Math. Log. Quart. 46 (2000) 1, 121 – 124
Mathematical Logic
Quarterly
c WILEY-VCH Verlag Berlin GmbH 2000
!
A Problem of Normal Form in Natural Deduction
Jan von Plato
Department of Philosophy, University of Helsinki,
Unioninkatu 40, Helsinki, Finland1)
Abstract. Recently Ekman gave a derivation in natural deduction such that it either
contains a substantial redundant part or else is not normal. It is shown that this problem is
caused by a non-normality inherent in the usual modus ponens rule.
Mathematics Subject Classification: 03F05.
Keywords: Natural deduction, Normal form, Elimination rules, Sequent calculus.
1 The problem
In Ekman [1], the following problem of normal form in intuitionistic natural deduction
is treated: Defining A ⊃⊂ B ≡ (A ⊃ B) & (B ⊃ A) and ∼ A ≡ A ⊃ ⊥, each derivation of the formula ∼ (P ⊃⊂ ∼ P ) either is not normal or else has a subderivation of
form
∼P ⊃ P ∼P
⊃E
P ⊃ ∼P
P
⊃E .
∼P
Such a derivation has the reduncancy, or is “indirect” in Ekman’s terminology, that
the derivation of the conclusion could be replaced by the derivation of the first occurrence of ∼ P . But this will produce a non-normal derivation, for the top occurrence
of ∼ P is the conclusion of ⊃ I, and the bottom occurrence the major premiss of ⊃ E.
In this note, we show that the above problem of normal derivability is caused by
the usual implication elimination rule of natural deduction, modus ponens,
A⊃B A
Mp .
B
This rule has a natural generalization into one that permits a normal derivation of
the formula ∼ (P ⊃⊂ ∼ P ).
2 Normal derivation with general elimination rules
In Schroeder-Heister [3], the use of a general elimination rule for conjunction
in natural deduction, analogous to the disjunction elimination rule, was suggested,
1) e-mail:
[email protected]
122
Jan von Plato
where [A, B] means that in the rule the auxiliary assumptions A, B are discharged:
[A, B]
..
.
A&B
C
&E
C
.
The special elimination rules
A&B
A
A&B
B
and
come out as special cases of the general rule by setting C = A and C = B:
A & B [B]
&E ,
B
A & B [A]
&E ,
A
respectively. In the other direction, if C is derivable from A, B, we have that C is
derivable from A & B by the special elimination rules
A&B
A
A&B
B
..
.
C.
But the special elimination rules confound the structure of derivations: The standard
way of translating cut-free sequent calculus derivations into natural deduction, when
applied to the two sequent calculus derivations
A, B ⇒ B
A, B ⇒ A
L&
A&B ⇒ A
A&B ⇒ B
A&B ⇒ A&B
L&
R& ,
produces the same natural deduction derivation
A&B
A&B
A
B
A&B
&I
A, B ⇒ A A, B ⇒ B
A, B ⇒ A&B
A&B ⇒ A&B
R&
L&
.
With the general elimination rules we have instead the two derivations
A&B
A
1.
[A]
A&B
&E,1.
B
A&B
2.
[B]
1.
&E,2.
&I
,
1.
[A] [B]
A&B
A&B
A&B
&I
&E,1. .
Thus, using the general rules, we can translate the sequent calculus derivations in a
way that reflects the order of application of the sequent calculus rules.
The use of modus ponens causes similar problems. To locate the correct general
elimination rule for implication, we note that the general conjunction elimination
rule and the disjunction elimination rule come out as translations of the left rules of
A Problem of Normal Form in Natural Deduction
123
sequent calculus, written here without contexts,
A, B ⇒ C
A&B ⇒ C
A&B
!
L&
[A, B]
..
.
C
&E
C
,
[A] [B]
..
..
.
.
A∨B C
C
A⇒C B⇒C
L∨
!
∨E .
A ∨ B⇒C
C
We now find the general elimination rule for implication through the translation
[B]
..
.
A⊃B A C
⇒A B⇒C
L⊃
!
⊃E .
A⊃B⇒C
C
The problem about normal form in Ekman [1] is solved by a derivation using the
general ⊃ E rule:
5.
4.
6.
[∼ P ] [P ] [⊥]
2.
4.
[P ⊃∼ P ] [P ]
2.
1.
[∼ P ⊃ P ]
[(P ⊃∼ P )&(∼ P ⊃ P )]
⊥
⊥
∼P
⊥
⊃I,4.
7.
⊃E,6.
⊃E,5.
3.
8.
[∼ P ] [P ] [⊥]
2.
3.
[P ⊃∼ P ] [P ]
⊥
⊥
⊃E,8.
⊃E,7.
⊃E,3.
&E,2.
⊥
⊃I,1.
∼ ((P ⊃ ∼ P ) & (∼ P ⊃ P ))
All major premisses of elimination rules in the derivation are assumptions, which
is the characteristic property of normal derivations with general elimination rules. In
particular, the conclusion ∼ P by ⊃ I is not a major premiss of ⊃ E.
The effect of general elimination rules is that permutation conversions, first found
by Prawitz [2] for disjunction elimination, apply uniformly.
3 Conclusion
In conclusion, it must be admitted that derivations of the complexity exemplified
above are not the easiest ones to find in natural deduction with general elimination
rules. They can be found by using the special rules followed by normalization, but an
easier way is to use sequent calculus that supports proof-search. Derivations can then
be compositionally translated into natural deduction: Starting from the endsequent,
if the last rule applied was a right rule, say R&, we translate
Γ⇒A ∆⇒B
Γ⇒A ∆⇒B
!
Γ, ∆ ⇒ A & B
A&B
and analogously for disjunction and implication. If the last rule was a left rule,
we translate in the way shown already. Axioms A ⇒ A become assumptions A in
124
Jan von Plato
⊥
natural deduction, and axioms ⊥ ⇒ C become applications of the rule . The
C
general result is:
Translation of sequent calculus left rules into general elimination rules
of natural deduction gives a natural deduction structure isomorphic
to sequent calculus structure.
The change to general elimination rules has no effect on the semantics of intuitionistic natural deduction that is based on the introduction rules. A detailed study of
normalizability and of the structure of normal derivations with general elimination
rules will be presented in a subsequent work.
References
[1] Ekman, J., Propositions in propositional logic provable only by indirect proofs. Math.
Logic Quarterly 44 (1998), 69 – 91.
[2] Prawitz, D., Natural Deduction: A Proof-Theoretical Study. Almqvist & Wicksell,
Stockholm 1965.
[3] Schroeder-Heister, P., A natural extension of natural deduction. J. Symbolic Logic
49 (1984), 1284 – 1300.
(Received: October 13, 1998; Revised: April 15, 1999)