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Shortened Truth Tables

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We want to know the truth-value for a molecular sentence on some ... If the negation is false, then the negated sentence is true. Shortened Truth Tables ... – PowerPoint PPT presentation

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Title: Shortened Truth Tables


1
Shortened Truth Tables
  • Sometimes we are not required to construct a
    full truth-table.
  • This would happen in two sorts of cases
  • We want to know the truth-value for a molecular
    sentence on some specified truth-value assignment
    of the atomic components.
  • We know the truth value of the molecular
    sentence, and we want to work backwards to
    determine the truth-value assignment of the
    atomic components that would lead to that
    truth-value.

2
Shortened Truth Tables
  • E.g. of 1
  • Given that A T, B F and C T, what is the
    truth-value of (A v (B É C))

3
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F T

4
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F T F

5
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F T T F

6
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F T T T F

7
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F T F T T F
  • This table says that (A v (B É C)) is F when A
    T, B F and C T.

8
Shortened Truth Tables
  • E.g. of 2.
  • Give a truth-value assignment for the atomic
    components that make (A v (B É C)) true.

9
Shortened Truth Tables
  • A B C (A v (B É C))
  • T

10
Shortened Truth Tables
  • A B C (A v (B É C))
  • T
  • Now we must work backwards.

11
Shortened Truth Tables
  • A B C (A v (B É C))
  • T
  • Given that (A v (B É C)) is T, what is the
    value of (A v (B É C))?

12
Shortened Truth Tables
  • A B C (A v (B É C))
  • T ?

13
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F

14
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F
  • Remember that a disjunction is only false when
    both disjuncts are false.

15
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F F F

16
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F F ? F ?
  • The material conditional is only false when the
    antecedent (B) is true and the consequent (C) is
    false.

17
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F F T F F

18
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F F T F F ?
  • If the negation is false, then the negated
    sentence is true.

19
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F F T F F T

20
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F F T F F T
  • What does this table tells us?
  • From the table we can read off the values of each
    of the atomic components

21
Shortened Truth Tables
  • A B C (A v (B É C))
  • T F F T F F T
  • What does this table tells us?
  • From the table we can read off the values of each
    of the atomic components

22
Shortened Truth Tables
  • A B C (A v (B É C))
  • F T T T F F T F F T
  • (A v (B É C)) is T when A F, B T and C T

23
Examples
  • Given that A T, B T and C F, what is the
    value of
  • A É (B C)

24
Examples
  • On what truth-value assignment is the following
    sentence true?
  • (A B) (C É B)

25
Examples
  • On what truth-value assignment is the following
    sentence true?
  • (B v C) (B A)

26
Truth-Functional Truth
  • Defn A sentence P of SL is truth-functionally
    true iff P is true on every truth-value
    assignment.

27
Truth-Functional Falsity
  • Defn A sentence P of SL is truth-functionally
    false iff P is false on every truth-value
    assignment.

28
Truth-Functional Indeterminacy
  • Defn A sentence P of SL is truth-functionally
    indeterminate iff P is neither truth-functionally
    true nor truth-functionally false.

29
Example
  • Using a full-truth table, determine whether the
    following sentence is truth-functionally true,
    false or indeterminate.
  • A É (B É A)
  • (B v C) (B A)
  • (A B) (C v A)

30
Shortened T-Tables
  • We can also use shortened truth-tables to
    determine
  • whether a sentence is not truth-functionally true
  • whether a sentence is not truth-functionally
    false
  • whether a sentence is truth-functionally
    indeterminate

31
Example
  • Use a shortened truth-table to show that the
    following sentence is not truth-functionally
    true
  • (G É E) v (M N)

32
Examples
  • Use a shortened truth-table to show that the
    following sentence is not truth-functionally
    false
  • L É (M L)

33
Examples
  • Use a shortened truth-table to show that the
    following sentence is truth-functionally
    indeterminate
  • (A v (B A))

34
Truth-Functional Truth and Falsity
  • Shortened tables cannot be used, however, to
    determine whether a sentence is
    truth-functionally true or false.
  • Why?
  • By definition, a sentence is truth-functionally
    true (or false) iff it is true (or false) on all
    truth-value assignments.

35
Truth-Functional Equivalence
  • Defn Sentences P and Q of SL are
    truth-functionally equivalent iff P and Q have
    the same truth-value on all truth-value
    assignments.
  • To determine that a sentence is
    truth-functionally equivalent, we must construct
    a full truth-table.
  • We can use a shortened table to establish that
    two sentences are not truth-functionally
    equivalent.

36
Examples
  • Use a full truth-table to show that the following
    pair of sentences are truth-functionally
    equivalent.
  • (A B) É C A É (B É C)

37
Examples
  • Use a shortened truth-table to show that the
    following pair of sentences are not
    truth-functionally equivalent.
  • J É K K É J
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