I. Overview

The English word logic comes from the Greek word "logos" usually translated as "word", but with the implication of an underlying structure or purpose. Hence its use as a synonym for God in the New Testament Gospel of John. Logic is often defined as the process of "correct" reasoning. A more precise definition might be the study of the structures of arguments that guarantees correct or true conclusions from correct or true premises. There are generally speaking two "kinds" of logic: deductive and inductive.

Inductive logic is the body of methods used to generate "correct" conclusions based on observation or data. It is the type of reasoning used in the natural sciences and statistics where general principles are "inferred" from many particular facts. The use of the methods of inductive logic always carries with it the risk of incorrect generalizations, so that the validity of this kind of argument is essentially probabilistic in nature. We will consider this type of reasoning later this semester in the Probability Unit.

Deductive logic is the type of reasoning used in mathematics where we start from general principles and derive from these principles particular facts and relationships. Deductive logic usually denotes the process of proving true statements (theorems) within an "axiomatic system". If one accepts the validity of the axiomatic system, one is "forced" to accept the validity of the derived theorems. Their "truth" is beyond dispute unless the whole axiomatic system is inconsistent. These notes are primarily concerned with deductive logic.

An axiomatic system can be thought of as consisting of the following four components.

1. The set of allowed symbols. These are sometimes called the "primitives" or undefined terms of the system.

2. The well-formed formulas. These are sequences of the allowed symbols constructed according to some allowed rules. Definitions of new symbols are allowed as well-formed formulas of old symbols.

3. The axioms or set of "self-evident" truths of the system. These are well-formed formulas which are taken as statements of fact which can not be proven within the system. In some sense, the axioms must be "accepted on faith".

4. The rules of inference. These are rules which allow or license moves from certain well-formed formulas to other well formed formulas. As with the axioms the rules of inference are accepted as being self-evidently valid.

To some people one of the disturbing aspects of deductive logic is the difference between the syntactic and semantic content of a conclusion. An argument is syntactically valid if it "follows" the correct form or syntax of the language. The semantic content of an argument is related to its meaning or interpretation. By and large deductive logic is concerned with the syntax of an argument and ignores the semantics of the sentences in the argument. For example, consider the following two arguments.

Premise : If a triangle is an equilateral triangle, then it is an isosceles triangle.

Premise : Triangle ABC is an equilateral triangle.

Conclusion : Triangle ABC is an isosceles triangle.

Premise : If a quixod is a rauncos quixod, then it is a kalonker quixod.

Premise : Sortz is a rauncos quixod.

Conclusion : Sortz is a kalonker quixod.

Both arguments are logically correct and, from the syntactical point of view, identical. Note: the correctness of the conclusion of Argument 1 does not really depend on any knowledge of geometry or triangles. It is the structure or syntax of the argument, not its content or interpretation, which forces us to the correct conclusion. An understanding or interpretation of the meaning of the sentences (i.e., the semantics) is not really required, and maybe not even desired, in a logical analysis of an argument.