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