3/22/2016
• 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?
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
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
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 argument’s validity is not effected by the truth of the
premises.
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.
Logical Reasoning
• Categorical syllogisms: Venn diagrams
» All A are B
A
B
All circles are red
Logical Reasoning
• Set Unions
Logical Reasoning
• Syllogisms
» Set Unions
◊ Some A are B
A
B
Some Squares are blue
Logical Reasoning
• Mutually exclusive sets
» No A are B
A
B
No circles are blue
Logical Reasoning
• Categorical syllogisms using Venn diagrams
All A are B
All B are C
Therefore, All A are C (valid conclusion)
C B A
Logical Reasoning
• Categorical syllogisms using Venn diagrams
All A are B
Some B are C
Therefore, Some A are C (Indeterminant)
B A
C
Confirmatory
B A
C
Contradictory
Logical Reasoning
• Categorical syllogisms using Venn diagrams
No A are B
No B are C
Therefore, no As are Cs?
A
B
C A
C
Confirmatory
B
Contradictory
Logical Reasoning
• Categorical syllogisms using Venn diagrams
Some A are B
Some B are C
Therefore, Some As are Cs?
A
A
B
C
Confirmatory
B
C
Contradictory
Logical Reasoning
• Categorical syllogisms using Venn diagrams
Some A are B
No B are C
Therefore, No As are Cs?
A
A
B
C
Confirmatory
B
C
Contradictory
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 -> 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
Logical Reasoning
• Conditional Reasoning
Valid Arguments
If P -> Q If it is an apple, it a fruit
If P -> Q If it is an apple, it a fruit
P It is an apple
~Q It is not a fruit
therefore, Q It is a fruit
therefore, ~P
It is not an apple
Modus Ponens
Modus Tollens
Invalid Arguments
If P -> Q If it is an apple, it a fruit
If P -> Q If it is an apple, it a fruit
Q It is a fruit
~P It is not an apple
therefore, P It is an apple
therefore, ~ Q It is not a fruit
Confirming the consequence
Denying the antecedent
Logical Reasoning
• Conditional Reasoning: A test
1)
Therefore,
2)
Therefore,
3)
Therefore,
4)
Therefore,
E -> V
~E
??
Nothing!
E -> V
~V
??
~E
E -> V
V
??
Nothing!
E -> V
E
??
V
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
A
B
1
2
Logical Reasoning
• The Wason selection task: another test
» Concrete with content knowledge
DRY
WET
Logical Reasoning
• Why do we make errors?
» Conditional vs. biconditional (form error)
◊ If and only if.
– E.g.. If you don’t 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
Logical Reasoning
• Hypothesis testing
» Science as a process of disconfirmation
» Statistical testing
◊ The null hypothesis
◊ If Null then No effect (if P -> Q)
◊ Is an effect
(~Q)
◊ We reject the null
(~P)
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 -> within a JND
– 50 people clap + 1 more -> not within a JND
Decisions
• Psychophsyics
» The Weber Fraction
I = c
I
◊ The Weber fraction for loudness = 1/10
– If 10 people clap, how many more must be added to notice the difference?
I = 1
10
10
– If 50 people clap, how many more must be added to notice the difference?
I = 5
10
50
Decisions
• Psychophysics
» Other Weber Fractions:
◊ Vision:
1/60
◊ Kinesthesia: 1/50
◊ Pain:
1/30
◊ Pressure
1/7
◊ Smell
1/4
◊ Taste
1/3
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
Decisions
• Psychophysics
» Signal detection theory
◊ Sensitivity
}
d
Library noises
Library noises plus
someone talking
Loudness
Library noises plus
a gunshot
Decisions
• Psychophysics
» Signal detection theory
◊ Response Bias: Criteria setting
Responds
Does not responds
radar noise
plus signal
Radar noise
Brightness
b
Decisions
• Psychophysics
» Signal detection theory
◊ Response Bias: Lax criterion
Responds
Does not responds
radar noise
plus signal
Radar noise
Brightness
Correct rejection
rate = 50 %
Miss rate = 15 %
Hit rate = 85 %
b
False Alarm rate = 50 %
Decisions
• Psychophysics
» Signal detection theory
◊ Response Bias: Lax criterion
Actual Events
Noise Signal+noise
Noise
Correct
rejection
Miss
Signal
False Alarm
50%
Hits
85%
Hit Rate
Receiver Operator
Chooses
1.0
d
b
0.5
0
0.5
False Alarm Rate
1.0
Decisions
• Psychophysics
» Signal detection theory
◊ Response Bias: Strict criterion
Does not responds
Responds
radar noise
plus signal
Radar noise
Brightness
Correct rejection
rate = 85 %
Miss rate = 50 %
b
Hit rate = 50 %
False Alarm rate = 50 %
Decisions
• Psychophysics
» Signal detection theory
◊ Response Bias: Lax criterion
Actual Events
Noise Signal+noise
Noise
Correct
rejection
Miss
Signal
False Alarm
15%
Hits
50%
Hit Rate
Receiver Operator
Chooses
1.0
b
0.5
d
0
0.5
False Alarm Rate
1.0
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
vs.
Which line is longer?
vs.
Which dot is higher?
Decisions
• The symbolic distance effect
RT
» Distance (descriminability) effect
Near
Distance
Far
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
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
vs.
Which balloon is higher?
Which balloon is lower?
vs.
Which yo-yo is higher?
Which yo-yo is lower?
Decisions
• The symbolic distance effect
» Semantic congruency effect
Higher
RT
Lower
Balloon
Position
Yo-yo
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
Decisions
• The symbolic distance effect
» Judging geographical distances
◊ Holyoak’s 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?
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 6:00
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 6:00 PM. The next morning, at 6:00 AM, he descends the
mountain, again travelling at varying paces and with breaks. He arrives at 6:00
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?
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?