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CS1502 Formal Methods in Computer Science

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

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CS1502 Formal Methods in Computer Science

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  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 Logically possible Not TW-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. TW-Necessity Logical-Necessity Logical Necessity • A sentence S is a logical necessity(logicaltruth) if and only if S is true in every logical circumstance. • A sentence S is a logical necessity(logicaltruth) if and only if S is true in every non-spurious row of its truth table.

  11. Example Not a tautology Logical Necessity TW-Necessity

  12. Example Not a tautology Not a Logical Necessity Not a TW-Necessity 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 won’t 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) a=a Cube(a) v Cube(b) Cube(a)  Small(a)

  14. Tautological Equivalence • Two sentences S and S’ aretautologically equivalentif and only if every row of their joint truth table assigns the same values to S and S’.

  15. S S’ Example S and S’ are Tautologically Equivalent

  16. Logical Equivalence • Two sentences S and S’ are logically equivalentif 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 |Qis 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-spuriousrow 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 P1…Pn • 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 P1…Pn

  24. Logical Equivalences Tautological Equivalences Summary • Every tautological equivalence is a logical equivalence. • Not every logical equivalence is a tautological equivalence.

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

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