Propositional Logic

Abstract

Before starting on the basic material of this book, we introduce a general representation scheme which is one of the most important types of structures in logic and computer science: Trees. We expect that most readers will be familiar with this type of structure at least informally. A tree is something that looks like the following: It has nodes (in this example, binary sequences) arranged in a partial order (extension as sequences means lower down in the picture of the tree). There is typically a single node (the empty set, Ø) at the top (above all the others) which is called the root of the tree. (We will draw our trees growing downwards to conform to common practice but the ordering on the tree will be arranged to mirror the situation given by extension of sequences. Thus the root will be the smallest (first) element in the ordering and the nodes will get larger as we travel down the tree.) A node on a tree may have one or more incomparable immediate successors. If it has more than one, we say the tree branches at that node. If each node has at most n immediate successors, the tree is n-ary or n-branching. (The one in the picture is a 2-ary, or as we usually say, a binary tree.) A terminal node, that is one with no successors, is called a leaf. Other terminology such as a path on a tree or the levels of a tree should have intuitively clear meanings. For those who wish to be formal, we give precise definitions.