D E F I N I T I O N S
|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 (pq), 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:
|If I study hard, then I will get an A.||p q|
|I will study hard.||p|
|Therefore, I will get an A.||q|
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.
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.
Other Valid Argument Forms
The following additional argument forms are valid.
|Example 1||Example 2|
|Conjunctive Simplification||p q||p q|
|Disjunctive Syllogism||p q||p q|
|Hypothetical Syllogism||p q|
|Proof by Division into Cases||p q|
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:
Converse error resembles modus ponens (which is why it's such an easy mistake to make!). Consider the following absurd example:
If someone kicks me, I will yell "ouch!"
I just yelled "ouch!".
Therefore, someone kicked me.
It is logical to presume that, if someone decides to kick you, your reaction would be a pained yelp. However, the fact that you yelled "ouch" does not necessarily mean that the nearest bystander walked up and kicked you. (You also might yell if you pricked yourself with a pin, dropped a hammer on your foot, or received a "D" on a test paper.)
Like the above converse error, inverse error also appears very similar to a valid argument form - the contrapositive.
If I hit my professor with a cream pie, he will flunk me.
I will not hit my professor with a cream pie.
Therefore, he will not flunk me.
Again, it is intuitively obvious that this reasoning does not work. While many professors may consider being nailed with a cream pie a sufficient reason to assign a grade of "F" to a student, there are an overwhelming number of other reasons for which you might flunk (cheating, not studying, not showing up for tests, etc.).
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:
All even integers are divisible by 2.
But suppose not; suppose there is a number that is even and is not divisible by 2.
There is no such number. (The contradiction.)
Therefore, the original premise is true.