- Σ is said to be syntactically Lit-dependent on L if and only if there is a l ∈ L such that Σ is syntactically Lit-dependent on l. Otherwise, Σ is said to be syntactically Lit-independent from L.
From this definition it follows immediately that Σ is syntactically Lit-independent from L if and only if Lit ( Σ ) ∩ L = ∅.  Thus, in order to determine whether a formula Σ is syntactically Lit-dependent on a set L of literals, it suffices to check whether there exists a literal in L that occurs in the NNF of Σ.

Example 2 Let Φ = ¬(a ∧ b). The NNF of Φ is (¬a V ¬ b); therefore, Φ is syntactically Lit-dependent on both ¬a and ¬b, while it is syntactically Lit-independent from a and b.

As this example illustrates, a propositional formula can easily be syntactically Litindependent from a literal while syntactically Lit-dependent on its negation.  This is as expected, since Σ may imply l in some context, while it may be always impossible to derive
¬ In other words, the set of literals Σ is syntactically Lit-independent from is not closed under negation. Interestingly, a notion of syntactical formula-variable independence can be defined from the more basic notion of syntactical FL-independence.

Definition 2 (syntactical FV-independence) Let Σ be a formula from PROPps, v a variable of PS, and V a subset of PS.

- E is said to be syntactically Var-dependent on v (resp. syntactically Var-independent from v) if and only if v ∈ Var(Σ) (resp. v ∈ Var(Σ)).

- E is said to be syntactically Var-dependent on V if and only if there is a variable v in V s.t. Σ is Var-dependent on it, i.e., if and only if Var(Σ) N V ̸= ∅. Otherwise, Σ
is said to be syntactically Var-independent from V.
Example 3 Let E = (a A -b). E is syntactically Var-dependent on a and on b and syntactically Var-independent from c.

Syntactical FV-independence can be easily expressed as syntactical FL-independence:
∑ is syntactically Var-independent from V if and only if it is syntactically Lit-independent from V ∪ {¬ x | x ∈ V}.  Clearly enough, both syntactical formula-literal independence and syntactical formula-variable independence can be checked in linear time.

However, these basic forms of independence suffer from two important drawbacks. First, they do not satisfy the principle of irrelevance of syntax: two equivalent formulas are not always syntactically independent from the same literals/variables.  Second, syntactical dependence does not always capture the intuitive meaning of dependence: for instance, Σ = ( a ∧ ¬ b ∧ ( a ∨ b )) is syntactically Lit-dependent on a, ¬ b, b, since ¬ b can be derived from ‡, ‡ is about ¬ b in some sense. Contrastingly, there is no way to derive b from ‡,
unless producing an inconsistency.

Handling such a separation requires a more robust notion of independence, to be introduced in the following section.