1. Variables
2. Incomplete Statements
3. Symbols for Incomplete Statements
4. Universal Quantifiers
5. Ways of Writing Universal Quantifiers
6. Universal Quantifiers and the Logical AND
7. Existential Quantifiers
8. Ways of Writing Existential Quantifiers
9. Existential Quantifiers and the Logical OR
10. Law of Reasoning with Universal Quantifiers
11. Law of Reasoning with Existential Quantifiers
12. Negation of the Universal Quantifier
13. Universal Quantifier and De Morgan's Law
14. Negation of the Existential Quantifier
15. Existential Quantifier and De Morgan's Law
16. Universal Quantifier with the Logical AND
17. Existential Quantifier with the Logical AND
18. Universal Quantifier with the Logical OR
19. Existential Quantifier with the Logical OR
x = January, x = February, etc.
We normally use letters at the end of the alphabet as variables, such as x, y, z.
For example, suppose the incomplete statement
x has 30 days
is about months. When we replace the variable x by April we obtain the true statement: April has 30 days.
When we replace the variable x by July we obtain the false statement:
July has 30 days.
x = A horse
x = A bird
x = A snake
x = A tiger.
Say in each case whether the statement is true or false.
For example, when x stands for a month we can let
P(x) = x has 30 days.
Then
P(September) = September has 30 days.
As another example, let x stand for a boat. Then we can let
P(x) = x has a mast.
This is an incomplete statement about boats. If x = my boat, then we obtain the statement:
P(my boat) = My boat has a mast.
For example, let x stand for a month, and let
P(x) = x has less than 32 days.
Then
For all x: x has less than 32 days
is a true statement about all months. It has the same meaning as the ordinary sentence
Every month has less than 32 days.
The phrase For all x: is called the universal quantifier.
P(x) = x is made of steel and concrete.
Give the statement made by the universal quantifier and this incomplete statement, and give an ordinary sentence with the same meaning.
Every box is empty
may be written with the universal quantifier in various ways with the same meaning as follows:
For all x: x is empty
For every x: x is empty
For any x: x is empty.
We write all these statements briefly in symbols as follows:
Ax: P(x),
where Ax: stands for the universal quantifier For all x: and P(x) stands for the incomplete statement x is empty.
P(x) = x needs water to grow.
Write the statement Ax: P(x) in various ways, and give the ordinary sentence with the same meaning.
For example, we can make a list of all the months, as follows:
January, February, ..., December.
Let P(x) = x has less than 32 days.
Now we can make a list of statements:
January has less than 32 days
February has less than 32 days
...
December has less than 32 days.
All these statements are true. Therefore we obtain a true composite statement when we join them all together by the word and, as follows:
January has less than 32 days, and February has less than 32 days, and ..., and December has less than 32 days.
In ordinary writing we may shorten this to:
January, February, ..., and December have less than 32 days.
The composite statement has the same meaning as:
For all x: x has less than 32 days.
This is an example of a general fact that may be summarized in symbols as follows:
If we can make a list a1, a2, ..., an of the individuals in a set, and if P(x) is an incomplete statement about these individuals, then
Ax: P(x) = P(a1) and P(a2) and ... and P(an).
For example, let x stand for a month, and let
P(x) = x has less than 30 days.
Then the statement
There exists an x such that: x has less than 30 days
means the same as the ordinary sentence
At least one month has less than 30 days.
The statement is true because we can put x = February and obtain the true statement: February has less than 30 days.
The phrase There exists an x such that: is called the existential quantifier.
There exists an x such that: x is empty
There is an x such that: x is empty
For some x: x is empty
For at least one x: x is empty.
We may write all these statements in symbols as follows:
Ex: P(x),
where Ex: stands for the existential quantifier, and P(x) stands for the incomplete statement x is empty.
P(x) = x has no legs.
Write the statement Ex: P(x) with the existential quantifier in various ways, and give the ordinary sentence with the same meaning. Name an animal for which the incomplete statement gives a true statement.
Let x stand for a hotel in the town, and let
P(x) = x is open.
The list of statements made by replacing x by each hotel is as follows:
The Hill View Hotel is open, which is true.
The Riverside Hotel is open, which is false.
The Central Hotel is open, which is true.
Because at least one of these statements is true (actually two of them are true), we obtain a true composite statement by joining all the statements in the list by the word or. Then the following statement is true:
The Hill View Hotel is open, or the Riverside Hotel is open, or the Central Hotel is open.
This has the same meaning as:
There exists an x such that: x is open.
This is an example of a general fact that may be summarized as follows:
If we can make a list a1, a2, ..., an of the individuals in a set, and if P(x) is an incomplete statement about these individuals, then
Ex: P(x) = P(a1) or P(a2) or ... or P(an).
All plants need water to grow
A lettuce is a plant.
We may conclude from these two statements that:
A lettuce needs water to grow.
This is an example of the law of reasoning with the universal quantifier as follows:
If the statement Ax: P(x) is true, and if "a" is a particular individual in the set represented by the variable x, then the statement P(a) is true.
In the above example x stands for a plant, and we have:
P(x) = x needs water to grow
and
a = A lettuce.
We may summarize this law in symbols as follows:
From Ax: P(x) we conclude P(a).
An elephant is an animal
An elephant has a trunk.
We may conclude from these two statements that:
There exists an animal with a trunk.
This is an example of the law of reasoning with the existential quantifier as follows:
If the statement P(a) is true, where "a" is a particular individual in the set represented by the variable x, then the statement Ex: P(x) is true.
In the above example x stands for an animal, a = An elephant, and P(x) = x has a trunk. Then we conclude: There exists an x such that: x has a trunk.
We may summarize this law in symbols as follows:
From P(a) we conclude Ex: P(x).
not-Ax: P(x).
This means It is not true that for all x: P(x). This has the same meaning as: There exists an x such that not-P(x). For example, let x stand for an animal, and let P(x) = x has four legs. Then not-Ax: P(x) means:
It is not true that every animal has four legs. In other words:
There exists an animal which does not have four legs.
We summarize this law in symbols as follows:
not-Ax: P(x) = Ex: not-P(x).
not-Ax: P(x) = not-(P(a1) and ... and P(an)).
By De Morgan's law,
not-(P(a1) and ... and P(an)) = not-P(a1) or ... or not-P(an).
But not-P(a1) or ... or not-P(an) has the same meaning as Ex: not-P(x).
For example, suppose the three hotels in a town are:
a1 = The Hill View Hotel
a2 = The Riverside Hotel
a3 = The Central Hotel.
Let x stand for a hotel, and let P(x) = x has a swimming pool.
Then the statement:
It is not true that every hotel has a swimming pool
may be written in symbols as follows:
not-Ax: P(x).
This may also be written:
not-P(a1) or not-P(a2) or not-P(a3),
which means:
The Hill View Hotel or the Riverside Hotel or the Central Hotel does not have a swimming pool.
not-Ex: P(x).
This means It is not true that there exists an x such that P(x), in other words: There does not exist an x such that P(x). This has the same meaning as: For all x: not-P(x).
For example, let x stand for a book in a collection of books, and let P(x) = x is missing. Then not-Ex: P(x) means It is not true that there is a book missing, in other words: Every book is not missing, or No book is missing.
We summarize this law in symbols as follows:
not-Ex: P(x) = Ax: not-P(x).
not-Ex: P(x) = not-[P(a1) or ... or P(an)].
By De Morgan's law,
not-[P(a1) or ... or P(an)] = not-P(a1) and ... and not-P(an).
But not-P(a1) and ... and not-P(an) has the same meaning as Ax: not-P(x).
For example, let x stand for one of the days at the weekend, and put: a1 = Saturday, a2 = Sunday. Let P(x) = x is a working day. Then the statement There does not exist an x such that x is a working day may be written in symbols as follows:
not-Ex: P(x).
This may also be written:
not-P(a1) and not-P(a2),
which means
Saturday is not a working day, and Sunday is not a working day.
For example, let x stand for a tree, let P(x) = x has green leaves and let Q(x) = x needs water to grow. Then Ax:[P(x) and Q(x)] means Every tree has green leaves and needs water to grow. This has the same meaning as the statement Every tree has green leaves, and every tree needs water to grow, which may be written in symbols as follows: Ax: P(x) and Ax: Q(x).
This is an example of the general law:
Ax:[P(x) and Q(x)] = Ax: P(x) and Ax: Q(x).
For example, let x stand for a man in a group, let P(x) = x has black hair, and let Q(x) = x is tall. Then Ex:[P(x) and Q(x)] means In the group there exists a man who has black hair and is tall. From this we may conclude that: In the group there exists a man who has black hair, and in the group there exists a man who is tall.
This is an example of the general law:
From Ex:[P(x) and Q(x)] we conclude [Ex: P(x) and Ex: Q(x)].The converse of this law is not true.
For example, let x stand for a man in a group, let P(x) = x has black hair, and let Q(x) = x is tall. Then Ex: P(x) and Ex: Q(x) means In the group there exists a man who has black hair, and in the group there exists a man who is tall. But we cannot conclude from this that: In the group there exists a man who has black hair and is tall. The man with black hair need not be tall, and the man who is tall need not have black hair; they may be two different men.
For example, let x stand for a man in a group, let P(x) = x has black hair, and let Q(x) = x has white hair. Then Ax:[P(x) or Q(x)] means Every man in the group has black hair or white hair. From this we cannot conclude: [Ax: P(x) or Ax: Q(x)], which means Every man in the group has black hair, or every man in the group has white hair.
However, the converse is true. From the statement Every man in the group has black hair, or every man in the group has white hair, we can conclude: Every man has black hair or white hair.
The general law is summarized in symbols as follows:
From [Ax: P(x) or Ax: Q(x)] we conclude Ax:[P(x) or Q(x)].
For example, let x stand for an electric light bulb in a room, let P(x) = x is missing, and let Q(x) = x is burned out. Then Ex:[P(x) or Q(x)] means At least one electric light bulb is missing or burned out. This has the same meaning as the statement At least one electric light bulb is missing, or at least one electric light bulb is burned out, which may be written in symbols as follows:
Ex: P(x) or Ex: Q(x).
This is an example of the general law:
Ex:[P(x) or Q(x)] = Ex: P(x) or Ex: Q(x).
Backward links: Statements I Statements II
Forward links: Quantifiers II Sets I Sets II Relations I Relations II
By R. H. B. Exell, 1998. King Mongkut's University of Technology Thonburi.
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