Saturday, January 27, 2007

Better Reasoning IV: Valid Argument Forms I

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[N.B. To quickly link to all the posts in this series, use the URL 'http://combatphilosopher.blogspot.com#reasoning'.]

In the last post in this series, the slightly technical concepts of the validity and soundness of deductive arguments were introduced. Although these ideas may initially appear a little abstract, they turn out to be extremely useful.

It turns out that it is possible to determine the validity of some arguments, merely in virtue of their 'form'. By 'form' here is meant roughly the arrangement of various components of the argument and their relations with respect to certain words which perform special logical functions. Here I will introduce a method for representing these forms in a manner that makes their structure explicit. I will then illustrate the use of this method, with some examples.

It turns out that certain words have a rather precise and predictable effect upon the sentences over which they operate. An important subset of these word are so-called 'logical connectives'. Although a detailed discussion of the connectives will be put to one side for now, we can get a rough and ready idea about logical connectives, by considering the logical role played by the word 'not', in a sentence. 'Not' is one of the more familiar logical connectives.

Suppose we have a sentence like "The cat is on the mat". This sentence will be true in cases when the proverbial cat is indeed on the mat. However, if we add the word 'not' to the sentence, thereby transforming it to "The cat is not on the mat", should the sentence have been true previously, it would now be false. Conversely, had the sentence previously been false (say, due to cat sitting nowhere near the mat), then the reformulation would make the sentence true. This demonstrates how the effect of adding 'not' to a sentence has a fairly obvious and predictable effect upon the truth of the sentence.

Now, by this point, you might be wondering why it is worth bothering to consider such a blindingly obvious example. One reason it is useful to consider this example is because it offers an intuitive method to introduce a rather useful little technical 'trick', known as 'symbolization'.

The first thing to notice about the sentence "The cat is on not the mat" is that it is exactly the same, in terms of it's truth and falsehood conditions, as the sentence "It is not the case that the cat is on the mat." Now, if we suppose that we will let the letter 'S' stand for the entire sentence "The cat is on the mat", then we can rewrite the sentence, after we have added the not (in the second formulation) as simply 'Not S'.

At first, this move might look kind of silly and trivial, however it is not. It makes it possible to state abstract truths about whole classes of sentences. This is because we could change the interpretation of S to something completely different, for example, "Summer is a coming in." and the logical facts would still remain the same. Consider the following abstract logical claim,

"Whenever a sentence 'S' is true, the sentence 'Not S' will be false, and when a sentence 'S' is false, the sentence 'Not S' will be true."

Notice how this claim remains true, regardless whether 'S' is interpreted as being "The cat is on the mat", or as being "Summer is a coming in." This trick makes it possible to make abstract statements about entire patterns of arguments. Moreover, this trick provides us with a handy shorthand with which we can identify valid patterns of inference.

Before proceeding any further, a few more words about this process of symbolization are in order. Traditionally, philosophers use the letters P, Q and sometimes R, when identifying patterns of inference. I have never seen an explanation for why this is the case, but it is useful to know the convention, so that other sources will be compatible with what is said here. I will follow this convention here. The second point to note is that these letters stand for entire propositions. That is to say, things that can be said as complete statements.

Now, we are ready to start looking at some simple patterns of valid inference. Let us begin by considering the way another special logical word functions, the word 'and'. Let us suppose that we happen to know that the sentence "Peas contain chlorophyll" is true (as I believe it to be), and we will also symbolize it with the letter 'P'. Let us also suppose that we also know that the sentence "Tonic water contains quinine" is true (as, again it is), and symbolize it with the letter 'Q'. Under these circumstances, we could validly infer "Peas contain chlorophyll and tonic water contains quinine.", or in symbolic form 'P and Q'. We can express this inference a little more clearly, if we put each premise in symbolic form on a separate line. Doing this results in the inference looking like this,

P
Q
Thus,
P and Q

This is a valid inference, no matter what sentences are substituted for the letters P and Q. This inference is sometime called 'conjunction'.

On the face of it, this may seem a little bit on the trivial side. However, we can imagine an investigator looking into the cause of an odd chemical reaction hypothesising that the reaction had been caused be the interaction between chlorophyll, quinine. Under such circumstances this inference might be made to show that both compounds were present in some mixture containing both peas and tonic water.

It turns out that this inference also works 'the other way'. That is to say that the two inferences,

P and Q
Thus,
P

P and Q
Thus,
Q

are both also valid. Inferences of this kind are called 'simplification inferences'. The really neat thing here (as with the previous case), is that the propositions we substitute for 'P' and 'Q' have no influence on the validity of the inference. To demonstrate this with a slightly silly example, the inference 'bong bongo and pielie pielie, thus bongo bongo', is also valid!

In this post, we have been introduced to a method of representing inferences in a manner which makes making statements about whole classes of inferences relatively easy. We have also seen two examples of valid argument forms concerning the word 'and'. In the next posting in this series, we will look at some further examples of valid argument forms.

The CP

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