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Computational Logic and Cognitive Science: An Overview

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## Computational Logic and Cognitive Science: An Overview

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### Computational Logic and Cognitive Science: An Overview

### Thank you very much!!

Session 2: Cognitive Challenges

ICCL Summer School 2008

Technical University of Dresden

26th of August, 2008

Helmar Gust & Kai-Uwe Kühnberger

University of Osnabrück

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Overview

- A Bunch of Cognitive Findings / Cognitive Challenges
- Wason Selection Task
- Remarks on Natural Language
- Sizes of Cities
- Theories of Mind
- Creativity
- Neuro-Symbolic Integration
- Causality
- Types of Reasoning
- Cognitive Architectures

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Wason Selection Task

- The Wason selection task
- 4 cards are given: On one side there is a number and on the other a letter printed.
- Rule: If there is a vowel at one side, there will be an even number at the other side.
- The following situation is given:

A D 4 7

- The task is: Turn as few cards as possible to prove the rule.
- The correct answer is to turn A and 7.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Wason Selection Task

- The experiment was executed in various versions.
- One showed the following results:
- A and 4: 46 %
- A: 33 %
- A and 7: 3 %
- Others: 18 %

Modus Tollens:

- If p, then q. And: not q. Therefore: not p.
- It seems to be the case that humans do not think logically…

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Wason Selection Task

- New rule: Only people over 18 are allowed to drink alcohol.
- Meaning: If for someone it is allowed to drink alcohol he/she must be over 18.
- The new situation:

15 Water Beer 22

- The solution is to turn Beer and 17.
- This version of the Wason selection task seems to be much easier to solve for humans.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Wason Selection Task

- Some proposals for an explanation of these results:
- Humans do not think logical at all (Gigerenzer).
- Humans think in models not in terms of logical deductions (Johnson-Laird).
- Humans need to embed their reasoning in concrete situations. They have problems in reasoning in idealized situations, i.e. mental models do not reduce the problem to the idealized (abstracted) situation.
- Humans can solve such problems, if it is placed in a social context (evolutionary psychology).
- Many theories were proposed to model these data.
- There are logic-based solutions as well as model-based solutions.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Wason Selection Task

- Another important point to mention is the way to describe the task in natural language.
- As a matter of fact, many logical connectives in natural language require a “more complex” interpretation than in classical logic.
- “Peter is in the living room or in the kitchen.”
- “Paul went to the university and gave a speech.” vs.“Paul gave a speech and went to the university.”
- “If Jim works hard for the exam he will pass it.”
- The standard version of the Wason selection task makes it plausible that a certain number of subjects interpret the implication as an equivalence.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Natural Language

- Natural language shows many features that cannot be easily modeled with classical logical approaches. Here are some examples:
- “Many students read different books.”
- Generalized quantifiers require an extension of classical logic.
- “Could you tell me what time is it?”
- Implicatures require a non-literal interpretation.
- “Yesterday John told me that in 150 years Germany will have a Mediterranean climate.”
- Temporal aspects require an extension of classical logic.
- “If I had been on holidays two weeks ago, I would not have a burnout now.”
- Counterfactuals
- “The king of France is bald.”
- Presuppositions extend the context in a non-trivial way, although there is nothing stated literally.
- “I am here.”
- Indexicals

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

San Diego vs. San Antonio

- An experiment due to Goldstein & Gigerenzer (having to do with knowledge and rationality in general):
- “Which city has more inhabitants: San Diego or San Antonio?”
- This question was asked American students and German students.
- Clearly German students knew little of San Diego, and many had never heard of San Antonio.
- Results:
- 62% of the American students answered correctly: San Diego.
- 100% of the German students answered correctly: San Diego.
- Gigerenzer proposes to use heuristics and cues to answer such questions resulting in a form of bounded rationality.
- In any case, there is a certain tension between bounded rationality and classical logic and knowledge representation.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Theories of Mind

- Theories of Mind
- Wise men problem (a variation of the famous muddy children problem).
- “Three wise men know there are three red hats and two blue hats (and they know that all three know that). The king placed a hat on each wise man, such that no wise man knows which color his hat has. Then he asks each wise man in a row which color his hat has.”
- Assume the first man says: “I don’t know.” and the second man says “I don’t know.” Why is it possible that the third man knows the color of his hat?

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Theories of Mind

- BBB is impossible (there are only two blue hats).
- P1 says: “I don’t know.”
- P2 and P3 infer that P1 sees a red hat: RBB is impossible.
- P2 says: “I don’t know.”
- P3 infers that P2 sees a read hat: BRB is impossible.
- P3 infers: P2 knows that P1 sees a red hat. In the remaining models there is only one where P3 has a blue hat: RRB. In this case P2 would know that she has a red hat.
- Therefore P3 answers that he has a red hat.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Theories of Mind

- Reasoning about the knowledge of other agents in a multi-agent systems seems to be natural to us.
- Maybe this is controversial. Nevertheless, if put into a reasonable situation, probably we are quite good in solving such puzzles…
- The frameworks proposed for representing and solving such puzzles are rather complicated.
- Modal logic / epistemic logic
- Situation theory
- Game theory
- In any case, classical logic needs to be extended in order to model reasoning about the beliefs of other agents.
- It is quite plausible to assume that humans do not apply game theory or perform deductions according to a modal logic calculus in order to solve this problem. They probably solve such problems differently.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Creativity: Examples

Jan van Eyck: The Arnolfini Marriage

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Creativity

- Creativity
- It seems to be unquestionable that humans show creative behavior.
- In particular, in problem solving, but also in using language productively (in particular, semantic productivity), in using metaphoric expressions, in generating theories, interpreting visual input, or making sense out of situations, humans show a remarkable ability of creativity.
- There are no really good theories that can describe this kind of creativity. One candidate may be analogical reasoning.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Neuro-Symbolic Integration

- Symbolic-subsymbolic distinction
- There is an obvious tension between symbolic and subsymbolic representations.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Neuro-Symbolic Integration

- Some interesting facts about the symbolic-subsymbolic distinction and cognitive science
- Classically natural language is considered to be a domain for symbolic theories.
- Chomsky’s claim was that natural language cannot be learned without assuming a universal grammar.
- His classical example was auxiliary inversion.
- Ecuador is in South America.
- Is Ecuador in South America?
- That woman who is walking her dog is Tom’s neighbor.
- *Is that woman who walking her dog is Tom’s neighbor?
- Is that woman who is walking her dog Tom’s neighbor?
- Nevertheless important insights were provided by Elman who showed how rather simple recurrent networks (Elman networks) can learn correctly auxiliary inversion.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Neuro-Symbolic Integration

- Some further remarks about the symbolic-subsymbolic distinction and cognitive science
- A further interesting fact is that one of the currently most influential theories in linguistics was developed by the neuroscientist Paul Smolensky.
- Optimality theory.
- Perhaps linguistics is a good testbed for neural modeling of complex data structures.
- In total, the integration of symbolic theories (in particular logic) into neural networks is an ongoing challenge.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Causality

- Causality seems to play an important role in human reasoning.
- Although the concept of causality is complicated and not very well understood, humans tend to structure the dynamics of the world by causes and effects.
- Reduction of causality to logical relations:
- Mackie: Causality can be explained by insufficient and non-redundant parts of unnecessary but sufficient causes (INUS condition).
- Example
- Short circuit is the cause of the house burning down (plus side conditions): together these events are unnecessary but sufficient for the destruction; the short circuit is insufficient but non-redundant.
- Nevertheless there are many different proposals for a logical reduction of causality, e.g. counterfactuals.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Reasoning Aspects

- Manifold of reasoning abilities:
- Deductions, inductions, abductions, analogical reasoning, associations, non-monotonic reasoning etc.
- An integration of these reasoning abilities is desirable.
- From a pure logical approach this does not seem to be a straightforward task.
- Even worse reasoning abilities are highly context dependent:
- Humans have the ability to jump easily from one context to another context, finding re-interpretations of a given input, and applying different types of reasoning types.
- Classical logical theories have their problems in modeling such situations.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Context Dependencies

Suppose you are in a forest and you want to heat some water. You do not have a container of any kind. You can cut a vessel of wood, but it would burn in the fire. How can you heat the water in this wooden vessel? Kokinov & Petrov (2001)

Davies & Goel (2001)

“I am here.”

“Oh, it’s raining.”

“Every student answered every question.”

Indurkhya (1992)

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Non-Monotonicity

Axioms

Birds can usually fly.

Penguins are birds.

Tweety is a Penguin.

Theorem

Tweety can fly.

Axioms

Birds can usually fly.

Penguins are birds.

Tweety is a Penguin.

Penguins can’t fly.

Theorem

Tweety cannot fly.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Non-Monotonicity

monotonic

extension

Theorems

Theorems

without p

Axioms

+ p

Axioms

new theorems

because of + p

Theorems

without p

non-monotonic

extension

Axioms

Theorems

incl. p

+ p

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Analogical Reasoning

“Electrons are the planets of the atom.”

“Current is the water in an electric circuit.”

?

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Analogical Reasoning

- Some statements about analogical reasoning right at the beginning:
- Analogy making is in general not case-based reasoning.
- Most interesting cases of analogies are cross-domain analogies.
- Analogical reasoning can be modeled with logical means.
- Analogical reasoning requires but cannot be reduced to deductions, inductions, and abductions.
- Analogical reasoning is the core of human creativity.
- A logical framework modeling analogical reasoning requires some non-standard techniques.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Cognitive Architectures

- The attempt to model cognitive behavior currently results in an inflationary number of different cognitive architectures.
- Examples are: ACT-R (Anderson), SOAR (Laird), AMBR (Kokinov), Clarion (Sun), NARS (Wang), Icarus (Langely), PSI (Dörner, Bach) etc.
- Some features of several (not of all) of these architectures:
- Integration of different reasoning types.
- “Non-rational” behaviors (associations, emotions etc.).
- Hybrid (neuro-symbolic) representations.
- Remark: not in the sense of neuro-symbolic integration, but more in the sense of “semantic networks + activation potentials”.
- Integration of various cognitive abilities.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

What Do We Have so Far?

- Wason selection task
- Remarks on Natural Language
- San Diego vs. San Antonio
- Theories of mind
- Creativity
- Symbolic-subsymbolic distinction
- Causality
- Reasoning
- Context, non-monotonicity, analogy
- Cognitive architectures

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Conclusion

- The mentioned cognitive capacities (or deficiencies) are relatively hard to model with standard logic techniques.
- The aim is to build intelligent systems that can come up with solutions of such problems.
- This requires non-classical forms of reasoning, extensions of classical logic into various directions, and the integration of different reasoning mechanisms.

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

Helmar Gust & Kai-Uwe Kühnberger

Universität Osnabrück

ICCL Summer School 2008

Technical University of Dresden, August 25th – August 29th, 2008

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