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CS1502 Formal Methods in Computer Science

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Tautologies and Logical Truth. 2. Constructing a Truth Table. Write down sentence ... S is a tautology. Tautological. All rows in truth table. Consequence ... – PowerPoint PPT presentation

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Title: CS1502 Formal Methods in Computer Science


1
CS1502 Formal Methods in Computer Science
  • Lecture Notes 4
  • Tautologies and Logical Truth

2
Constructing a Truth Table
  • Write down sentence
  • Create the reference columns
  • Until you are done
  • Pick the next connective to work on
  • Identify the columns to consider
  • Fill in truth values in the column
  • EG (A (A v (B C))) v B
  • (in Boole and on board)

3
Tautology
  • A sentence S is a tautology if and only if every
    row of its truth table assigns true to S.

4
Example
  • Is ?(A ? (?A ? (B ? C))) ? B a tautology?

5
Example
6
Logical Possibility
  • A sentence S is logically possible if it could be
    true (i.e., it is true in some world)
  • It is TW-possible if it is true in some world
    that can be built using the program

7
Examples
  • Cube(b) ? Large(b)
  • ?(Tet(c) ? Cube(c) ? Dodec(c))
  • e ? e

Logically possible
TW-possible
Not TW-possible
Logically possible
Not Logically possible
8
Spurious Rows
  • A spurious row in a truth table is a row whose
    reference columns describe a situation or
    circumstance that is impossible to realize on
    logical grounds.

9
Example
Spurious!
Spurious!
10
Logical Necessity
  • A sentence S is a logical necessity (logical
    truth) if and only if S is true in every logical
    circumstance.
  • A sentence S is a logical necessity (logical
    truth) if and only if S is true in every
    non-spurious row of its truth table.

11
Example
Logical Necessity
TW-Necessity
Not a tautology
12
Example
Not a TW-Necessity
Not a Logical Necessity
Not a tautology
According to the book, the first row is spurious,
because a cannot be both larger and smaller than
b. Technically, though, Larger and Smaller
might mean any relation between objects. So, the
first row is really only TW-spurious. This
issue wont come up with any exam questions based
on this part of the book. (The book refines
this later.)
13
Tet(b) ? Cube(b) ? Dodec(b)
Tet(b) ??Tet(b)
aa
Cube(a) v Cube(b)
Cube(a) ? Small(a)
14
Tautological Equivalence
  • Two sentences S and S are tautologically
    equivalent if and only if every row of their
    joint truth table assigns the same values to S
    and S.

15
Example
S and S are Tautologically Equivalent
16
Logical Equivalence
  • Two sentences S and S are logically equivalent
    if and only if every non-spurious row of their
    joint truth table assigns the same values to S
    and S.

17
Example
S
S
Not Tautologically equivalent
Logically Equivalent
18
Tautological Consequence
  • Sentence Q is a tautological consequence of P1,
    P2, , Pn if and only if every row that assigns
    true to all of the premises also assigns true to
    Q.
  • Remind you of anything?
  • P1,P2,,Pn Q is also a valid argument!
  • A Con Rule Tautological Consequence

19
Example
premises
conclusion
Tautological consequence
20
Logical Consequence
  • Sentence Q is a logical consequence of P1, P2, ,
    Pn if and only if every non-spurious row that
    assigns true to all of the premises also assigns
    true to Q.

21
premise
conclusion
Not a tautological consequence
Is a logical consequence
22
Summary
23
Summary
Logical-Consequences of P1Pn
  • Every tautological consequence of a set of
    premises is a logical consequence of these
    premises.
  • Not every logical consequence of a set of
    premises is a tautological consequence of these
    premises.

Tautological- Consequences of P1Pn
24
Summary
  • Every tautological equivalence is a logical
    equivalence.
  • Not every logical equivalence is a tautological
    equivalence.

25
Summary
  • Every tautology is a logical necessity.
  • Not every logical necessity is a tautology.
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