A Little Logic

 

1. Philosophy is about arguments

Philosophy is not not about the world. Philosophy is not about ideas or theories. Philosophy is not about books, or individual men and women.

Philosophy is about arguments.

True, arguments are often about the world. And arguments usually involve ideas or theories. And arguments can normally be found in books. And books, to our knowledge, are all written by men and women. But the subject-matter, the métier, of philosophy, is arguments. More particularly, philosophy is about arguments concerning the most general claims to be made about anything.

 

2. Kinds of Arguments

For the purposes of an introductory philosophy course, you need to know that:

(1) All arguments are either invalid or valid;

(2) All valid arguments are either unsound or sound.

 

3. What is an Argument?

An argument is a finite set of premises followed by a conclusion.

(This definition is not without problems, but I shall ignore them here).

Consider the following:

 

(1) Never!

(2) Green is green is yellow.

(3) Hooooommmm.

––––> I'm at home now, give me a call.

 

This is not an invalid argument. This is simply not an argument at all.

Premises and conclusions in arguments must be full, grammatically correct sentences (i.e. not words, or half-sentences) that are declarative (i.e. not questions or exclamations) and that can be true or false.

Consider the following:

 

(1) He saw the movie.

(2) It rained yesterday.

---> All animals have tails.

 

Is this an invalid argument, or not an argument at all? Most of us would want to say that this is not an argument at all. There is no relation, and no attempt at showing a relation, between the two premises and the conclusion. So let us stipulate that in arguments the conclusion must be related to, or must be intended to be related to, the premises.

We could transform it into an argument, as follows:

 

(1) He saw the movie.

(2) If he saw the movie, then it rained yesterday and all mammals have tails.

---> All animals have tails.

This is an argument because the conclusion is related to, or is intended to be related to, the premises.  

However, it is an invalid argument. The conclusion does not follow from the premises. The conclusion is about animals, and none of the premises mentions animals. The second premise only mentions mammals.

 

4. Valid Arguments

However, we can turn the above invalid argument into a valid argument, as follows:

 

(1) He saw the movie.

(2) All animals are mammals.

(3) If he saw the movie, then it rained yesterday and all mammals have tails.

---> All animals have tails.

 

A valid argument is an argument in which, IF you accept the premises, THEN you must accept the conclusion, or else you are contradicting yourself.

In the case of a valid argument, if you grant the premises, then you must grant the conclusion. The conclusion does indeed follow from the premises.

Hence, in the case of a VALID argument, IF the premises be true, then the conclusion MUST be true.

An argument may be valid even if every single premise, as well as the conclusion, is false. The following is a valid argument:

 

(1) All dogs are immortal.

(2) Socrates is a dog.

--> Socrates is immortal.

 

However, the above argument is not a sound argument.

A sound argument is a valid argument in which each premise is true. And since the argument is valid, the conclusion must be true, also, because it is impossible, in the case of a valid argument, for the premises to be true and the conclusion be false.

An example of a sound argument (assuming, that is, that these premises are indeed true) is:

 

(1) Everything that is actual is possible.

(2) Human beings are actual.

––> Human beings are possible.

 

What is important about a sound argument is that you have no choice but to agree with it. Sound arguments put an end to all possible debate and discussion. They are the ultimate debate-stoppers.

Consider the argument above again. Remember that if this argument is sound, then there is nothing in the world that can alter the truth that human beings are possible.

Non-trivial sound arguments are incredibly rare in philosophy. They are like gold dust.

In general, the problem with an argument is that at least one premise is false, and hence, that the (valid) argument is unsound. Here is an example of an unsound argument:

 

(1) No swans are black.

(2) All ravens are black.

–––> No swans are ravens.

 

This argument is unsound, even though the conclusion is true, because one premise is also false ((1)). There exist black swans.

(Who knows, premise (2) may be false also. Maybe some ravens are not black. Maybe there are albino ravens. However, you only need one false premise to make an argument unsound.)

Some arguments however will be so bad that they are invalid. The conclusion doesn't follow from the premises. Even if you did believe all the premises, you could still reject the conclusion and not contradict yourself. For example:

 

(1) If God does not exist, then there is no right or wrong.

(2) God exists.

---> There is right and wrong.

 

This argument is invalid because even if one agreed with the first two premises, one could still reject the conclusion. The conclusion does not follow from the premises. It might be the case, for example, that even if God exists, there is no right or wrong.

The argument, as it were, assumes this, but never states it. But it must be stated, otherwise the conclusion does not follow.

 

Be careful, in general, never to say that arguments are true or false. Arguments cannot be true or false. They can only be invalid or valid, and in the case of valid arguments, only sound or unsound.

Be careful, in general, never to say that premises (i.e. particular sentences or claims) are valid or invalid or sound or unsound. Premises cannot be valid or invalid. They can only be true or false.