Slides / Bullets

• Tautologies
  • A tautology is a sentence whose truth table contains nothing but T’s in the column under the main connective.
• Logical truth
  • A sentence is logically true or logically necessary if it must be true (as a matter of logic); i.e. if it’s true in all possible circumstances.
    • Any argument whose conclusion is logically true is valid.
    • Even an argument with no premises at all.
    • Conversely: any sentence that follows from the null set of premises is a logical truth.
• Tautology and logical truth
  • All tautologies are logical truths.
  • Not all logical truths are tautologies.
    • SameRow(a, a)
    • b = b
    • ¬Between(a, b, b)
    • ¬(Large(a) ∧ Small(a))
• TT-possibility
  • A sentence is TT-possible if its truth table contains at least one T under the main connective.
    • What is the relation between the following claims:
      • P is TT-possible
      • ¬P is TT-possible
      • P is a tautology
      • ¬P is a tautology
    • Answer: P is TT-possible if and only if ¬P is not a tautology; ¬P is TT-possible if and only if P is not a tautology.
• Logical possibility and TT-possibility
  • A sentence is logically possible if it might (as far as logic is concerned) be true; if it is true in some possible circumstance.
    • P is logically possible if and only if ¬P is not logically necessary.
  • All logically possible sentences are TT-possible, but not all TT-possible sentences are logically possible.
• For next week:
  • Read: 4.1; optionally, 4.2-4.4
    • Do the ‘You try it’ exercises.
    • If you’ve never done so, you might also try getting the hang of the ‘Boolean search’ mechanism of some Internet search engine - see exercise ***.
  • Do exercises 3.21 (48%), 4.2 (24%), 4.4-4.7 (7% each).