Formal Logic

Other Kinds of Syllogisms

We covered Categorical Syllogisms last post (A is in B, B is in C, thus A is in C, etc.). There's two other major types.

Disjunctive Syllogism

Either A or B.
A.
Therefore not B.

You can interchange the letters or the 'not's. 

Example:

Either Joe is a college student, or he is not one.
Joe is a college student.
Therefore, he is not not a college student.

That one's pretty obvious. Note that the two parts of the major premise cannot both be true, because they are mutually exclusive. 

Contradictions versus Contraries

Two contrary premises cannot both be true. At least one must be false, but both may be false.

Two contradictory (mutually exclusive) premises cannot both be true or false. One must be true and the other must be false.

Example:

Either the light is red or the light is green.

This is a false premise because it is in the either/or format, but the two possibilities are not mutually exclusive, so the statement is false. There is a possibility that the light is neither red or green, but yellow (or off, or some other color).

Example of how to turn the claims from contrary to mutually exclusive:

Either the light is red or the light is not red.

Note that if you claim that the light is red AND the light is not red, then this is a contradiction, and false. 

Conditional Syllogism

This typically follows the if-then format.

Modus Ponens

If P, then Q.
P.
Therefore Q.

Example:

If it is less than 0 degrees Celsius outside, the water on the pond's surface will be frozen.
It is less than 0 degrees Celsius outside.
Therefore, the water on the pond's surface will be frozen.

This logic is valid. If additional information can be found (such as high turbidity preventing the water from freezing, or a local source of heat, or different pressures, etc.), then the argument may become unsound by virtue of the major premise (the 1st one) being proven false.

Modus Tollens

It is also true that...

If P, then Q.
Not Q.
Therefore not P. 

Example:

If it is nighttime, the sun is not visible in the sky.
It is not the case that the sun is not visible in the sky (i.e. the sun is visible in the sky).
Therefore, it is not nighttime.

The following two arguments are fallacious.

Affirming the Consequent

If P, then Q.
Q.
Therefore P.

Example:

If the sun is high in the sky, then the stars are not visible.
The stars are not visible.Therefore, the sun is high in the sky.
One possible alternative is that it is nighttime, but it is cloudy.

Denying the Antecedent

If P, then Q.
Not P.
Therefore, not Q.

Example:

If your body has been cremated, then you are dead.
Your body has not been cremated.
Therefore, you are not dead.

Obviously, you can still be dead but not cremated--you could be buried, for example, or eaten by wild animals.

Part 1: http://my.umbc.edu/discussions/435
Part 3: http://my.umbc.edu/discussions/455