Constructive semantics and the validity of Peirce's law

monotonicity (B ′ extends B) k ′ also establishes �B ′(p ⊃ q) ⊃ p. Therefore, �B ′ p
by construction k ′ (k ′′ ). As p is basic, we have ⊢B ′ p according to clause (1′ ) (this
corresponds to Prawitz’s clause (i)), that is, the construction k ′ (k ′′ ) is a derivation of
p in basis B ′ .
There are two cases: First, if the derivation k ′ (k ′′ ) does not use the basic rule p
q ,
then the derivation is already a derivation of p in B. Second, if the derivation k ′ (k ′′ )
uses it, then there is a topmost occurrence of this rule, and the subderivation of its
premiss is already a derivation of p in B. In both cases ⊢B p by a derivation k∗ and,
consequently, �B p by construction k∗ .
The construction k establishing p in B is the following procedure: We first apply
k ′ to a fixed construction k ′′ . This construction k ′ (k ′′ ) is a derivation using only basic
rules of B ′ . We then extract construction k∗ from construction k ′ (k ′′ ) by inspection.
We start with the topmost application of a basic rule and do a downward search
until we find a conclusion p. This is bound to occur, since the derivation k ′ (k ′′ ) is a
construction of p. �
Moreover, by using (⊃I) and (⊃E), which are supposed to be validated by the
semantics, Peirce’s law ((ϕ ⊃ �) ⊃ ϕ) ⊃ ϕ can by induction be shown to be valid for
atoms ϕ and any � in the fragment {⊃}. And the validity of Peirce’s law for any
ϕ and � can then be obtained by induction too, taking into account the remarks at
the end of section 2. We take this result as implying that, from a constructivist point
of view, the use of bases containing only production rules (i.e., basic rules without
the discharging of assumptions) has to be revised. Otherwise a non-constructive
principle would be validated, rendering the fragment {⊃} of NM and NI incomplete
with respect to the semantics.
Of course, this argument is not sufficient for the positive claim that the constructive
semantics obtained by permitting atomic bases with discharge of assumptions
is appropriate for intuitionistic logic. There may be other defects in this kind of
proof-theoretic semantics, in particular when logical constants beyond implication
are concerned.
5 Funding
This work was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES) and Deutscher Akademischer Austauschdienst (DAAD), grant
[1110-11-0 CAPES/DAAD to W.d.C.S.], by Agence National de la Recherche (ANR)
and Deutsche Forschungsgemeinschaft (DFG) within the French-German ANR-
DFG project “Hypothetical Reasoning”, grant [DFG Schr 275/16-1/2 to T.P. and
P.S.-H.]; and by a German-Brazilian exchange grant [444 BRA-113/66/0-1 to T.P.
and P.S.-H.].
References
[1] Dummett, M. (1991). The Logical Basis of Metaphysics. London: Duckworth.
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