Lecture 2: Argument Forms

8/03/04, University of Wyoming

Logic Key: Last time we looked at the following truth-functional operators: not, if-then, and, or, if-and-only-if. For the sake of brevity I will begin to use a simpler symbolic notation for them. Here is a key:

· “not” is symbolized as Ø

· “if-then” is symbolized as ®

· “and” is symbolized as Ù

· “or” is symbolized as Ú

· “if and only if” is symbolized as «

Deductively Valid Arguments

Modus ponens. Consider the valid argument given above:

1. If the moon is made of blue cheese, then pigs fly.

2. The moon is made of blue cheese.

_________

3. Therefore, pigs fly.

Now consider a second deductively valid argument:

1. If there are no chance factors in chess, then chess is a game of pure skill.

2. There are no chance factors in chess.

_________

3. Therefore, chess is a game of pure skill.

These arguments bear an obvious similarity to one another. You might say that they both have the same form. Roughly:

1. If this, then that.

2. This.

_________

3. Therefore, that.

We can make the general form of this argument even more explicit by using letters (p, q, r, s, t, … ) to stand for sentences. Using this device we can write the common form of the above arguments as follows:

1. p ® q

2. p

_________

3. Therefore, q.

If we substitute the sentence “the moon is made of blue cheese” for p and the sentence “pigs fly” for q, we get the first of the above arguments. Similarly, if we substitute the sentence “there are no chance factors in chess” for p and the sentence “chess is a game of pure skill” for q, we get the second of the above arguments.

What is important to notice is that any argument having this same general form will be valid (though not necessarily sound) no matter what sentences you stick in for p and q.

This form of argument is known as modus ponens.

Modus tollens. Now consider a different argument:

1. If the moon is made out of blue cheese, then pigs fly.

2. Pigs don’t fly.

__________

3. Therefore, the moon is not made out of blue cheese.

Is this argument valid? Could the premises be true and the conclusion false? Well, consider. The first premises says that if it is true that the moon is made out of blue cheese, then it will also be true that pigs fly. The second premise, however, says that it is not true that pigs fly. But then it can’t be true that the moon is made out of blue cheese. For if it were true, then we could conclude by modus ponens that pigs fly and we would contradict ourselves. Therefore, given premises (1) and (2), it must follow that the moon is not made out of blue cheese. The argument is valid.

Do such arguments also have a general form? Yes. Using our trick of letting letters stand for sentences, the general form of the above argument is as follows:

1. p ® q

2. Øq

__________

3. Therefore, Øp

This form of argument is known as modus tollens. And as with modus ponens, no matter what sentences you stick in for p and q, the result will be a valid argument. Try it.

Some Other Deductively Valid Argument Forms

OK. I assume that by now you’ve got the hang of writing out the general form of an argument, so I will dispense with examples. Here are a few other deductively valid forms of argument.

Hypothetical Syllogism.

1. p ® q

2. q ® r

________

3. Therefore, p ® r

Disjunctive Syllogism.

1. p Ú q

2. Øp

________

3. Therefore, q

Dilemma.

1. p Ú q

2. p ® r

3. q ® s

_________

4. Therefore, r Ú s

Simplification (Conjunction Elimination)

1. p Ù q

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2. Therefore, p

Addition (Disjunction Introduction)

1. p

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2. Therefore, p Ú q