Methods of Proof

Methods of Proof

1. Truth Trees

   • Hook Rule
     • Negated - Stack
       • 1st = Same, 2nd = Negated
     • 1st - Negated, 2nd - Same
     • P HOOK Q; P is antecedent, Q is consequent.
     • **Conditional
   • Wedge Rule
     • Negated - Stack
     • INCLUSIVE disjunction
   • Tribar rule
     • Negated - Branch
     • **Material equivalence
   • Ampersand Rule
     • Negated - Branch
     • **Conjunction
   • Slashed tribar rule
     • Negated - Branch
     • EXCLUSIVE disjunction
   • ***Number every line, on left and right

2. Truth Tables

   • 2^n = No. of worlds, where n = No. of variables

3. Validity of Arguments

   • Counterexample
     • Truth values for the variables in the world where the premises are true, but conclusion is false.
     • Only exist in invalid arguments.
   • Premises and conclusion
     • Invalid - When ALL premises are true, but conclusion is false.
       • If 1+ premise is false, argument is automatically valid.
   • Ways to test for validity:
     • Truth Tree - Negate the conclusion (thus main operator), find if there is a counter-argument to it being false.
       • All closed branches = Valid
         • 1 + open branch = Invalid
     • Truth Table - Write premises with normal conclusion
       • Always do the conclusion first --find instances of true premises when conclusion is false.
   • Sound Argument
     • MUST be valid logically and factually
     • An unsound argument is logically valid, but not factually.

4. Types of Propositions

   • Tautology
     • True in all possible worlds
   • Contingency
     • True in at least one world, and false in at least one world.
     • Truth Tree: Do both tautology and contradiction test. If neither, it's a contingency.
     • Truth Table: Main column of proposition has at least one true and one false.
   • Contradiction
     • False in all possible worlds