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Lecture 3

August 29, 2011

Review:

Defined: argument, proposition, premises, conclusions.

Today:
* Look at conditionals.
* look at an argument containing conditionals.
* Truth generating machines

Conditionals

If…  is the ANTECEDENT.   …then Is the CONSEQUENT.

Example: If Socrates is a man, Socrates is mortal.

The truth value depends on the relationship between the antecedent and the consequent..

Examples of Arguments (more sophisticated).

Better example: (from Descartes, Meditation I)

1) If you are certain that an empirical proposition is true then, there is no way to doubt its truth.
2) There are ways to doubt the truth of empirical propositions.
3) Therefore, you can’t be certain about empirical propositions

Example of 2) “I believe that it is true that there is a classroom in front of me”.  If you are dreaming then you can believe in the truth of the presence of a classroom, but your proposition would be false.

Validity

A valid argument has the following property: if the premises are true, the conclusion must be true.

An argument is valid if and only if it is not possible for the all of its premises to be true and its conclusion false.

Abstract: form of the argument, not its content.

All fish swim.
All sharks are fish.
Therefore, all sharks swim.

All particles have mass.
All electrons are particles.
Therefore, all electrons have mass.

All As are Bs
All Cs are As
Therefore, all Cs are Bs.

A = fish or particle; B = swim(miming) or having mass; C = sharks or electrons.

So, what have we learned about validity?
* An argument is valid solely by virtue of the logical form it has.
* The subject matter is irrelevant.

To see the last point clearly, consider the following argument:

All plants have minds.
All ladders are plants.
Therefore, all ladders have minds.

This is a strange argument indeed! Surely the premises and the conclusion are false! But notice the argument form. It is (S). That means the argument is valid even though the premises and the conclusion are known to be false. What’s going on here?
Well, first, the plant, ladder, and minds argument demonstrates the last bullet point: validity is not attributed to an argument’s subject matter. Rather, validity describes a particular relationship between premises and conclusions. See the definition again (either version).

I’m sure by now you are wondering why I have the gall to call valid arguments “good” when they (depending on the form) apply equally to cases in which the premises and the conclusion are true and those in which they are false. To understand what is going on, remember what I said about validity being a truth-generating machine. Here’s the way this machine works: feed it true premises, and it will automatically spit out true conclusions! Not all argument forms have that special property. Consider the following:

All As are Bs.
x is a B.
Therefore, x is an A.

This argument form is invalid. That means it is possible for one to come up with an argument whereby the premises are true but the conclusion false. Here is an example:

All senators are paid.
Sam is paid.
Therefore, Sam is a senator.

Ask yourself this: just because all senators are paid and Sam is paid, does it “follow” that Sam is a senator? No, obviously not. Teachers are paid, and it is possible that Sam is a teacher.

So, you see, only some argument forms are valid, and only those are truth-generating machines. I’m sure you can see why philosophers are so hyped up about valid arguments. Wouldn’t you want to own a truth-generating machine.

For all its wonder, notice that this truth-generating machine, which I’ve been calling “validity,” has a bit of a downside. In order to generate true conclusions, you must give it true premises. Otherwise, if you give it false premises, then there’s no telling what it will produce for you. For validity, garbage in, garbage out (as it were). In some cases, false premises will spit out a true conclusion; in other cases, false premises will spit out a false conclusion (as in the case with the plants, ladders, and minds). Without true conclusions, the machine doesn’t function well. So, I recognize that validity is a peculiar machine. But, still, it is a wonderful thing, isn’t it?

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