1.2 Relevance

Let's consider Example 1.1.4 one more time: Let p be, "The winning ticket starts with 3-7-9 and Mary's ticket starts with 3-7-9", and let q be the statement, "Mary's ticket is the winning ticket". It is easy to see that p does not imply q, but knowing the truth of p does give one reason to hope that Mary has won. This is a case where the truth of one statement gives support to the truth of another statement but falls short of guaranteeing it. This relation between the statements p and q gives rise to our next important concept, that of relevance.

Definition: Let p and q be two propositions, then p is said to be relevant to q if the truth of p increases the probability that q is true but p does not imply q.

The first thing we should note is that implication is not a form of relevance, since the truth of p guarantees the truth of q whenever p implies q. For this textbook we have included in the definition of relevance the condition that p does not guarantee q. Some textbooks which discuss relevance do not include this clause, so if reading other sources, always check their definition. We include this clause so as to be able to use these concepts later when we discuss different types of arguments - which are classified easily by the relationship between the argument's premises and conclusion.

Perhaps the best way to understand relevance is to consider several examples.

Example 1.2.1

Let p be, "Jane knows the capital of Arizona"and let q be, "Jane knows all the capitals of the United States" , then the truth of p is relevant to the truth of q.

Here q could be made true by the conjunction of several cases; Jane knows the capital of California, and Jane knows the capital of Nevada, and Jane knows the capital of and so forth, and clearly Jane knowing the capital of Arizona is one such case. In this case the relevance results in the fact that p is one of the many ways which together make q true. Note that on one hand p is relevant to q, but on the other hand q implies p. This observation tells us that order is important, since p can be relevant to q, but q may not be relevant to p because q implies p. This is one possibility. Many times p is relevant to q and q is relevant to p. The next few examples illustrate this.

Example 1.2.2

Let p be, "John ate uncooked meat" and let q be, "John will develop an e-coli infection", then p is relevant to q.

Clearly p alone does not establish q, but the fact that John ate uncooked meat and then came down with a case of e-coli is some evidence which supports the truth of q. In this case, knowing that John developed an e-coli infection is also some support for the proposition that John ate uncooked meat, hence in this case q is also relevant to p.

Example 1.2.3

Let p be, "Mr. Smith is a millionaire" and let q be, "The Smiths live in a mansion", then p is relevant to q, since being a millionaire increases the probability that q is true, but does not guarantee it.

In all of the above cases, we have two statements, p and q, where the truth of p is relevant to the truth of q. In other words, knowing that p is true lends some credence to the truth of q, but unlike implication, the truth of p statement does not guarantee the truth of q.

We might think of relevance in terms of degrees of evidence. When we want to establish that a certain statement is true, many times we offer evidence from various sources. Each single piece of evidence is related to the truth of the statement we want to establish (is relevant to it), but in many cases no single piece of evidence by itself firmly establishes what we wish to show.

It is now time to test your understanding of relevance by taking the following quiz where we explore some interesting cases concerning when p is relevant to q.

###### Test your understanding 2: relevance

Question [h] in the above quiz introduces us to the concept of context when determining if p is relevant to q.

In general this means that the judgment of p being relevant to the truth of q depends on the assumption of other factors in such a way that one assumption means p is very relevant to q (in the sense that the truth of p means q is likely) whereas another assumption means that p is not very relevant to q. Recall that for p to be relevant to q, we only require that the truth of p gives us some reason to believe that the probability of q being true has increased without guaranteeing its truth.

Here is another example:

Let p be, "Mary is over 18" and let q be the statement, "Mary has a license to drive". In the United States, the truth of p gives some reason to accept the truth of q, so p is relevant to q. However it could be that is some cultures women never drive, or if Mary is a person who lived in the past before cars were invented. So depending on the context the degree of relevancy between p and q will vary, or in some cases make p not relevant to q, since the truth of p guarantees the falsity of q.

So how does one treat context in determining whether p is relevant to q? The answer is pretty simple - if some context is known to affect the relevancy of p to q, then just state the contextual dependence. In our above example we could state:

If Mary lives in the United States, then p is relevant to q, but if Mary lives in a culture where women never drive, then p is not relevant to q or the degree of relevance is very low.

The reason for introducing contextual relevance is to help you start thinking about degrees of relevance. Since the bar is pretty low for p to be relevant to q, it is important to consider to what degree does the truth of p support the truth of q?

That is what we will do next.