Degrees of relevance
When considering the definition of relevance, we noted that we only require some connection between p being true and the determination of q being true. This is a very low bar, and it is natural to make distinctions between degrees of relevance. To that end, we introduce two more definitions which we will use in this course:
Our Definition: Suppose that p is relevant to q and that the truth of p means the truth of q is more probable than not, then we say p is strongly relevant to q.
Our Definition: Suppose that p is relevant to q and that the truth of p means the truth of q is less probable than probable, then we say p is weakly relevant to q.
Read the above definitions carefully! Pay special attention to the phrases, "more probable than not" and "less probable than probable".
What are we to make of such statements?
In general, when determining the probability of q being true given the fact that p is true, if that probability is greater than 0.5 (or 50%), then we say that p is strongly relevant to q. On the other hand, if the probability of q being true given the fact that p is true, is less than 0.5 (or 50%), then we say that p is weakly relevant to q.
If the probability of p is .50 given that q is true, we will say that p is as equally probable as not given that q is true, in which case we can use the term 'equally relevant'.
In the above division of weak versus strong relevance, the choice of 0.5 was chosen to conform to probabilistic uses of phrases like 'p is more probable than not'. This choice is not universal and is used here only to start the thinking process concerning degrees of relevance. In other disciplines, such as statistics, it is not uncommon to expect much higher levels of probability than what is given here. In other cases it may not even make sense to talk about the probability of an event, or it is far too difficult to actually calculate the probability of an event. In those cases we can still talk about weak and strong relevance but we need to justify why we think p is strongly relevant or weakly relevant to q by giving additional reasons, some of which may appeal to what is known as the principle of induction which we will discuss later. Thus the above definitions are given as a start to thinking about degrees of relevance, and should not be taken as standard or universal definitions.
What this does tell however is that a more complete understanding of weak and strong relevance requires an examination of probability, and in particular what is known as conditional probability. We examine this in Part 4 of this text and only make informal use of the terms now. A more complete treatment of this topic takes us away from logic to the mathematical study of probability and statistics.
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