Philosophy 160
Introduction to Ethics

Handout 2 - Logic and Truth


Logic is the study of correct reasoning.

A statement is a sentence that describes the world as being a certain way.

 Examples of statements:  Examples of sentences that are not statements:
 'The earth is round.'  'What time is it?'
 'The earth is flat.'  'Does God exist?'
 'God exists.'  'Cool, dude!'
 'I hope that God exists.'  'Holy @#$%!'
 'Abortion is wrong.'  'Tell me what time it is.'
 'Some people believe that abortion is wrong.'  'Close the door.'

A statement is true when the world is the way the statement says the world is.

An argument is a sequence of statements, the last of which (the conclusion) is supposed to follow from the others (the premises).

A valid argument is one with the following property: IF all of its premises are true, then its conclusion must also be true.

(said another way: an argument is valid when it is impossible for its premises to be true and its conclusion false.)

We said above that the conclusion of an argument is supposed to follow from the premises.  To say that an argument is valid is to say that the conclusion really does follow from the premises.

An argument is sound when it is valid and all of its premises are true.

Some Common Valid Argument Forms

 Modus Ponens (MP)    Multiple Modus Ponens (MMP)
 1. If P, then Q  1. P  1. P
 2. P  2. If P, then Q  2. If P, then Q
 3. Therefore, Q  3. Therefore, Q  3. If Q, then R
     4. Therefore, R
 Modus Tollens (MT)    Multiple Modus Tollens (MMT)
 1. If P, then Q.    1. If P, then Q.
 2. not-Q    2. If Q, then R.
 3. Therefore, not-P.    3. not-R
     4. Therefore, not-P.
 Disjunctive Syllogism (DS)   Categorical Syllogism 
 1. P or Q  1. P or Q  1. All A's are B.
 2. not-P  2. not-Q  2. x is an A.
 3. Therefore, Q  3. Therefore, P  3. Therefore, x is B.


Some Sample Valid Arguments

1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.

1. A fetus is a person.
2. If a fetus is a person, then abortion is wrong.
3. Therefore, abortion is wrong.

1. All cats are toaster ovens.
2. Heathwood is a cat.
3. Therefore, Heathwood is a toaster oven.

1. If Lincoln is alive, then Lincoln survived his assassination attempt.
2. Lincoln did not survive his assassination attempt.
3. Therefore, Lincoln is not alive.

1. If there is life on Mars, then there is extraterrestrial life.
2. There is life on Mars.
3. Therefore, there is extraterrestrial life.

1. Either Jack lives in Northampton or he lives in Amherst.
2. Jack does not live in Amherst.
3. Therefore, Jack lives in Northampton.

Some Sample Invalid Arguments

1. The environment is important.
2. Therefore, everyone should recycle.

1. All apples are fruits.
2. The thing in my hand is a fruit.
3. Therefore, the thing in my hand is an apple.

1. If John is a communist, then John opposes the status quo.
2. John opposes the status quo.
3. Therefore, John is a communist.

1. If O.J. was found guilty, then O.J. did it.
2. O.J. was not found guilty.
3. Therefore, O.J. did not do it.


Necessary and Sufficient Conditions

Let P and Q be states of affairs, or conditions.

examples of states of affairs:

snow's being white
your passing this course
your graduating from UMass
my walking to class
my moving my legs

P is a necessary condition of Q when Q can occur only if P also occurs.

(said another way: if Q occurs, then P also occurs)

P is a sufficient condition of Q when if P occurs, then Q also occurs.

(said another way: P can occur only if Q also occurs)

Some results of these definitions:

If P is both necessary and sufficient for Q, then P can occur if and only if Q occurs.

If P is necessary for Q, then Q is sufficient for P.

If P is sufficient for Q, then Q is necessary for P.

Some examples:

Being male is a necessary condition for being a bachelor.

Acing every exam is sufficient for passing this course.

Moving your legs is necessary, but not sufficient, for walking.