Truth versus Inference and the concept of Soundness

The concepts of validity and invalidity are so central to the study of logic that it is worthwhile discussing them more at some length, as long experience has shown that the different use of these terms outside of a logic course lingers in the minds of many learning philosophical logic. This can cause confusion and wrong inferences based on those different meanings.

Perhaps the most persistent confusion concerning the concepts of validity and invalidity is the tendency to think they are synonymous with the concepts of truth and falsehood, namely that valid is equivalent to true, and invalid is equivalent to false. This is not the case, and great effort should be taken to learn the difference between these concepts. Truth and falsity are properties of propositions, whereas validity and invalidity are properties of inference (or, as we often say, arguments). The inference of an argument is either valid or invalid, whereas the individual components of the argument - its premises and conclusion - are either true or false. The non equivalence of these terms leads to the possibility that all of the components which make up an argument (the premises and conclusion) might be false, whereas the inference made by the argument is valid. To illustrate this possibility consider the following argument:

 

Example 2.3.1

Today is Wednesday and the day that follows Wednesday is Thursday.

Therefore tomorrow is Thursday.


If you are reading this on any day other than Wednesday, both the premise and conclusion are false, but the inference is valid - since if the conclusion were false (tomorrow is not Thursday) it is impossible for the premise, "Today is Wednesday and the day that follows Wednesday is Thursday" to be true. If the above example still seems odd, review again the formal definition of validity.

Another common misconception is the incorrect view that invalid arguments are somehow bad arguments, in the sense that either the conclusions to invalid arguments must be false, or somehow any reasoning which is invalid is poor reasoning. This too is incorrect. Consider the following invalid argument:

 

Example 2.3.2

I parked my car in the west parking lot this afternoon, as I do every time I drive to school.

My car has always been exactly where I parked it at the end of the school day.

Therefore, after classes, my car will be parked in the west parking lot.


In this case, it is possible for the conclusion to be false (your car will be missing), even though the premises are true. What this means is that the argument is invalid, but the premises are strongly relevant to the conclusion (imagine drawing the opposite conclusion from the same premises, that your car will not be parked in the west parking lot). Again, this illustrates the difference between invalidity and poor reasoning. An inference may be invalid, but still present good reasons to accept the conclusion - the difference is that the conclusion to an invalid argument is not guaranteed to be true whenever all of the premises turn out to be true, whereas with a valid argument we have such a guarantee.

Two trivial cases.

Recall the two-step test for invalidity:

The two step method for determining whether an argument is not valid.

To determine whether an argument is valid or not, many times it is easier to check to see if it is not valid. An argument is not valid only if both of the conditions [a] and [b] are true:

 

[a] It is possible for the argument to have all true premises.

[b] Under the possibility above, the argument can have a false conclusion.

 

An argument is not valid only if both of these conditions hold. The failure of one or both of these conditions means the argument is valid.

Now consider the following two arguments.

I.

Today is Monday Jan. 5 and Sunday Feb. 3.

Hence I will win the lottery.

 

II.

If the afternoon traffic is heavy, then I may not make it home in time to see the game.

Hence tomorrow I will go to work or tomorrow I will not go to work.

Value: 5

By applying the two-step test for invalidity, we discover . . .

 
 
 
 
 
 

 

Note that for argument I, the only premise is never true, hence condition [a] can't be checked, and hence the argument is valid by our two step method. For argument II, we can't check box [b] since the conclusion is always true. Hence argument II is also valid. Clearly this type of validity is a special case, which we denote by the term 'trivially valid".

Definition: An argument that is valid by virtue of the fact that it is impossible to have all true premises or because the conclusion is always true, is called trivially valid.

 

Clearly arguments which are trivially valid are only of passing interest. What we really care about are arguments whose premises can be sometimes true and sometimes false, but when they are all true then the conclusion must be true. And among the subclass of all such arguments, we really care about those which actually have all true premises. These valid argument are so important they have a special classification.

Definition: An argument is sound if it is valid and has all true premises.

Notice that for an argument to be sound, two conditions must be meet:

  1. The argument must be valid
  2. The argument must actually have all true premises.

Think about what this means for a second, and check your understanding so far by answering the following question:

 

Value: 5

The conclusion to a sound argument is always true.

 
 

 

Hopefully, your reasoning went something like this: Since the argument is sound, then it is both valid and actually has all true premises, so the conclusion must be true, by definition of validity.

The following is an example of a sound argument:

 

Example 2.3.3

If a number is greater than 7 it is greater than 3.

8 is greater than 7.

Therefore 8 is greater than 3.


First verify that the argument is valid (check to see it is impossible to have all true premises and a false conclusion), and then check to make sure the premises are all true. Once this is done the truth of the conclusion follows by logical necessity.

Notice that sound arguments could still be trivially valid, since the conclusion of an argument might be a statement which is always true and the premises could be statements such that each one could be actually true. Indeed there have been attempts (which are still current) to avoid the issue of trivial validity by restricting the type of premises and conclusion an argument can have. These efforts have not really caught on, perhaps in part since it is so easy to just label the case as an example of a trivially valid argument (or a trivially sound argument) and move on - as there is nothing more here of interest.

We complete one branch of our argument classification scheme related to validity by introducing one last (easy) definition.

Definition: If an argument is valid, but has at least one premise which is false, then the argument is valid but not sound.

 

Equivalently, if a valid argument has a conclusion which is actually false, then the argument is valid but not sound.

We have actually encountered an argument which was valid but not sound already, which we repeat here as an example:

 

Example 2.3.4

The Cullinan diamond weighs more than 100 grams.

The diamond was worn by Queen Victoria when she was crowned in 1964.

Therefore the Cullinan diamond weighs more than a 2 gram feather.


As noted before, Queen Victoria was not crowned in 1964, so the argument has at least one false premise. However, the argument is valid since if both premises were true then the conclusion must be true. This means the argument is valid but not sound.

Before moving on to the concepts of weak and strong arguments, let's review what we have so far.

Our first step is to classify the inference of an argument as either valid or not. If valid, then we check to see if the argument has all true premises. If the answer is yes, then the argument is classified as sound. If the answer is no, then the argument is classified as valid but not sound. The following diagram illustrates these possibilities.

 

argument classification c

Test your understanding by taking the following quiz which asks you to classify arguments according to the most specific classification above (if an argument is valid and sound, then classify it as sound, in other words move to the right as far as possible in the classification tree).

 Test your understanding: argument classification 1

 

Classify the following arguments according to the most specific classification possible (move as far right as possible in the classification tree).