Argument Forms

 Testing For Argument Validity | Modus Ponens and Modus Tollens | Other Valid Argument Forms
Common Fallacies | The Contradiction Rule


Argument A series of statements .
Premises All statements in an argument except the final one.
Conclusion The final (or concluding) statement in an argument.
Symbol for "therefore", normally used to identify the conclusion of an argument.
Modus Ponens Latin for "method of affirming." A rule of inference used to draw logical conclusions, which states that if p is true, and if p implies q (pimpliesq), then q is true.
Modus Tollens Latin for "method of denying." A rule of inference drawn from the combination of modus ponens and the contrapositive. If q is false, and if p implies q (pq), then p is also false.
Fallacy An error in reasoning.
Contradiction Rule Given a statement p, if ~p leads logically to a contradiction, then p must be true.

Testing For Argument Validity

As shown in earlier, the content of a statement or argument can be represented in a logical form by the use of statement variables.  For example:

An argument form is valid if, no matter what statements are substituted for the premises statement variables, if the premises are all true, then the conclusion is also true. The truth of the conclusion must follow necessarily from the truth of the premises.

To determine an argument's validity:

If the conclusion is true for each critical row, then the argument form is valid. But if even one of the critical rows contains a false conclusion, the argument is invalid.

Practice Exercises

Modus Ponens and Modus Tollens

These 2 methods are used to prove or disprove arguments, Modus Ponens by affirming the truth of an argument (the conclusion becomes the affirmation), and Modus Tollens by denial (again, the conclusion is the denial). Consider the following argument:

If it is bright and sunny today, then I will wear my sunglasses.

Modus Ponens Modus Tollens
It is bright and sunny today. I will not wear my sunglasses.
Therefore, I will wear my sunglasses. Therefore, it is not bright and sunny today.

Construction of a truth table will show that these two argument forms are equivalent, and further demonstrates the fact that a conditional statement is logically equivalent to its contrapositive.

Practice Exercises

Other Valid Argument Forms

The following additional argument forms are valid.

Example 1                    Example 2
Disjunctive Addition p q
p q pq

Conjunctive Simplification p p q
p q

Disjunctive Syllogism  p q p q
~q ~p
p q

Hypothetical Syllogism p q
q r
p r

Proof by Division into Cases p
p r
q r

Common Fallacies

An error in logic or reasoning is called a fallacy, and the result of such errors is an invalid argument. The three most common fallacies are:

Two other reasoning errors which are common are:

Be aware that a valid argument may have a false conclusion, particularly if the premises are false. Likewise, an invalid argument may result in a true conclusion.

The Contradiction Rule

The contradiction rule is the basis of the proof by contradiction method. The logic is simple: given a premise or statement, presume that the statement is false. If this presumption leads to a contradiction, then the given statement must be true. Consider the following: