Constructive semantics and the validity of Peirce's law

(5) Let the subderivation having p as conclusion be �3 and let n := 3.
(6) If �n is open, then the open assumption must be p ⊃ q. If �n is closed, it is a
proof of p. In case �n is open, the last rule application in �n is either of a basic rule
or of (⊃E). Thus, again, �n has p ⊃ q as open assumption which is major premiss of
(⊃E), such that there is a subderivation �n+1 having as conclusion p:
�n:
�n+1
�
.
p p ⊃ q
(⊃E)
q.
basic or elimination rule
p
(7) We go back to the beginning of (6) for n := n + 1; the procedure is repeated until
we arrive at a closed derivation �3+m of p, for m ≥ 1, which must eventually be the
case, since every derivation is a finite tree. �
For atoms, the universal validity of Peirce’s rule follows from Theorems 3 and 4:
Theorem 5 For any atoms p and q, if Peirce’s rule is admissible in B, then it is valid
in B, that is (p ⊃ q) ⊃ p �B p.
Proof. Suppose that for an extension C of B it holds that �C (p ⊃ q) ⊃ p. By
clause (3) we have p ⊃ q �C p. From Theorem 3 we know that, if p ⊃ q �C p, then
p ⊃ q ⊢C p, and therefore ⊢C (p ⊃ q) ⊃ p. By Theorem 4 we have ⊢C p. By clause (1)
we have �C p. Hence (p ⊃ q) ⊃ p �B p. �
3.3 Admissibility and basic rules discharging assumptions
Theorem 4 holds for the fragment {⊃} of NM extended only by bases of production
rules (as in Sandqvist’s approach), and does not generalize to basic rules discharging
assumptions. For example, if B consisted solely of the rule
[p]
.
q (∗)
p
then there would be a closed derivation proving (p⊃q)⊃p in the fragment {⊃} of NM
extended by B. In using (∗), this derivation makes use of a basic rule which discharges
an assumption. If basic rules discharging assumptions—like rule (∗)—were allowed
in a basis, then admissibility of Peirce’s rule could not be shown. Considering as
atomic bases only production systems, whereas in the logical framework of natural
deduction rules which can discharge assumptions (such as implication introduction)
are a fundamental ingredient, can be seen as applying double standards to the concept
of deduction (see [12]). This is at least an issue that needs further clarification. As
it rests on this issue, Sandqvist’s intended justification of classical logic cannot be
considered conclusive without such further argument. 7
7 As mentioned in [12], dropping the reference to arbitrary extensions of bases would block
Sandqvist’s justification of classical logic as well as bases with assumption discharge. However, the
8