1 Relations between statements

Let's consider another example:

 

1. I have a dime, a quarter, and 11 pennies in my pocket.

2. I have some change in my pocket.

 

For simplicity's sake, let's call the first statement p and the second statement q. Now we pose our first question related to logical analysis:

Does the truth of p have anything to do with the truth of q?

By this, we simply mean, in the broadest of terms, whether the truth of the statement, "I have a dime, a quarter and 11 pennies in my pocket" lends credence to the statement, "I have some change in my pocket"?

Clearly the answer is yes, for if the first is true, then certainly the second must be true. Why is this so? The answer to this question has to do with the connection between the terms, "dime", quarter" and "pennies" and the word "change" when used in this context and the many possible ways one can have change in one's pocket.

Now let's turn the question around and ask:

Does the truth of q have anything to do with the truth of p, or more plainly, does the truth of, "I have some change in my pocket" increase the probability that, "I have a dime, a quarter, and 11 pennies in my pocket" is true?

To make this as clear as possible, what we are asking is whether the truth of q gives us any reason whatsoever to suspect that p might be true?

Again, the answer is yes, as one possible way to have change in one's pocket is indeed to have a dime, a quarter and 11 pennies, but as this is just one of many combinations of coins which makes the statement true, the truth of q does not guarantee the truth of p.

p see q and says, we might be related

Let's consider another example:

  1. Today is October 1.
  2. The word, "ostentatious" has 12 letters.

As above we will refer to the first sentence as p and the second as q , and ask the same questions:

Does the truth of p have anything to do with the truth of q , and similarly

Does the truth of q have anything to do with the truth of p?

If you are worried that statement p is not actually true, (since today is probably not October 1), don't be. To answer this question, we just assume that it is true - just like we assumed in the previous case that we had a dime, a quarter and 11 pennies in our pocket.

In both cases, we are inclined to say that the truth of p has nothing to do with the truth of q, and also that the truth of q has nothing to do with the truth of p.

The supposition that today is the first of October is unrelated to the fact that the word "ostentatious" has 12 letters, and the fact that the word "ostentatious" has 12 letters as nothing to do with today's date. As above, the reason is related to the connection between the meanings of the terms, but for now we will put aside that connection and simply record our observations.

Given two statements, p and q , then we have at least the following three cases:

  1. The truth of one statement guarantees the truth of another statement.
  2. The truth of one statement gives some support to the truth of another statement but does not guarantee it.
  3. The truth of one statement has nothing to do with the truth of another statement.

 

Before we continue, it might be helpful to point out that our use of the letters p and q is arbitrary. We could just as well use k , m , #, © or any other symbol, as long as it is understood what these symbols stand for. In our case, our symbols are just going to stand for sentences which have the property of being either true or false,where false is understood to mean the same thing as "not true". Sentences that have this property are known as propositions. Propositions are so important in our study of logic that we will enter the term in our list of formal definitions.

Definition: A sentence which has the property of being either true or false is called a proposition. Propositions are also called statements.

 

Are all sentences propositions?

 

As it turns out, using the letters t, f, or v to stand for propositions might cause confusion, so we will not use them in this textbook.  

Let us now consider the first type of relationship, where the truth of one statement guarantees the truth of another statement. This type of relationship is known as implication.