CS1502 Formal Methods in Computer Science

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CS1502 Formal Methods in Computer Science

• Lecture Notes 4
• Tautologies and Logical Truth

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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)

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Tautology

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

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Example

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

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Example
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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

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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
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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.

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Example
Spurious!
Spurious!
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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.

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Example
Logical Necessity
TW-Necessity
Not a tautology
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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.)
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Tet(b) ? Cube(b) ? Dodec(b)
Tet(b) ??Tet(b)
aa
Cube(a) v Cube(b)
Cube(a) ? Small(a)
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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.

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Example
S and S are Tautologically Equivalent
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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.

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Example
S
S
Not Tautologically equivalent
Logically Equivalent
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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

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Example
premises
conclusion
Tautological consequence
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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.

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premise
conclusion
Not a tautological consequence
Is a logical consequence
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Summary
Necessary S is always true Possible S could be true Equivalence S and S always have the same truth values Consequence Whenever P1Pn are true, Q is also true
Tautological All rows in truth table S is a tautology S is Tautologically possible S and S are Tautologically equivalent Q is a tautological consequence of P1Pn
Logical All non-spurious rows S is logically necessary (logical truth) S is logically possible S and S are logically equivalent Q is a logical consequence of P1Pn
TW Logic Tarskis World S is TW necessary S is TW possible S and S are TW equivalent Q is a TW-consequence of P1Pn
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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
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Summary

• Every tautological equivalence is a logical
  equivalence.
• Not every logical equivalence is a tautological
  equivalence.

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Summary

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