A well-formed formula (wff) is an expression made up of variables, parentheses, and connective symbols, which include negation, conjunction, disjunction, and implication. Atomic statements are simple statements without connectives, while molecular statements consist of one or more atomic statements, with specific rules that define valid wff structures. Parentheses must be used correctly in composite statements, as improper placement or ambiguity can lead to invalid formulas.

1. Well-Formed Formula(WFF)
BY
Ms. Komal V Rokade.

2. Well-Formed Formula(WFF)
Well-Formed Formula(WFF) is an expression
consisting of variables(capitalletters),parentheses,
and connective symbols. An expression is basically
a combination of operands & operators and here
operands and operators are the connective
symbols.

3. Below are the possible Connective Symbols:
• ¬ (Negation)
• ∧ (Conjunction)
• ∨ (Disjunction)
• ⇒ (Rightwards Arrow)
• ⇔ (Left-Right Arrow)

4. Statement Formulas
1. Statements that do not contain any connectives
are called Atomic or Simple statements and these
statements in themselves are WFFs.
• For example,
• P, Q, R, etc.
2. Statements that contain one or more primary
statements are
called Molecular or Composite statements.
• For example,

5. 3. If P and Q are two simple statements, then some of the Composite
statements which follow WFF standards can be formed are:
• -> ¬P
• -> ¬Q
• -> (P ∨ Q)
• -> (P ∧ Q)
• -> (¬P ∨ Q)
• -> ((P ∨ Q) ∧ Q)
• -> (P ⇒ Q)
• -> (P ⇔ Q)
• -> ¬(P ∨ Q)
• -> ¬(¬P ∨ ¬Q)

6. Rules of the Well-Formed Formulas
1 .A Statement variable standing alone is a Well-
Formed Formula(WFF).
For example– Statements like P, ∼P, Q, ∼Q are
themselves Well Formed Formulas.
2. If ‘P’ is a WFF then ∼P is a formula as well.
3. If P & Q are WFFs, then (P∨Q), (P∧Q), (P⇒Q),
(P⇔Q), etc. are also WFFs.

7. Example Of Well Formed Formulas:

8. Examples which may seem like a WFF
but they are not WFF
• (P), ‘P’ itself alone is considered as a WFF by
Rule 1 but placing that inside parenthesis is
not considered as a WFF by any rule.
• ¬P ∧ Q, this can be either (¬P∧Q) or ¬(P∧Q) so
we have ambiguity in this statement and
hence it will not be considered as a WFF.
Parentheses are mandatory to be included in
Composite Statements.

9. • ((P ⇒ Q)), We can say (P⇒Q) is a WFF and let
(P⇒Q) = A, now considering the outer
parentheses, we will be left with (A), which is not
a valid WFF. Parentheses play a really important
role in these types of questions.
• (P ⇒⇒ Q), connective symbol right after a
connective symbol is not considered to be valid
for a WFF.
• ((P ∧ Q) ∧)Q), conjunction operator after (P∧Q) is
not valid.

10. • ((P ∧ Q) ∧ PQ), invalid placement of
variables(PQ).
• (P ∨ Q) ⇒ (∧ Q), with the Conjunction
component, only one variable ‘Q’ is present. In
order to form an operation inside a
parentheses minimum of 2 variables are
required.