Reasoning

Reasoning

  • Reasoning Paradox: Human reasoning is often considered deficient when compared to logic and mathematics. Conversely, AI systems based on these standards are seen as deficient when compared to human standards.

Reasoning and the Brain

  • Different brain systems are used for different reasoning problems.
  • fMRI Study (Goel, Buchel, Frith, & Dolan, 2000):
    • Participants solved syllogisms.
    • Syllogisms consist of two premises and a conclusion.
      • Example of a congruent syllogism:
        • Premise 1: All poodles are pets.
        • Premise 2: All pets have names.
        • Conclusion: Therefore, all poodles have names.
      • Example of an incongruent but valid syllogism:
        • Premise 1: All pets are poodles.
        • Premise 2: All poodles are vicious.
        • Conclusion: Therefore, all pets are vicious.
    • Results:
      • 74% responded valid for incongruent syllogisms.
      • 84% responded valid for congruent syllogisms.

Syllogisms

  • Syllogisms can involve content that is incongruent with reality (e.g., poodles, dogs).
  • Abstract or content-free syllogisms also exist.
    • Example:
      • Premise 1: All P are B.
      • Premise 2: All B are C.
      • Conclusion: Therefore, all P are C. (Valid, content-free)

Brain Activation During Reasoning

  • Meaningful content: Activation in left ventral prefrontal and left parietal-temporal areas.
  • Content-free: Activation in posterior parietal reasoning areas. Similar activation is seen when solving algebra problems.

Reasoning about Conditionals

  • Deductive Reasoning:
    • Reasoning where conclusions follow with certainty from the premises.
    • Examples: valid/invalid syllogisms.
    • Deterministic.
  • Inductive Reasoning:
    • Reasoning where conclusions follow probabilistically from the premises.
    • Example: Predicting if it will rain today.
    • Probabilistic.

Conditional Statements

  • Conditional Statement: An assertion that if an antecedent is true, then a consequent must be true (If A, then B).
    • Antecedent: The condition (A in If A, then B).
    • Consequent: The result (B in If A, then B).
  • Example: If you read this chapter, then you will be wiser.
    • Antecedent (A): If you read this chapter.
    • Consequent (B): Then you will be wiser.
Valid Rules of Inference
  • Modus Ponens: Given A is true, infer B is true.
  • Modus Tollens: Given B is false, infer A is false.
Invalid Rules of Inference
  • Affirmation of the Consequent: Given B is true, infer A is true (invalid).
  • Denial of the Antecedent: Given A is false, infer B is false (invalid).
  • Deductive reasoning is often tested by asking people to state whether conditional statements (if-then) are valid/invalid.

Modus Ponens

  • Definition: Method for affirming.
  • Example:
    • If Joan understood the textbook, then she would get a good grade.
    • Joan understood the textbook.
    • Therefore, Joan got a good grade.

Modus Tollens

  • Definition: Method for denying or abolishing.
  • Example:
    • If Joan understood this book, then she would get a good grade.
    • Joan did not get a good grade.
    • Therefore, Joan did not understand this book.
  • Both modus ponens and modus tollens are valid inferences.

Affirmation of the Consequent

  • Example:
    • If Joan understood the textbook, then she would get a good grade.
    • Joan got a good grade.
    • Therefore, Joan understood the textbook. (Invalid)
  • Explanation: Joan could have gotten a good grade for other reasons (luck, class information).
  • There may be other routes to a good grade.

Denial of the Antecedent

  • Example:
    • If Joan understood the textbook, then she would get a good grade.
    • Joan did not understand the textbook.
    • Therefore, Joan did not get a good grade. (Invalid)
  • Explanation: This conditional statement is invalid as it doesn't exhaust other possibilities.
  • Similar to correlation does not equal causation – other possible causes/outcomes exist.

Evaluation of Conditional Arguments

  • People rarely fail to accept a modus ponens inference.
  • High levels of logical reasoning are primarily shown with modus ponens.
  • Acceptance of the valid modus tollens is only slightly greater than the frequencies with which they accept the invalid inferences.

Causal Reasoning

  • If-then statements can be interpreted as logical conditionals or causal statements.
  • Logical conditionals: Simply evaluating whether the conclusions follow from the premises.
  • Causal Interpretation: Leads people to reason about the statement based on their knowledge about causes in the real world.
    • Example: Less likely to agree with a statement that says “poodles are vicious” even if their goal is to say whether the conclusions follow from the premises.

Bayes Nets

  • Graphical formalism used in AI for reasoning about real-world knowledge.
  • A network can show that both viral pneumonia and bacterial pneumonia cause headache, cough, and fever (generative causes).
  • Antibiotics are given to cure bacterial pneumonia, and aspirin is administered to relieve headache (preventative causes).

Wason Selection Task

  • A subject is given a conditional statement of the form