Formal%20logic%20and%20reasoning

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7/30/2018

• Chapter 11
• Formal logic and reasoning
• Syllogisms
• Conditional reasoning
• Hypothesis testing
• Decisions
• Psychophsysics and Symbolic distance
• Cognitive maps

Study Question. .Describe the Wasson selection
task. What common type of logical errors are made
by people attempting this task? Compare and
contrast strict and lax criterion for responding.
How can bias effect accuracy rates?
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Logical Reasoning

• Deductive vs. Inductive reasoning
• Deductive Reasoning Drawing a conclusion from a
  list of premises by following the rules of logic.
• E.g., X has a better basketball team than SMU
• SMU has a better basketball team
  than Acadia
• therefore, X has a better basketball team than
  Acadia
• Inductive Reasoning Inferring a principle based
  on factual information.
• E.g., A store was robbed of 15 TVs
• John has no alibi and 15 TVs in his
  house therefore, John is probably involved
  in the robbery

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Logical Reasoning

• Syllogisms - A three-statement logical form, two
  premises followed by a conclusion.
• E.g., All sophomores are students.
• All students pay tuition.
• Therefore, All sophomores pay tuition.
• Abstract/general form
• All A are B
• All B are C
• Therefore, all A are C

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Logical Reasoning

• Syllogisms
• Try this
• All whales are fish
• All fish are insects
• Therefore, all whales are insects??
• Validity An argument is valid if the conclusion
  logically follows from the premises.
• Truth An arguments validity is not effected by
  the truth of the premises.

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Logical Reasoning

• Syllogisms
• Try this
• All whales are ocean dwellers
• Some ocean dwellers are orcas
• Therefore, some orcas are ocean dwellers
• Soundness An argument is sound if it is valid
  and the premise are true.

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Logical Reasoning

• Categorical syllogisms Venn diagrams
• All A are B

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Logical Reasoning

• Set Unions

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Logical Reasoning

• Syllogisms
• Set Unions
• Some A are B

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Logical Reasoning

• Mutually exclusive sets
• No A are B

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Logical Reasoning

• Categorical syllogisms using Venn diagrams
• All A are B
• All B are C
• Therefore, All A are C (valid conclusion)

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Logical Reasoning

• Categorical syllogisms using Venn diagrams
• All A are B
• Some B are C
• Therefore, Some A are C (Indeterminant)

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Logical Reasoning

• Categorical syllogisms using Venn diagrams
• No A are B
• No B are C
• Therefore, no As are Cs?

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Logical Reasoning

• Categorical syllogisms using Venn diagrams
• Some A are B
• Some B are C
• Therefore, Some As are Cs?

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Logical Reasoning

• Categorical syllogisms using Venn diagrams
• Some A are B
• No B are C
• Therefore, No As are Cs?

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Logical Reasoning

• Conditional Reasoning. Logical determination of
  whether the evidence supports, refutes, or is
  irrelevant to the stated conditional relationship
• A conditional reasoning approach to John and the
  TVs
• E.g., If P -gt Q If John is the robber, then he
  has 15 TVs
• Q John has 15 TVs
• therefore, P John is the robber
• Oops I forgot John is a TV repairer who works
  out of his home, and none of the TVs that he has
  are stolen.
• The above argument is not a valid argument
• Affirming the consequence
• This is one of the most common logical errors

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Logical Reasoning

• Conditional Reasoning

Valid Arguments
Invalid Arguments
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Logical Reasoning

• Conditional Reasoning A test
• 1) E -gt V
• E
• Therefore, ??
• Nothing!
• 2) E -gt V
• V
• Therefore, ??
• E
• 3) E -gt V
• V
• Therefore, ??
• Nothing!
• 4) E -gt V
• E
• Therefore, ??
• V

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Logical Reasoning

• The Wason selection task another test
• Each card has a letter on one side and a number
  on the other
• What are the fewest cards you need to turn over
  to confirm or deny the following hypothesis
• If it has a vowel on one side, there is an even
  number on the other side

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Logical Reasoning

• The Wason selection task another test
• Concrete with content knowledge

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Logical Reasoning

• Why do we make errors?
• Conditional vs. biconditional (form error)
• If and only if.
• E.g.. If you dont eat your supper, you get no
  ice cream
• We say or hear a conditional statement, but we
  think or mean a biconditional.
• Confimation Bias
• We search for positive evidence
• Matching hypothesis
• Memory load and Modus Tollens

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Logical Reasoning

• Hypothesis testing
• Science as a process of disconfirmation
• Statistical testing
• The null hypothesis
• If Null then No effect (if P -gt Q)
• Is an effect (Q)
• We reject the null (P)

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Decisions

• Psychophysics an experimental approach that
  attempts to relate psychological experience to
  physical stimuli.
• Fechner and the difference threshold
• Just Noticeable Difference (JND). The smallest
  difference between two similar stimuli that can
  be distinguished.
• Weber fraction
• Relates changes in stimulus intensity to sensory
  magnitude
• e.g., 3 people clap 1 more -gt within a JND
• 50 people clap 1 more -gt not within a JND

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Decisions

• Psychophsyics
• The Weber Fraction

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Decisions

• Psychophysics
• Other Weber Fractions
• Vision 1/60
• Kinesthesia 1/50
• Pain 1/30
• Pressure 1/7
• Smell 1/4
• Taste 1/3

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Decisions

• Psychophysics
• Absolute Threshold The critical level of
  intensity that gives rise to sensation.
• Problems with determining the absolute threshold
• The radar operator example
• Bias versus sensitivity
• Signal detection theory
• Noise and noise plus signal
• E.g., Library noise and library noise plus a
  gunshot

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Decisions

• Psychophysics
• Signal detection theory
• Sensitivity

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Decisions

• Psychophysics
• Signal detection theory
• Response Bias Criteria setting

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Decisions

• Psychophysics
• Signal detection theory
• Response Bias Lax criterion

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Decisions

• Psychophysics
• Signal detection theory
• Response Bias Lax criterion

Actual Events Noise Signalnoise
Correct rejection
Noise
Miss
Receiver Operator Chooses
False Alarm 50
Hits 85
Signal
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Decisions

• Psychophysics
• Signal detection theory
• Response Bias Strict criterion

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Decisions

• Psychophysics
• Signal detection theory
• Response Bias Lax criterion

Actual Events Noise Signalnoise
Correct rejection
Noise
Miss
Receiver Operator Chooses
False Alarm 15
Hits 50
Signal
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Decisions

• The symbolic distance effect
• Distance (descriminability) effect The greater
  the difference (or distance) between the two
  stimuli being compared, the faster the dexision
  that that they differ.
• E.g.s

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Decisions

• The symbolic distance effect
• Distance (descriminability) effect

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Decisions

• The symbolic distance effect
• The Symbolic Distance (descriminability) effect
  A distance (or descriminability) effect that is
  based on semantic or other long term memory
  knowledge.
• E.g., Symbolic imagery effects
• Which is larger a mouse or a horse?
• Which is larger a donkey or a horse?
• Effects mirror (physical) distance effects
• RT is a log function of perceived size discrepancy

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Decisions

• The symbolic distance effect
• The semantic congruency effect. Decisions are
  faster when the dimension being judged matches or
  is congruent with the implied semantic dimension

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Decisions

• The symbolic distance effect
• Semantic congruency effect

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Decisions

• The symbolic distance effect
• Banks et al. (1976)
• Distance and congruety
• Number magnitude estimates
• Which is larger? 1 or 2 vs. 1 or 5
  vs. 8 or 9 vs. 5 or 9

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Decisions

• The symbolic distance effect
• Judging geographical distances
• Holyoaks work
• People judge distances from their own perspective
• E.g., Which are further apart?
• Halifax to Fredericton vs. Calgary to
  Vancouver
• Semantic / propositional intrusions
• Which is further north, Edmonston, NB or
  Victoria, BC?

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Problems for upcoming lecture

• Complete the following Sequence
• O, T, T, F, F, S, S, E, N, .
• A Buddhist Monk leaves for a retreat atop a
  nearby mountain. He leaves at 600 AM and follows
  the only path that leads up the mountain. He
  travels quickly some of the way, he travels
  slowly, he stops for breaks. He arrives at the
  top of the mountain at 600 PM. The next morning,
  at 600 AM, he descends the mountain, again
  travelling at varying paces and with breaks. He
  arrives at 600 PM
• Is there a point on the trail that the monk
  would have passed at exactly the same time of day
  on the way up and on the way down the trail?
• Three hobbits and three orcs need to cross a
  river. There is only one boat, and it can only
  hold two creatures at a time. This presents a
  problem Orcs are vicious and whenever there are
  more orcs than hobbits they immediately attack
  and eat the hobbits. Thus, you can never let orcs
  outnumber hobbits on either side of the river.
• Can you schedule a series of crossing that will
  get everyone safely across the river?

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Problems for upcoming lecture

• Connect these nine dots with four connected
  straight lines.

Three people play a card game. Each player has
money in front of them (their ante). One each
hand of this game, one player loses and the other
two players win. The rules state that the loser
must use the money in front of them to double
the amount of money in front of each of the other
two players. They stake their antes and play
three hands. Each of them loses once and no one
goes bust. The each finish with 8.00. What were
the original antes (Hint it is not 2 each).
A landscaper has been instructed to plant four
new trees such that each one is exactly the same
distance away from each of the other trees. Is
this possible?