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
Identify which of the following sentences are truth-functional
conjunctions. If the sentence is a truth-functional conjunction, identify the
two conjuncts (by writing them down).
- Jack and Jill are coworkers.
- Tom is a fireman and a father.
- Ringo Starr and John Lennon were bandmates.
- Lucy loves steak and onion sandwiches.
- Cameron Dias has had several relationships, although she has never
married.
- Bob and Sally kissed.
- A person who plays both mandolin and guitar is a multi-
instrumentalist.
- No one has ever contracted rabies and lived.
- Jack and Jill are cowboys.
- Josiah is Amish; nevertheless, he is also a drug dealer.
- The Tigers are the best baseball team in the state, but they are not as
good as the Yankees.
- Bob went to the beach to enjoy some rest and relaxation.
- Lauren isn't the fastest runner on the team; still, she is fast enough to have made it to the national championship.
- The ring is beautiful, but expensive.
- It is sad, but true that many Americans do not know where their next meal will come from.