Conditional Reasoning

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Conditional Reasoning

• Kevin Paterson
• k.b.paterson_at_derby.ac.uk

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Topics

• nature of conditional reasoning
• inference made with conditionals
• Wason Selection Task and different effects with
  abstract thematic content
• different theories that account for performance
  on the selection task

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Why is reasoning interesting?!

• Everyday situations require us to reason.
• We receive seek information that affects
  immediate thoughts, actions, plans, decisions,
  and so on.
• How this information affects our thinking or
  behaviour depends on nature of the information
  the information we have stored in our memories.
• Reasoning is means by which we make use of this
  information draw conclusions or inferences from
  it.
• Rreasoning is fundamental to intelligence.

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Deductive versus inductive reasoning

• Can distinguish between two types of reasoning
• Inductive - conclusions that are probably true
  given the evidence
• Deductive - conclusions that must be true given
  the evidence

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Example

• Simon is the brother of Julie
• Julie is the mother of Emma
• Simon is the uncle of Emma
• Simon is older than Emma

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Deductive reasoning

• Three key questions
• The competence question If people are able to
  reason logically, then what mental events allow
  this to happen?
• The bias question Why do logical errors occur in
  a systematic and non-random manner, and what do
  these biases tell us about deductive reasoning
  processes?
• The content question When reasoning is applied
  to real-world situations why does thematic
  content affect reasoning ?

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• Propositional reasoning
• uses connectives not, and, or and if
• Sentence with if conditional statement
• Statements using construction
• if..then.
• Example
• If Erica is in Derby, then Kevin is in Edinburgh
• antecedent consequent
• According to propositional calculus
• A proposition can be true or false

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Truth Tables

• p q If p then q
• True True
• True False False True
• False False
• p Erica is in Derby
• q Kevin is in Edinburgh
• If p then q
• If Erica is in Derby, then Kevin is in Edinburgh
• p implies q, q must occur if p does

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• To summarise
• a conditional statement is true when its
  antecedent and consequent are both true
• a conditional statement is false when its
  antecedent is true and its consequent is false

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Inferences in conditionals

• 2 valid inferences that we can make with
  conditionals
• Modus Ponens
• (p) (q)
• If Erica is in Derby, then Kevin is in Edinburgh
• Erica is in Derby (p)
• Therefore Kevin is in Edinburgh (q)
• If p, then q
• p,
• therefore q

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• Modus Tollens
• (p) (q)
• If Erica is in Derby, then Kevin is in Edinburgh
• Kevin is not in Edinburgh ( not-q)
• Therefore Erica is not in Derby (not-p)
• If p, then q
• not-q,
• therefore not-p

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• Denial of the Antecedent
• (p) (q)
• If Erica is in Derby, then Kevin is in Edinburgh
• Erica is not in Derby ( not-p)
• Therefore Kevin is not in Edinburgh (not-q)
• If p, then q
• not-p,
• therefore not-q

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• Affirmation of the consequent
• (p) (q)
• If Erica is in Derby, then Kevin is in Edinburgh
• Kevin is in Edinburgh (q)
• Therefore Erica is in Derby (p)
• If p, then q
• q,
• therefore p

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Percentage of inferences drawn in conditional
reasoning experiments

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Studying conditionals

• 3 classic ways to study conditionals
• 1. Conditional inference tasks
• Conditional statement and premise
• conclusion evaluation
• conclusion generation
• select conclusion from list
• 2. Truth table tasks
• asked to construct or evaluate truth tables

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Experiment

• RULE if there is an even number on one side of
  the card, then there is a vowel on the other
  side.
• What cards MUST you turn over to see if the rule
  is true?

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7
D
O
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• 3. The Wason Selection Task (Wason, 1966)
• Abstract Version
• 4 cards with a letter on one side and a number on
  the other.
• You can see one side of each card
• A (p), J (not-p), 3 (q), 7 (not-q)
• Conditional rule that relates to those 4 cards
• According to logical analysis the correct answer
  is
• A (p card) and 7 (not-q card)
• Few as 10 get correct answers (original abstract
  form)

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• Two types of error
• 1. Selection of the unnecessary q card
• 2. Failure to select the not-q card
• Why do people get it wrong?
• Evans Lynch (1973) - matching bias
• Summary - abstract selection task
• 1. Few people get it right
• 2. Make 2 types of error
• 3. Exhibit something called matching bias

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• Thematic version
• Thematic facilitation effect - easier to reason
  with problems of concrete content
• Wason Shapiro (1971) - Towns Transport
• Every time I go to Manchester, I go by train
• found facilitation
• Johnson-Laird, Legrenzi Legrenzi (1972) -Postal
  Rule
• If a letter is sealed then it has a 50 lire
  stamp on it
• 81 facilitation effect

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• Manktelow Evans (1979) Food Drinks
• Problem - failed to get facilitation with
  thematic content
• Every time I eat haddock then I drink gin
• No facilitation on Towns Transport
• Suggested - postal rule material was test of
  memory.
• Rule in England in PO higher value stamps were
  required on sealed letters, therefore Ss read
  rule as if a letter is sealed then it must have
  a 50 lire stamp on it
• Memory-cue hypothesis (Griggs Cox, 1982)

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• Griggs Cox (1982) Drinking age rule
• Clarified issues
• Towns Transport - no facilitation
• Postal rule on Florida Ss - no facilitation
• Drinking age rule - 74 facilitation
• Evans Pollard (1987) showed importance of
  police officer scenario
• Summary - thematic content
• 1. Early studies showed high facilitation
• 2. Later contradicted, cant conclude that
  thematic materials make us reason better, other
  explanations put forward.

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Is there an explanation for performance of the
selection task?

• Theories fall into 2 categories
• 1. General theories of reasoning
• Formal rule theory (Rips, Braine, OBrien)
• have mental inference rules (modus ponens), apply
  to tasks
• have better representation of some of rules than
  others (e.g. modus tollens)
• Mental models (Johnson-Laird, Byrne)
• represent statement as model
• due to incomplete model that lead to errors on
  task A 2 A 2
• . . . 2

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• Heuristics and biases (Evans, Over)
• Heuristic-analytic theory (Evans 1984,1989)
• 2 stages of reasoning
• heuristic stage - certain information is deemed
  relevant and selected for further processing
• analytic stage - inference is generated from the
  selected information
• Relevance theory (Evans and Over,1996) matching
  bias - cases more relevant when features match
  those named in conditional
• attributed to the function of the If and
  not-heuristics
• if - enhances relevance, directs attention to
  antecedent condition.
• not - directs the attention to proposition it
  denies.

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• Expected Information Gain (Oaksford and Chater)
• Cognitive behaviour is adapted to the structure
  of the environment, it does not depend on rule or
  models
• Ss arent testing rules but seeking information
  that will help them update their beliefs
• Gain information that decreases uncertainty
• Card selections are based on information value of
  each card in the form of expected information
  gain
• Choices are rational

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• 2. Domain specific theories
• Pragmatic reasoning schemas (Cheng, Holyoak)
• Schemas - generalised knowledge learnt from
  experience, can include rules for action or
  inference
• Propose permission schema learnt from
  situations in which require permission to perform
  actions
• can be applied to conditional in form
• If action is to be taken, then the precondition
  must be satisfied

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• Permission schema produces 4 production rules
• R1 If the action is to be taken then the
  precondition must be satisfied
• R2 If the action is not to be taken then the
  precondition need not be satisfied
• R3 If the precondition is satisfied then the
  action may be taken
• R4 If the precondition is not satisfied then the
  action must not be taken
• In selection task - p is action and q is
  condition
• Drinking age - rules indicate selection of p
  (drinking beer) not-q (16 yrs old) (R1, R4)
• No mistake of selecting q - R3 says can only
  drink if over age limit

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Summary

• Various different accounts have been put forward
  to try and give an understanding of how people
  reason on the selection task
• they need to account for
• competence
• bias
• content
• Not all do this
• References
• Evans, J.ST.B.T., Newstead, S.E. Byrne, R.M.J.
  (1993). Human reasoning The psychology of
  deduction. Hove LEA
• Evans, J.St.B.T. (1984). Heuristic and analytic
  processes in reasoning. British Journal of
  Psychology, 75, 451-468.
• Evans, J.St.B.T. (1996). Deciding before you
  think Relevance and reasoning in the selection
  task. British Journal of Psychology, 87, 223-240.