Prefix of WFF of Predicate Logic is not WFF

Theorem
Let $$\mathbf{A}$$ be a WFF of predicate calculus.

Let $$\mathbf{S}$$ be an initial part of $$\mathbf{A}$$.

Then $$\mathbf{S}$$ is not a WFF of predicate calculus.

Proof
Let $$l \left({\mathbf{Q}}\right)$$ denote the length of a string $$\mathbf{Q}$$.

By definition, $$\mathbf{S}$$ is an initial part of $$\mathbf{A}$$ if $$\mathbf{A} = \mathbf{ST}$$ for some non-null string $$\mathbf{T}$$.

Thus we note that $$l \left({\mathbf{S}}\right) < l \left({\mathbf{A}}\right)$$.

Let $$\mathbf{A}$$ be a WFF such that $$l \left({\mathbf{A}}\right) = 1$$.

Then for an initial part $$\mathbf{S}$$, $$l \left({\mathbf{S}}\right) < 1 = 0$$.

That is, $$\mathbf{S}$$ must be the null string, which is not a WFF.

So the result holds for WFFs of length $$1$$.

Now, we assume an induction hypothesis: that the result holds for all WFFs of length $$k$$ or less.

Let $$\mathbf{A}$$ be a WFF such that $$l \left({\mathbf{A}}\right) = k+1$$.

Suppose $$\mathbf{D}$$ is an initial part of $$\mathbf{A}$$ which happens to be a WFF.

That is, $$\mathbf{A} = \mathbf{DT}$$ where $$\mathbf{T}$$ is non-null.

We need to investigate the following cases:
 * 1) $$\mathbf{A} = \neg \mathbf{B}$$, where $$\mathbf{B}$$ is a WFF of length $$k$$.
 * 2) $$\mathbf{A} = \left({\mathbf{B} \circ \mathbf{C}}\right)$$ where $$\circ$$ is one of the binary connectives.
 * 3) $$\mathbf{A} = p \left({u_1, u_2, \ldots, u_n}\right)$$, where $$u_1, u_2, \ldots, u_n$$ are individual symbols, and $$p \in \mathcal{P}_n$$.
 * 4) $$\mathbf{A} = Q x: \mathbf{B}$$, where $$\mathbf{B}$$ is a WFF of length $$k-3$$, $$Q$$ is a quantifier ($$\forall$$ or $$\exists$$) and $$x$$ is a variable.

We deal with these one by one.

Cases $$1$$ and $$2$$ are covered by the argument in No Initial Part of a WFF of PropCalc is a WFF.


 * $$3:$$ $$\mathbf{A} = p \left({u_1, u_2, \ldots, u_n}\right)$$:

Here we have that $$\mathbf{D}$$ is a string of the form:
 * $$p$$ where $$p$$ is an $n$-ary predicate symbol. This can not be a WFF.
 * $$p ($$ which is also not a WFF.
 * $$p (u_1, u_2, \ldots, u_k$$ which can not be a WFF.
 * $$p (u_1, u_2, \ldots, u_k,$$ which can not be a WFFs.


 * $$4:$$ $$\mathbf{A} = Q x: \mathbf{B}$$:

$$\mathbf{D}$$ can not be $$Q$$, $$Q x$$ or $$Q x:$$ as none of these are WFFs.

So $$\mathbf{D}$$ is a WFF starting with $$Q x: $$, so $$\mathbf{D} = Q x: \mathbf{E}$$ where $$\mathbf{E}$$ is also a WFF.

We remove the initial $$Q x: $$ from $$\mathbf{A} = \mathbf{DT}$$ to get $$\mathbf{B} = \mathbf{ET}$$.

But then $$\mathbf{B}$$ is a WFF of length $$k-2$$ which has $$\mathbf{E}$$ as an initial part which is itself a WFF.

This contradicts the induction hypothesis.

Therefore no initial part of $$\mathbf{A} = Q x: \mathbf{B}$$ can be a WFF.

Thus all four cases have been investigated, and we have found that no initial part of any WFF of length $$k+1$$ can be a WFF.

The result follows by strong induction.