Introduction to Symbolic Logic

Abstract: Conventions for translating ordinary language statements into symbolic notation are outlined.

We are going to set up an artificial "language" to avoid the difficulties of vagueness, equivocation, amphiboly, and confusion from emotive significance.
1. The first thing we are going to do is to learn the elements of this "new language."
2. The second is to learn to translate ordinary language grammar into symbolic notation.
3. The third thing is to consider arguments in this "new language."
Symbolic logic is by far the simplest kind of logic—it is a great time-saver in argumentation. Additionally, it helps prevent logical confusion when dealing with complex arguments..
1. The modern development of symbolic logic begin with George Boole in the 19th century.
2. Symbolic logic can be thought of as a simple and flexible shorthand:
  1. Consider the symbols:
    
    [(p q) (q r)] (p r).
  2. This rule was well known to the Stoics, but they expressed it this way:
    
    "If, if the first then the second and if the second then the third, then, if the first then the third."
  3. We will find that all of the essential manipulations in symbolic logic are about as complex and working with numbers made up on ones and zeros.
We begin with the simplest part of propositional logic: combining simple propositions into compound propositions and determining the truth value of the resulting compounds.
1. Propositions can be thought of as the "atoms" of propositional logic.
  1. Simple propositions are statements which cannot be broken down without a loss in meaning.
    1. E.g., "John and Charles are brothers" cannot be broken down without a change in the meaning of the statement. Note the change in meaning from "John and Charles are brothers" to the mistranslation "John is a brother" and "Charles is a brother."
    2. On the other hand, "John and Charles work diligently" can be broken down without a change in meaning:
      
      "John works diligently." "Charles works diligently." (It is assumed contextually that the meaning of the original statement is not that John and Charles work diligently together.)
  2. Conventionally, capital letters (usually towards the beginning of the alphabet) may be used as abbreviations for propositions.
    
    E.g., "John and Charles are brothers" can be symbolized as B.
    
    and "John and Charles work diligently" can be symbolized as the two statements: J and C.
    
    The logical operator "and," as we will see, will be symbolized in these notes as " " although other symbols are often used elsewhere.
2. In addition to propositions, propositional logic uses operators on propositions.
  1. Propositions can be thought of like the sticks of a tinker-toy set.
  2. Operators are like the connecting blocks. Typical operators include "and," "or," and "implies."
  3. By adding more and more operators, we get more complex structures.
3. For evaluation of statements, there is only one condition to be learned:
  
  "In order to know the truth value of the proposition which results from applying an operator to propositions, all that need be known is the definition of the operator and the truth value of the propositions used."