Reasoning and Meta-reasoning Sonia Marin IT-University of - - PowerPoint PPT Presentation

Reasoning and “Meta”-reasoning

Sonia Marin IT-University of Copenhagen, Denmark

85-211 Cognitive Psychology, Guest lecture

CMU Qatar November 28, 2018

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What is reasoning?

Reasoning is the process of “drawing” conclusions from principles and from evidence

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What is reasoning?

Reasoning is the process of “drawing” conclusions from principles and from evidence

• deduce new conclusion → top-down
• evaluate proposed conclusion → bottom-up

2 / 15

What is reasoning?

Reasoning is the process of “drawing” conclusions from principles and from evidence

• deduce new conclusion → top-down
• evaluate proposed conclusion → bottom-up

Two main forms of reasoning:

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What is reasoning?

Reasoning is the process of “drawing” conclusions from principles and from evidence

• deduce new conclusion → top-down
• evaluate proposed conclusion → bottom-up

Two main forms of reasoning:

• 1. deductive: from the general to the specific = no new information

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What is reasoning?

Reasoning is the process of “drawing” conclusions from principles and from evidence

• deduce new conclusion → top-down
• evaluate proposed conclusion → bottom-up

Two main forms of reasoning:

• 1. deductive: from the general to the specific = no new information
• 2. inductive: from the specific to the general = new information

2 / 15

What is reasoning?

Reasoning is the process of “drawing” conclusions from principles and from evidence

• deduce new conclusion → top-down
• evaluate proposed conclusion → bottom-up

Two main forms of reasoning:

• 1. deductive: from the general to the specific = no new information
• 2. inductive: from the specific to the general = new information

When and where do you use each type?

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What is reasoning?

Reasoning is the process of “drawing” conclusions from principles and from evidence

• deduce new conclusion → top-down
• evaluate proposed conclusion → bottom-up

Two main forms of reasoning:

• 1. deductive: from the general to the specific = no new information

◮ e.g. mathematics

• 2. inductive: from the specific to the general = new information

When and where do you use each type?

2 / 15

What is reasoning?

Reasoning is the process of “drawing” conclusions from principles and from evidence

• deduce new conclusion → top-down
• evaluate proposed conclusion → bottom-up

Two main forms of reasoning:

• 1. deductive: from the general to the specific = no new information

◮ e.g. mathematics

• 2. inductive: from the specific to the general = new information

◮ e.g. experimental sciences

When and where do you use each type?

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Deductive reasoning

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Propositional logic

Syntax: p : it is raining ✜

atomic propositions

¯ p : it is not raining ✣ s : she is sad ¯ s : she is not sad Semantics: Observed world

¯ p : it is not raining ✣ s : she is sad

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Propositional logic

Syntax: p : it is raining ✜

atomic propositions

¯ p : it is not raining ✣ s : she is sad ¯ s : she is not sad Semantics: Observed world

V (p) = 0 V (s) = 1

¯ p : it is not raining ✣ s : she is sad

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Propositional logic

Syntax: p : it is raining ✜

atomic propositions

¯ p : it is not raining ✣ s : she is sad ¯ s : she is not sad p ∨ s : it is raining or she is sad Semantics: Observed world

V (p) = 0 V (s) = 1 V (p ∨ s) = 1

¯ p : it is not raining ✣ s : she is sad

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Propositional logic

Syntax: p : it is raining ✜

atomic propositions

¯ p : it is not raining ✣ s : she is sad ¯ s : she is not sad p ∨ s : it is raining or she is sad p ∧ s : it is raining and she is sad Semantics: Observed world

V (p) = 0 V (s) = 1 V (p ∨ s) = 1 V (p ∧ s) = 0

¯ p : it is not raining ✣ s : she is sad

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Propositional logic

Syntax: p : it is raining ✜

atomic propositions

¯ p : it is not raining ✣ s : she is sad ¯ s : she is not sad p ∨ s : it is raining or she is sad p ∧ s : it is raining and she is sad p ⊃ s : if it is raining then she is sad Semantics: Observed world

V (p) = 0 V (s) = 1 V (p ∨ s) = 1 V (p ∧ s) = 0 V (p ⊃ s) = ?

¯ p : it is not raining ✣ s : she is sad

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Wason selection task

https://www.youtube.com/watch?v=qNBzwwLiOUc

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Deduction and fallacies

p ⊃ s p

− − − − − − − − −

s p ⊃ s ¯ s

− − − − − − − − −

¯ p p ⊃ s ¯ p

− − − − − − − − −

¯ s p ⊃ s s

− − − − − − − − −

p

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Deduction and fallacies

p ⊃ s p

− − − − − − − − −

s p ⊃ s ¯ s

− − − − − − − − −

¯ p Modus ponens p ⊃ s ¯ p

− − − − − − − − −

¯ s p ⊃ s s

− − − − − − − − −

p

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Deduction and fallacies

p ⊃ s p

− − − − − − − − −

s p ⊃ s ¯ s

− − − − − − − − −

¯ p Modus ponens p ⊃ s ¯ p

− − − − − − − − −

¯ s p ⊃ s s

− − − − − − − − −

p Denying the antecedent ×

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Deduction and fallacies

p ⊃ s p

− − − − − − − − −

s p ⊃ s ¯ s

− − − − − − − − −

¯ p Modus ponens Modus tollens p ⊃ s ¯ p

− − − − − − − − −

¯ s p ⊃ s s

− − − − − − − − −

p Denying the antecedent ×

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Deduction and fallacies

p ⊃ s p

− − − − − − − − −

s p ⊃ s ¯ s

− − − − − − − − −

¯ p Modus ponens Modus tollens p ⊃ s ¯ p

− − − − − − − − −

¯ s p ⊃ s s

− − − − − − − − −

p Denying the antecedent × Affirming the consequent ×

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Deduction and fallacies

p ⊃ s p

− − − − − − − − −

s p ⊃ s ¯ s

− − − − − − − − −

¯ p Inference rules Modus ponens Modus tollens p ⊃ s ¯ p

− − − − − − − − −

¯ s p ⊃ s s

− − − − − − − − −

p Denying the antecedent × Affirming the consequent ×

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Deduction and fallacies

p ⊃ s p

− − − − − − − − −

s p ⊃ s ¯ s

− − − − − − − − −

¯ p Inference rules Modus ponens Modus tollens p ⊃ s ¯ p

− − − − − − − − −

¯ s p ⊃ s s

− − − − − − − − −

p Logical fallacies Denying the antecedent × Affirming the consequent ×

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Theory of reasoning

Dual-process theory: Two systems in one brain

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Theory of reasoning

Dual-process theory: Two systems in one brain

• System 1: implicit, automatic, unconscious
• System 2: explicit, controlled, conscious

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Theory of reasoning

Dual-process theory: Two systems in one brain

• System 1: implicit, automatic, unconscious
• System 2: explicit, controlled, conscious

Discussion:

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Theory of reasoning

Dual-process theory: Two systems in one brain

• System 1: implicit, automatic, unconscious
• System 2: explicit, controlled, conscious

Discussion: In which contexts are you using either of these systems? Separately or in parallel?

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Theory of reasoning

Dual-process theory: Two systems in one brain

• System 1: implicit, automatic, unconscious
• System 2: explicit, controlled, conscious

Discussion: In which contexts are you using either of these systems? Separately or in parallel? Some cognitive psychologists question the merits of studying logical formalisms.

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Theory of reasoning

Dual-process theory: Two systems in one brain

• System 1: implicit, automatic, unconscious
• System 2: explicit, controlled, conscious

Discussion: In which contexts are you using either of these systems? Separately or in parallel? Some cognitive psychologists question the merits of studying logical formalisms. What do you think can be gained by studying how people reason wrt. logical rules? Would it seem more “scientific” to study intuitive reasoning?

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What about real life?

“Pure” logic is a structural description of what a valid statement is but...

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What about real life?

“Pure” logic is a structural description of what a valid statement is but... for the analysis of daily language and arguments, it lacks certain operators.

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What about real life?

“Pure” logic is a structural description of what a valid statement is but... for the analysis of daily language and arguments, it lacks certain operators. There are many sentences that you cannot express in classic logic

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What about real life?

“Pure” logic is a structural description of what a valid statement is but... for the analysis of daily language and arguments, it lacks certain operators. There are many sentences that you cannot express in classic logic but can be expressed in modal logic.

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What about real life?

“Pure” logic is a structural description of what a valid statement is but... for the analysis of daily language and arguments, it lacks certain operators. There are many sentences that you cannot express in classic logic but can be expressed in modal logic. Example: ”I may get burned if I lie in the sun for too long”.

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What about real life?

“Pure” logic is a structural description of what a valid statement is but... for the analysis of daily language and arguments, it lacks certain operators. There are many sentences that you cannot express in classic logic but can be expressed in modal logic. Example: ”I may get burned if I lie in the sun for too long”. In classic logic, you can say: ”I get burned if I lie in the sun for too long”

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What about real life?

“Pure” logic is a structural description of what a valid statement is but... for the analysis of daily language and arguments, it lacks certain operators. There are many sentences that you cannot express in classic logic but can be expressed in modal logic. Example: ”I may get burned if I lie in the sun for too long”. In classic logic, you can say: ”I get burned if I lie in the sun for too long” but not express the possibility of maybe getting burned.

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What about real life?

“Pure” logic is a structural description of what a valid statement is but... for the analysis of daily language and arguments, it lacks certain operators. There are many sentences that you cannot express in classic logic but can be expressed in modal logic. Example: ”I may get burned if I lie in the sun for too long”. In classic logic, you can say: ”I get burned if I lie in the sun for too long” but not express the possibility of maybe getting burned. Modal logic is an extension of classic propositional logic with modal operators

8 / 15

What about real life?

“Pure” logic is a structural description of what a valid statement is but... for the analysis of daily language and arguments, it lacks certain operators. There are many sentences that you cannot express in classic logic but can be expressed in modal logic. Example: ”I may get burned if I lie in the sun for too long”. In classic logic, you can say: ”I get burned if I lie in the sun for too long” but not express the possibility of maybe getting burned. Modal logic is an extension of classic propositional logic with modal operators

• riginally expressing possibility and necessity of a proposition.

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“Meta”-reasoning

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Modal logic

Propositional logic: p : it is raining ✜

atomic propositions

¯ p : it is not raining ✣ s : she is sad ¯ s : she is not sad p ⊃ s : if it is raining then she is sad p ∧ s : it is raining and she is sad p ∨ s : it is raining or she is sad

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Modal logic

Propositional logic: p : it is raining ✜

atomic propositions

¯ p : it is not raining ✣ s : she is sad ¯ s : she is not sad p ⊃ s : if it is raining then she is sad p ∧ s : it is raining and she is sad p ∨ s : it is raining or she is sad Modal logic:

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Modal logic

Propositional logic: p : it is raining ✜

atomic propositions

¯ p : it is not raining ✣ s : she is sad ¯ s : she is not sad p ⊃ s : if it is raining then she is sad p ∧ s : it is raining and she is sad p ∨ s : it is raining or she is sad Modal logic:

◇p

: it is possible that it is raining

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Modal logic

Propositional logic: p : it is raining ✜

atomic propositions

¯ p : it is not raining ✣ s : she is sad ¯ s : she is not sad p ⊃ s : if it is raining then she is sad p ∧ s : it is raining and she is sad p ∨ s : it is raining or she is sad Modal logic:

◇p

: it is possible that it is raining ◻s : it is necessary that she is sad

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Possible world semantics

✣ ✜ ✜

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Possible world semantics

Observed world

¯ p : it is not raining ✣ s : she is sad

✜ ✜

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Possible world semantics

Observed world

V (p) = 0 V (s) = 1 V (p ∧ s) = 0 V (p ∨ s) = 1

¯ p : it is not raining ✣ s : she is sad

✜ ✜

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Possible world semantics

Observed world

V (p) = 0 V (s) = 1 V (p ∧ s) = 0 V (p ∨ s) = 1 V (◻p) = V (◇s) =

¯ p : it is not raining ✣ s : she is sad

✜ ✜

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Possible world semantics

Observed world

V (p) = 0 V (s) = 1 V (p ∧ s) = 0 V (p ∨ s) = 1 V (◻p) = V (◇s) =

¯ p : it is not raining ✣ s : she is sad

Possible worlds

p : it is raining ✜ ¯ s : she is not sad p : it is raining ✜ s : she is sad

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Possible world semantics

◻p is true as for all possible worlds p is true Observed world

V (p) = 0 V (s) = 1 V (p ∧ s) = 0 V (p ∨ s) = 1 V (◻p) = 1 V (◇s) =

¯ p : it is not raining ✣ s : she is sad

Possible worlds

V (p) = 1 V (p) = 1

p : it is raining ✜ ¯ s : she is not sad p : it is raining ✜ s : she is sad

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Possible world semantics

◻p is true as for all possible worlds p is true

◇s is true as there exists a possible world such that s is true

Observed world

V (p) = 0 V (s) = 1 V (p ∧ s) = 0 V (p ∨ s) = 1 V (◻p) = 1 V (◇s) = 1

¯ p : it is not raining ✣ s : she is sad

Possible worlds

V (p) = 1 V (s) = 0 V (p) = 1 V (s) = 1

p : it is raining ✜ ¯ s : she is not sad p : it is raining ✜ s : she is sad

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Epistemic reasoning

The term modal logic also includes other extensions, for instance:

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Epistemic reasoning

The term modal logic also includes other extensions, for instance:

• temporal logic for the expression of past or future truths;

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Epistemic reasoning

The term modal logic also includes other extensions, for instance:

• temporal logic for the expression of past or future truths;
• deontic logic for the expression of obligations;

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Epistemic reasoning

The term modal logic also includes other extensions, for instance:

• temporal logic for the expression of past or future truths;
• deontic logic for the expression of obligations;
• epistemic logic for the expression of cognitive truth like belief and knowledge.

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Muddy children puzzle

http://sierra.nmsu.edu/morandi/coursematerials/MuddyChildren.html

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Deduction in modal logic

How do we reason about such structure?

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Deduction in modal logic

How do we reason about such structure? Inference rules: A

∨1 −

− − − − −

A ∨ B B

∨2 −

− − − − −

A ∨ B A B

∧ −

− − − − − − −

A ∧ B A ⇒ B

⊃ −

− − − − − − − − − −

A ⊃ B

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Deduction in modal logic

How do we reason about such structure? Inference rules: A

∨1 −

− − − − −

A ∨ B B

∨2 −

− − − − −

A ∨ B A B

∧ −

− − − − − − −

A ∧ B A ⇒ B

⊃ −

− − − − − − − − − −

A ⊃ B ? ◻ −

− −

◻A ?

◇ −

− −

◇A

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Questions?

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