The Main Operator
Every wff has one main operator. Any wff of PL is the result of a finite applications of the definition of a wff. We saw that you may apply the definition as many (finite) times as you like in order to form wffs. For any wff formed, the last application of the definition determines the main operator.
To see this, let's form a few wffs.
| wffs | clause of definition |
| A | 1 |
| B | 1 |
| (A & B) | 4 |
| ((A & B) v C) | 3 |
| ~((A & B) v C) | 2 |
We formed the wff ~((A & B) v C) by successive applications of the definition of a wff. In the final step, we formed the wff by using clause 2 and placing a tilde in front of ((A & B) v C). Thus the tilde is the main operator. We call a wff with a tilde as the main operator a negation.
Another example:
| wffs | clause of definition |
| C | 1 |
| D | 1 |
| ~C | 2 |
| ~D | 2 |
| (~C ⊃ ~D) | 5 |
The last step invokes clause 5 which means that the main operator is the horseshoe. We call a wff whose main operator is the horseshoe a conditional. We call the left half of a conditional the antecendent and the right half the consequent.
A conjunction is a wff whose main operator is the ampersand. Its parts are called conjuncts.
A disjunction is a wff whose main operator is the wedge. Its parts are called disjuncts.
A biconditional is a wff whose main operator is the triple-bar. Its parts are called, clumsily, the parts of the biconditional.
You can completely describe a wff in these terms. Here are some wffs and
their descriptions:
| wff | description |
| ~(A ⊃ B) | A negation of a conditional whose parts are simple propositions. |
| (A v B) v (C ⊃ D) | A disjunction whose left disjunct is a disjunction of simple propositions and whose right disjunct is a conditional of simple propositions. |
| (~A & ~B) ⊃ (A ≡ B) | A conditional whose antecedent is a conjunction of negated simple propositions and whose consequent is a biconditional of simple propositions. |
Note that each of these wffs is described first in terms of its main connective. The first is a negation, the second a disjunction, and the third a conditional.
There are two exercises for this section. The first will have you locate the main operator of a wff. The second will ask you to describe those same wffs using the terms introduced above.
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| table of contents | List of Exercises |