Chapter 11 »Formal logic and reasoning ◊Syllogisms ◊Conditional reasoning ◊Hypothesis testing »Decisions ◊Psychophsysics and Symbolic distance ◊Cognitive

• 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?

04/19/23

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

AB

All circles are red

Logical Reasoning

• Set Unions

Logical Reasoning

• Syllogisms» Set Unions

◊ Some A are B

AB

Some Squares are blue

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

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

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

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

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

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

• 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

DR

Y

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 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 I10

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

Correct rejectionrate = 85 %

Miss rate = 50 %

Hit rate = 50 %

False Alarm 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

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

Higher

Lower


» 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


» 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?

Documents

Chapter 11 »Formal logic and reasoning ◊Syllogisms ◊Conditional reasoning ◊Hypothesis testing »Decisions ◊Psychophsysics and Symbolic distance ◊Cognitive