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Meta-logical problems: Knight, 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 problems: Knight, knaves, and Rips Overview: Summarize the puzzle & Rips’s 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 assertor’s 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 K&K problems, however: The analytic introspection provided in the study’s 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 K&K 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 K&K 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 K&K 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 DeMorgan’s 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 Rips’s 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, Rips’s 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 Rips’s 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 Rips’s 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? Rips’s theory lacks the framework needed to address this scenario, even though the role of C is captured procedurally.

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Problems with Rips’s 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 Rips’s 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 Rips’s 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 Rips’s 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 Rips’s 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 Rips’s 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 (don’t 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 DeMorgan’s 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. Rips’s 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 Rips’s 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 Rips’s 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.

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