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 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.