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
For each of the following, write out what atomic proposition
each constant stands for. Then translate the sentences using the
constants you have defined. Finally, after you have translated the
sentence, identify which truth-functional connective is the main operator
of the sentence.
- Coral is not both a plant and an animal. (P, A)
- Although protozoa and chimpanzees are both eukaryotes, they are
not both animals. (There are four atomic propositions here; just use A,
B, C, and D for each different proposition).
- Neither chimpanzees nor protozoa are prokaryotes. (C, P)
- China has not signed the Kyoto Protocol and neither has the United
States. (C, U)
- Either Chevrolet or McDonald's will support the Olympic team, but
they won't both support it. (C, M)
- Peter Jennings is either a liar or has a really bad memory. (L, M)
- Peter Jennings is neither a liar nor has a really bad memory. (L, M)
- Peter Jennings is both a liar and has a really bad memory. (L, M)
- Peter Jennings is not both a liar and a person with a really bad
memory. (L, M)
- Chevrolet won't support the Olympic team this year, and McDonald's
won't either. (C, M)
- Mother Theresa may be a saint. Even so, she has not been canonized
yet by the Catholic Church. (S, C)
- The best distance runner of the last two decades is either Paul Tergat
or Haile Gebrselassie, but it certainly isn't Jim Ryun. (T, G, R)
- Jim Ryun was the best high school miler of all time, but he ran a
slower time than Alan Webb. (R, W)
- Neither Paul Tergat nor Haile Gebrselassie knows how to play hockey, but they both know how to play soccer. (A, B, C,
- Ethiopians are neither good bobsledders nor tennis players, but they are excellent distance runners. (B, T,
- Before Helen Keller met Annie Sullivan, she could neither speak, read,
nor communicate. (S, R, C)
- Although Helen Keller learned to communicate, she never learned to play soccer or baseball. (C, S, B)
- Tom is allowed to play football or soccer, but not both. (F, S)
- Tom will major in either engineering and physics, or business and
sociology. (E, P, B, S)
- Cartman is both xenophobic and racist, but he isn't a murderer or a thief. (X, R, M, T)