Propositional Logic Functions
Read these four sections to learn how to identify and apply propositional (or sentential) logic functions. Using these symbols, you should be able to turn statements into symbolic formulas to more clearly see the logical connections taking place and determine when the conclusions are valid. It can look confusing at first, but moving slowly through these units will allow you to make valid logical proofs.
As you go, complete the exercises, then check your answers against the answer keys.
Note that the symbols used in some places can differ slightly from those used elsewhere. This is because there is not one standard set of symbols used for sentential logic, but a few. This table shows you the differences and helps translate between them.
In the resources in this course, the symbols for disjunction and negation are the same in both systems, but the symbols for conjunction, conditional, and biconditional are different.
| Name | Meaning | Symbol 1 | Symbol 2 |
| Conjunction | and | & | • |
| Disjunction | or | v | v |
| Negation | not | ~ | ~ |
| Conditional | if/then | → | ⊃ |
| Biconditional | if and only if | ↔ | ≡ |
Exercise
Answers
- a. P = Coral is a plant; A = Coral is an animal
- b. ~(P ⋅ A)
- c. Main operator is the negation
- a. A = Protozoa are eukaryotes; B = Chimpanzees are eukaryotes; C
- = Protozoa are animals; D = Chimpanzees are animals
- b. (A ⋅ B) ⋅ ~(C ⋅ D)
- c. Main operator is the second dot
- a. C = Chimpanzees are prokaryotes; P = Protozoa are prokaryotes
- b. ~(C v P)
- c. Main operator is the negation
- a. C = China has not signed the Kyoto Protocol; U = The United States has not signed the Kyoto Protocol
- b. ~(C v U)
- c. Main operator is the negation
- a. C = Chevrolet will support the Olympic team; M = McDonald’s will support the Olympic team
- b. (C v M) ⋅ ~(C ⋅ M)
- c. Main operator is the first dot
- a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad memory
- b. L v M
- c. Main (and only) operator is the wedge
- a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad memory
- b. ~(L v M)
- c. Main operator is the negation
- a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad memory
- b. L ⋅ M
- c. Main (and only) operator is the dot
- a. L = Peter Jennings is a liar; M = Peter Jennings has a really bad memory
- b. ~(L ⋅ M)
- c. Main operator is the negation
- a. C = Chevrolet will support the Olympic team this year; M = McDonald’s will support the Olympic team this year
- b. ~(C v M)
- c. Main operator is the negation
- a. S = Mother Theresa is a saint; C = Mother Theresa has been canonized by the Catholic Church
- b. S ⋅ ~C
- c. Main operator is the dot
- a. T = Paul Tergat was the best distance runner of the last two decades; G = Haile Gebrselassie was the best distance runner of the last two decades; R = Jim Ryun was the best distance runner of the last two decades
- b. (T v G) ⋅ ~R
- c. Main operator is the dot
- a. R = Jim Ryun was the best high school miler of all time; W = Jim Ryun ran a slower time than Alan Webb
- b. R ⋅ W
- c. Main (and only) operator is the dot
- a. A = Paul Tergat knows how to play hockey; B = Haile Gebrselassie knows how to play hockey; C = Paul Tergat knows how to play soccer; D = Haile Gebrselassie knows how to play soccer
- b. ~(A v B) ⋅ (C ⋅ D)
- c. Main operator is the first dot
- a. B = Ethiopians are good bobsledders; T = Ethiopians are good tennis players; D = Ethiopians are good distance runners
- b. ~(B v T) ⋅ D
- c. Main operator is the dot
- a. S = Before Helen Keller met Annie Sullivan she could speak; R = Before Helen Keller met Annie Sullivan she could read; C = Before Helen Keller met Annie Sullivan she could communicate
- b. (~S ⋅ ~R) ⋅ ~C
- c. Main operator is the second dot
- a. C = Helen Keller learned to communicate; S = Helen Keller learned to play soccer; B = Helen Keller learned to play baseball
- b. C ⋅ (~S ⋅ ~B)
- c. Main operator is the first dot
- a. F = Tom is allowed to play football; S = Tom is allowed to play soccer
- b. (F v S) ⋅ ~(F ⋅ S)
- c. Main operator is the first dot
- a. E = Tom will major in engineering; P = Tom will major in physics; B = Tom will major in business; S = Tom will major in sociology
- b. (E ⋅ P) v (B ⋅ S)
- c. Main operator is the wedge
- a. X = Cartman is xenophobic; R = Cartman is a racist; M = Cartman is a murderer; T = Cartman is the thief
- b. (X ⋅ R) ⋅ (~M ⋅ ~T)
- c. Main operator is the second dot