On the reiteration (or repetition) rule in natural deduction: is the rule "obvious" or does it need a proof?

Question

• I think the " reiteration rule" could be stated as follows : anything that is true under no assumption or under assumption $A_n$ is also true under any subordinate assumption $A_{n+k}$ with k greater than 0.

• Is there a tautology that guarantees this rule?

• Is it possible to prove this rule?

• Are there logical systems in which the reiteration/repetition rule does not hold?

Answer 1

The general meta-logical principle that justifies Reiteration is:

If $\Gamma_1 \vDash \psi$, then $\Gamma_1 \cup \Gamma_2 \vDash \psi$

That is: if $\psi$ is a logical consequence of a set of statements (assumptions) $\Gamma_1$, then $\psi$ is also a logical consequence of those very assumptions $\Gamma_1$ to which we added a further set of assumptions $\Gamma_2$

We can prove this using formal semantics. The general definition of $\Gamma \vDash \psi$ is that there is no truth-assignment (valuation) $h$ such that $h(\phi)=True$ for all $\phi \in \Gamma$ and $h(\psi)=False$.

So, take any $\Gamma_1$, $\Gamma_2$, and $\psi$. Assume $\Gamma_1 \vDash \psi$. By definition, that means that there is no truth-assignment (valuation) $h$ such that $h(\phi)=True$ for all $\phi \in \Gamma_1$ and $h(\psi)=False$. Now let's do a proof by Contradiciton, and let's assume that $\Gamma_1 \cup \Gamma_2 \not \vDash \psi$. That means that there is some $h$ such that $h(\phi)=True$ for all $\phi \in \Gamma_1 \cup \Gamma_2 $ and $h(\psi)=False$. But if $h(\phi)=True$ for all $\phi \in \Gamma_1 \cup \Gamma_2 $, then it is certainly true that $h(\phi)=True$ for all $\phi \in \Gamma_1$. Hence, we have that there is some $h$ for which $h(\phi)=True$ for all $\phi \in \Gamma_1$ and $h(\psi)=False$. This contradicts the earlier claim that there was no such $h$. So, it follows that $\Gamma_1 \cup \Gamma_2 \vDash \psi$. Hence, we have proven our desired result: If $\Gamma_1 \vDash \psi$, then $\Gamma_1 \cup \Gamma_2 \vDash \psi$

Answer 2

Yes, the so-called Weakening rule:

if $\Gamma \vdash B$, then $\Gamma, A \vdash B$.

From an "Hilbert-style" point of view, the relevant tautology is:

$\vDash A \to (B \to A)$.

Answer 3

There are a lot of slightly different versions of natural deduction.

In many of them, this principle is included as a rule or axiom — in that case, it is usually known as “weakening”, as noted in Mauro Allegranza’s answer.

In some variants, it’s not included as an axiom or rule. So in these versions, if you want to know or use this fact, then yes, it requires proof. As with anything in maths, being “obvious” is never a substitute for proof — it just means that (hopefully) the proof should be straightforward. One such proof (assuming that the completeness theorem has already been shown) is given in Bram28’s answer. Other more elementary proofs are usually possible, working just with derivations and not involving models; but these will unavoidably depend more on which specific version of natural deduction you’re using.

Answer 4

The reiteration rule follows from a natural deductive proof system which uses an inference rule called "Addition". Specifically the addition rule says from $P$ we can infer $P \lor Q $, for any statement $Q$.

In some cases reiteration is trivial, for instance if the statement is a premise you can cite the same premise as many times as you want, citing 'premise' as justification. In other cases reiteration is unavoidable, for instance in the proof of $~ \vdash P \to P $ we need to reiterate an assumption. $$ ~~~~~~~~~~~~~~~~~~~~~~~~~~1.~ P ~~~~~~~~~~~~~~~~~~~~~~~\rm{(Some~Inference ~Rule)} \\ ~~~~~~~2. ~P \lor \neg P~~~~~~~~~~\rm {~~~Addition ~1. } \\ ~~~~~~~~~~~~~~~~~~~~~3. ~\neg \neg P ~~~~~~~~~~~~~~~~~~~\rm{Double~ Negation ~1} \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4. ~P ~~~~~~~~~~~~~~~~~~~~~~\rm{Disjunctive ~Syllogism~2, 3} $$ Alternatively you could cite double negation (introduction and then elimination) .