Until now, we have avoided mention of an important fact about the study of logic which causes considerable difficulty to a beginner. We have taken proposition as a fundamental,  undefined term and defined three ways of forming propositional functions from simple propositions. The reader must be cautioned to restrict the formation of propositional functions to just those rules of combination specifically defined. The difficulty is that in discussing propositions, their algebra, and methods of deductive reasoning from these propositions, we are forced to make many statements about propositions. Since these statements seem to satisfy our intuitive definition of what a proposition should be, it is easy to fall into the trap of treating such statements as propositions. To clarify the situation, it is necessary to distinguish between the logic which we construct out of our propositions, called object logic, and the logic which we are using when we talk about the propositions,  called syntax logic. In particular, if p and q are propositions in the object logic, p', p + q, and pq are also propositions in object logic. However,  the statements "p is false" and "p = q" are propositions in syntax logic,  since they are statements about the propositions.

With this warning to the student about the construction of suitable propositional functions, we will end any formal consideration of syntax logic. All remarks concerning logic will from now on refer to object logic. This restriction will, of course, limit the rigor of our treatment,  but it is felt that this is excusable in an introductory text.