Formal Methods 101: Zero-Order (Propositional) Logic

Introductions:

  • Zero-Order Logic (also known as propositional logic or propositional calculus) is the foundation of most formal logic systems.
  • In Zero-Order Logic there is no notion of variables, but merely TRUE or FALSE propositions (statements), represented by symbols (usually A, B, C….), with logical operators (Negations, Binary Connectives) which describe an argument flow.
  • Proposition:
    • A proposition is simply a TRUE or FALSE statement.
    • Ex. Proposition A: All men are mortal
      • Proposition A is TRUE
    • Ex. Proposition B: My cat is a dog
      • Proposition B is FALSE
  • Negation (NOT): ¬
    • Ex. Predicate: ¬A
    • This negation relation means the negation (opposite) of the proposition A is TRUE. (If proposition A is TRUE, the expression ¬A is FALSE. If proposition A is FALSE, the expression ¬A is TRUE)
  • Conjunction (AND): ∧
    • Ex. Expression (Argument): A ∧ B
    • This conjunction operator means the expression is TRUE if and only if (iff) propositions A AND B are TRUE.
  • Disjunction (OR): ∨
    • Ex. Expression (Argument) : A ∨”B”
    • This conjunction operator means the expression is TRUE if and only if (iff) propositions A OR B are TRUE.
  • Material Implication: →
    • Ex. Expression (Argument) : A→B
    • This statement means, if A is TRUE, then B is TRUE. (Note: it gives no information about what B is if A is not TRUE.)
  • Material Bijection: ↔
    • Ex. Expression (Argument) : A↔B
    • This is statement means if A is TRUE, then B is TRUE, and if B is TRUE, then A is TRUE, if A is FALSE, then B is FALSE, and if B is FALSE, then A is FALSE. (In other words, the Boolean value for both A and B is never different.)
    • Note: This operator is equivalent to implication both ways; in other words (A→ B) ∧ (B ← A).

Order of operations for FOL:

  1. ¬ is evaluated first
  2. ∧ is evaluated next
  3. ∨ is evaluated next
  4. → is evaluated last (thus so is ↔)

Argument Construction:

  • A proposition is simply a TRUE or FALSE statement.
    • Ex.
      • Proposition A: All men are mortal.
      • Proposition B: Socrates is a Man.
      • Propositional Argument written Formally: A ∧ B → C
        • Propositional Argument: All men are mortal and Socrates is a man; therefore (implies), Socrates is mortal.
      • New Proposition C: Socrates is mortal.

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