Well-formed Formulas (WFFs) of Propositional Logic
Propositional logic uses a symbolic “language” to represent the logical structure, or form, of a compound proposition. Like any language, this symbolic language has rules of syntax—grammatical rules for putting symbols together in the right way. Any expression that obeys the syntactic rules of propositional logic is called a well-formed formula, or WFF.
Fortunately, the syntax of propositional logic is easy to learn. It has only three rules:
- Any capital letter by itself is a WFF.
- Any WFF can be prefixed with “~”. (The result will be a WFF too.)
- Any two WFFs can be put together with “•”, “∨”, “⊃”, or “≡” between them, enclosing the result in parentheses. (This will be a WFF too.)
Some logic textbooks add a 4th rule: Parentheses may be omitted when doing so doesn’t result in any ambiguity. This convention makes some formulas slightly easier to read and write, but complicates the rules of syntax. To avoid unnecessary complications, I’ll adhere to the stricter convention of keeping all parentheses.
Here are some examples of well-formed formulas, along with brief explanations how these formulas are formed in accordance with the three rules of syntax:
|A||by rule 1|
|~A||by rule 2, since A is a WFF|
|~~A||by rule 2 again, since ~A is a WFF|
|(~A • B)||by rule 3, joining ~A and B|
|((~A • B) ⊃ ~~C)||by rule 3, joining (~A • B) and ~~C|
|~((~A • B) ⊃ ~~C)||by rule 2, since ((~A • B) ⊃ ~~C) is a WFF|
In contrast, here are a few formulas that are not
|A~||the ~ belongs on the left side of the negated proposition|
|(A)||parentheses are only introduced when joining two WFFs with •, ∨, ⊃, or ≡|
|(A • )||there’s no WFF on the right side of the •|
|(A • B) ⊃ C)||missing paranthesis on the left side|
|(A • ⊃ B)||cannot be formed by the rules of syntax|
The last connective introduced by rule 2 or 3 is called the main connective of the WFF. The main connective represents the logical structure of the compound proposition as a whole. For example, if the main connective is a “~”, the proposition as a whole is a negation. If the main connective is a “•”, the proposition is a conjunction, and so on.
Propositions joined by the main connective are called its components or component propositions. Components may themselves be compound propositions, made up of simpler components. Some components have special names. In a conjunction, the components joined by the “•” (dot) are called its conjuncts. In a disjunction, the propositions joined by the “∨” (wedge) are called disjuncts. In a conditional, the component to the left of the “⊃” (horseshoe) is called the antecedent and the component to the right is called the consequent.
The proposition (A • ~B)
is a conjunction
because its main connective is the dot. The propositions A and ~B are its conjuncts
The proposition (~A ∨ (B ≡ C)) is a disjunction because its main connective is the wedge. The propositions ~A and (B ≡ C) are its disjuncts.
The proposition ((A ∨ B) ⊃ (C • D)) is a conditional because its main connective is the horseshoe. The proposition (A ∨ B) is its antecedent and (C • D) is its consequent.