Table of Contents
|1. Statements and Logical Operators||Exercises for Section 1|
|2. Logical Equivalence, Tautologies and Contradictions||Exercises for Section 2|
|3. The Conditional and the Biconditional||Exercises for Section 3|
|4. Tautological Implications and Tautological Equivalences||Exercises for Section 4|
|5. Rules of Inference||Exercises for Section 5|
|6. Arguments and Proofs||Exercises for Section 6|
|7. Predicate Calculus||Exercises for Section 7|
You have been assigned the job of evaluating the attempts of mortals to prove the existence of God. And many attempts there have been. Three in particular have caught your attention: they are known as the cosmological argument, the teleological argument, and the ontological argument.
Teleological Argument (St. Thomas Aquinas): All things in the world act towards an end. They could not do this without their being an intelligence that directs them. This intelligence is God.
Ontological Argument (St. Anselm): God is a being than which none greater can be thought. A being thought of as existing is greater than one thought of as not existing. Therefore, one cannot think of God as not existing, so God must exist.
Are these arguments valid?
Logic is the underpinning of all reasoned argument. The Greeks recognized its role in mathematics and philosophy, and studied it extensively. Aristotle, in his Organon, wrote the first systematic treatise on logic. His work in particular had a heavy influence on philosophy, science and religion through the Middle Ages. But Aristotle's logic was logic expressed in ordinary language, so was still subject to the ambiguities of natural languages. Philosophers began to want to express logic more formally and symbolically, in the way that mathematics is written (Leibniz, in the 17th century, was probably the first to envision and call for such a formalism). It was with the publication in 1847 of G. Boole's The Mathematical Analysis of Logic and A. DeMorgan's Formal Logic that symbolic logic came into being, and logic became recognized as part of mathematics. This also marked the recognition that mathematics is not just about numbers (arithmetic) and shapes (geometry), but encompasses any subject that can be expressed symbolically with precise rules of manipulation of those symbols. It is symbolic logic that we shall study in this chapter.
Since Boole and DeMorgan, logic and mathematics have been inextricably intertwined. Logic is part of mathematics, but at the same time it is the language of mathematics. In the late 19th and early 20th century it was believed that all of mathematics could be reduced to symbolic logic and made purely formal. This belief, though still held in modified form today, was shaken by K. Gödel in the 1930's, when he showed that there would always remain truths that could not be derived in any such formal system. We'll mention more about this as we go along.
The study of symbolic logic is usually broken into several parts. The first and most fundamental is the propositional calculus, and this is the subject of most of this web text. Built on top of this is the predicate calculus, which is the language of mathematics. We shall study the propositional calculus in the first six sections and look at the predicate calculus briefly in the last two.
Waner and Costenoble are extremely grateful to the many reviewers who read drafts of the original notes, to David Knee, Bill McKeough, and Aileen Micheals for numerous suggestions, and to Barbara Bohannon and Rorbert Bumcrot for their supplement which insipred them.