Computational Logic in Human Reasoning

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
Computational Logic in Human ReasoningRobert
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

• 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

• 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

• 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

• 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

• 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.

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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
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
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.
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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
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
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

• 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

• 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

• 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

• 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

• 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 doesnt work.

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.

D
F
3
7
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 dont 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

• 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.

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• 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.

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