The tools of our trade

Philosophers advance their discipline through logical argument.  So as philosophers we want to be able to tell apart good and bad arguments.  Three questions naturally arise:

These questions are of particular interest to philosophers, who not only make extensive use of argumentation, but who are also typically interested in the theory of knowledge.  And a good argument is a way of extending our knowledge.
 

1   What is an argument?

Let's start with a very simple example.  Suppose I'm trying to convince you that ice-cream is good for you.  Then I am going to try to support this claim, by citing some other claims, claims which I hope you already accept.  Suppose you and I both accept:
 

 Either ice-cream is fattening or ice-cream is good for you.

And suppose we also the following:
 

 Ice-cream isn't fattening.

Then I might say to you:
 

"Look:  Either ice-cream is fattening or ice-cream is good for you. But ice-cream isn't fattening . So it is good for you."

This is a really stupid little example, I know.  But still, it does have the structure of an argument (a rather stupid argument, but the stupidity does not reside in its structure).
 

The conclusion of the argument is the claim that
 

 Ice-cream is good for you.

In support of this conclusion I adduce two other claims, which we will call the premises of the argument.  The argument consists of two premises and a conclusion, which we will write thus:
 

(Argument 1)

Obvious point:  the premises and the conclusion are claims about the way the world is.  They are the sorts of things that can be either true or false, depending on the way the world turns out to be.

So the answer to our first question is this:

(Who says you don't get any answers in a philosophy course?)

2    What is a good argument? 

2.1   Validity

To evaluate the goodness of an argument we have to know what arguments are supposed to be for.

The premises of an argument are supposed to support the conclusion.

Consider our little ice-cream argument.  Do the premises support the conclusion? 

At least one of the premises is false in fact (ice-cream really is fattening) but there is still a clear and obvious sense in which the premises provide maximal possible support for the conclusion.

What is that sense? It is this:  Anyone who accepted both premises should also accept the conclusion.

Why?  The reason is that the truth of the premises guarantees the truth of the conclusion.  And that is what we mean by a valid  argument.  (Maybe this isn't the way you are used to using the term "valid".  Don't worry about that at the moment.)

What is the nature of this guarantee?  The argument does not by itself guarantee that the conclusion is true.  Rather, it guarantees that if the premises are true, then the conclusion is true.  It is impossible in a very strong sense of impossibility for the premises to be true and the conclusion false.  Alternatively put, there is no possible way for the premises to be true and the conclusion false.  Or, there is no possible case in which the premises would be true and the conclusion false.
 

Note: this is the single most important piece of information in this course.  It is the heart and soul of it.

We can now see the virtue of offering a valid argument.  It is a perfect truth preservative.  Put truth in (the premises)  and you must get truth out (the conclusion).  Thus if someone accepts the premises of a valid argument they are rationally obliged to accept the conclusion.

Question: what if you don't accept the conclusion?

Answer: then you are rationally obliged to give up at least one of the premises.

 That is to say:

(If you think ice-cream isn't good for you then you have to reject one of the following: ice-cream isn't fattening or  Either ice-cream is fattening or it's good for you.)

Given our account of validity, what does invalidity amount to?  That shouldn't be too hard to work out.

An argument is valid if it is impossible for the conclusion to be false while the premises true .  An argument is invalid if it is possible for the conclusion to be false while the premises are true.
 
 

Example:  Argument 2

Premise 1	If ice-cream is fattening then it isn't good for you.
Premise 2	Ice-cream isn't good for you.
Conclusion	Ice-cream is fattening.

The form of Argument 2

Example:  Argument 3

Suppose that the premise is true, because the Republicans will enact big tax cuts which will make the deficit grow.  Suppose that if the Democrats get in then the deficit will grow for some other reason — perhaps because they will increase welfare entitlements without corresponding tax increases.  So whichever party wins the deficit will grow.  So we can imagine both premises being true and the Democrats winning.  This is a case in which the premises are true and the conclusion is false.  The argument is invalid.

A possible case which renders the premises true and the conclusion false is called a counterexample to the validity of the argument.

2.2   The truth of the premises

Argument 4

3   How can we tell whether  an argument is valid or invalid?

We have seen what invalidity amounts to.
 

There is a counterexample—a possible case in which the premises are true and the conclusion is false.

So to show that an argument is invalid we try to construct a counterexample.  We try to tell a story (not necessarily a true story, just a possibly true story, a coherent story) in which both premises are true and the conclusion is false. If we can construct such a story, a case, then that will demonstrate the invalidity of the argument.

Much of philosophy consists in taking someone's argument and constructing such a counterexample.  Somebody might argue, for example, that the existence of evil proves the non-existence of an all powerful, all-good being.  The argument from evil goes like this:
 

Argument 5 — the (simple) argument from evil:

Premise	Evil exists.
Conclusion	God does not exist.

If you accept the premise and the argument is valid, then you ought to accept the conclusion.  Those who accept the premise and don't want to accept the conclusion have to show that the argument is invalid.  They have to tell a story (a possible story will do!) in which the premise is true and in which an all-powerful, all-good being also exists.  If that is a possibility, then the argument is deductively invalid.  They must tell a story in which God (an all-powerful all-good being) allows evil to exist, without that compromising either his power (his ability to destroy evil) or his  goodness (his desire to maximize the good).

Take an argument that involves less controversial subject matter:
 

Argument 6

Premise 1	Either ice-cream has lots of fat or it isn't good for you.
Premise 2	Ice-cream has lots of fat.
Conclusion	Ice-cream isn't good for you.

This sounds very strange(even though the conclusion is probably true), because the first premise sounds obviously false.  We all know (or think we know) that fatty foods are bad for us, and the first premise tells us effectively that unless ice-cream has lots of fat it isn't good for you.  So we are inclined to think that the argument is a total non-starter.  But in asking about validity we are not asking about in whether in fact the premises are true or false, or whether the conclusion is true or false. We are asking what if  the premises were true.  Would the conclusion also have to be true? Or can we think of a possible case in which the premises are true and the conclusion false?

Suspend your attachment to the idea that fatty foods are bad.  (Who knows, this might be the next big breakthrough in the science of nutrition.)  Imagine a situation in which all fatty foods, and only fatty foods, are good for you (This is one of the suppositions of Woody Allen's film Sleeper.)  In that possible situation the first premise is true.   Now we imagine, also, that ice-cream retains its fattiness in that imagined situation.  That's not hard to do.  So both premises are true in that possible situation.  Is the conclusion true?  NO. So we have a counterexample.  The argument is invalid.

Look back out our first example,

Premise 1	Either ice-cream is fattening or it is good for you.
Premise 2	Ice-cream isn't fattening
Conclusion	Ice-cream is good for you

For consider the following argument:
 

Argument 7

These two arguments share something in common.  A certain form.  Both involve some basic sentences and are built up from them using not and or.

For example, suppose we use the following abbreviations:
 

F: ice-cream is fattening.
G: ice-cream is good for you.

Let's agree to use bold capital letters like X and Y for arbitrary sentences (just as you use x and y for arbitrary numbers when we do algebra).  Then both arguments are instances of the following argument schema:

Every instance of the schema disjunctive syllogism is valid.

5    Non-deductive arguments (or inductive arguments)

So far we have been talking about arguments that are supposed to be deductively valid.    But not all arguments that are not deductively valid are without merit.  There are also arguments in which the premises lend some support to the conclusion even though the arguments are not deductively valid in this sense (that is to say, the truth of the premises does not guarantee the truth of the conclusion).  Here's an example (perhaps):

Argument 8

 

Premise 1	Butter is fattening.
Premise 2	Cheese is fattening.
Premise 3	Cream is fattening.
Premise 4	Butter, cream and cheese are dairy products.
Conclusion	Dairy products are fattening.

Maybe the premises lend some support to the conclusion.  (Do they?)  Well, even if they do, and even if they are true, they don't guarantee the truth of the conclusion.  We can think of a case in which all the premises are true and the conclusion is false.

(In fact the actual case is one such, since non-fat milk is a non-fattening dairy product.  So it turns out that, in fact, all the premises are true but the conclusion is false.  The actual circumstance is thus a counterexample to the validity of the argument.)

Inductive arguments can be good or bad.  Roughly, an inductive argument is good if the premises render the conclusion probable.  How probable?  Well, the more probable the better.  The premises of a deductive argument render the conclusion maximally probable.  The probability of the conclusionof a deductive argument , given the premises, is maximal:  one.  The probability of the conclusion of an inductive argument , given the premises, is high, but less than maximal.

One very common mode of inductive reasoning—in everyday life, in science, and also in philosophy, is inference to the best explanation.  Suppose you know the following: 

Jones is lying on the ground, dying, with a knife in his back, groaning and muttering.  Smith's hands are covered in blood.  As Smith flees the scene of the crime you hear Jones's dying words, which are  "Smith was the one......" 

What is the best explanation of this little bunch of facts?  Perhaps it is the following hypothesis:

Hypothesis: Smith is Jones's killer. 

If that really is the best explanation of the data then it seems we are justified in inferring, from the data, that Smith is Jones's killer.

Argument 9

Is this argument logically valid?  (Apply  the definition of validity.  Search for a possible case in which premises are true and conclusion is false.)

Can you think of a counterexample - a possible scenario in which all the premises are true and the conclusion is false.

Of course you can! 

So the argument is not  valid.  But still, in the light of the premises (and no others)  we seem to be reasonably justified in affirming that Smith killed Jones.  That's because it explains the data (premises 1 through 3) and that, barrring any other evidence, it seems to be the best available explanation of the data.

So argument 4 is really best construed as an inference to the best explanation, not a failed attempt to construct a logically valid argument.  As such it isn't reasonable to criticize it simply because it fails to be valid. 

Is this valid? Clearly not.  People might be mistaken about whether or not a miracle has occurred.  (  They might be lying.  The mere fact that people report miracles is logically compatible with the non-existence of God.  But one cannot criticize an argument like this simply by pointing out that it is not valid.  Rather, I suspect that people are putting forward the hypothesis that God exists as the best explanation for the data: that thousands of people have reported that miracles occur.

The wishful thinking hypothesis also explains the data here (i.e. lots of sincere miracle reports).  Is it better than the God hypothesis.  Well, at least it's a candidate explanation, and we can now have a debate about the merits of the two rival explanations.