20. Two Variables
21. Symbols for Incomplete Statements with Two Variables
22. Quantifiers with One Variable
23. Two Universal Quantifiers
24. One Universal Quantifier and One Existential Quantifier
25. Law of Reasoning with Two Quantifiers
26. Two Existential Quantifiers
27. Negation of Statements with Two Universal Quantifiers
28. Negation of Statements with a Universal Quantifier Followed by an Existential Quantifier
29. Negation of Statements with an Existential Quantifier Followed by a Universal Quantifier
30. Negation of Statements with Two Existential Quantifiers
For example, suppose we have a number of men and women in a group. Let x stand for a man in the group, and let y stand for a woman in the group. Then
x is older than y
is an incomplete statement about the men and women in the group.
When we replace one of the variables by an individual, we obtain an incomplete statement containing the other variable. For example, if the men are John, Peter and George, and the women are Anne, Mary and Susan, then
x is older than Anne
John is older than y
are incomplete statements with one variable replaced and the other variable remaining.
When we replace both variables by individuals we obtain a statement. For example,
Peter is older than Susan
is a statement obtained by replacing both x and y by members of the sets of men and women in the group.
x is the capital of y.
For example, let
P(x, y) = x is older than y,
where x stands for a man and y stands for a woman in a group of people.
As another example, let
P(x, y) = x is the capital of y,
where x stands for a city and y stands for a country.
Let P(x, y) be an incomplete statement with the two variables x and y. Let
Ax: P(x, y) = Q(y)
and
Ex: P(x, y) = R(y).
Then Q(y) and R(y) are new incomplete statements containing only the variable y.
For example, let
P(x, y) = x is taller than y,
where x is a man and y is a woman in a group of people. Then Ax: P(x, y) is the incomplete statement:
Q(y) = Every man is taller than y,
where y is a woman in the group. If we now let y = Anne, we obtain the statement
For all x, P(x, Anne),
which means
Every man is taller than Anne.
Similarly, Ex: P(x, y) is the incomplete statement:
R(y) = At least one man is taller than y,
where y is a woman in the group. If we now let y = Anne, we obtain the statement
There exists x such that P(x, Anne),
which means
At least one man is taller than Anne.
In the same way we may put
Ay: P(x, y) = Q(x)
and
Ey: P(x, y) = R(x),
where Q(x) and R(x) are incomplete statements containing the variable y.
For example, when
P(x, y) = x is taller than y, we may interpret the following symbols as follows:
Ax: Ay: P(x, y),
and
Ay: Ax: P(x, y).
These two statements have the same meaning. Therefore it does not matter which universal quantifier is first, and we have the general law:
Ax: Ay: P(x, y) = Ay: Ax: P(x, y).For example, let x stand for a man and let y stand for a woman in a group of people. Let P(x, y) = x is taller than y. Then Ax: Ay: P(x, y) means Every man is taller than every woman, and Ay: Ax: P(x, y) means For every woman, every man is taller.
For example, let x stand for a fruit shop in a town, and let y stand for a kind of fruit. Let P(x, y) = x has y. Then Ex: Ay: P(x, y) means: There exists a fruit shop which has every kind of fruit. This implies that: For every kind of fruit there exists a fruit shop which has it; this is the meaning of the symbols: Ay: Ex: P(x, y).
The general law is written in symbols as follows:
From Ex: Ay: P(x, y) we conclude Ay: Ex: P(x, y).The converse of this implication is not true. For example, suppose it is true that For every kind of fruit there is at least one shop that has it. It may be necessary to go to different shops for different kinds of fruit. Then it is not true that At least one shop has every kind of fruit.
Ex: Ey: P(x, y)
and
Ey: Ex: P(x, y).
These two statements have the same meaning. Therefore it does not matter which existential quantifier is first, and we have the general law:
Ex: Ey: P(x, y) = Ey: Ex: P(x, y).For example, let x stand for a computer operator in an office, and let y stand for a word processing program. Let P(x, y) = x knows how to use y. Then Ex: Ey: P(x, y) means At least one computer operator knows how to use at least one word processing program, and Ey: Ex: P(x, y) means At least one word processing program can be used by at least one computer operator.
not-Ax: Ay: P(x, y).
Applying the laws for negating statements with quantifiers we obtain first
Ex: not-Ax: P(x, y),
and then
Ex: Ey: not-P(x, y).
For example, the statement Every man is taller than every woman may be written in symbols as follows: Ax: Ay: P(x, y), where x stands for a man, y stands for a woman, and P(x, y) = x is taller than y.
The negation of this statement is written in symbols as follows: Ex: Ey: not-P(x, y). This means At least one man is not taller than at least one woman.
We summarize this law in symbols as follows:
not-Ax: Ay: P(x, y) = Ex: Ey: not-P(x, y).
not-Ax: Ey: P(x, y).
Applying the laws for negating statements with quantifiers we obtain first
Ex: not-Ey: P(x, y),
and then
Ex: Ay: not-P(x, y).
For example, the statement Every man is taller than at least one woman may be written in symbols as follows: Ax: Ey: P(x, y), where x stands for a man, y stands for a woman, and P(x, y) = x is taller than y.
The negation of this statement is written in symbols as follows: Ex: Ay: not-P(x, y). This means There is at least one man who is not taller than any woman.
We summarize this law in symbols as follows:
not-Ax: Ey: P(x, y) = Ex: Ay: not-P(x, y).
not-Ex: Ay: P(x, y).
Applying the laws for negating statements with quantifiers we obtain first
Ax: not-Ay: P(x, y),
and then
Ax: Ey: not-P(x, y).
For example, the statement At least one fruit shop in the town has every kind of fruit may be written in symbols as follows: Ex: Ay: P(x, y), where x stands for a fruit shop, y stands for a kind of fruit, and P(x, y) = x has y.
The negation of this statement may be written in symbols as follows: Ax: Ey: not-P(x, y). This means Every fruit shop does not have at least one kind of fruit.
We summarize this law in symbols as follows:
not-Ax: Ey: P(x, y) = Ex: Ay: not-P(x, y).
not-Ex: Ey: P(x, y).
Applying the laws for negating quantifiers we obtain first
Ax: not-Ey: P(x, y),
and then
Ax: Ay: not-P(x, y).
For example, the statement At least one person in the village owns at least one kind of motor vehicle may be written in symbols as follows: Ex: Ey: P(x, y), where x stands for a person in the village, y stands for a kind of motor vehicle, and P(x, y) = x owns y.
The negation of this statement is written in symbols as follows: Ax: Ay: not-P(x, y). This means For any person in the village and any kind of motor vehicle, that person does not own that kind of motor vehicle, in other words No-one in the village has a motor vehicle.
We summarize this law in symbols as follows:
not-Ex: Ey: P(x, y) = Ax: Ay: not-P(x, y).
Backward links: Statements I Statements II Quantifiers I
Forward links: 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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