Argument Forms

For a valid argument, the truth of its conclusion follows necessarily (by logical form alone) from the truth of its premises. It is impossible to have a valid argument with true premises and a false conclusion. When an argument is valid, the truth of the conclusion is said to be inferred or deduced from the truth of the premises.

Consider the following argument form: top

If p, then q.

\ q.

The fact that this argument form is valid is called modus ponens (method of affirming) since the conclusion is an affirmation.

components		premises		conclusion
p	q	p ®q	p	q
T	T	T	T	T
T	F	F	T	F
F	T	T	F	T
F	F	T	F	F

The first row is the only row in which both premises are true (critical row), and the conclusion in that critical row is also true. Hence the argument form is valid.

Now consider another argument form: top

If p, then q.

Ø q

\ Ø p.

The fact that this argument form is valid is called modus tollens (method of denying) since the conclusion is a denial.

components		premises		conclusion
p	q	p ®q	Ø q	Ø p
T	T	T	F	F
T	F	F	T	F
F	T	T	F	T
F	F	T	T	T

The last row is the only critical row, and the conclusion in that critical row is also true. Hence the argument form is valid.

Example:
Use modus ponens or modus tollens to fill in the blanks of the following arguments so they become valid inferences.

If there are more pigeons than there are pigeonholes, then two pigeons roost in the same hole.
There are more pigeons than there are pigeonholes.
\ ____________________________________________________

If the sum of a number's digits is divisible by 9, then the number is divisible by 9.
The number is not divisible by 9.
\ ____________________________________________________

Disjunctive Syllogism top
When there are only two possibilities and you can rule one out, the other must be the case. Symbolically we have

a) p Ú q	b) p Ú q
Ø q	Ø p
\ p	\ q

Fallacies top
A fallacy is an error in reasoning that results in an invalid argument. Common fallacies are using ambiguous premises, assuming what is to be proved, and jumping to conclusions without adequate grounds. We again use truth tables to demonstrate two additional fallacies, called converse error and inverse error.

Converse Error

p ® q.

\ p.

components		premises		conclusion
p	q	p ®q	q	p
T	T	T	T	T
T	F	F	F	T
F	T	T	T	F
F	F	T	F	F

The third row is a critical row, and the conclusion in that critical row is false. Hence the argument form is invalid.

Inverse Error

p ® q.

Ø p

\ Ø q.

components		premises		conclusion
p	q	p ®q	Ø p	Ø q
T	T	T	F	F
T	F	F	F	T
F	T	T	T	F
F	F	T	T	T

The third row is a critical row, and the conclusion in that critical row is false. Hence this argument form is also invalid.

Proof by Division into Cases top
It often happens that you know one thing or another is true. If you can show that in either case a certain conclusion follows, then this conclusion must also be true. This argument form is:
p Ú q
p ® r
q ® r
\ r.

Example:
x is positive or x is negative.
If x is positive, then x² > 0.
If x is negative, then x² > 0.
\ x² > 0.

Contradictions and Valid Arguments top
If you can show that the supposition that statement p is false leads logically to a contradiction, then you can conclude that p is true. We will use truth tables to verify the contradiction rule.

Ø p ® c, where c is a contradiction

\ p

components			premise	conclusion
p	Ø p	c	Ø p ®c	p
T	F	F	T	T
F	T	F	F	F

Row one is the only critical row, and the conclusion is true so this argument form is valid. Note that the contradiction rule is the logical heart of the method of proof by contradiction. A slight variation also provides the basis for solving many logical puzzles by eliminating contradictory answers: If an assumption leads to a contradiction, then that assumption must be false.