Reasoning for Humans: Clear Thinking in an Uncertain World PHIL 171 - - PowerPoint PPT Presentation

reasoning for humans clear thinking in an uncertain world
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Reasoning for Humans: Clear Thinking in an Uncertain World PHIL 171 - - PowerPoint PPT Presentation

Reasoning for Humans: Clear Thinking in an Uncertain World PHIL 171 Eric Pacuit Department of Philosophy University of Maryland pacuit.org Truth-Value Assignment A truth-value assignment specifies a unique truth-value (either T or F) for


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Reasoning for Humans: Clear Thinking in an Uncertain World

PHIL 171

Eric Pacuit

Department of Philosophy University of Maryland pacuit.org

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Truth-Value Assignment

A truth-value assignment specifies a unique truth-value (either T or F) for each atomic formula.

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Consider the formula (A → (A ∨ B)).

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Consider the formula (A → (A ∨ B)). The atomic subformulas are A and B

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Consider the formula (A → (A ∨ B)). The atomic subformulas are A and B There are 4 truth-value assignments for this formula:

  • 1. A is T, B is T
  • 2. A is T, B is F
  • 3. A is F, B is T
  • 4. A is F, B is F

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How many truth value assignments are there for a single atomic proposition A?

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How many truth value assignments are there for a single atomic proposition A? 2

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How many truth value assignments are there for a single atomic proposition A? 2 How many truth value assignments are there for two atomic propositions A and B?

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How many truth value assignments are there for a single atomic proposition A? 2 How many truth value assignments are there for two atomic propositions A and B? 4

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How many truth value assignments are there for a single atomic proposition A? 2 How many truth value assignments are there for two atomic propositions A and B? 4 How many truth value assignments are there for three atomic propositions A, B, and C?

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How many truth value assignments are there for a single atomic proposition A? 2 How many truth value assignments are there for two atomic propositions A and B? 4 How many truth value assignments are there for three atomic propositions A, B, and C? 8

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How many truth value assignments are there for a single atomic proposition A? 2 How many truth value assignments are there for two atomic propositions A and B? 4 How many truth value assignments are there for three atomic propositions A, B, and C? 8 How many truth value assignments are there for four atomic propositions A, B, C and D?

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How many truth value assignments are there for a single atomic proposition A? 2 How many truth value assignments are there for two atomic propositions A and B? 4 How many truth value assignments are there for three atomic propositions A, B, and C? 8 How many truth value assignments are there for four atomic propositions A, B, C and D? 16

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How many truth value assignments are there for a single atomic proposition A? 2 How many truth value assignments are there for two atomic propositions A and B? 4 How many truth value assignments are there for three atomic propositions A, B, and C? 8 How many truth value assignments are there for four atomic propositions A, B, C and D? 16 How many truth value assignments are there for n atomic propositions A1, A2, . . . , An?

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How many truth value assignments are there for a single atomic proposition A? 2 How many truth value assignments are there for two atomic propositions A and B? 4 How many truth value assignments are there for three atomic propositions A, B, and C? 8 How many truth value assignments are there for four atomic propositions A, B, C and D? 16 How many truth value assignments are there for n atomic propositions A1, A2, . . . , An? 2n

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Truth Assignments

Given a truth assignment for all the atomic propositions in ϕ, how do we determine the truth value of ϕ?

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Conjunction

Eric had steak and wine. (S ∧ W ) S W T T T F F T F F

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Conjunction

Eric had steak and wine. (S ∧ W ) S W T T T F F T F F

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Conjunction

Eric had steak and wine. (S ∧ W ) S W (S ∧ W ) T T T T F F F T F F F F

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Conjunction

Eric had steak and wine. (S ∧ W ) S W (S ∧ W ) T T T T F F F T F F F F

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Conjunction

Eric had steak and wine. (S ∧ W ) S W (S ∧ W ) T T T T F F F T F F F F

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Conjunction

Eric had steak and wine. (S ∧ W ) S W (S ∧ W ) T T T T F F F T F F F F

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Conjunction

Eric had steak and wine. (S ∧ W ) S W (S ∧ W ) T T T T F F F T F F F F

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Conjunction

Eric had steak and wine. (S ∧ W ) S W (S ∧ W ) T T T T F F F T F F F F

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Truth-Table for Conjunction

ϕ ψ (ϕ ∧ ψ) T T T T F F F T F F F F

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Disjunction

Eric had steak or wine. (S ∨ W ) S W (S ∨ W ) T T T T F T F T T F F F

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Disjunction

Eric had steak or wine. (S ∨ W ) S W (S ∨ W ) T T T T F T F T T F F F

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Disjunction

Eric had steak or wine. (S ∨ W ) S W (S ∨ W ) T T T T F T F T T F F F

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Disjunction

Eric had steak or wine. (S ∨ W ) S W (S ∨ W ) T T T T F T F T T F F F

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Disjunction

Eric had steak or wine. (S ∨ W ) S W (S ∨ W ) T T T T F T F T T F F F

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Truth-Table for Disjunction

ϕ ψ (ϕ ∨ ψ) T T T T F T F T T F F F

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Negation

Eric didn’t have steak. ¬S S ¬S T F F T

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Negation

Eric didn’t have steak. ¬S S ¬S T F F T

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Negation

Eric didn’t have steak. ¬S S ¬S T F F T

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Truth-Table for Negation

ϕ ¬ϕ T F F T

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P ∧ ¬(Q ∨ R) T

P ¬(Q ∨ R) P ∧ ¬(Q ∨ R) T T T T F F F T F F F F

P T ¬(Q ∨ R) T

Q ∨ R ¬(Q ∨ R) T F F T

Q ∨ R F

Q R Q ∨ R T T T T F T F T T F F F

Q F R F

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P ∧ ¬(Q ∨ R) T

P ¬(Q ∨ R) P ∧ ¬(Q ∨ R) T T T T F F F T F F F F

P T ¬(Q ∨ R) T

Q ∨ R ¬(Q ∨ R) T F F T

Q ∨ R F

Q R Q ∨ R T T T T F T F T T F F F

Q F R F

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P ∧ ¬(Q ∨ R) T

P ¬(Q ∨ R) P ∧ ¬(Q ∨ R) T T T T F F F T F F F F

P T ¬(Q ∨ R) T

Q ∨ R ¬(Q ∨ R) T F F T

Q ∨ R F

Q R Q ∨ R T T T T F T F T T F F F

Q F R F

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P ∧ ¬(Q ∨ R) T

P ¬(Q ∨ R) P ∧ ¬(Q ∨ R) T T T T F F F T F F F F

P T ¬(Q ∨ R) T

Q ∨ R ¬(Q ∨ R) T F F T

Q ∨ R F

Q R Q ∨ R T T T T F T F T T F F F

Q F R F

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P ∧ ¬(Q ∨ R) T

P ¬(Q ∨ R) P ∧ ¬(Q ∨ R) T T T T F F F T F F F F

P T ¬(Q ∨ R) T

Q ∨ R ¬(Q ∨ R) T F F T

Q ∨ R F

Q R Q ∨ R T T T T F T F T T F F F

Q F R F

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P ∧ ¬(Q ∨ R) T

P ¬(Q ∨ R) P ∧ ¬(Q ∨ R) T T T T F F F T F F F F

P T ¬(Q ∨ R) T

Q ∨ R ¬(Q ∨ R) T F F T

Q ∨ R F

Q R Q ∨ R T T T T F T F T T F F F

Q F R F

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Recap: Truth Tables

ϕ ψ (ϕ ∧ ψ) T T T T F F F T F F F F ϕ ψ (ϕ ∨ ψ) T T T T F T F T T F F F ϕ ¬ϕ T F F T

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Find truth tables for the formulas

  • P ∧ Q
  • ¬(P ∧ Q)
  • ¬P ∨ ¬Q
  • ¬P ∧ ¬Q

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P Q (P ∧ Q) ¬(P ∧ Q) (¬P ∨ ¬Q) (¬P ∧ ¬Q) T T T F F F T F F T T F F T F T T F F F F T T T (P ∧ Q) and ¬(P ∧ Q) are contradictory: they always have opposite truth values

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Material Conditional

If Eric had steak, then he had wine. (S → W ) S W (S → W ) T T T T F F F T T F F T

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Material Conditional

If Eric had steak, then he had wine. (S → W ) S W (S → W ) T T T T F F F T T F F T

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Material Conditional

If Eric had steak, then he had wine. (S → W ) S W (S → W ) T T T T F F F T T F F T

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Material Conditional

If Eric had steak, then he had wine. (S → W ) S W (S → W ) T T T T F F F T T F F T

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Material Conditional

If Eric had steak, then he had wine. (S → W ) S W (S → W ) T T T T F F F T T F F T

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Truth-Table for the Conditional

ϕ ψ (ϕ → ψ) T T T T F F F T T F F T

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Recap: Truth Tables

ϕ ψ (ϕ ∧ ψ) T T T T F F F T F F F F ϕ ψ (ϕ ∨ ψ) T T T T F T F T T F F F ϕ ψ (ϕ → ψ) T T T T F F F T T F F T ϕ ¬ϕ T F F T

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A truth table for a formula ϕ is a table, where each row is a truth assignment for the atomic propositions in ϕ and there is a column for ϕ (and possible subformulas of ϕ) list the truth values of ϕ for each truth assignment.

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