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
Translate the following English sentences into our formal
language using conjunction (the dot), negation (the tilde), or disjunction
(the wedge). Use the suggested constants to stand for the atomic
propositions.
- Either Bob will mop or Tom will mop. (B = Bob will mop; T = Tom will
mop)
- It is not sunny today. (S = it is sunny today)
- It is not the case that Bob is a burglar. (B = Bob is a burglar)
- Harry is arriving either tonight or tomorrow night. (A = Harry is
arriving tonight; B = Harry is arriving tomorrow night)
- Gareth does not like his name. (G = Gareth likes his name)
- Either it will not rain on Monday or it will not rain on Tuesday. (M = It
will rain on Monday; T = It will rain on Tuesday)
- Tom does not like cheesecake. (T = Tom likes cheesecake)
- Bob would like to have both a large cat and a small dog as a pet. (C =
Bob would like to have a large cat as a pet; D = Bob would like to
have a small dog as a pet)
- Bob Saget is not actually very funny. (B = Bob Saget is very funny)
- Albert Einstein did not believe in God. (A = Albert Einstein believed
in God)