Meta-logical problems: Knight, knaves, and Rips

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Meta-logical problemsKnight, knaves, and Rips

• P.N. Johnson-Laird
• Princeton University
• Ruth M.J. Byrne
• University of Wales College of Cardiff
• Presented by Rob Janousek

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Meta-logical problemsKnight, knaves, and Rips

• Overview
• Summarize the puzzle Ripss theory
• Problems for the natural deduction approach
• Explore mental models reasoning
• Compare predictions experiment data
• Concluding thoughts questions

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Knights Knaves

• In the world of Knights Knaves
• Knights always tell the truth
• Knaves always tell falsehoods
• Example Two inhabitants A and B
• A says I am a knave and B is a knave.
• B says A is a knave.

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Knights Knaves

• In the world of Knights Knaves
• Knights always tell the truth
• Knaves always tell falsehoods
• Example Two inhabitants A and B
• A says I am a knave and B is a knave.
• B says A is a knave.
• Conclusion A is a knave and B is a knight

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A response to The Psychology of Knights and
Knaves by Lance J. Rips - University of Chicago

• Rips proposes a theory that the cognitive process
  involved in solving the knight-knave brain
  teasers is accounted for under a natural
  deductive logic framework.
• Rules defining the properties and relationships
  between knights and knaves
• Example
• Rule 3 NOT knave(x) entails knight(x)

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A response to The Psychology of Knights and
Knaves by Lance J. Rips - University of Chicago

• Rips proposes a theory that the cognitive process
  involved in solving the knight-knave brain
  teasers is accounted for under a natural
  deductive logic framework.
• Rules for manipulating formulae in propositional
  logic
• Example
• Rule 8 (DeMorgan-2)
• NOT(p AND q) entails NOT p OR NOT q

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A response to The Psychology of Knights and
Knaves by Lance J. Rips - University of Chicago

• Rips proposes a theory that the cognitive process
  involved in solving the knight-knave brain
  teasers is accounted for under a natural
  deductive logic framework.
• Rules for commencing and progressing through
  examination of all logical contingencies
  contradictions
• Example
• Assume the first encountered assertors assertion
  as a premise, and iteratively proceed to
  follow-up on all consequences.
• Then assume the negation of this assertion as the
  premise and do likewise.
• Then repeat this procedure for each assertor.

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Observations Intuitions

• The formal PROLOG procedure outlined by Rips is
  an effective algorithm for solving many KK
  problems, however
• The analytic introspection provided in the
  studys initial protocol evidence points to a
  less rigorous problem solving method.

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Observations Intuitions

• The formal PROLOG procedure outlined by Rips is
  an effective algorithm for solving many KK
  problems, however
• This algorithm only functions by reasoning
  forward on assumptions, even when solutions may
  more readily derived from backwards progression
  (using reducto ad absurdum).

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Observations Intuitions

• The formal PROLOG procedure outlined by Rips is
  an effective algorithm for solving many KK
  problems, however
• Many of the steps performed by the algorithm are
  redundant or test trivial cases. Considering
  irrelevant options is unduly burdensome on
  conceptual bookkeeping for humans.

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Observations Intuitions

• The formal PROLOG procedure outlined by Rips is
  an effective algorithm for solving many KK
  problems, however
• There is a peculiar linguistic issue that
  promotes confusion when a knave produces an AND
  statement (x AND y)
• (NOT x) OR (NOT y) by DeMorgans
• NOTx AND NOTy in the context of a liar

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The Challenge for Mental Models

• Rips concludes by requesting an explicit account
  of knight-knave reasoning that is
• Theoretically explicit (not ambiguous in its
  account)
• Empirically adequate (effectively explains the
  real world observations collected from experiment
  data)
• More than a mere notational reassignment of the
  same formal inference rules (not a mental models
  version of strict natural deduction)

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Problems with Ripss theory

• Rips overlooks the meta-logical nature of the
  problem domain
• The truth theoretic analysis of statements is
  foregone by adopting propositional logic formulas
  and appropriate relations.
• Taken in isolation, Ripss theory lacks the
  notion of validity as there is no truth
  assignment (and be shown to be complete through
  such).

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Problems with Ripss theory

• The knight-knave example is only one type of
  meta-logical puzzle
• The formal natural deduction procedure used
  cannot accommodate the switch from knight-knave
  truth telling to logician-politician deduction
  applying
• Example In the world of Logicians Politicians
• Logicians always make valid deductions
• Politicians never make valid deductions

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Problems with Ripss theory

• A says
• either B is telling the truth or else B is a
  politician
• (but not both)
• B says
• A is lying
• C deduces
• that B is a politician
• Is C a logician?
• Ripss theory lacks the framework needed to
  address this scenario, even though the role of C
  is captured procedurally.

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Problems with Ripss theory

• There is only a single procedure/algorithm
  supplied to solve the meta-logical problems
• Human reasoning is far less systematic and varies
  with the particular configuration of the problem
  statement.
• While Ripss procedure will yield the correct
  result, pragmatic considerations make it a poor
  model of human reasoning once the number of
  deduction steps grows large.

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Problems with Ripss theory

• The theory places too large a burden on human
  faculties
• Too much is required on the part of working
  memory.
• Protocol evidence prior to Ripss experiments
  shows difficulty in juggling propositional
  formulae without written aid for even simple
  examples.

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Meta-logical Reasoning with Mental Models

• The mental models approach assumes the ability to
  make simple propositional declarations not based
  on formal inference rules, but rather on modeling
  and revising possible states of the the involved
  entities/tokens.
• Example A or B (or both)
• not A
• Therefore, B

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Meta-logical Reasoning with Mental Models

• Example A or B (or both)
• not A
• Therefore, B
• First all possible states of the first premise
  are considered A, B, A, B, A, B
• Next the information in the second premise is
  incorporated, and inconsistencies are removed
  from consideration A, B, A, B, A, B
• Of the information under consideration in the
  single remaining model, the conclusion is
  extracted as not corresponding to any premise
  Therefore B

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Strategies for Meta-logical Reasoning

• The Full Chain
• A notational variant of Ripss procedure.
• Mental models replace the formal inference rule
  notation.
• Assume that an assertor tells the truth, and
  follow up the consequences, and the consequences
  of the consequences, and so on.
• Then assume that an assertor tells a lie and
  proceed likewise.
• Then repreat both these processes for all
  premises (eliminating contradiction assignment)

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Strategies for Meta-logical Reasoning

• The Full Chain
• Problems for human reasoning when traversing the
  branches of disjunctions.
• Limited capacity of working memory results in
  experiment participants needing to start over or
  guess about token status in mental models.
• While functional in basic cases, this mental
  models version of Ripss framework suffers the
  same flaws once the problem complexity is
  increased.

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Strategies for Meta-logical Reasoning

• The Simple Chain
• Assumes that the disjunctive consequences are too
  difficult to reliably formulate.
• Assume that the assertor in the first premise
  tells the truth and follow up the consequences
  until completed, or until it becomes necessary to
  follow up disjunctive consequences. Assume the
  first assertor is then lying and continue
  likewise (dont examine consequences of other
  premises)

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Strategies for Meta-logical Reasoning

• The Simple Chain
• Consistent with limits on working memory as one
  can continue a search for solutions without
  getting bogged down testing multiple conditions
  at each disjunction.
• This strategy does not guarantee a solution will
  be found, but functions well as a worst case
  default to work with until other heuristics
  strategies can be applied.

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Strategies for Meta-logical Reasoning

• The Circular Strategy
• A heuristic type rule for dealing with self
  referential claims of the form
• A asserts that A is false and B is true
• This also relates to an important observation
  that neither a knight nor a knave can claim (in
  isolation) that he is a knave.

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Strategies for Meta-logical Reasoning

• The Circular Strategy
• If a premise is circular, follow up the immediate
  consequences of assuming that it is true, and
  then follow up the immediate consequences of
  assuming that it is false.
• A asserts that A is false and B is true
• Since this statement refutes itself, A cannot be
  true. However if A is false, then (A is false
  and B is true) is a false assertion. Since the
  first conjunct is satisfied, it must be the
  second that is false, and thus B must be false

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Strategies for Meta-logical Reasoning

• The Hypothesize-and-Match Strategy
• More flexible than the Simple Chain and Circular
  Strategy as it provides a useful out when a
  contradiction arises.
• If the assumption that the first assertor A is
  telling the truth leads to a contradiction, try
  to match A with the content of the other
  assertions and proceed to follow up consequences
  under the A assumption.

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Strategies for Meta-logical Reasoning

• The Hypothesize-and-Match Strategy
• Example A asserts that A and B
• B asserts that not A
• Model assuming A A,B
• Add second premise A,A,B (contradiction)
• Now match A A,B (consistent)

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Strategies for Meta-logical Reasoning

• The Same-Assertion-and-Match Strategy
• Example
• A asserts that not C
• B asserts that not C
• C asserts that A and not B
• Both A and B make the same claim, so are either
  both true or both false. Consequently, C cannot
  be satisfied and therefore must be false.

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Strategies for Meta-logical Reasoning

• The Same-Assertion-and-Match Strategy
• If two assertions make the same (different)
  claims and a third assertor, C, assigns the two
  assertors to different (the same) types, then
  attempt to match C with the content of the other
  assertions and follow up the consequences
• (Alternatively)
• A asserts that C
• B asserts that C
• C asserts that A and B

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Predictions from Mental Models

• The simple strategies assume the capacity to
  process premises is limited.
• Negated conjunctions (by DeMorgans Law) force
  the consideration of a disjunctive model set.
• Positive matches are easier to deal with than
  negative mismatches (loosing track of multiple
  negations).
• Given these strategies and limitations, several
  predictions follow

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Predictions from Mental Models

• Problems that can be solved using the simple
  strategies are easier than those requiring the
  Full Chain approach.
• Ripss first experiment data supports this
  prediction with 28 correct conclusions for the
  simple strategy accessible problems and only 14
  correct conclusions in problems requiring the
  Full Chain.

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Predictions from Mental Models

• The difficultly of the problem will be related to
  the number of clauses that need to be examined to
  solve it.
• The number of links that need to be traversed in
  the application of the simple strategies relates
  to the number of steps needed by Ripss program
  (corresponding to results of the second
  experiment).
• However the simple strategies vary in the number
  of links they introduce (i.e. the circular
  strategy is less costly than hypothesize-and-match
  )
• The parsing order of the premises can influence
  which strategies are available and thus the
  number of links traversed.

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Predictions from Mental Models

• A hypothesis of an assertion being true is easier
  to process than one which is false assuming all
  else in the problem is equal.
• The process of negating mental models requires
  some cognitive resources.
• Example
• A says I am a knave or B is a knight A, B
• B says I am a knight B
• Versus
• A says I am a knave or B is a knave A,B
• B says I am a knight B

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Deducing Conclusions

• The use of mental models and the four simple
  strategies account for more of Ripss results
  than the natural deductive strategy.
• Some problems in the experiments were not able to
  be solved by any strategy given aside from the
  Full Chain.
• However, the percent of correct responses on
  these hard problems, while low, was still
  statistically above that of mere chance
  (guessing).

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Deducing Conclusions

• Is natural deduction necessitated despite
  resource limitations in memory?
• Other options may include an expanded Simple
  Chain
• Only continue to follow up on consequences of of
  disjunctive consequences to a certain threshold
  level.
• Continue the Simple Chain approach beyond the
  truth values of the first assessor.