Symbolic Arguments

• A symbolic argument consists of a number of premises followed by a conclusion.
• For example:

  If today is Wednesday then snow will fall.
  Snow is falling
  Therefore today is Wednesday
     

• The example is an invalid argument.
  • It is not valid because it can snow on other days.
  • This is called the Fallacy of the Converse

  If today is Wednesday then snow will fall.
  Today is Wednesday.
  Therefore Snow is falling.
     

• This example is a valid argument.
  • This is called the Law of Detachment
• A symbolic argument can be shown to be valid if the conjunction of the premises implies the conclusion is a tautology.

• If today is Wednesday then snow will fall.
  Today is Wednesday.
  Therefore Snow is falling.
  
  p: Today is Wednesday
  q: Snow is falling
  
  P1:  p → q
  P2: p
  C: ∴ 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 last line is all T, the statement is a tautology and the argument is valid.

  If today is Wednesday then snow will fall.
  Snow is falling
  Therefore today is Wednesday.
  
  p: Today is Wednesday
  q: Snow is falling
  
  P1:  p → q
  P2: q
  C: ∴ p 
  
  show [(p → q) ∧ q] → p is a tautology
  
  
   p
  q
   p → q
  (p → q) ∧ q
   (p → q) ∧ q → p
  
   T
  T
  T
  T
  T
  
   T
  F
  F
  F
  T
  
   F
  T
  T
  T
  F
  
   F
  F
  T
  F
  T
  
  
     

• The last line does not contain all T and the statement is not a tautology. The argument is not valid.
• In Fact, the argument fails when it is not Wednesday but it is snowing.
• There are four standard valid arguments and two standard invalid arguments:
• 
• Do problems page 147 13-64.