Phil 10 study note-Ch.2

Chapter 2 of formal sentential logic

  • The purpose of truth function and table
    • To find the validity of the argument easier
  • Limitation
    • 1) How do we represent other complicated statements
    • 2) What if there is no row where all premises are true?
  • Answer to limitation 1)
    • Enlarge the truth table
      • (This is easy)
    • After adding the truth columns find the row where all premises are true
    • If there is any row where all premises are true, and conclusion is false, then argument is invalid
  • Answer to limitation 2)
    • Terminology to answer limitation 2)
      • Types of statements (concerns single statement)
        • tautologies
          • statement that is always true
          • applies only to compound statement
          • nothing follows from tautology
            • if you tell your roommate “I took a shower or I didn’t”, you actually informed him nothing
        • contradictions
          • statement that is always false
          • applies only to compound statement
          • everything follows from contradiction
        • contingencies
          • atomic statements are always contingent
      • Relations between statements (concerns more than 2 statements)
        • equivalency
          • truth columns of two statements are exactly the same
        • consistency
          • there exists one row where the statements are true
          • limitation 2) is when the premises are not consistent
            • how can we tell if the argument is valid?
        • implication
          • An argument is valid iff the premises jointly imply the conclusion
            • which means there is no counter-example row
    • Answer to limitation 2)
      • Argument is valid where the premises are inconsistent
      • So before, we had to find a row where premises are all true and conclusion is also true. However, in the case where there is no such row, we can now tell the argument valid if there is no counter-example row.
      • Counter-example row
        • row where premises are all true and conclusion is false
        • when there is no counter-example row the argument is valid even if there is no true where both all premises are true and conclusion is true.
        • Basically, all you need to do to figure out validity of argument by the truth table is to find a counter-example row
          • if you find one, then the argument is invalid
          • if you don’t, then the argument is valid
          • esayyyyyyyyyy
      • The conclusion is that the test for joint implication is same as the test for validity
        • if s1, s2 jointly imply s3, then s1, s2 are premises for conclusion s3 of the valid argument.
Advertisements