Section 1.1 Overview

Statement: a sentence possessing truth value.
Logical connectives join statements into formulas:
- conjunction
- disjunction
- implication (does its table seem weird to you?)
- equivalence
- negation (unary)
These are not independent - some of these derive from the others!
Well-formed formula (wff - "whiff") - What makes one well-formed?
- Order of precedence:
  - parentheses
  - '
  - conjunction, disjunction
  - implication
  - equivalence
- main connective (last to be applied)
Truth table for a wff with n statement letters: 2ⁿ rows
tautology: wff which is always true (represented by 1).
contradiction: wff which is always false (represented by 0).
equivalent wffs: two wffs are equivalent if the wff

A <-> B
is a tautology. (How can we prove that?)
De Morgan's Laws:
- (A or B)' <=> A' and B'
- (A and B)' <=> A' or B'
Algorithm: a set of instructions that can be mechanically executed in a finite amount of time in order to solve some problem.
Often written out in pseudocode, the authors provide us an example: the algorithm TautologyTest makes use of a special form of demonstration that an implication is, in fact, always true (a tautology). That is, they proceed by contradiction: assume that the implication P -> Q is false. Then P must be true, and Q false (the only scenario which makes an implication false).
Building a truth table for the implication also constitutes an algorithm to test to see if it is true, but, although the truth table algorithm may be more powerful (as more general, working for all would-be tautologies), tautology may be faster when applied to an implication.

longa@nku.edu