Formal logic and reasoning ◊Syllogisms & Conditional reasoning –Hypothesis testing »Judgement...

• Formal logic and reasoning◊ Syllogisms & Conditional reasoning

– Hypothesis testing

» Judgement◊ Psychophysics

• Decisions I» Physical and Symbolic distance

◊ Cognitive maps

» Algorithms & Heuristics◊ Representiveness

– The hot hand debate

Study Question.•.Describe the Wasson selection task. What common type of

logical errors are made by people attempting this task?• What is the symbolic distance effect and why is it important in

understanding the notion of representation?

04/19/23Chapter 11, Part 1

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 CBU

CBU has a better basketball team than SMU

therefore, X has a better basketball team than SMU

» 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 (content-free)

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 whales are orcas» Soundness: An argument is sound if it is valid and the premise

are true.

Logical Reasoning

• Categorical syllogisms: Venn diagrams» All A are B

AB

All circles are red

Logical Reasoning

• Set Unions

Logical Reasoning

• Syllogisms» Set Unions

◊ Some A are B

AB

Some Squares are red

Logical Reasoning

• Mutually exclusive sets» No A are B

AB

No circles are blue

Logical Reasoning

• Categorical syllogisms using Venn diagramsAll A are B

All B are C

Therefore, All A are C (valid conclusion)

ABC

Logical Reasoning

• Categorical syllogisms using Venn diagramsAll A are B

Some B are C

Therefore, Some A are C (Indeterminant)

AB

C

Contradictory

AB

C

Confirmatory

(All whales are ocean dwellers)

(Some ocean dwellers are orcas)

(Some whales are orcas)

All apples are fruitsSome fruits are bananasTherefore, Some apples are bananas

Logical Reasoning

• Categorical syllogisms using Venn diagramsNo A are B

No B are C

Therefore, no As are Cs?

AC

B

Contradictory

A

Confirmatory

B C

No fruit are purple polka-dotted foodNo purple polka-dotted food are mangosTherefore, No fruit are mangos

Logical Reasoning

• Categorical syllogisms using Venn diagramsSome A are B

Some B are C

Therefore, Some As are Cs?

A

C

B

Contradictory

A

Confirmatory

B

C

Some apples are red thingsSome red things are fire trucks Therefore, Some apples are fire trucks

Logical Reasoning

• Categorical syllogisms using Venn diagramsSome A are B

No B are C

Therefore, No As are Cs?

A

C

B

Contradictory

B

Confirmatory

CA

Some apples are green thingsNo green things are redTherefore, No apples are red

Logical Reasoning

• Cognitive Neuroscience» Three cognitive theories

1. Mental model theory (Johnson-Laird)◊ Draws on spatial processes

◊ Right hemisphere

2. Mental Logic (Rips)◊ Language-based

◊ Left Hemisphere

3. Dual Mechanism◊ Implicit: Unschooled and automatic

◊ Explicit: Formal, deliberate

◊ Two unspecified regions of processing

Cf. Double dissociation procedures

Logical Reasoning

• Cognitive Neuroscience» Goel’s neuroimaging studies

All dogs are petsAll poodles are dogs

Content No content

All D are PAll N are D

Therefore, All fish are scaly All poodles are pets All N are P

Control Experimental

Logical Reasoning

• Cognitive Neuroscience» Results

◊ Reasoning in the presence of content was related to activation in Wernicke’s area

– Left hemishere

◊ Reasoning in the absence of content activated regions of the perietal lobe associated with visuo-spatial processing.

– Right hemisphere

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

– BTW: 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

If P -> Q If it is an apple, it a fruit

~Q It is not a fruit

therefore, ~P It is not an apple

Modus Tollens

Valid Arguments


P It is an apple

therefore, Q It is a fruit

Modus Ponens

Invalid Arguments


~P It is not an apple

therefore, ~ Q It is not a fruit

Denying the antecedent


Q It is a fruit

therefore, P It is an apple

Confirming the consequence

Logical Reasoning

• Conditional Reasoning» A test1) E -> V

~ETherefore, ??

Nothing!

2) E -> V ~V

Therefore, ??~E

3) E -> V V

Therefore, ?? Nothing!

4) E -> V E

Therefore, ??V

Logical Reasoning

• Conditional Reasoning» Rips & Marcus’ (1977) results

AlwaysSometimes Never

P -> QP~Q

P -> QPQ

100 0 0

0 0 100

P -> Q~PQP -> Q~P~Q

5 79 16

21 77 2

Logical Reasoning

• Conditional Reasoning» Rips & Marcus’ (1977) results

AlwaysSometimes Never

P -> QQ~P

P -> QQP

23 77 0

4 82 18

P -> Q~QPP -> Q~Q~P

0 23 77

57 39 4

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» Content knowledge

Only patrons with a “wet” stamp are allowed to drink alcohol.

DR

YWET

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)

» Research hypotheses◊ If we split attention then we have reduced resources

◊ We had reduced resources vs. we have no reduced resources

– Confirming the consequence vs. modus tollens

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 FractionΔ II

= c

◊ The Weber fraction for loudness = 1/10

– If 10 people clap, how many more must be added to notice the difference?

Δ I10

1=10

– If 50 people clap, how many more must be added to notice the difference?

Δ I50

5=50

Decisions• Psychophysics

» Other Weber Fractions:◊ Vision: 1/60

◊ Kinesthesia: 1/50

◊ Pain: 1/30

◊ Pressure 1/7

◊ Smell 1/4

◊ Taste 1/3


» 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


» Signal detection theory◊ Sensitivity

Loudness

Library noisesLibrary noises plus someone talking

Library noises plus a gunshot

}d


» Signal detection theory◊ Response Bias: Criteria setting

Brightness

Radar noise radar noiseplus signal

RespondsDoes not responds

Correct rejectionrate = 50 %

Miss rate = 15 %

False Alarm rate = 50 %

Hit rate = 85 %


» Signal detection theory◊ Response Bias: Lax criterion

Brightness





Actual EventsNoise Signal+noise

Rec

eive

r O

pera

tor

Cho

oses

Noise

Signal

Correctrejection

Miss

False Alarm50%

Hits85%

0.5 1.0

0.5

1.0

0

False Alarm Rate

Hit

Rat

e

d


Miss rate = 50 %

Hit rate = 50 %



» Signal detection theory◊ Response Bias: Strict criterion

Brightness





Actual EventsNoise Signal+noise

Rec

eive

r O

pera

tor

Cho

oses

Noise

Signal

Correctrejection

Miss

False Alarm15%

Hits50%

False Alarm Rate0.5 1.0

0.5

1.0

0

Hit

Rat

e

d


» Signal detection theory◊ Memory operating characteristics


Miss rate = 50 %

Hit rate = 50 %


Familiarity

New words Old words


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

Which line is longer?

vs.

Which dot is higher?

vs.


» Distance (descriminability) effect

DistanceNear Far

RT


» 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


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

Which balloon is higher?Which balloon is lower?

vs.

Which yo-yo is higher?Which yo-yo is lower?

vs.


» Semantic congruency effect

PositionBalloon Yo-yo

RT

Lower

Higher


» Banks et al. (1976)◊ Distance and congruety

– Number magnitude estimates

Which is larger? 1 or 2

vs. 1 or 5

vs. 8 or 9


» Judging geographical distances◊ Holyoak’s work

– People judge distances from their own perspective

– E.g., Which are further apart?

Antigonish to Fredericton vs. Sault Ste. Marie to Thunder Bay

455 km458 km


» Judging geographical distances◊ Semantic / propositional intrusions

– Which is further north, Edmonston, NB or Victoria, BC?


» Judging geographical distances– Which is further south: Detriot, MI, or Windsor, ON?

N


» Judging geographical distances– Which is further east: Florida or Chile?


» Judging geographical distances– Which is further south: Montréal or Paris?


» Judging geographical distances– Which is further west Reno or San Diego?


» Roper (2002) Global Geographic Literacy Survey– American youth’s estimates of USA population


» Roper (2002) Global Geographic Literacy Survey◊ Only 37% of young Americans can find Iraq on a map—though

U.S. troops have been there since 2003.

◊ 6 in 10 young Americans don't speak a foreign language fluently.

◊ 20% of young Americans think Sudan is in Asia. (It's the largest country in Africa.)

◊ 48% of young Americans believe the majority population in India is Muslim. (It's Hindu—by a landslide.)

◊ Half of young Americans can't find New York on a map.

Decisions

• Algorithms and Heuristics» Reasoning under uncertainty: Inductive reasoning

◊ Algorithms: A specific rule or solution procedure that is guaranteed to furnish the correct answer if it is followed.

– E.g., finding a forgotten phone number

◊ Heuristics: A strategy or approach that works under some circumstances, but is not guaranteed to produce the correct answer.

» Kahneman and Tversky’s work◊ Behavioural decision work◊ Ups and downs of heuristics

– Cf., Visual illusions

Decisions

• Algorithms and Heuristics» The representiveness heuristic

◊ E.g., Flip a coin 6 times, which is more likely– HHHHHH or HHTHTT

◊ Which lottery ticket is most likely to win the next 6-49?– 04-11-19-29-33-39 or 01-02-03-04-05-06

◊ The representativeness heuristic - samples are like the populations that they are pulled from.

– The representativeness heuristic leads to a number of decision biases

Prize PayoutsPrize Matches Amount Winners1st 6 $3,500,000.00 02nd 5 + Bonus $1,193.70 2393rd 5 $2,223.40 1064th 4 $85.10 5,2455th 3 $10.00 91,9356th 2 + Bonus $5.00 65,843

Lotto 6/49 - The winning numbers were….23 40 41 42 44 45 Bonus 43

Decisions


◊ The 649 Controversy (Wed, March 19, 2008) – Too many second place winners?

Decisions


◊ The law of small numbers– Who is more likely to have days where more than 60% of the births are male? St.

Martha’s or the IWK?

◊ Ignoring base rates– John: Truck driver or classics professor at Dalhousie?

– Cancer Screening example

1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive results. 9.6% of women without breast cancer will also get positive results. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?

--> Baysian probabilities

◊ The Gambler’s fallacy– Superbowl ‘09 coin flip

Decisions


◊ The birthday bet– If you bet against the birthday bet, what is P(winning)?

Person 2 -> 364/365 = .99

Person 3 cannot have the same birthday as 1 or 2

& Person 2 cannot have the same birthday as 1

Multiplicative Rule: The joint probability of two independent events is the product of their individual probabilities

Person 3 -> 363/365 X .99 = .99

Person 4 -> 362/365 X .99 = .98

Person 5 -> 361/365 X .98 = .97

Person 6 -> 360/365 X .97 = .95

Decisions


◊ The birthday betPerson 10 -> 356/365 X .90 = .88

Person 15 -> 351/365 X .77 = .75

Person 20 -> 346/365 X .62 = .59

Person 25 -> 341/365 X .46 = .43

Person 30 -> 336/365 X .32 = .29

Person 35 -> 331/365 X .21 = .19

Person 40 -> 326/365 X .12 = .11

Person 45 -> 321/365 X .07 = .06

Person 50 -> 316/365 X .03 = .03

Decisions


◊ Misperception of random events

◊ Is the following a random sequence? (Gilovich, 1991)– OXXXOXXXOXXOOOXOOXXOO

◊ “Maximally” indicative of randomness– Same number of X’s and O’s

– Same number of X’s follow O’s as X’s following X’s

(and vice versa)

– P(alteration) = .5

– Most people expect P(alteration) =.7

P(hit | 3 misses) P(hit | 2 misses) P(hit | 1 misses) P(hit) P(hit | 1 hit) P(hit | 2 hits) P(hit | 3 hits))Julius Irving .52 .51 .51 .52 . 53 .52 .48Andrew Tony .52 .53 .51 .46 . 43 .40 .34Team .56 .53 .54 .52 . 51 .50 .46

Decisions


◊ The hot hand in basketball– Study 1: 100 fans completed survey

91 % : a player has a better chance of making a shot after hitting the last two or three, a lower chance of hitting a shot if they have missed the last two or three

68 % for free throws

85 % believed that it was important to pass the ball to someone who had just made several shots in row.

◊ Study 2: 76ers home games from 80-81 season– Conditional probabilities

Decisions


◊ The hot hand in basketball– Study 3: Free throws

Shooter & shot Obs. & shot Shooter & last shot Obs. & last shot Correlation .02 .04 .40 .42

Decisions


◊ The hot hand in basketball– Predicting:

Bet either “high” (5 cents for a hit, lose 4 cents for a miss)

Or “low” (2 cents for a hit, lose 1 cent for a miss)

Shooters and observers bet

◊ Similar misconceptions (The clustering illusion)– Winning streaks as team momentum

– Hitting slumps in baseball

Formal logic and reasoning ◊Syllogisms & Conditional reasoning –Hypothesis testing »Judgement...

Documents

Transcript of Formal logic and reasoning ◊Syllogisms & Conditional reasoning –Hypothesis testing »Judgement...