P-Reading: Deductive Arguments: Validity & Soundness

P-reading: Deductive Arguments: Validity & Soundness

(Moore & Parker, 44-46; Guttenplan, 17-19, 28-31)

Earlier in the course we defined a deductive argument as one in which the premises are claimed to support the conclusion in such a way that it is impossible for the premises to be true and the conclusion false. If the premises do in fact support the conclusion in this way, the argument is said to be valid. Thus, a valid deductive argument is an argument such that it is impossible for the premises to be true and the conclusion false. In these arguments the conclusion follows with strict necessity from the premises. Conversely, an invalid deductive argument is a deductive argument such that it is possible for the premises to be true and the conclusion false. In invalid arguments the conclusion does not follow with strict necessity from the premises, even though it is claimed to.

An immediate consequence of these definitions is that there is no middle ground between valid and invalid. There are no arguments that are "almost" valid and "almost" invalid. If the conclusion follows with strict necessity from the premises, the argument is valid: if not, it is invalid. To test an argument for validity we begin by assuming that all premises are true, and then we determine if it is possible, in light of that assumption, for the conclusion to be false. Here is an example:

All television networks are media companies.

NBC is a television network.

Therefore, NBC is a media company.

In this argument both premises are actually true, so it is easy to assume that they are true. Next we determine, in light of this assumption, if it is possible for the conclusion to be false. Clearly this is not possible. If NBC is included in the group of television networks (second premise) and if the group of television networks is included in the group of media companies (first premise), it necessarily follows that NBC is included in the group of media companies (conclusion). In other words, assuming the premises true and the conclusion false entails a strict contradiction. Thus the argument is valid. Here is another example:

All banks are financial institutions.

Wells Fargo is a financial institution.

Therefore, Wells Fargo is a bank.

As in the first example, both premises of this argument are true, so it is easy to assume they are true. Next we determine, in light of this assumption, if it is possible for the conclusion to be false, In this case it is possible. If banks were included in one part of the group of financial institutions and Wells Fargo were included in another part, then Wells Fargo would not be a bank. In other words, assuming the premises true and the conclusion false does not involve any contradiction, and so the argument is invalid. In addition to illustrating the basic idea of validity, these examples suggest an important point about validity and truth, In general, validity is not something that is determined by the actual truth or falsity of the premises and conclusion. Both the NBC example and the Wells Fargo example have actually true premises and an actually true conclusion, yet one is valid and the other invalid. Validity is something that is determined by the relationship between premises and conclusion. The question is not whether premises and conclusion are true or false, but whether the premises necessarily, logically producethe conclusion.

(1) If the Earth is round, then many things would just fall off it.

(2) Things do not just fall off the Earth.

The Earth is not round.

This is a valid argument, but its conclusion is certainly false. How can this happen? Validity requires that it not be possible for the premises to be true and the conclusion false. It does not require that the premises and conclusion actually be true. In this case, since the conclusion is false and the argument is valid, at least one of the premises must be false. The culprit is quite clearly premise (1).

Nevertheless, there is one arrangement of truth and falsity in the premises and conclusion that does determine the issue of validity: any deductive argument having actually true premises and an actually false conclusion is invalid. The reasoning behind this fact is fairly obvious: If the premises are actually true and the conclusion is actually false, then it certainly is possible for the premises to be true and the conclusion false. Thus, by the definition of invalidity, the argument is invalid. This table presents examples of deductive arguments that illustrate the various combinations of truth and falsity in the premises and conclusion:

A sound argument is a deductive argument that is valid and has all true premises. Both conditions must be met for an argument to be sound, and if either is missing the argument is unsound. Because a valid argument is one such that it is impossible for the premises to be true and the conclusion false, and because a sound argument does in fact have true premises, it follows that every sound argument, by definition, will have a true conclusion as well. A sound argument, therefore, is what is meant by a perfect deductive argument in the fullest sense of the term. Sometimes they are even called “proofs”.

Validity in Everyday Life

Below is a sample argument. I want you to say whether it is valid and ask yourself how you arrived at your answer:

Cats have backbones.

All pets have backbones.

All pets are cats.

The argument is transparently invalid. What is more interesting is the question of how we come to know this. There seem to be two ways in which we manage the task. One can be called Visualizing, and the other Analogizing.

Visualizing

The Visualizing method works like this: you imagine what it would be like for the premises of an argument to be true and, at the same time, you try to imagine that the conclusion is false. Applied to the above argument, the method might come out as follows:

I picture cats as having backbones. This is easy since they do have backbones. I next picture various pets - goldfish, dogs, cats, etc. as having backbones and I try not to picture too many pets as lacking backbones. This is also fairly easy since it is likely that most pets do have backbones. The truth is often, but not always, easier to picture. Finally, I picture people having pets other than cats. I find that I can picture them this way without upsetting my previous efforts to picture the premises of the argument. It is this that convinces me that the argument is invalid - that the premises can be true and the conclusion false.

The method is called 'Visualizing' and I used the word 'picture' in applying it, but it is not strictly and literally visual. It is more a method based on our sensitivity to possibilities. My justification for the use of the visual metaphor is that such words as 'see' and 'picture' are not solely visual: I can see a reason as well as a landscape.

Analogizing

When required to say why they think the SAMPLE argument is invalid, many people are apt to say something like this: if the SAMPLE argument were valid, then I could just as well argue in the following way:

People have fathers.

All frogs have fathers.

All frogs are people.

But, it is as plain as can be that the premises of this argument are true and its conclusion is false - it is blatantly invalid. Since the argument about frogs has the same form as the argument about cats, the invalidity of the frog argument shows the invalidity of the original one. Crucial to this procedure is the idea of certain arguments being just like or analogous to one another. What the arguments share is not a content, but a form or structure. Implicit in Analogizing is the idea that what makes the above argument invalid is that they are of the same structure, and that this structure is in some way defective. We can spell out what goes on in Analogizing as follows. First, a structure for the argument is isolated. In the SAMPLE argument case; this structure can be displayed as follows:

As have Bs.

All Cs have Bs.

All Cs are As.

Next, there is a search for an argument which has just this structure, but which has true premises and a clearly false conclusion. This is where the argument about frogs came in. It is analogous in structure to the SAMPLE argument, but it doesn't require any effort to see that its premises are true and its conclusion is false.