Constructive semantics and the validity of Peirce's law

Draft, submitted.
Constructive semantics
and the validity of Peirce’s law ∗
Wagner de Campos Sanz
Universidade Federal de Goiás
Faculdade de Filosofia
Campus II, Goiânia, GO
74001-970 Brasil
wsanz@uol.com.br
Thomas Piecha
Universität Tübingen
Wilhelm-Schickard-Institut für Informatik
Peter Schroeder-Heister
Universität Tübingen
Sand 13, 72076 Tübingen
Germany
piecha@informatik.uni-tuebingen.de
Wilhelm-Schickard-Institut für Informatik
Sand 13, 72076 Tübingen
Germany
psh@uni-tuebingen.de
February 29, 2012
Abstract
In his approach to proof-theoretic semantics, Sandqvist claims to provide a
justification of classical logic without using the principle of bivalence. Following
ideas by Prawitz, his semantics relies on the idea that logical systems extend
atomic systems, so-called “bases”. We argue that the restriction of bases to
systems of production rules, which cannot discharge assumptions, is a central
feature of Sandqvist’s approach, which leads to the validation of classical principles
such as double negation elimination or Peirce’s rule. We also show that this
analysis applies to a more abstract form of constructive semantics as considered
by Prawitz. This demonstrates that the choice of the format of atomic bases is of
crucial importance for proof-theoretic semantics, and that the common choice of
production systems as bases renders intuitionistic logic incomplete with respect
to this semantics.
Keywords Proof-theoretic semantics, constructive semantics, classical logic, intuitionistic
logic, double negation elimination, Peirce’s law, Peirce’s rule, admissibility,
derivability
∗ We are grateful to Dag Prawitz of notifying us of Tor Sandqvist’s work and for leaving to us his
notes on this work. Special thanks go to Tor Sandqvist for discussing with us his ideas on various
occasions in detail.
1

1 Introduction
In Dummett-Prawitz-style proof-theoretic semantics the meaning of a proposition is
given in terms of what conditions must be fulfilled in order to assert the proposition.
If the condition to assert a proposition is the possession of a proof, then a constructive
semantics requires a description of what are proofs of basic propositions and of what
are proofs of logically complex propositions. The description is usually given as an
inductive definition. Among others, Dummett [1] and Prawitz [5, 6, 7] are references
for this approach (for an overview see [11, 13]). Recently, Sandqvist [10] proposed a
semantics for logically complex propositions, which is closely related to constructive
semantics, and which takes the form of an inductive definition too. It is of special
interest as it is intended to yield a justification of classical logic without making use
of the principle of bivalence.
We focus on the logical constant of implication. The implicational fragment {⊃}
of natural deduction for minimal logic NM (and also for intuitionistic logic NI) is
given by the following introduction and elimination rules:
[ϕ]
.
� (⊃I)
ϕ ⊃ �
ϕ ϕ ⊃ � (⊃E)
�
These rules also hold in the fragment {⊃} of natural deduction for classical logic NK,
although they do not suffice to obtain all classical theorems in this fragment. Classical
laws like Peirce’s law ((ϕ ⊃ �) ⊃ ϕ) ⊃ ϕ are provable when one of the following two
versions of Peirce’s rule is added:
[ϕ ⊃ �]
.
ϕ
ϕ
(ϕ ⊃ �) ⊃ ϕ
ϕ
Each version can be derived from the other by using (⊃I) and (⊃E). Let NP be NM
plus (one version of) Peirce’s rule. The addition of ex falso quodlibet
⊥
ϕ
to NM gives NI, and the addition of it to NP gives NK. Peirce’s rule is not justifiable
constructively, since its addition to NI allows to prove tertium non datur, which is
normally rejected as being non-constructive 1 .
2 Sandqvist’s semantics for classical logic
Sandqvist [10] has proposed a semantic justification of classical logic, which does not
make use of the principle of bivalence, for the rule of double negation elimination
for the fragment {⊃, ⊥, ∀}. Once double negation elimination is established, the
1 See Heyting [2], p. 103f.
2

other logical constants can be defined, and the justification becomes a justification for
classical logic without using the principle of bivalence. Sandqvist considers bases B
which are sets of basic sequents, that is, relations between finite sets of basic sentences
and basic sentences, where basic sentences are formulas containing neither logical
constants nor free variables. In the following, basic sentences are also called atoms.
We are not considering quantifiers, as they are not relevant to the arguments put
forward here. This means that we can consider atoms to be propositional variables.
The language of all bases is the same, that is, the set of all atoms. A basis B can thus
be viewed as a set of atomic inferences, that is, of production rules
p1 . . . pn
pn+1
where the pi are atoms. For a given basis B Sandqvist then defines inductively a
relation �B, called ‘valid inferability’, between finite sets 2 of sentences in general and
sentences in general (ibid., p. 213, our numbering; we omit the clause for ∀):
Definition 1.
Where p is a basic sentence: �B p ⇐⇒ Every set of basic sentences
closed under B contains p
Where Γ is non-empty: Γ �B ϕ ⇐⇒ �C ϕ for every C ⊇ B
such that �C � for every � ∈ Γ
(1)
(2)
�B ϕ ⊃ � ⇐⇒ ϕ �B � (3)
�B ⊥ ⇐⇒ �B p for every basic sentence p (4)
An inference ϕ1 . . . ϕn
ϕ
holds for B if ϕ1, . . . , ϕn �B ϕ. The inference is considered
logically valid if ϕ1, . . . , ϕn �B ϕ for every B. If Γ �B ϕ holds for every B, then we
also speak of logical consequence, denoted by Γ � ϕ. A justification of the inference
of double negation elimination consists thus in showing that this inference is a logical
consequence. This is stated in his main lemma as follows (ibid., p. 216), where the
choice of the basis B is arbitrary:
Lemma 4 (Double negation). (ϕ ⊃ ⊥) ⊃ ⊥ �B ϕ. 3
The proof of Lemma 4 begins as follows (ibid., p. 216):
Proof. It suffices to prove the result for basic ϕ; given Lemma 3, non-basic
applications of double-negation elimination can be reduced to basic ones
in the manner, mutatis mutandis, of Prawitz [4, pp. 39–40].
Lemma 3 (ibid., p. 216) states that the usual introduction and elimination rules for
the fragment {⊃, ∀} hold for the inferential relation �B (as given in his Definition 1).
Lemmas 3 and 4 establish that Γ �B ϕ whenever ϕ is a classical consequence of Γ
(ibid., p. 217), from which Sandqvist’s Theorem 2 (expressing the justification of
classical logic) follows directly (ibid., p. 214):
2 Sandqvist considers only finite sets of assumptions.
3 More precisely, he uses a consequence relation referring to substitutions for free variables, which
is not significant in the context of propositional logic considered here.
3

Theorem 2 (Classical logic). If ϕ is a classical consequence of Γ, then
Γ �B ϕ.
This justification is built upon monotonic extensions of bases, that is, if B ′ ⊇ B, then
�B ϕ ⇒ �B ′ ϕ. What Sandqvist would therefore have achieved is a justification of
classical logic in the context delineated. 4 He says (ibid., p. 214):
Thus, whatever your attitude towards particular inferences among atoms,
in so far as your use of logical compounds is governed by the semantics
we have formulated, you have no choice but to accept all classically valid
sentences and inferences.
Remark on intuitionistic versus classical disjunction
It should be emphasized that Sandqvist’s Theorem 2 applies without restriction only
to the fragment of classical logic he is considering, that is, the fragment based on
{⊃, ⊥, ∀}. If we include disjunction (or analogously existential quantification), then
the validity of double negation elimination can no longer be reduced to the atomic
case, provided disjunction is given its intuitionistic interpretation according to the
following semantical rule:
�B ϕ ∨ � ⇐⇒ �B ϕ or �B � (5)
In particular, Lemma 4 does not necessarily hold, if ϕ has the form � ∨ ¬�. Thus
Sandqvist has not given an example of a valid law which is not derivable in intuitionistic
logic, if disjunction is understood intuitionistically, since of a valid law we would
expect that its substitution instances, including those containing disjunction, are valid
as well. Of course, if we understand disjunction ϕ ∨� not in its intuitionistic sense (5)
but in its classical sense by its de Morgan equivalent ¬(¬ϕ ∧ ¬�), then Lemma 4
provides such an example. Sandqvist is fully aware of this fact and mentions this
point explicitly (ibid., p. 215).
This restriction concerning intuitionistic disjunction affects the significance of
Sandqvist’s result only marginally. The fact that Sandqvist’s semantics validates
the laws of classical logic for intuitionistically understood implication is a crucial
point which is against basic intuitions of intuitionism, in particular, as he does not
presuppose that atomic formulas behave classically, but instead proves that this is
the case. Taking this into account we can interpret Sandqvist’s result as follows: If
disjunction is understood classically, then the intuitionistic semantics proposed in
his Definition 1 renders all classical laws valid. This is definitely not something an
intuitionist would accept and a very remarkable conceptual result. It can be viewed
either as a constructive justification of classical logic (this is how Sandqvist reads it),
or as pointing to certain deficiencies of the underlying semantics (as we read it).
3 Admissibility versus derivability
We want to point out that Sandqvist’s semantics is inadequate. It relies on a too narrow
notion of what is a basic rule, since the discharging of assumptions is excluded. That
4 This context is quite similar to constructivist approaches.
4

is, basic rules of the more general form
[Γ1]
.
p1
. . .
pn+1
where the Γi are (possibly empty) sets of atoms that can be discharged, are excluded;
only basic rules that are production rules are considered in Sandqvist’s semantics.
If we only allow for such inferences between atoms that can be given by production
rules, then it is possible to produce a justification of double negation elimination and
of Peirce’s law. However, the restriction of basic rules to production rules is something
that Sandqvist takes for granted and that would have to be justified.
In order to put Sandqvist’s semantics into a wider perspective, we relate his notion
of validity to the more common notion of admissibility. We show that Peirce’s rule
is universally admissible, that is, admissible in every basis B, and show that this
suffices to establish its validity in Sandqvist’s sense, provided bases are restricted to
production rules which cannot discharge assumptions.
3.1 Admissible rules and validity
A rule of the form
ϕ1 . . . ϕn
�
is admissible when it is guaranteed that there is a formal proof of the conclusion if
there are formal proofs of the premisses. It is derivable if there is a derivation in the
system, having as open assumptions no more than the premisses ϕ1, . . . , ϕn of the rule
and as endformula its conclusion �.
We will show that Peirce’s rule is admissible in NM. An admissible but nonderivable
rule can only be added to a given system of natural deduction if its application
is restricted to premisses which do not depend on any assumptions. Otherwise,
Peirce’s law would be provable by implication introduction (⊃I) in NM:
[Γn]
.
pn
[(ϕ ⊃ �) ⊃ ϕ]
(Peirce’s rule)
ϕ
(⊃I)
((ϕ ⊃ �) ⊃ ϕ) ⊃ ϕ
It can be seen that the universal validity of implications and admissibility in
Sandqvist’s sense coincide: Let ⊢B denote the derivability relation given by the basic
rules of a basis B. Then Sandqvist’s clause (1) can be reformulated into the following
equivalent clause:
Where p is an atom: �B p ⇐⇒ ⊢B p
Let Γ = {p1, . . . , pn} be a set of atoms and q an atom. For the empty basis, Sandqvist’s
clause (2) implies
Γ � q ⇐⇒ For all bases B: If ⊢B p1, . . . , ⊢B pn, then ⊢B q
5
(†)

In other words, Γ � q holds if and only if the basic rule p1 . . . pn
q
is admissible in
all bases B. That is, for atoms and the empty basis the notion of valid inferability is
equivalent to the notion of admissibility in all bases. Therefore, for the empty basis
and for atoms p and q, we have
� p ⊃ q ⇐⇒ For all bases B: If ⊢B p, then ⊢B q
In other words, p ⊃ q is valid if and only if the basic rule p
q
is admissible in every
basis. But can admissibility be taken as a sufficient condition for validity of other
sentences more complex than implications p ⊃ q ?
The answer is positive, at least for a special case of Peirce’s rule considered below.
We first show a completeness theorem for the derivability of atoms from assumptions
which are either atoms or implications between atoms.
Theorem 3 (Atomic completeness) Suppose ϕ1, . . . , ϕn �B p, where each ϕi (1 ≤ i ≤
n) is either an atom pi or an implication pi ⊃ qi between atoms. Then ϕ1, . . . , ϕn ⊢B p.
Proof. We know that p is derivable in every extension C of B, for which �C ϕi holds
for every i (1 ≤ i ≤ n). Now consider the extension C ′ of B, which is obtained from
B in the following way: For every i we add pi as an axiom, if ϕi is an atom pi, and
we add
pi
qi
as a basic rule, if ϕi is an implication pi ⊃ qi. Then �C ′ ϕi for every i (1 ≤ i ≤ n).
Therefore p is derivable in C ′ . If we now replace every application of a rule
by an application of (⊃E)
pi
qi
pi pi ⊃ qi
qi
we obtain a derivation of p from ϕ1, . . . , ϕn over B. �
This (restricted) completeness theorem is an immediate consequence of the fact
that in the semantical clause (2) of consequence and thus of implication (clause (3))
arbitrary extensions of bases are considered.
3.2 Admissibility of Peirce’s rule
Instead of reconsidering the rule of double negation elimination, we concentrate on
Peirce’s rule and on Peirce’s law. This has the advantage that only the clauses (1), (2)
and (3) of Sandqvist’s Definition 1 are relevant, whereas the justification of double
negation elimination depends on clause (4) for ⊥ in addition. Thus any possible
reservations one might have about clause (4) do not interfere with the following
discussion. As a justification of Peirce’s rule amounts to a justification of classical
logic just as a justification of double negation elimination does, using Peirce’s rule to
this effect is not an undue choice. Indeed, as Sandqvist points out (ibid., p. 215):
6

[. . .] although for the sake of convenience proof of Theorem 2 [. . .] does
proceed by way of double-negation elimination (a rule involving both ⊃
and ⊥), the validity of, say, Peirce’s law (((ϕ ⊃ �) ⊃ ϕ) ⊃ ϕ) in no way
depends on the clause for ⊥.
This should also be clear from the fact that Sandqvist’s Definition 1 must be considered
to be “not a system of deduction, but a compositional meaning theory according to
which the semantic properties of any sentence [. . .] are fully determined by those of
its subsentences” (ibid., p. 215).
In the context of natural deduction, concentrating on Peirce’s rule and on Peirce’s
law means that only the fragment {⊃} must be taken into account. Peirce’s law is
not provable in the fragment {⊃} of NM. Consequently, Peirce’s rule is not derivable
either.
Although Peirce’s rule for atoms is not derivable in the fragment {⊃} of NM, it
can be shown to be admissible for any basis of production rules, using only prooftheoretical
means. 5
Theorem 4 (Admissibility of Peirce’s rule) For all atoms p, if there is a closed derivation
(i.e. a proof) of (p ⊃ q) ⊃ p in the fragment {⊃} of NM over any basis B, then there is
a closed derivation of p in this fragment.
Proof. (1) A closed derivation � of (p⊃q)⊃p for p in the fragment {⊃} of NM over
any basis B can be transformed into a closed derivation �1 in normal form, according
to theorem 2 in Prawitz [4, chapter III, p. 40] 6 .
(2) The last rule application in �1 is neither the application of a basic rule nor of
(⊃E); otherwise the derivation would not be closed. The application has to be (⊃I):
�1:
[p ⊃ q]
.
p
(p ⊃ q) ⊃ p
(⊃I)
(in case no assumption p ⊃ q is discharged in the application of (⊃I), we already have
a closed derivation of p).
(3) The subderivation
of p in �1 is also in normal form.
�2:
p ⊃ q
.
p
(4) The last rule application in �2 is not of (⊃I). It can only be of a basic rule or of
(⊃E). In both cases there is a branch starting with an open assumption that is the
major premiss of (⊃E) and must be p ⊃ q. Thus �2 has the form:
�2:
.
p p ⊃ q (⊃E)
q. basic or elimination rule
p
5 This has been pointed out similarly for double negation elimination by Prawitz [8].
6 This theorem holds likewise for extensions by basic rules.
7

(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

4 Constructive semantics and incompleteness of minimal
logic
Sandqvist’s semantics resembles other forms of constructive semantics. Prawitz [5,
p. 278] gives a semantic clause for implication that has the same structure as Sandqvist’s
clauses (2) and (3) when combined. He defines constructions of sentences over
a Post system S as basis by the following induction:
(i) k is a construction of an atomic sentence ϕ over S if and only if k is a
derivation of ϕ in S.
(ii) k is a construction of a sentence ϕ ⊃ � over S if and only if k is a
constructive object of the type of ϕ ⊃� and for each extension S ′ of S and
for each construction k ′ of ϕ over S ′ , k(k ′ ) is a construction of � over S ′ .
As Post systems are given as sets of production rules, the bases used in Prawitz’s
semantics are the same as in Sandqvist’s semantics. The noticeable difference to
Sandqvist’s semantics is that Prawitz’s semantic clause (ii) for implication contains
the additional requirement of having a construction k transforming any proof of ϕ
into a proof of � in order to establish the validity of ϕ ⊃ �. The fact that a derivation
in a Post system is considered to be a construction for an atom is a reason to consider
a rule belonging to the basis as a construction; this is implicit in clause (i). As basic
rules are one-step derivations, basic rules are constructions too.
Adding the requirement of having such a construction k to Sandqvist’s clauses (1)
and (2) yields the following clauses:
Where p is an atom: �B p by a construction k
⇐⇒ ⊢B p by a derivation k in B (1 ′ )
Where Γ = {�1, . . . , �n} is non-empty: Γ �B ϕ by a construction k
⇐⇒ �C ϕ by construction k applied to
constructions k1, . . . , kn for every C ⊇ B
such that �C �i by construction ki
We prove that for the thus modified clauses Peirce’s law is valid. This shows that our
criticism of Sandqvist’s approach has a more general range of application, pertaining
to all constructive semantics that can be given this or a similar form.
Theorem 6 For basic p and q, �B((p ⊃q)⊃p)⊃p holds constructively, for all bases B.
Proof. Suppose that k ′ establishes �B(p ⊃ q) ⊃ p, where B is a basis of production
rules. We show that there is a construction k such that k(k ′ ) establishes �B p.
We consider the extension B ′ =
� p
q
�
(2 ′ )
∪ B. Thus �B ′ p ⊃ q. In other words, the
basic rule p
q is a construction k′′ transforming proofs of p into a proof of q. By
resulting nonmonotonicity in the semantics of implication appears to us too great a sacrifice. Actually,
Lorenzen’s [3] admissibility semantics of implication is of this kind. Interestingly, Lorenzen shows
that in his semantics (which does not consider extensions of bases), Peirce’s law can be justified if one
admits classical means of reasoning in the metalanguage (ibid., p. 52).
9

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.].
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