Exercise 7.12 [propositional-validity-exercise]

Decide whether each of the following sentences is valid, unsatisfiable, or neither. Verify your decisions using truth tables or the equivalence rules of Table [logical-equivalence-table] (page logical-equivalence-table).

1. ${Smoke} {:\;{\Rightarrow}:\;}{Smoke}$

2. ${Smoke} {:\;{\Rightarrow}:\;}{Fire}$

3. $({Smoke} {:\;{\Rightarrow}:\;}{Fire}) {:\;{\Rightarrow}:\;}(\lnot {Smoke} {:\;{\Rightarrow}:\;}\lnot {Fire})$

4. ${Smoke} \lor {Fire} \lor \lnot {Fire}$

5. $(({Smoke} \land {Heat}) {:\;{\Rightarrow}:\;}{Fire}) {\;\;{\Leftrightarrow}\;\;}(({Smoke} {:\;{\Rightarrow}:\;}{Fire}) \lor ({Heat} {:\;{\Rightarrow}:\;}{Fire}))$

6. $({Smoke} {:\;{\Rightarrow}:\;}{Fire}) {:\;{\Rightarrow}:\;}(({Smoke} \land {Heat}) {:\;{\Rightarrow}:\;}{Fire})$

7. ${Big} \lor {Dumb} \lor ({Big} {:\;{\Rightarrow}:\;}{Dumb})$