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 possibleTW-possible Not TW-possibleLogically 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!

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. Logical-Necessity TW-Necessity

11 Example Logical NecessityTW-NecessityNot a tautology

12 Example Not a TW-NecessityNot a Logical NecessityNot 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 won’t come up with any exam questions based on this part of the book. (The book refines this later.)

13 Tet(b)   Tet(b) a=a Tet(b)  Cube(b)  Dodec(b) Cube(a)  Small(a) Cube(a) v Cube(b)

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 SS’

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 Not Tautologically equivalentLogically Equivalent SS’

18 Tautological Consequence Sentence Q is a tautological consequence of P 1, P 2, …, P n 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 Tautological consequence premisesconclusion

20 Logical Consequence Sentence Q is a logical consequence of P 1, P 2, …, P n if and only if every non- spurious row that assigns true to all of the premises also assigns true to Q.

21 Not a tautological consequence Is a logical consequence premiseconclusion

22 Summary Necessary S is always true Possible S could be true Equivalence S and S’ always have the same truth values Consequence Whenever P1…Pn 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 P1…Pn 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 P1…Pn TW Logic + Tarski’s World S is TW necessary S is TW possible S and S’ are TW equivalent Q is a TW- consequence of P1…Pn

23 Summary 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 Logical-Consequences of P1…Pn

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

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