Equivalent Statements

• Two statements are equivalent (S₁ ↓ S₂ ) if the have the same truth values in the answer columns of the truth tables.
• ~(p ∧ q) <=> ~p ∨ ~q
• ~(p ∨ q) <=> ~p ∧ ~q
• p → q <=> ~p ∨ q
• ~(p → q) <=> p ∧ ~q

Symbolic Arguments

• An argument takes the following form:

  • P1 
    P2
    ...
    Pn
    ----
    ∴ C
    

  • If it rains, we will not go to the park
    It is raining
    ---------------------------------------
    ∴ We will not go to the park.
           

  • P_i is called a premise
  • C is called the conclusion.
• An argument is valid if [P₁ ∧ P₂ ∧ … P_n ) → C
• There are a number of known valid arguments:
  • Law of Detachment

    p → q
    p
    ----
    ∴ q
    	

  • Law of Contraposition

    p → q
    ~q
    ----
    ∴ ~p
    	

  • Law of Syllogism

    p → q
    q → r
    -------
    ∴ p → r
    	

  • Disjunctive Syllogism

    p ∨ q
    ~p
    -----
    ∴ q
    	

• There are two named fallacies
  • Fallacy of Converse

    p → q
    q
    ----
    ∴ p
    	

  • Fallacy of Inverse

    p → q
    ~p
    ----
    ∴ ~q