LOGIC Chapter 2

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

2. Review of Ch 1:Representation and Computation • Cognitive Modelling • Computational Representational Understanding of Mind (CRUM) framework • History of Cognitive Science FCAC, University of Hyderabad

3. Review: CogSci? • Cognition can be understood as computation • Therefore, • theories should be specified in terms of • formal mental representations and • the computational steps performed on these. • Cognitive modelling follows quite naturally from this basic assumption FCAC, University of Hyderabad

4. Cognitive Modelling • is the characteristic research methodology of cognitive science (CogSci) • is theoretically grounded and empirically guided • results in generative theories • is not confined to 'effects', but specifies mental representations and processes FCAC, University of Hyderabad

5. Review: CogSci Integrates Multiple Research Traditions • Formal analysis of tasks and systems • using techniques from philosophy & logic, mathematics & physics, and the foundations of computer science • Empirical methods • from experimental psychology & neuroscience, and from anthropology, used for model testing • Computational (Programming) techniques • developed in Artificial Intelligence, used for model construction FCAC, University of Hyderabad

6. Analogy with a Computer FCAC, University of Hyderabad

7. Six Approaches to Modelling Mind • Each of the following has a particular kind of representation and a corresponding set of computational procedures • Logic • Rules • Concepts • Analogies • Images • Neural Connections FCAC, University of Hyderabad

8. Evaluation Criteria • Criteria for Evaluating theories of mental representation • Representational Power • Computational Power • Problem Solving: Planning, Decision, Explanation • Learning • Language • Psychological Plausibility • Neurological Plausibility • Practical Applicability • Education, Design, Intelligent Systems, Mental Illness FCAC, University of Hyderabad

9. Logic FCAC, University of Hyderabad

10. Outline • Introduction • Characteristics of a Logical Representation Scheme • Validity, Soundness • Prepositional Logic • Syntax, Semantics • Inference • Predicate (First-order) Logic • Syntax, Semantics • Inference • Other Logics • Modal, Temporal, Fuzzy, Deontic FCAC, University of Hyderabad

11. Introduction • Formal Logic provides some powerful tools for looking at the nature of representation and computation • Main thesis: • People have mental representations similar to sentences in logic • People have deductive and inductive procedures that operate on those sentences • Deductive and inductive procedures, applied to the sentences, produce inference FCAC, University of Hyderabad

12. Thought as motion • In the following we will give three descriptions of an episode of thinking. One could characterise them as movements of thought. • The subject in each case has one, and then other thoughts, which together form a series through which there is a kind of motion. • Logic could be seen as containing laws of this sort of motion. • Not all movements of thought are the concern of logic. FCAC, University of Hyderabad

13. How do thoughts differ? • Example 1: Brown is sitting at his desk gazing out of the window. He notices that the buds are just beginning to open on the trees. This reminds him of the unusually warm weather which was experienced last year at this time. That thought prompts the further thought that he must have his central heating boiler seen to as soon as possible. FCAC, University of Hyderabad

14. How do thoughts differ? • Example 2: Smith finds that her car won’t start. She remembers that when Jones’s car failed to start, it was because the distributor was wet. She also recalls reading that distributor problems are common in the sort of car she has. She is aware of how damp it is today. She concludes, therefore, that a wet distributor is the cause of the trouble. FCAC, University of Hyderabad

15. How do thoughts differ? • Example 3: Green is planning his summer holiday. He knows that he can go by aeroplane or car. If he goes by aeroplane, he will get there faster, but will be unable to take much luggage. If he goes by car, he can take much more. He recognises that the success of his holiday depends on his having the right sort of clothing for the unpredictable weather. He could not take the needed clothing on the aeroplane. He concludes that if the holiday is to be successful, he will have to go by car. FCAC, University of Hyderabad

16. Questions to ponder about the three descriptions of thought • What is the difference between Brown’s thoughts, and the other two kinds of thoughts? • Which thoughts might be described as making arguments? • How do Smith and Green’s thoughts differ? • Which of these three kinds of thought shows reasoning? • Which of these kinds of thought shows deductive reasoning? FCAC, University of Hyderabad

17. How do thoughts differ? • Example 1: Brown is sitting at his desk gazing out of the window. He notices that the buds are just beginning to open on the trees. This reminds him of the unusually warm weather which was experienced last year at this time. That thought prompts the further thought that he must have his central heating boiler seen to as soon as possible. ASSOCIATIVE THINKING FCAC, University of Hyderabad

18. Describing the three styles of thought • Brown’s thoughts: We might describe Brown’s thoughts as associative thinking or thought by spreading activation. Patterns in his perceptual stimuli gives rise to a chain of thoughts which activate further thoughts. This kind of pattern recognition is not usually studied as part of logic reasoning, but may be studied with other AI representations, such as Semantic Networks or Artificial Neural Networks. FCAC, University of Hyderabad

19. How do thoughts differ? • Example 2: Smith finds that her car won’t start. She remembers that when Jones’s car failed to start, it was because the distributor was wet. She also recalls reading that distributor problems are common in the sort of car she has. She is aware of how damp it is today. She concludes, therefore, that a wet distributor is the cause of the trouble. INDUCTIVE LOGIC FCAC, University of Hyderabad

20. Describing the three styles of thought • Smith’s thoughts: We might describe Smith’s thoughts as a form of inductive reasoning. Smith relies upon past events and consideration of probabilities to inform her decisions. This kind of thought is termed inductive logic. FCAC, University of Hyderabad

21. How do thoughts differ? • Example 3: Green is planning his summer holiday. He knows that he can go by aeroplane or car. If he goes by aeroplane, he will get there faster, but will be unable to take much luggage. If he goes by car, he can take much more. He recognises that the success of his holiday depends on his having the right sort of clothing for the unpredictable weather. He could not take the needed clothing on the aeroplane. He concludes that if the holiday is to be successful, he will have to go by car. DEDUCTIVE REASONING FCAC, University of Hyderabad

22. Describing the three styles of thought • Green’s thoughts: Green’s thoughts might be described as using deductive reasoning. In the description of Green’s thought has `If then’ statements and facts, and makes conclusions after reasoning about these conditional statements and facts. FCAC, University of Hyderabad

23. Inductive vs Deductive Reasoning • Inductive reasoning is going from true statements to true statements but it is a shaky process. • Thagard talks about “broad sense” induction as any inference that introduces uncertainty (leading to shakyness) and the “narrow sense” induction (inductive generalization) where general conclusions are reached from particular examples. • Deductive reasoning assumes the premises to true. Then if the premises are true the conclusions have to be true. There is nothing shaky about the transition. For deductive reasoning we can see that the process of going from premise to conclusions is independent of the truth of any assumptions. • Arguments have premises and conclusions that may be true or false. When reasoning is deductive there is only one case that can be ruled out: that is when the premises for the argument are all true and the conclusion is false. FCAC, University of Hyderabad

24. Valid and invalid deductive arguments • Validity = An argument is VALID if and only if it is necessary that if all it premises are true, its conclusion is true. • The intuitive idea captured by this definition is this: If it is possible for the conclusion of an argument to be false when its premises are all true, then the argument is not reliable (that is, it is invalid) • If true premises guarantee a true conclusion then the argument is valid. Alternatively, an argument is VALID if and only if it is impossible for all the premises to be true while the conclusion is false. • When an argument is valid its premises ENTAIL its conclusion. FCAC, University of Hyderabad

25. Examples of deductive arguments • The sun has shone every day for thousands of years, therefore the sun will shine tomorrow. • The sun is using up fuel which has a finite supply, therefore one day the sun will no longer shine. • If he is Napolean Bonaparte, then he is the Emperor of France. • He believes he is Napolean Bonaparte, therefore he believes he is Emperor of France (modal logic which can represent beliefs) FCAC, University of Hyderabad

26. Sound Argument • An argument is SOUND if and only if it is valid and all its premises are true. • All sound arguments have true conclusions. • An argument may be unsound in two ways: • if it is invalid, or • it has one or more false premises. FCAC, University of Hyderabad

27. Representing arguments in formal systems • Logical deductive arguments have been represented in natural language since the time of Classical Greek philosophers (Aristotle, etc). • However, there are drawbacks in using natural language to automate reasoning? • Natural language is ambiguous • Automated processing of natural language is a difficult challenge in its own right! • An answer is to represent argument and reasoning within formal systems FCAC, University of Hyderabad

28. Formal System • Formal systems in mathematics possess a number of elements: • 1. A finite set of symbols • 2. A grammar (syntax), • that is, a way of constructing well-formed formulae (wff) out of the symbols. (There should exist a decision procedure that can always tell whether a formulae is a wff) • 3. A set of axioms • 4. A set of inference rules • 5. A set of theorems. • This set includes all the axioms, plus all wffs which can be derived from previous-derived theorems by means of rules of inference. (There may not exist a decision procedure for deciding whether a wff is a theorem or not) FCAC, University of Hyderabad

29. Formal System (contd.) • Both Propositional Logic and Predicate Logic are types of formal system • In the next few lectures we will see how to form and use expressions in these logics. • What all these different logics possess in addition to syntax and rules of inference is semantics • Semantics • has to do with associating elements of a logical language with aspects of the real world. In propositional logic, logical atoms are associated with propositions about the world. • An association of atoms with propositions is called an interpretation. (Russel and Norvig page 203) FCAC, University of Hyderabad

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31. Propositional Logic FCAC, University of Hyderabad

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37. 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

38. Next Class: Logic Continued • 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