Summary Section 1
In this section we have introduced the concepts of propositions and explored when two propositions, which we have arbitrarily called p and q, are related to each other in terms of truth.
We defined formally the following concepts:
- A sentence which has the property of being either true or false is called a proposition.
- Statement variables are letters used to stand for propositions, just like letters are used to stand for numbers in algebra.
- If the truth of a statement p guarantees that another statement q must be true, then we say that p implies q, or that q is implied by p.
- A proposition p is said to be relevant to another proposition q if the truth of p counts for the truth of q but does not guarantee it.
- 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.
- 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.
- 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.
- 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.
- If propositions p and q are not truth functionally related, then we say p and q are independent.
When a relation between two statements is not one of implication new information can change the strength or weakness of an inference, but if a relation between two statements is one of implication new information can not change the nature of the inference.
The relations of implication, relevance, and independence are mutually exclusive if we restrict the scope of our propositions to those propositions which are contingent.
There are other relations between statements than the ones discussed in this section which will be covered in future sections.