New Page 1

Argument with Quantifiers

Universal instantiation is the fundamental tool of deductive reasoning. The rule of universal instantiation says that

If some property is true of everything in a domain, then it is true of any particular thing in a domain.

The validity of "universal instantiation argument" form follows immediately from the definition of truth values for a universal statement. One of the most famous examples of universal instantiation is as follows:

	All human beings are mortal.
	Socrates is a human being.
∴	Socrates is mortal.

The reasoning behind this example is outlined below:

	All human beings are mortal.	-- This is a universal truth
	Socrates is a human being.	-- This is a particular instant
∴	Socrates is mortal.	-- Conclusion

Universal Modus Ponens

The rule of universal instantiation can be combined with modus ponens to obtain the rule called universal modus ponens.

The following argument form is valid:

	∀ x, if P(x) then Q(x)
	P(a) for a particular 'a'
∴	Q(a)

Informally,

	If x makes P(x) true, then x makes Q(x) true.
	'a' makes P(x) true.
∴	'a' makes Q(x) true.

The above argument form consists of two premises and a conclusion. Furthermore, at least one premise is quantified. An argument of this form is called a Syllogism. For example,

	All human beings are mortal.	-- Major premise
	Socrates is a human being.	-- Minor premise
∴	Socrates is mortal.	-- Conclusion

Note that in the above example major premise is quantified.

As an example, show that the following quantified argument is valid.

	If a number is even, then its square is even.
	k is a particular number that is even.
∴	k² is even.

We start by writing major premise with quantifier.

	∀ x, If x is even, then x² is even.
	k is a particular even number.
∴	k² is even.

Now suppose:

E(x) stands for "x is even."
S(x) stands for "x² is even."
k stands for a particular even number.

Then the given argument has the following form:

	∀ x, If E(x) then S(x)
	E(k) for a particular k.
∴	S(k)

And this argument form is the same as that of universal modus ponens. Therefore, this form of argument is valid (by universal modus ponens.)

Application of Universal Modus Ponens

Lemma: The sum of any two even integers is even.

Proof.

Suppose m and n are particular but arbitrarily chosen even integers.

Step 1.	Then	m = 2r for some integer r.
Step 2.	and	n = 2s for some integer s.
Step 3.	Hence,	m + n = 2r + 2s	by substitution
		= 2(r + s)	by factoring out the 2.
Step 4.	Now,	r + s is an integer.
	and so,	2(r + s) is even.
Step 5.	Therefore,	m + n is even.

And, this completes the proof.

Now, we show how each of the above step is justified by argument are valid by universal modus ponens.

Step 1.		If an integer is even, then it equals twice some integer.
		m is a particular even integer.
	∴	m equals twice some integer r.

Step 2.		If an integer is even, then it equals twice some integer.
		n is a particular even integer.
	∴	n equals twice some integer s.

Step 3.		If a quantity is an integer, then it is real number.
		r and s are particular integers.
	∴	r and s are real numbers.

		For all a, b, c, if a, b, c are real numbers, then ab + ac = a(b + c).
		a = 2, b = r, c = s are particular real numbers.
	∴	2r + 2s = 2(r + s)

Step 4.		For all m and n, if m and n are integers then m + n is an integer.
		m = r and n = s are two particular integers.
	∴	r + s is an integer.

Step 5.		If a number equals twice some integer, then that number is even.
		2(r + s) equals twice the integer r + s.
	∴	2(r + s) is even.

Universal Modus Tollens

The rule of universal instantiation can be combined with modus tollens to obtain the rule called universal modus tollens. This consider be one of most important rules (or agrument form) of inference.

The following argument form is valid:

	∀ x, if P(x) then Q(x)
	~Q(a) for a particular 'a'
∴	~P(a)

Informally,

	If x makes P(x) true, then x makes Q(x) true.
	'a' does not make Q(x) true.
∴	'a' does not make P(x) true.

Again, the above argument form consists of two premises and a conclusion. Furthermore, at least one premise is quantified. For example,

	All human beings are mortal.	-- Major premise
	Zeus is not mortal.	-- Minor premise
∴	Zeus is not a human being.	-- Conclusion

Note that in the above example major premise is quantified.

As an example, show that the following quantified argument is valid.

	All human beings are mortal.
	Zeus is not mortal.
∴	Zeus is not human being.

We start by writing major premise with quantifier.

	∀ x, If x is human being, then x is mortal.
	Z is a particular being that is not mortal.
∴	Z is not a human being.

Now suppose:

H(x) stands for "x is a human being."
S(x) stands for "x is a mortal."
Z stands for Zeus.

Then the given argument has the following form:

	∀ x, If H(x) then M(x)
	~M(Z)
∴	~H(Z)

And this argument form is the same as that of universal modus tollens. Therefore, this form of argument is valid (by universal modus tollens.)

_______________________________________________

in ∈

for all ∀

there exists ∃

greater equal ≥

less equal ≤

not equal ≠

equivalent ≡

implication →

and ∧

or ∨

therefore ∴