Symbolic Arguments

• We will consider arguments of the form

•     Premise 1:  statement
      Premise 2:  statement
      ...
      Premise n:  statement
      ---------------------
      Conclusion: statement
      

• For example

  If we make home made ice cream then we will get a headache.
  We made home made ice cream
  Therefore we got a headache.
      

• An argument is valid when the conclusion follows from a given set of premises.
• This is true if the conjunction of the premises imply the conclusion.
• To show this
  1. Write the premises and the conclusion in symbolic form
  2. show p₁ ∧ p₂ ... ∧ p_n → c is a tautology.
• For our example:

  If we make home made ice cream then we will get a headache.
  We made home made ice cream
  Therefore we got a headache.
  
  p: we make home made ice cream
  q: we get a headache
  
    p → q
    p
  ∴ q
  
  show (p → q ) ∧ p → q is a tautology.
  
   p
   q
   p → q 
   p → q ∧ p
    p → q ∧ p → q
  
   T
   T
   T
   T 
   T
   T
   F
   F
   F 
   T
   F
   T
   T
   F 
   T
   F
   F
   T
   F 
   T
  
      Since the final row is all T, the statement is a tautology and the argument is valid.
      

• How about

  If we make home made ice cream then we will get a headache.
  We got a headache.
  Therefore we made home made ice cream
      

• Note, the truth table tells us where the argument is not valid.