TITLE: About theorem's proof length in propositional calculus
QUESTION [5 upvotes]: In PC(propositional calculus) system, how long will a formula's proof be?
That is to say if there exists a computable function $f$ such that for any formula $A$, if $\vdash_{\mathrm{PC}}A$ then $A$ has a proof with length less than $f(A)$.

REPLY [2 votes]: In propositional logic, a formula is provable if and only if it is a tautology, and the set of tautologies is decidable. Moreover, for any complete and effective proof system, if we know a formula is a tautology, we can simply enumerate proofs until we find a proof of the formula. Therefore, given a formula $\phi$, we can compute the following function: if $\phi$ is not a tautology, return $1$. Otherwise, search for the first proof of $\phi$ that you find, and return the length of that proof. This function will be computable and give an affirmative answer to the question. Notice that this answer is also independent of how the "length" of a proof is defined; the answer works for any definition of "length" as long as the length of the proof is computable from the proof itself. 
The situation would be very different in first-order logic. In that context, the set of provable formulas is no longer decidable, and the definition of "length" becomes very important.