Symbolic Arguments

• A symbolic argument consists of a set of premises and a conclusion

• P1: If it is cold then it will snow.
  P2: It is cold
  ------------------------------------
  C:  Therefore it will snow.
  
  
  p: It is cold
  q: It will snow
  
    p → q
    p
   ---
  ∴ q
      

• To show a symbolic argument is valid show:
• p₁ ∧ p₂ → c is a tautology
• So to show the above is a valid argument we need to show

  ((p → q) ∧ p)  → q
  
  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
  
      

• This argument is called The law of detachment

•    If you study then you will get an "A"
     You did not get an "A"
     -------------------------------------
     Therefore you did not study.
     

• Show that this is a valid argument.
• This is called The law of contraposition

•   p ∨ q
   ~p
  ----
  ∴ ~q
     

• Show that this is true
• This called Disjunctive Syllogism

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

• Show that this is true
• This called The law of Syllogism

•  p → q
   q
  ---
  ∴p
     

• This is not true, and is called The fallacy of the Converse

•   p → q
   ~p
  ---
  ∴~q
     

• This is not true, and is called The fallacy of the Inverse