LOGIC-3 Chapter 2

1. LOGIC-3Chapter 2 Paul Thagard (2005). Mind: An Introduction to Cognitive Science. 2nd Edition. MIT Press FCAC, University of Hyderabad

2. Summary last class… • Inference rules for Propositional Logic • Inference rules, validity and tautologies in Truth Tables • Equivalences in Propositional Logic • Limitations of Propositional Logic • First Order Predicate Logic • Predicates and quantification • Equivalences in Predicate Logic • English to Predicate Logic FCAC, University of Hyderabad

3. Russel & Norvig, Ch 8 FCAC, University of Hyderabad

4. Russel & Norvig, Ch 8 FCAC, University of Hyderabad

5. First Order Predicate Logic (FOPL) continued… FCAC, University of Hyderabad

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12. Russel & Norvig, Ch 9 FCAC, University of Hyderabad

13. Russel & Norvig, Ch 9 FCAC, University of Hyderabad

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24. Russel & Norvig, Ch 8 FCAC, University of Hyderabad

25. Logic as a Model of Cognition FCAC, University of Hyderabad

26. Logic vs Cognition • Evaluation of Logic as a Representation scheme • Representational power • Computational power • Problem Solving: Planning, Decision, Explanation • Learning • Language • Psychological Plausibility • Neurological Plausibility • Practical Applicability FCAC, University of Hyderabad

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28. Representational Power • Formal Logics do not capture everything that is necessary for mental representation • We saw the power and limitations of Propositional and Predicted Logics. Additionally, they do not allow us to deal with uncertainty (only truth or falsity) • For this, we need to go to Probability or Fuzzy Logic FCAC, University of Hyderabad

29. Computational Power • Deductive reasoning can be implemented by applying formal inference rules. • Planning: How to accomplish a goal? • Logical deduction can be used to solve planning problems. • Computationally slow • Purely deductive planning is monotonic (can only draw new conclusions, but can not reject previous ones). • Can not learn from experience FCAC, University of Hyderabad

30. Computational Power (contd.) • Decision: Selection of best means to get to the goal. • Deductive planning can be used to find a logical path from an initial state to a goal state (forward chaining). • What if there are more paths? Which path to choose? Logic doesn’t give the answer, you need probabilities • => Probabilistic reasoning systems. FCAC, University of Hyderabad

31. Computational Power (contd.) • Explanation: In this, you are trying to understand why something happened. • Logical deduction can be used to generate explanation. Useful in physics and sciences. • Not all explanations are deductive and not all deductions are explanations! • We can deduce the height of a flagpole from the length of the shadow, trigonometry and laws of optics. • It seems odd to say that the length of a flagpole’s shadow explains the flagpole’s height!! • Sometimes, abduction (abductive inference) can be used to generate explanation. Form a hypothesis about the truth of a proposition to do inferencing. FCAC, University of Hyderabad

32. Computational Power (contd.) • Learning: Ability to use experience to improve performance. • Not much work done in Learning with Logics. • Inductive generalization: • easy(comp545), easy(comp747) • Therefore, (for-all x) (comp-course(x)=>easy(x)) • Risky generalization!! • Abduction is another way to implement learning in logical systems. Indispensable in science and everyday life, but a risky inference process! FCAC, University of Hyderabad

33. Computational Power (contd.) • Language: • Linguists have sometimes taken logic to be a natural tool for understanding the structure of (natural) language. • Natural language is too ambiguous sometimes to be represented by logical systems. FCAC, University of Hyderabad

34. Psychological Plausibility • Three positions regarding formal logic vs psychology • Formal logic is an important part of human reasoning. • Experimental evidence that people use rules like modus ponens. • Wason’s selection task shows why Logic doesn’t work • Formal logic is only distantly related to psychology, but a useful framework for understanding optimal reasoning • Formal Logic is only distantly related to human reasoning, so CogSci should pursue other approaches • Dominant view in psychology. FCAC, University of Hyderabad

35. Wason’s selection task Four cards, letters on one side, numbers on the other. D F 3 7 Determine whether the following rule holds: If D is on one side, then 3 is on the other side. By turning which cards can we ascertain the rule? (Modus Ponens P=>Q and Modus Tollens ~Q=>~P) Only 5-10% of all people select the right cards. FCAC, University of Hyderabad

36. Wason selection task Determine whether the following rule holds: If a person is drinking beer in a bar, then the person should be over eighteen. Most people get the right answer!! Conclusion? People don’t use logic, but have evolved a cheater detection scheme: If you receive a benefit, you must meet its requirement. Alternatively,subjects interpret descriptive and deontic (obligations, permissions, etc.) conditionals differently. Johnson-Laird talks about mental images as a better representation scheme. FCAC, University of Hyderabad

37. Psychological Plausibility (contd.) • Alternative models • Pragmatic reasoning based on rules argument of Cheng and Holoyok: people are good at concrete bar-and-age cases than abstract letter-and-number case. • Mental models of Johnson-Laird • Ex: Football-player strong lifts-heavy-objects • Good for representing “some”, “for-all” • Tversky and Kahneman challenged the notion that people’s reasoning follows rules of Probability • Ex: people violate the rule P(p&q)  P(p), where p is “Frank is college-educated” and q is “Frank is a carpenter” FCAC, University of Hyderabad

38. Neurological Plausibility • Simplistic view: neurons perform logical operations at the synapses • Brain scanning experiments show • Since deductive reasoning is purely lingustic, left hemisphere is important (Goel et al, 1998 showed syllogism didn’t activate Rt. Hemisphere) • Recent views: Goel, 2003 survey says two pathways, one for syllogistic reasoning involves both linguistic (Lt.) and visual-spatial systems (Rt.) FCAC, University of Hyderabad

39. Practical Applicability • Logic is very important in education • Undergraduate Logic and Critical thinking courses are very popular in the US. • They suggest ways that people should think and reason better. • Engineering design based on Logic-based programming languages, like Prolog • Ex: If a beam is simply supported, its depth shall be grater than one-thirtieth of its clear span. FCAC, University of Hyderabad

40. References • Paul Thagard (2005). Mind: An Introduction to Cognitive Science. 2nd Edition. MIT Press. • Slides from Susse, Dept of Philosophy, Michigan State University. • Slides from Logic in AI course: D.D. Peters, Computer Science, Birmingham Univ, UK • AIMA slides from Russel and Norvig FCAC, University of Hyderabad