1. 1 Computational Logic in Human Reasoning Robert Kowalski (Imperial College, United Kingdom) Formal logic was originally developed as a normative model of human reasoning. However, numerous psychological experiments, including the Wason selection task, suggest that logic plays little role in human reasoning. As a consequence, other, more computationally oriented approaches, such as production systems, have had a greater impact in cognitive science. In this tutorial, I will review some of the psychological and philosophical literature on human thinking and argue that computational logic can reconcile logical and computational models of human thinking. Computational logic is a wide-spectrum language used in computing for both high-level specifications and low-level implementations. Moreover, the same kinds of reasoning that are used to execute logic programs at run time can also be used to transform specifications into implementations at compile time, and sometimes to decompile lower- level programs into higher-level form. This use of logic at multiple levels is analogous to the use of different levels of thinking in the dual process model. In cognitive psychology, the dual process model hypothesizes that two kinds of thinking operate in tandem. Intuitive thinking operates automatically, effortlessly and subconsciously, while deliberative thinking operates serially, effortfully and consciously. Logic is normally associated only with deliberative thinking. However, I will argue that computational logic can be used to model both intuitive and deliberative thinking, as well as many of the relationships between them. Logic is a general-purpose, domain-independent reasoning mechanism. In cognitive science, on the other hand, it is generally held that the mind is composed of special- purpose modules, in which general-purpose thinking is problematic and relatively unimportant. However, in computational logic, modularity and special-purpose knowledge are compatible with general-purpose reasoning. I will argue that this compatibility can also model the way in which special-purpose knowledge and general-purpose reasoning interact in the human mind. Lectures 1, 2, 3, 4 Lecture 5 Logic and Modularity (on my webpage)

2. 2 Computational Logic in Human Reasoning Robert Kowalski Lecture 1 Arguments against and alternatives to logic Lecture 2 Logic programming as a hybrid of declarative and procedural representations Lecture 2/3 The confusion between logic and production systems Lecture 3 Abductive Logic Programming Agents as a combination of logic programming, logic, and production systems Lecture 4 Computational logic as both a descriptive model and a normative model of human thinking Lecture 5 Computational logic as a model of both conscious and subconscious thinking and a solution to the problem of the meaning of life

3. 3 Arguments against Logic and alternatives to Logic as a model of human thinking “Mind: Introduction to cognitive science” by Paul Thagard, 1996 (A popular overview of the state of the art.) “How the mind works” by Steven Pinker, 1997 (The Wason Selection Task) “Review of MIT Encyclopedia of the Cognitive Sciences” by George Lakoff in AIJ, 2001

4. 4 Mind: Introduction to cognitive science by Paul Thagard Mental phenomena can be understood in terms of representation and computation Logic Rules Concepts (frames, scripts, schemata) Analogy (case-based reasoning) Images Connections (neural nets)

5. 5 Logic (according to Thagard) “Formal logic is only distantly related to human reasoning”. But “logic is, however, useful …in that it can suggest ways that people should reason better.” (Logic is prescriptive rather than descriptive.) But Thagard does not elaborate or explain how this could be possible.

6. 6 Rules (according to Thagard) “Rules are if-then structures …very similar to conditionals…, but they have very different representational and computational properties.” “One of the advantages is that rules can be interpreted as defaults.” But logic has been extended to deal with default reasoning. E.g. default logic, non-monotonic modal logic, circumscription, negation as failure in logic programming, argumentation. E.G. All birds can fly.

7. 7 Rules In logic-based systems the fundamental operation is logical deduction. In rule-based systems, it is search. But in logic it is also necessary to search for deductions. “Rule-based problem solving sounds a lot like logical deduction, but it differs in that much more attention is paid to strategies for applying the right rules at the right time.” But in logic search strategies are also needed to explore the search space of deductions. Of all the approaches, “rules have the most psychological applications.”

8. 8 Rules (page 45) “unlike logic, rule-based systems can also easily represent strategic information about what to do”: If you want to go home and you have the bus fare, then you can catch a bus. Forward reasoning with the rule simulates backward reasoning with the belief in logic programming form: You go home if you have the bus fare and you catch a bus.

9. Thagard confuses the relationship between production rules: If conditions then do actions. and logical implications (also called conditionals): If conditions then conclusions. Thagard writes (page 47): “rules can be used to reason either forward or backward.” But this is not a true property of production rules, but rather a characteristic feature of logical implications.

10. 10 Confusion about the relationship between logic programming and production rules. Simon (Production Systems.The MIT Encyclopedia of the Cognitive Sciences) includes Prolog, along with ACT-R, “among the production systems widely used in cognitive simulation”. Russell and Norvig (Artificial Intelligence: A Modern Approach) view production rules as logical implications used to reason forward.

11. 11 Logic programs look like rules Logic programs are sets of conditionals: If B1 and … and Bn then H In Horn clause logic programming, B1 and … and Bn and H are restricted to atomic formulae. In normal logic programming, the conditions B1 and … and Bn can be negations of atomic formulae. Negation in conditions makes normal logic programming a non-monotonic logic for default reasoning, e.g. X can fly if X is a bird and not X is flightless X is flightless if X is a penguin, etc.

12. 12 Inference in logic programming All inference is backward reasoning, using conditionals: If B1 and … and Bn then H as goal-reduction procedures: to show H by showing B1 and … and Bn. Because conditionals are used only backwards, they are normally written backwards: H if B1 & … & Bn. or H :- B1,…, Bn (in Prolog notation) Traditional logic is normally associated with forward reasoning: From B1 & … & Bn, conclude H.

13. 13 Negation as failure (NAF) Negative conditions of the form: not A are solved by trying to show A (using backward reasoning) and failing. Example X can fly if X is a bird and not X is flightless X is flightless if X is a penguin, X is a bird if X is a penguin tweety is a penguin

14. 14 Oaksford, M. & Chater, N. (2002). Commonsense reasoning, logic and human rationality. But logic programs can represent not only declarative problem specifications, but also efficient goal-reduction procedures.

15. 15 Concepts (also known as “frames” or “schemata”) Concepts are representations of typical entities, and are not strict definitions. Concepts are associated with default inheritance in hierarchies. “Concepts can be translated into rules, but they bundle information differently than sets of rules, making possible different computational procedures.” Arguably, concepts can be formalised by means of default reasoning

16. 16 Analogy Reasoning by analogy compares a new case with an old case. But analogical reasoning is compatible with deductive, abductive and inductive reasoning. Legal reasoning is a typical example of analogical reasoning.

17. 17 Images A picture is worth a thousand words. A picture can explicitly represent information that would normally need to be inferred from a linguistic representation. Perhaps thinking in terms of images is related to thinking by means of atomic sentences, representing examples.

18. 18 Connections (neural networks) Two kinds of representations: In local representations, the units have specific interpretations as concepts or propositions. Can be expressed as logic programs with weights on conditions and conclusions. In distributed representations, “hidden units” have no specific interpretation. Can be expressed as logic programs that argue for and against a conclusion? Connectionist representations can be used to perform parallel constraint satisfaction. Logic programs can also be executed in parallel.

19. 19 How the mind works (Steven Pinker) Condition-action rule production systems as the main example of how the mind works. Wason selection task as the main example of why logic doesn’t work.

20. 20 Wason selection task Four cards, letters on one side, numbers on the other. Determine whether the following rule holds: If D is on one side, then 3 is on the other side. Only 5-10% of all people select the right cards. DF37

21. 21 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.

22. 22 A Review of The MIT Encyclopedia of the Cognitive Sciences” (George Lakoff in Journal of Artificial Intelligence) “Concepts are shaped by the sensory-motor system, by neural structures, and by bodily experience in the world.” “These results contradict the idea … that thought is disembodied symbol manipulation.” “Conceptual metaphors are cross-domain mappings that permit abstract concepts to import most of their inference structure from concepts with a direct sensory- motor basis.”

23. 23 Some challenges Logic needs to be embodied in an observation- thought-action cycle. Logic needs to include both goal-reduction rules (beliefs) and condition-action rules (goals). We need to explain the Wason selection task. (By distinguishing beliefs from goals?) We need to relate logic and neural networks. Logic needs to combine general rules and typical examples. We need to exploit Logic as a prescriptive theory of communication and reasoning.

24. 24 Logic needs to be put in its place in the thinking component of an intelligent agent embedded in the world observe act An agent Perceptual processing Motor processing The world think decide