A First Course in Mathematical Logic and Set Theory

CHAPTER 1

PROPOSITIONAL LOGIC

1.1 SYMBOLIC LOGIC

Let us define mathematics as the study of number and space. Although representations can be found in the physical world, the subject of mathematics is not physical. Instead, mathematical objects are abstract, such as equations in algebra or points and lines in geometry. They are found only as ideas in minds. These ideas sometimes lead to the discovery of other ideas that do not manifest themselves in the physical world as when studying various magnitudes of infinity, while others lead to the creation of tangible objects, such as bridges or computers.

Let us define logic as the study of arguments. In other words, logic attempts to codify what counts as legitimate means by which to draw conclusions from given information. There are many variations of logic, but they all can be classified into one of two types. There is inductive logic in which if the argument is good, the conclusion will probably follow from the hypotheses. This is because inductive logic rests on evidence and observation, so there can never be complete certainty whether the conclusions reached do indeed describe the universe. An example of an inductive argument is:

A red sky in the morning means that a storm is coming.

We see a red sky this morning.

Therefore, there will be a storm today.

Whether this is a trust-worthy argument or not rests on the strength of the predictive abilities of a red sky, and we know about that by past observations. Thus, the argument is ...