# Math 10130 Lecture 32 by wku77463

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```									                                    Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Math 10130 Lecture 32

9 April 2008

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Last time

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Last time

Last time:

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Last time

Last time: We considered quantiﬁed formulas in several diﬀerent
contexts.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Last time

Last time: We considered quantiﬁed formulas in several diﬀerent
contexts.
Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Last time

Last time: We considered quantiﬁed formulas in several diﬀerent
contexts.
Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ”
- true when the domain of discourse is the set of natural
numbers N

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Last time

Last time: We considered quantiﬁed formulas in several diﬀerent
contexts.
Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ”
- true when the domain of discourse is the set of natural
numbers N
- false when the domain of discourse is the set of integers Z

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Last time

Last time: We considered quantiﬁed formulas in several diﬀerent
contexts.
Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ”
- true when the domain of discourse is the set of natural
numbers N
- false when the domain of discourse is the set of integers Z
Principle: Truth or falsity of a statement depends on the
context.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Alternation of Quantiﬁers

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Alternation of Quantiﬁers

Consider the following structure:

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Alternation of Quantiﬁers

Consider the following structure:
Domain of discourse: people

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Alternation of Quantiﬁers

Consider the following structure:
Domain of discourse: people
Mxy stands for “x loves y ”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Alternation of Quantiﬁers

Consider the following structure:
Domain of discourse: people
Mxy stands for “x loves y ”

Q: Do ∀x ∃y Mxy and ∃y ∀x Mxy express the same proposition?

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Comparing the two formulas

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Comparing the two formulas

∀x ∃y Mxy says, “For each person x, there is a y such that x loves
y .”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Comparing the two formulas

∀x ∃y Mxy says, “For each person x, there is a y such that x loves
y .”

In other words, “Everybody loves somebody.”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Comparing the two formulas

∀x ∃y Mxy says, “For each person x, there is a y such that x loves
y .”

In other words, “Everybody loves somebody.”

∃y ∀x Mxy says, “There is a person y such that for all persons x,
x loves y .”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Comparing the two formulas

∀x ∃y Mxy says, “For each person x, there is a y such that x loves
y .”

In other words, “Everybody loves somebody.”

∃y ∀x Mxy says, “There is a person y such that for all persons x,
x loves y .”

In other words, “Somebody is loved by everybody.”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A tough example

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A tough example

Domain of discourse: the set of natural numbers, N

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A tough example

Domain of discourse: the set of natural numbers, N
Mxy stands for “x ≤ y ”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A tough example

Domain of discourse: the set of natural numbers, N
Mxy stands for “x ≤ y ”

Let’s discuss this beast of a formula:

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A tough example

Domain of discourse: the set of natural numbers, N
Mxy stands for “x ≤ y ”

Let’s discuss this beast of a formula:

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A huge syntactic tree

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A huge syntactic tree

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A huge syntactic tree

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A huge syntactic tree

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

(Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A huge syntactic tree

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

(Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))

Mxy & ¬x = y                 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A huge syntactic tree

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

(Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))

Mxy & ¬x = y                 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))

(Mxz & ¬x = z) &(Mzy & ¬z = y )

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A huge syntactic tree

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

(Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))

Mxy & ¬x = y                 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))

(Mxz & ¬x = z) &(Mzy & ¬z = y )

Mxz & ¬x = z         Mzy & ¬z = y

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A huge syntactic tree

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) 6

∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

(Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))

Mxy & ¬x = y 1                   ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) 5

(Mxz & ¬x = z) &(Mzy & ¬z = y ) 4

Mxz & ¬x = z 2       Mzy & ¬z = y 3
10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula

1 Mxy & ¬x = y says “x < y .”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula

1 Mxy & ¬x = y says “x < y .”
2 Mxz & ¬x = z says “x < z.”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula

1 Mxy & ¬x = y says “x < y .”
2 Mxz & ¬x = z says “x < z.”
3 Mzy & ¬z = y says “z < y .”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula

1 Mxy & ¬x = y says “x < y .”
2 Mxz & ¬x = z says “x < z.”
3 Mzy & ¬z = y says “z < y .”
4 (Mxz & ¬x = z) &(Mzy & ¬z = y ) says “x < z and z < y ” or
“x < z < y .”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula

1 Mxy & ¬x = y says “x < y .”
2 Mxz & ¬x = z says “x < z.”
3 Mzy & ¬z = y says “z < y .”
4 (Mxz & ¬x = z) &(Mzy & ¬z = y ) says “x < z and z < y ” or
“x < z < y .”
5 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) says “There is a z such
that x < z < y .”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula

1 Mxy & ¬x = y says “x < y .”
2 Mxz & ¬x = z says “x < z.”
3 Mzy & ¬z = y says “z < y .”
4 (Mxz & ¬x = z) &(Mzy & ¬z = y ) says “x < z and z < y ” or
“x < z < y .”
5 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) says “There is a z such
that x < z < y .”
6 (the big formula) says “For all x and y , if x < y , then there is a
z such that x < z < y .”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula, continued

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula, continued

“For all x and y , if x < y , then there is a z such that x < z < y .”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula, continued

“For all x and y , if x < y , then there is a z such that x < z < y .”
Is this true of all pairs of natural numbers?

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula, continued

“For all x and y , if x < y , then there is a z such that x < z < y .”
Is this true of all pairs of natural numbers?
Consider the pair (0, 1).

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula, continued

“For all x and y , if x < y , then there is a z such that x < z < y .”
Is this true of all pairs of natural numbers?
Consider the pair (0, 1).
0 < 1, but there is no z such that 0 < z < 1.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Breaking down the formula, continued

“For all x and y , if x < y , then there is a z such that x < z < y .”
Is this true of all pairs of natural numbers?
Consider the pair (0, 1).
0 < 1, but there is no z such that 0 < z < 1.
Thus, the formula is false.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Changing the context

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Changing the context
Let’s change the domain of discourse to the set of rational
numbers Q, i.e. the set of postive and negative fractions.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Changing the context
Let’s change the domain of discourse to the set of rational
numbers Q, i.e. the set of postive and negative fractions.
Is the formula

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

true or false in this new context?

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Changing the context
Let’s change the domain of discourse to the set of rational
numbers Q, i.e. the set of postive and negative fractions.
Is the formula

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

true or false in this new context?
Remember, the formula is translated “For all x and y , if x < y ,
then there is a z such that x < z < y .”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Changing the context
Let’s change the domain of discourse to the set of rational
numbers Q, i.e. the set of postive and negative fractions.
Is the formula

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

true or false in this new context?
Remember, the formula is translated “For all x and y , if x < y ,
then there is a z such that x < z < y .”
Suppose we have rational numbers x and y such that x < y .

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Changing the context
Let’s change the domain of discourse to the set of rational
numbers Q, i.e. the set of postive and negative fractions.
Is the formula

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

true or false in this new context?
Remember, the formula is translated “For all x and y , if x < y ,
then there is a z such that x < z < y .”
Suppose we have rational numbers x and y such that x < y .

Q: Is there a z such that x < z < y ?

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Changing the context
Let’s change the domain of discourse to the set of rational
numbers Q, i.e. the set of postive and negative fractions.
Is the formula

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

true or false in this new context?
Remember, the formula is translated “For all x and y , if x < y ,
then there is a z such that x < z < y .”
Suppose we have rational numbers x and y such that x < y .

Q: Is there a z such that x < z < y ?
x+y
A: Yes. Let z =            2 .
10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Changing the context
Let’s change the domain of discourse to the set of rational
numbers Q, i.e. the set of postive and negative fractions.
Is the formula

∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

true or false in this new context?
Remember, the formula is translated “For all x and y , if x < y ,
then there is a z such that x < z < y .”
Suppose we have rational numbers x and y such that x < y .

Q: Is there a z such that x < z < y ?
x+y
A: Yes. Let z =            2 .      Thus, the formula is true.
10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Recall. . .

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Recall. . .

“logically true” means true in every situation

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Recall. . .

“logically true” means true in every situation
“consistent” means true in some situation

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Recall. . .

“logically true” means true in every situation
“consistent” means true in some situation
“inconsistent” means true in no situation

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Recall. . .

“logically true” means true in every situation
“consistent” means true in some situation
“inconsistent” means true in no situation

Now, we have the following new deﬁnitions:

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Recall. . .

“logically true” means true in every situation
“consistent” means true in some situation
“inconsistent” means true in no situation

Now, we have the following new deﬁnitions:
A closed predicate formula is

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Recall. . .

“logically true” means true in every situation
“consistent” means true in some situation
“inconsistent” means true in no situation

Now, we have the following new deﬁnitions:
A closed predicate formula is
logically true if it is true in every structure,

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Recall. . .

“logically true” means true in every situation
“consistent” means true in some situation
“inconsistent” means true in no situation

Now, we have the following new deﬁnitions:
A closed predicate formula is
logically true if it is true in every structure,
consistent if it is true in some structure, and

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

Recall. . .

“logically true” means true in every situation
“consistent” means true in some situation
“inconsistent” means true in no situation

Now, we have the following new deﬁnitions:
A closed predicate formula is
logically true if it is true in every structure,
consistent if it is true in some structure, and
inconsistent if it is true in no structure.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula

We will work with a vocabulary consisting of a 1-place predicate
Gx and a 2-place predicate Mxy .

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula

We will work with a vocabulary consisting of a 1-place predicate
Gx and a 2-place predicate Mxy .
To show: ∀x (¬Gx ⇒ ∃y Mxy ) is consistent.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula

We will work with a vocabulary consisting of a 1-place predicate
Gx and a 2-place predicate Mxy .
To show: ∀x (¬Gx ⇒ ∃y Mxy ) is consistent.
That is, we need to ﬁnd a structure in which ∀x (¬Gx ⇒ ∃y Mxy )
is true.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people
Gx: “x is single”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people
Gx: “x is single”
Mxy : “x is married to y ”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people
Gx: “x is single”
Mxy : “x is married to y ”
In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
x is not single, then x is married.”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people
Gx: “x is single”
Mxy : “x is married to y ”
In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
x is not single, then x is married.” True.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people
Gx: “x is single”
Mxy : “x is married to y ”
In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
x is not single, then x is married.” True.
Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people
Gx: “x is single”
Mxy : “x is married to y ”
In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
x is not single, then x is married.” True.
Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
No. It is false in the following structure:

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people
Gx: “x is single”
Mxy : “x is married to y ”
In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
x is not single, then x is married.” True.
Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
No. It is false in the following structure:
Domain: people

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people
Gx: “x is single”
Mxy : “x is married to y ”
In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
x is not single, then x is married.” True.
Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
No. It is false in the following structure:
Domain: people
Gx: “x is male”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people
Gx: “x is single”
Mxy : “x is married to y ”
In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
x is not single, then x is married.” True.
Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
No. It is false in the following structure:
Domain: people
Gx: “x is male”
Mxy : “x is married to y ”

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people
Gx: “x is single”
Mxy : “x is married to y ”
In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
x is not single, then x is married.” True.
Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
No. It is false in the following structure:
Domain: people
Gx: “x is male”
Mxy : “x is married to y ”
Then the formula says, “For every person x, if x is not male, then
x is married.”
10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A consistent formula, continued
Here’s the desired structure:
Domain: people
Gx: “x is single”
Mxy : “x is married to y ”
In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
x is not single, then x is married.” True.
Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
No. It is false in the following structure:
Domain: people
Gx: “x is male”
Mxy : “x is married to y ”
Then the formula says, “For every person x, if x is not male, then
x is married.” False.
10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula
Vocabulary: 1-place predicate Fx.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula
Vocabulary: 1-place predicate Fx.
To show: ∀x Fx ⇒ ∃x Fx is logically true.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula
Vocabulary: 1-place predicate Fx.
To show: ∀x Fx ⇒ ∃x Fx is logically true.
Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula
Vocabulary: 1-place predicate Fx.
To show: ∀x Fx ⇒ ∃x Fx is logically true.
Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
Let S be any structure.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula
Vocabulary: 1-place predicate Fx.
To show: ∀x Fx ⇒ ∃x Fx is logically true.
Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
Let S be any structure.
Assume that ∀x Fx ⇒ ∃x Fx is false in S.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula
Vocabulary: 1-place predicate Fx.
To show: ∀x Fx ⇒ ∃x Fx is logically true.
Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
Let S be any structure.
Assume that ∀x Fx ⇒ ∃x Fx is false in S.
Thus, ∀x Fx is true in S, but ∃x Fx is false in S.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula
Vocabulary: 1-place predicate Fx.
To show: ∀x Fx ⇒ ∃x Fx is logically true.
Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
Let S be any structure.
Assume that ∀x Fx ⇒ ∃x Fx is false in S.
Thus, ∀x Fx is true in S, but ∃x Fx is false in S.
∀x Fx is true in S: Every object in the domain of S has the
property F .

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula
Vocabulary: 1-place predicate Fx.
To show: ∀x Fx ⇒ ∃x Fx is logically true.
Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
Let S be any structure.
Assume that ∀x Fx ⇒ ∃x Fx is false in S.
Thus, ∀x Fx is true in S, but ∃x Fx is false in S.
∀x Fx is true in S: Every object in the domain of S has the
property F .
∃x Fx is false in S: There is no object in the domain of S has
the property F .

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula
Vocabulary: 1-place predicate Fx.
To show: ∀x Fx ⇒ ∃x Fx is logically true.
Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
Let S be any structure.
Assume that ∀x Fx ⇒ ∃x Fx is false in S.
Thus, ∀x Fx is true in S, but ∃x Fx is false in S.
∀x Fx is true in S: Every object in the domain of S has the
property F .
∃x Fx is false in S: There is no object in the domain of S has
the property F .
Impossible!

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula
Vocabulary: 1-place predicate Fx.
To show: ∀x Fx ⇒ ∃x Fx is logically true.
Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
Let S be any structure.
Assume that ∀x Fx ⇒ ∃x Fx is false in S.
Thus, ∀x Fx is true in S, but ∃x Fx is false in S.
∀x Fx is true in S: Every object in the domain of S has the
property F .
∃x Fx is false in S: There is no object in the domain of S has
the property F .
Impossible!
Thus, ∀x Fx ⇒ ∃x Fx is true in S.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula
Vocabulary: 1-place predicate Fx.
To show: ∀x Fx ⇒ ∃x Fx is logically true.
Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
Let S be any structure.
Assume that ∀x Fx ⇒ ∃x Fx is false in S.
Thus, ∀x Fx is true in S, but ∃x Fx is false in S.
∀x Fx is true in S: Every object in the domain of S has the
property F .
∃x Fx is false in S: There is no object in the domain of S has
the property F .
Impossible!
Thus, ∀x Fx ⇒ ∃x Fx is true in S.
But S is an arbitrary structure, so this holds for all structures.
10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula, continued

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula, continued

Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula, continued

Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

1.                                                  ∀x Fx               (assume)

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula, continued

Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

1.                                                  ∀x Fx               (assume)
2.                                                  ???                   (∀E 1)

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula, continued

Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

1.                                                  ∀x Fx               (assume)
2.                                                  ???                   (∀E 1)
.
.                                                     .
.
.                                                     .
.
.                                                     .
.
.                                                     .

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula, continued

Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

1.                                                  ∀x Fx                (assume)
2.                                                  ???                    (∀E 1)
.
.                                                     .
.
.                                                     .
.
.                                                     .
.
.                                                     .
?                                                   ∃x Fx                 (∃I, ??)

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula, continued

Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

1.                                                  ∀x Fx                 (assume)
2.                                                  ???                     (∀E 1)
.
.                                                     .
.
.                                                     .
.
.                                                     .
.
.                                                     .
?                                                   ∃x Fx                  (∃I, ??)
? ∀x Fx ⇒ ∃x Fx                                                          (⇒I, 1-??)

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula, continued

Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

1.                                                 ∀x Fx                (assume)
2.                                                 ???                    (∀E 1)
.
.                                                    .
.
.                                                    .
.
.                                                    .
.
.                                                    .
?                                                  ∃x Fx                 (∃I, ??)
? ∀x Fx ⇒ ∃x Fx                                                         (⇒I, 1-??)

10130 Lecture 32
Review
A bit more on the alternation of quantiﬁers
One last example
Logical Theory for Predicate Logic

A logically true formula, continued

Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

1.                                                  ∀x Fx                (assume)
2.                                                  ???                    (∀E 1)
.
.                                                     .
.
.                                                     .
.
.                                                     .
.
.                                                     .
?                                                   ∃x Fx                 (∃I, ??)
? ∀x Fx ⇒ ∃x Fx                                                          (⇒I, 1-??)

We’ll do this later. . .

10130 Lecture 32

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