Logic and Set Theory/Fundamental principles of logical reasoning

Logic is the art of taking true statements and concluding from them other statements which are also true. For instance, one may know that whenever it rains, the sky is cloudy. This statement is an example for what mathematicians call an implication. An implication is a statement of the form "if so-and-so, then so-and-so". In order to denote an implication, mathematicians use the double arrow symbol .

Definition (implication):

Let and be statements which are either true or false. Then the statement " implies ", commonly written as

,

is called an implication.

For example, for the statement "whenever electric current flows through a coil, that coil glows" a mathematician might write

.

Another example for an implication might be

.

This statement is meant to say "If a coil glows, then that coil is hot."

Now from these two statements, we may infer that if electric current flows through a coil, then that coil is hot. For, if electric current flows through a coil, then that coil glows by our first statement. Yet the second statement then implies that that coil is hot.

Often, one derives statements such as the above ones from empirical observation. Yet the above example already hints at the possibility of discovering new true statements through logic, without making any further empirical observations. Instead, logic proceeds by applying a set of deduction rules to existing statements.