How to discuss an argument’s content if you have a propositional argument.
How to discuss an argument’s content if you have a propositional argument.
Example 1:
The 3rd premise of our argument is this: V→S. This says that if an argument is valid, then it is sound. This is false. Consider the following argument:
All dogs are cats
All mice are dogs
All mice are cats
This argument is valid but it has false premises. Thus, it is not a sound argument.
Example 2
The 43th premise of our argument says this:
C→(V&~S)
This says that if an argument has a contradiction as a premise, then it is valid and unsound. This is true. A contradiction is a statement that is false in every interpretation. An argument is invalid if and only if it has an interpretation where the premises are all true and the conclusion is false. If one premise in an argument is a contradiction, then the argument can never have an interpretation where the premises are all true, much less an interpretation where the premises are all true and the conclusion false. But while the argument is valid, it cannot be sound. Because a contradiction is always false, an argument with a contradiction as a premise cannot have all true premises. And a sound argument not only has a valid form but it must have true premises as well.
Discussion: when we have a conditional statement, the antecedent of this statement is the subject. You must make this condition hold. For example (1), the antecedent was V (and presumably V=an argument is valid according to the dictionary. This means that we must deal with a valid argument. We make the antecedent hold and then we see if the consequent holds as well)
Example 3
The first premise of our argument says this: ~C→T. This says that if an argument does not have a contradiction as a premise, then it has true premises. This is false. Even though an argument may not have a contradiction as a premise, it may still have false premises. While a contradiction is false in virtue of its form, other statements can be false in virtue of their content. These statements are said to be contingently false. consider this argument:
No animals are cats
Some cats are not vertebrates
No vertebrates are animals
Every statement in this argument is false but not a single statement is a contradiction.
Example 4
The 16th premise says this:
(P&C)→~V
This says that if an argument has consistent premises and a contradiction as its conclusion, then it is invalid. This is true. If an argument has consistent premises, then there is an interpretation where they are all true. If the conclusion is a contradiction, then it is false in every interpretation. This means that we have at least one interpretation where the premises are all true and the conclusion is false. We can picture this by means of the following truth table;
P1 P2 … Pn C
: : : F
T T T F * invalid
: : : F
:
Example 5
The 3rd premise says this: ~Pà~T. This says that if the premises are inconsistent, then the premises cannot have a tautology. This is false.
P1 P2 P3 P4
T F T T
T T F T
T T F F
T T T F
When the premises are inconsistent, there is no interpretation where they are all true. But this does preclude tautologies. As we can see from this table, P1 is a tautology, it is true in every interpretation. But the premises are still inconsistent. There is no interpretation where they are all true.
Additional hints:
1.When you explain why the premises are true or false in your paper, there is one common mistake that students make. And this mistake will cost lots of points. Students will write something like this:
The first premise says this; V→S. As you can see from my truth table, this statement is contingent. it has both true and false interpretations.
Don’t do this. I already know V→S is contingent. I don’t you to tell me this. I don’t even need to see a stupid truth table. If a statement is contingent, it means it is not true or false in virtue of its form. It means it has both true and false interpretations. I want to know which interpretation actually holds. To determine this, you need to tell me what V stands for and what S stands for. You go back to your dictionary, plug in values, and treat the statement as a short answer question.
2. Most of your premises will be conditional statements. Translate them back into English as ‘If_______, then_________’ statements. If you have V→S where V=an argument is valid and S=an argument is sound, rewrite V→S as this: If an argument is valid, then it is sound. It doesn’t matter if the original statement said this: An argument is valid only if it is sound.
Why should you do this? You’re less likely to make a mistake in how to focus your discussion. Students get hung on on ‘only if’ expressions and other sorts of expressions. They’re less likely to make mistakes if they discuss ‘if_____, then _______’ statements. You make what follows the ‘If’ hold’ and then you see if what follows the ‘then’ also holds.
© September, 2008
C. Bolton
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