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 ⋅ ~S) ⋅ M (The main operator is the second dot - in this case it doesn't actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator).
- ~S ⋅ ~H (The main operator is the conjunction).
- P ⋅ ~N (The main operator is the conjunction).
- (~S ⋅ ~F) ⋅ A (The main operator is the second dot - in this case it doesn't actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator).
- ~C ⋅ (F ⋅ B) (The main operator is the first dot - in this case it doesn't actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator).
- (T ⋅ ~S) ⋅ L (The main operator is the second dot - in this case it doesn't actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator).
- E ⋅ W (There is only one truth functional operator, the conjunction. So that is by default the main operator!)
- (T v L) ⋅ ~S (The main operator is the conjunction).
- (B v F) ⋅ ~(P v D) (The main operator is the conjunction).
- 10.(A ⋅ J) ⋅ E (The main operator is the second dot - in this case it doesn't actually matter which dot since the sentence has the same meaning whichever of the conjuncts you treat as the main operator).
- (A ⋅ J) v (C ⋅ M) (The wedge is the main operator).
- S ⋅ ~H (The main operator is the conjunction).