Download P325 L18-Reasoning

Reasoning
n
Deductive Reasoning
Deductive reasoning vs. Inductive reasoning
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Deductive reasoning
n If
valid argument (follows logical rules)
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Logic
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Facts -- items that may be true or false
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Operators -- truth value depends on facts
and
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n If
premises are true
n Then conclusions are true (guaranteed)
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Have truth table
AND, OR, NOT
AND(A,B)
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Inductive reasoning
n No
certainty about truth
n Assess
probability of conclusion being true
OR(A,B)
A is FALSE A is TRUE
B is FALSE
FALSE
FALSE
B is FALSE
FALSE
TRUE
B is TRUE
FALSE
TRUE
B is TRUE
TRUE
TRUE
Schemas
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Templates for combining operators and facts
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P OR Q
NOT P
∴Q
Deductive Reasoning
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If Premises true and argument valid ---Conclusion
Conclusion true
Domain general
Apply schemas to any situation
If premises true and valid form
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First part of conditional -- antecedent (e.g,. P)
Second part of conditional -- consequent (e.g., Q)
Modus Ponens (affirming the antecedent)
P→Q
P
n ∴Q
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Valid?
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Truth – statement about world (premises or conclusion)
Validity – form of the argument
Conditional Schemas
Conditional statements
Propositional Calculus
reasoning from "if then" statements
P → Q (if P then Q)
P
(P is TRUE)
∴Q
(therefore Q is TRUE)
Conclusions guaranteed to be true
Note:
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A is FALSE A is TRUE
Yes
Example
If it is sunny then Dr. Mounts will play golf
It is sunny
n ∴ Dr. Mounts will play golf
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Conditional Schemas
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Conditional Schemas
Modus Tollens (denying the consequent)
Denying the Antecedent
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P→Q
n NOT Q
n ∴NOT P
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Valid?
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P→Q
NOT P
n ∴NOT Q
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Yes
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Valid?
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Example
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Example
If it is sunny then Dr. Mounts will play golf
Dr. Mounts will not play golf
n ∴ It is not sunny
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No
If it is sunny then Dr. Mounts will play golf
It is not sunny
n ∴ Dr. Mounts will not play golf
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Conditional Schemas
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Human Performance
Affirming the Consequent
P→Q
Q
n∴P
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Modus Ponens -- affirming the antecedent
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Valid?
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Modus Tollens -- denying the consequent
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No
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Example
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If it is sunny then Dr. Mounts will play golf
n Dr. Mounts will play golf
n ∴ It is sunny
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People are good at
Will use spontaneously
People are not good at
Typically don’t use spontaneously
Affirming the consequent
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Examples
Often tempting to people
Will use -- even though invalid
Wason (1968)
(1968)--- 4 card problem
If it rains today, the game will be cancelled
n It rained today
n ∴The game was cancelled
n
If it rains today, the game will be cancelled
The game was not cancelled
n ∴It did not rain today
E
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If it rains today, the game will be cancelled
The game was cancelled
n ∴It rained today
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F
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If a card has a vowel on one side, it will have an
even number on the back
Which cards allow you to evaluate hypothesis?
E -- modus ponens (affirming the antecedent) [valid]
n 2 -- affirming the consequent [invalid]
n F -- denying the antecedent [invalid]
n 5 -- modus tollens (denying the consequent) [valid]
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Wason (1968) results
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Vowel (E) only -- 33%
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Vowel: Modus ponens
ponens--- valid
Vowel (E) + Even (2) -- 46%
Why so bad
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Confirmation bias
Matching
Content / Familiarity
Vowel: Modus ponens
ponens--- valid
n Even: affirming the consequent -- invalid
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Vowel (E) + Odd (5) -- 4% -- correct answer
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Vowel: Modus ponens -- valid
Odd: Modus tollens -- valid
Confirmation bias
n
People readily use modus ponens
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Look for evidence consistent with beliefs
Mistakenly affirm the consequent
People don't use modus tollens
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Matching
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People assume all information conveyed relevant
People don’t process all information fully
Surface structure of hypothesis
If Vowel then Even
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Looking for evidence inconsistent with beliefs
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Check Vowel
Check Even
Performance much better on:
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If a card has a vowel on one side, then it will not
have an even number on the other side.
Familiarity
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Familiar examples help reasoning
Aids in finding appropriate reasoning schema
Pragmatic reasoning schemas
n schema or outline for interacting with world
n different schemas for different situations
n fairly concrete "rules" to follow
n obligation schema
n permission schema
Familiarity Example
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Everyone drinking beer must be 21
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Who do you check?
Ned
Beverage:
Age:
Beer
??
Ted
Soda
??
Jed
??
22
Zed
??
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Hypothesis Testing
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If my hypothesis is correct, then I will obtain
specific pattern of results
Hypothesis → Results
Hypothesis
∴ Results
Modus Ponens -- Valid
Not applicable (Can’
(Can ’t know truth of hypothesis)
Hypothesis Testing
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Hypothesis → Results
Not Results
∴ Not Hypothesis
Affirming the consequent
Invalid
The Null Hypothesis
Hypothesis Testing
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Hypothesis → Results
Results
∴ Hypothesis
Null Hypothesis
Null Hypothesis → Null Results
Not Null Results
n ∴ Not Null Hypothesis
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Combined with:
Null Hypothesis OR Alternative Hypothesis
Not Null Hypothesis
n ∴ Alternative Hypothesis
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Modus Tollens -- Valid
Conclude hypothesis is wrong
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Why is this initially difficult
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Uses Modus Tollens
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