• How hypotheses are established
  • A Posteriori Reasoning: on the basis of evidence
    • Perception → Current, nearby events
    • Memory → Past events
    • Testimony → Truth of what is said or written
    • Symptoms → Disease
    • Statistics → Causal connection
    • Forecasting Model → Future Events
    • Evidence in a criminal trial → Guilt of defendant
    • Confirmed prediction of a scientific theory → Truth of the theory
  • A Priori Reasoning : logical deduction from self-evident truths
    • Proof that the sum of two odd integers is even
    • Proof that the base angles of an isosceles triangle are equal

• Rationalism is the view that
  • Important truths about the world can be established by a priori reasoning, i.e. by logical deduction from self-evident truths.
    • Such truths would be contingently true and knowable a priori
    • A priori means independent of experience.
• Empiricism is the view that
  • Important truths about the world can be established only by a posteriori reasoning.
    • Such truths would be contingently true and knowable only a posteriori.
    • A posteriori means based on experience.

• Logical Necessity vs Contingent Truth
  • A statement is self-contradictory if it’s internally inconsistent.
    • Examples:
      • Some people are taller than themselves
      • Dallas is in Texas though it’s isn’t
      • 2 + 2 = 22
    • Such propositions are contradictions-in-terms, false by definition, false by virtue of their meaning, false in every possible world.
  • A statement is logically necessary if it’s negation is self-contradictory.  
    • Examples:
      • 2 + 2 = 4
      • Everything is what it is and not something else.
      • All fathers are parents.
    • Such propositions are true by definition, true by virtue of their meaning, true in every possible world.
  • A statement is contingently true if it’s true but not logically necessary.  
    • Examples:
      • Some molecules have two atoms
      • Harper Lee wrote To Kill a Mockingbird
      • The total energy of a closed physical system remains constant
    • Such propositions are true in the actual world but false in some possible worlds.
  • Leibniz and Hume distinguished between logical necessity and contingent truth, using different terminology
    • Leibniz spoke of truths of reason and truths of fact.
    • Hume distinguished relations of ideas and matters of fact.
  • Silly Example:
    • A person wants to prove that a gallon equals four quarts. So he buys ten gallon-containers and measures the number of quarts each holds.
    • Silly because a gallon equals four quarts by definition.
  • Are the following logically necessary, contingent, or logically impossible (self-contradictory)?
    • 1 + 1 = 2
    • All events are caused.
    • All effects are caused.
    • The total energy of a closed physical remains constant
    • Human beings exist.
    • There’s a greatest integer.
    • No object can be accelerated beyond the speed of light.

• A Priori vs A Posteriori
  • A statement is knowable a priori if it can be known independently of experience.
  • A statement is knowable only a posteriori if it can be known only based on experience.
  • The existence of infinitely many primes is knowable a priori, by working through the proof. 
  • That Neil Armstrong walked on the moon is knowable only a posteriori, through newspaper reports, TV clips, encyclopedias, and websites such as nasa.gov

• In sum:
  • Rationalism holds that some important contingent truths are knowable a priori
  • Empiricism holds that all important contingent truths are knowable only a posteriori.