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 symbolic
language using any of the three truth functional operators (i.e.,
conjunction, negation, and disjunction). Use the constants at the end of
each sentence to represent the atomic propositions they are obviously
meant for. After you have translated the sentence, identify which truth-
functional connective is the main operator of the sentence. (Note: not
every sentence requires parentheses; a sentence requires parentheses
only if it contains more than two atomic propositions).
- Bob does not know how to fly an airplane or pilot a ship, but he does
know how to ride a motorcycle. (A, S, M)
- Tom does not know how to swim or how to ride a horse. (S, H)
- Theresa writes poems, not novels. (P, N)
- Bob does not like Sally or Felicia, but he does like Alice. (S, F, A)
- Cricket is not widely played in the United States, but both football and
baseball are. (C, F, B)
- Tom and Linda are friends, but Tom and Susan aren't - although Linda and Susan are. (T, S, L)
- Lansing is east of Grand Rapids but west of Detroit. (E, W)
- Either Tom or Linda brought David home after his surgery; but it
wasn't Steve. (T, L, S)
- Next year, Steve will be living in either Boulder or Flagstaff, but not
Phoenix or Denver. (B, F, P, D)
- Henry VII of England was married to Anne Boleyn and Jane Seymour,
but he only executed Anne Boleyn. (A, J, E)
- Henry VII of England executed either Anne Boleyn and Jane Boleyn or
Thomas Cromwell and Thomas More. (A, J, C, M)
- Children should be seen, but not heard. (S, H)