Problem

The problem is that the validity of this argument

comes from the internal structure of these

sentences--which Propositional Logic cannot “see”:



_______________________


Solution

• We need to split the atom!

• To display the internal structure of those “atomic”

sentences by adding new categories of vocabulary

items

• To give a semantic account of these new vocabulary

items and

• To introduce rules for operating with them.

Singular statements

Make assertions about persons, places, things or

times

Examples:

• Socrates is a man.

• Athens is in Greece.

• Thomas Aquinas preferred Aristotle to Plato

To display the internal structure of such sentences

we need two new categories of vocabulary items:

• Individual constants (“names”)

• Predicates

Individual Constants Individual Constants & PredicatesPredicates

SocratesSocrates is a manis a man.

The name of a person—which we’ll express by an individual constant

The name of a person—which we’ll express by an individual constant

A predicate—assigns a property (being-a-man) to the person named

More Vocabulary

• So, to the vocabulary of Propositional Logic we add:

• Individual Constants: lower case letters of the Individual Constants: lower case letters of the

alphabet (a, b, c,…, u, v, w)alphabet (a, b, c,…, u, v, w)

• Predicates: upper case letters (A, B, C,…, X, Y, Z)Predicates: upper case letters (A, B, C,…, X, Y, Z)

– These represent predicates like “__ is human,” “__

the teacher of __,” “__ is between __ and __,” etc.

– Note: Predicates express properties which a single

individual may have and relations which may hold

on more than one individual

Predicates

• 1-place predicates assign properties to individuals:

__ is a man

__ is red

• 2-place predicates assign relations to pairs of individuals

__ is the teacher of __

__ is to the north of __

• 3-place predicates assign relations to triples of individuals

__ preferred __ to __

• And so on…

Translation

SocratesSocrates is a man.

Hss

SocratesSocrates is mortal.

Mss

AthensAthens is in GreeceGreece

Iagag

ThomasThomas preferred AristotleAristotle to Plato.Plato.

Ptaptap

We always put the predicate first followed by the names of the things to which it applies.

What are properties & relations?

Are they ideas in people’s heads?

Are they Forms in Plato’s heaven?

All this is controversial so we resolve to treat properties and relations as things that aren’t controversial, viz. sets.

Predicates designate sets!

• Individual constants name individuals—persons,

places, things, times, etc.

• We resolve to understand properties and relations as

sets—of individuals or ordered n-tuples of individuals

(pairs, triples, quadruples, etc.)

Fido

{ brown things }

is brown

Predicate Logic Vocabulary• Individual Constants: lower case letters of the alphabet (a,

b, c,…, u, v, w)

• Predicates: upper case letters (A, B, C,…, X, Y, Z)

• Connectives: , , • , , and

• Variables: x, y and z

• Quantifiers:

– Existential ( [variable]), e.g. (x), (y)…

– Universal ( [variable]) or just ([variable] ), e.g. (x), (y), (x), (y)…

Sets

• A set is a well defined collection of objects.

• The elements of a set, also called its members, can be anything: numbers, people, letters of the alphabet, other sets, and so on.

• Sets A and B are equal iff they have precisely the same elements.

• Sets are “abstract objects”: they don’t occupy time or space, or have causal powers even if their members do.

Venn Diagrams

• Set theoretical concepts, like union and intersection can be represented visually by Venn Diagrams

• Venn Diagrams represent sets as circles and

• Represent the emptiness of a set by shading out the region represented by it and

• Represent the non-emptiness of a set by putting an “X” in it.

Set Membership & Set Inclusion

Mama GrandmaThe Kids

The Babushka Family: C is the daughter of D and D is the daughter of E, but C is not the daughter of E!


• x S : x is a member of the set S

– 2 {1, 2, 3}

– C {daughters of D}

• Anything can be a member of a set…including

another set!

– {1, 2, 3} {{1, 2, 3}, 4}

• But members of members aren’t members!

– 4 {1, 2, 3}, 4}, but 2 {{1, 2, 3}, 4} !

• C is a daughter of D and D is the daughter of E but C

is not the daughter of D!


S1 ⊆ S2 : S1 included in (is a subset of) S2

•This means every member of S1 is a member of S2

– {1, 2, 3} {1, 2, 3, 4}⊆

The numbers 1, 2, and 3 are members of {1, 2, 3, 4},

but the set {1, 2, 3} is not a member!

– {1, 2, 3} {1, 2, 3, 4}

•It is however a member of this set:

– {1, 2, 3} {{1, 2, 3}, 4}

•Sets are identical if they have the same members so

– {1, 2, 3, 4} ≠ {{1, 2, 3}, 4}

Ordered Sets• We use curly-brackets to designate sets so, e.g. the

set of odd numbers between 1 and 10 is {3, 5, 7, 9}

• Order doesn’t matter for sets so, e.g.

– {3, 5, 7, 9} = {3, 9, 5, 7}

• But sometimes order does matter: to signal that we

use pointy-brackets to designate ordered n-tuples—

ordered pairs, triples, quadruples, quintuples, etc.

– <Adam, Eve> ≠ <Eve, Adam>

– <1, 2, 3> ≠ <3, 2, 1>

The Set of Russian Doll Sets

• C {Babushkas}• {Babushkas} {Russian Doll Sets}• C {Russian Doll Sets}

• C is a doll—not a set of dolls!

Having a property• An individual’s having a property is understood as its

being a member of a set.

• “Fido is brown” says that Fido is a member of the set of brown things.

Fido { brown things }ε

What Singular Sentences Say

• Singular sentences are about set membership.

• Ascribing a property to an individual is saying that it’s a member of the set of individuals that have that property, e.g.

– “Descartes is a philosopher” says that Descartes is a member of {Plato, Aristotle, Locke, Berkely, Hume, Quine…}

• Saying that 2 or more individuals stand in a relation is saying that some ordered n-tuple is a member of a set so, e.g.

– “San Diego is north of Chula Vista” says that <San Diego,Chula Vista> is a member of {<LA,San Diego>, <Philadelphia,Baltimore>…}

Singular Sentences



_______________________


• We can now understand what 2

and 3 say:

• 2 says that Socrates ε {humans}

• 3 says that Socrates ε {mortals}

Singular statementsTranslation

Ducati is brown.

Bd

Tweety bopped Sylvester

Bts

General Statements

General Sentences



_______________________


• But we still don’t have an account

of what 1 says

• Because it doesn’t ascribe a

property or relation to any

individual or individuals.

Emptiness

Shading in a region says that the set it represents is empty.

A

This says that there are no As.

There are no unicorns.

unicorns

(x) Ux

Non-emptiness

‘X’ in a region says that the set it the region represents is non-empty, i.e. there’s something there

A

This says that at least one thing is an A.

There are horses.

horses

X

(x) Hx

Intersection

Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B.

This is the intersection of sets A and B

A B

No horses are carnivores

(x) (Hx • Cx)

horses carnivores

Intersection & Conjunction

This region represents the set of things that are both A and B.

A B

X

Some horses are brown

brown things

(x)(Hx • Bx)

X

horses

Set Complement

The complement of a set if everything in the universe that’s not in the set.

The shaded area is the complement of A—the set of things that are not A.

A

Set Complement

This says that something is in both A but not B.

A B

X

Some horses are not brown

brown thingshorses

X

(x)(Hx • Bx)

All horses are mammals

(x)(Hx Mx)

horses mammalshorses

All horses are mammals

horses mammals

There are no non-mammalian horseshorses ∩ non-mammals = { } (the empty set)

Set Union

Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both.

This is the union of sets A and B

A B

Union and Disjunction• If we say something is in the union of A and B we’re

saying that it’s either in A or in B.

A B

General Sentences



_______________________


• We can’t treat “all men” as the

name of an individual as, e.g. the

sum or collection of all humans

• Consider: “All men weigh under

1000 pounds.”

• We want to ascribe properties to

every man individually.

The Socrates Argument is Valid!



_______________________


• 1 says that there are no men that

don’t belong to the set of mortals

• 2 says that Socrates ε {men}

• 3 says that, therefore, Socrates ε

{mortals}

All men are mortals

There’s nothing in the set of men outside of the set of mortals;

the set of immortal men is empty.

men mortals

immortal men: nothing here!

Socrates is a man.

• This is the only place in the men circle Socrates can go since we’ve already said that there are no immortal men.

men mortals

Therefore, Socrates is a mortal.

• So Socrates automatically ends up in the mortals circle: the argument is, therefore, valid!

men mortals

Arguments Involving Relations

1. All dogs love their people

2. Bo is Obama’s Dog

____________________

3. Bo loves Obama

If we treat the predicates in these 3 sentences as one-place predicates, we can’t explain the validity of this argument!

It looks like we’re ascribing 3 different properties that don’t have anything to do with one another: people-loving, being-Obama’s-dog and Obama-loving

Relations

• We understand relations as sets of ordered n-tuples,

e.g.

• being to the north of is {<Los Angeles, San Diego>,

<Philadelphia, Baltimore>, <San Diego, Chula

Vista>…}

• being the quotient of one number divided by another:

x into y is {<2,2,1>, <2,4,2>, <5,45,9>…}

Singular sentences about relations

• Standing in a relation is also understood in terms of

set membership.

• It’s a matter of being a member of an ordered n-tuple

which is a member of a set.

Michelle is Barak’s wife.

ε {<Eve, Adam>, <Hillary, Bill>…}

Validity of the Dog Argument



____________________

3. Bo loves Obama

We can consider being the dog of and loves as relations and designate them by the 2-place predicates “D”and “L”

Now the 3 sentences have something in common--they don’t involve 3 different properties; they involve 2 relations: being the dog of and loves

So we can link them to show validity

Bo is Obama’s dog: Dbo

All dogs love their people

(x)(y)(Dxy Lxy)For all x, y, if x is the dog of y then x loves y

Validity of the Dog Argument



____________________

3. Bo loves Obama

1. (x)(y)(Dxy Lxy)

2. Dbo______________

3. Lbo

1 says that for any x and y, if x is the dog of y then x loves y.

2 says that Bo is the dog of Obama.

So it follows that Bo loves Obama!

All dogs love their people

(x)(y)(Dxy Lxy)

__ is the dog of __ __ loves __

Bo is Obama’s Dog

Dbo


<Bo,Obama>

Bo loves Obama

Lbo


<Bo,Obama>

There are dogs and there are cats.

(x)Dx (x)Cx : There exists an x such that x is a dog and

there exists an x such that x is a cat

There are dog-cats

(x)(Dx Cx): There exists an x such that x is a dog-and-a-

cat

Overlapping quantifiers

Everybody loves somebody

(x)(y)Lxy: for all x, there exists a y such that x loves y

Everybody is loved by somebody

(x)(y)Lyx: for all x, there exists a y such that y loves x

There’s somebody everybody loves

(x)(y)Lyx: There exists an x such that for all y, y loves x

There’s somebody who loves everybody

(x)(y)Lxy: there exists an x such that for all y, x loves y

The EndThe End

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