1.3 Independence

After considering the concept of relevance, it is natural to ask about cases where p does not imply q and where p is also not relevant to q. For example, if p is the statement, "There are 8.4 million inhabitants of New York City" and q the statement, "Dogs are mammals", the truth of p is not relevant to the truth of q, and also the truth of q is not relevant to the truth of p, nor does p imply q or q imply p. In these cases we say that p is independent of q. We will formally define this notion, but before we do so we will need two preliminary definitions.

Definition: A proposition p is said to be contingent if it is not always true or not always false. In other words, contingent propositions are true under some conditions and false under others.

Definition. Let p and q be contingent propositions, then if the truth or falsity of a statement p counts toward the truth or falsity of another statement q and if the truth or falsity of q counts toward the truth or falsity of p, then we say that p and q are truth functionally related.

Definition: If propositions p and q are not truth functionally related, then we say p and q are independent.

Notice that the two-way nature of truth functional relations means that if p is independent of q, then q is also independent of p.

The notion of truth independence will play a central role in formal logic, but for now we will just examine some examples to get a feel for what it means for two propositions

to be be independent.

Example 1.3.1. Let p be, "Tom's first child was born October 9, 2010", and let q be, "Pianos have 88 keys", then the truth of q is independent of the truth p.

Example 1.3.2. Let p be, "At normal pressure water freezes at 0 degrees centigrade", and let q be, "February has 28 days except for leap years", then q is independent of p.

Example 1.3.3. Let p be, "Columbus sailed for the Americas in 1492", and let q be, "The concert ended at 10:15pm", then q is independent of p.

Sometimes it is difficult to say whether two statements are logically independent or relevant, as relevance allows for the truth of two propositions to be connected even in remote ways. This possibility poses no problem for the logician who is primarily interested in hypothetical cases where the question, "If. . .then" is of more interest many times than the question of "is actually". As a result, when questions concerning relevance versus independence arise the logician can just consider what follows if p is relevant to q, and similarly what follows if q is independent of p. A similar problem arises between the relations of implication versus relevance. Exploring these different possibilities has been one of the most fruitful areas of logical investigation, and in a sense has defined the direction of such investigations since at least Aristotle's time. In particular the question of implication versus very strong relevance characterizes what is known as the problem of induction, which we will encounter in some detail in Part 1 Section 3.