Citation for published version (APA):
Bruijn, de, N. G. (1985). The rule of the superfluous third. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8503). Technische Hogeschool Eindhoven.
Document status and date: Published: 01/01/1985
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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics
and Computer Science
Memorandum 1985-03 Issued February 1985
The rule of the superfluous third
by
N.G. de Bruijn
Eindh~ven University of Technology, Department of Mathematics and
Computer Science,
P .0. Box 513, 5600 MB Eindhoven, The Netherlands.
by N. G. de Bruijn
1. INTRODUCTION.
We consider minimal propositional calculus M(-),~} with the usual rules for introduction and elimination of
implication and conjunction. From here we get to intuitionistic calculus by adding the axiom F -) a for all propositions a; F stands for a particular proposition called 'falsum' or
'contradiction'. We can get from ML(-)J\} to the classical calculus by adding the stronger rule that «a-)F}-)F}-)a for all a (the 'double negation rule').
In the minimal calculus with two letters a and F there are 18 equivalence classes of expressions (see[l]). equivalence of two expressions being defined by the derivability of A from B and B from A. In that sYHtem of 18 classes there 1s a natural 'orthogonal' decomposition, with orthogonality of u and v defined by
«u -) v) -) v) 1\ «v -) u) -) u)
We recall from [I) that all 18 classes are obtained from the orthogonal dissections u A v Aw where u is taken from
F -) F • «a -) F} -) a) -) a, a -) F,
v from
F -) F • «F -) a) -) F) -) F. F -) a
and w from
F -) F , «a -) F) ~ (F -) a» -) a.
In particular we mention that
a :: «(a -) F) -) a) -) a) 1\ (F -) a) 1\ «a -) F) (F -)a» -) a,
«a -) F) -) F) -) a ;;;: «(a -) F) -) a) -) a) 1\ (F -) a).
The latter formula shows the 'natural' way to split the
double negation rule into two parts: one is the intuitionistic F -) a, the other one is «a -) F) -) a) -) a (a particular case of Pierce's rule). It is this rule «a -) F) -) a) -) a that forms the subject of this note.
As stations between the minimal and the classical
calculus, systems can be studied characterized by the following rules:
(i) The intuitionistic rule F -) a.
(ii) The weak intuitionistic rule «F -) a) -) F) -) F. (iii) The rule «a -) F) -) a) -) a.
(iv) The conjunction of (ii) and (iii): let us call that the 'weak classical rule'.
gives the double negation rule {(a -) F) -) F) -) at and that means that we have classical logic.
2. NOTATION.
As in [1] we simplify formulas by writing implication as concatenation with associativ1ty from the left. Example: (ab)c(de) stands for «a -) b) -) c) -) (d -) e).
As usual, negation-, a is considered to abbreviate a -) F.
3. THE SUPERFLUOUS THIRD.
The princ1ple of the excluded third is the rule a
V
(.a). In combination with the elimination rule for v we get the schemea -) b
(3.1)
b
which is independent of the notion of disjunction.
Let us call this the scheme of the 'superfluous third'. It does not say that there is no other possibility beside a and -, a, but it just says we need not consider other possibilities. For the derivation of b they are superfluous.
The scheme of the superfluous third is directly legitimatized by the rule (1i1). at least the one for b, wh1ch is bFbb. This
Is obvious from the following derivation:
2 (aF)b assumption 2 3 bFbb superfluous third 4 bF 5
I-F
6 by modus ponens 7 by modus ponens 8 aF introd. implication 9 b by modus ponens 10 bFb introd. implication 11 b by modus ponens.Conversely. if we accept the scheme (3.1) we get bFbb at once, just applying the special case with a
=
b.4. PROPAGATION OF THE RULE.
If we have n letters al • • • • • a~ plus the letter F, and if we assume (iil) for a, •••• ,a'V'-:
a Fa,a a ,
I t l a"(\ F a .... a .... ' (4.1)
then we can derive the rule for all expressions of the calculus: for all expressions
if'
we get ~ F $I1.
This is easy to show by the following steps: (i) the rule holds for F. i.e., we can derive FFFF. (ii) i f we have t;Fcti/ and tfF4f'cf then we can derive
THEOREM. If (G(a, , ••• ,a~) is an expression in ML( -), A) that
is derivable in classical logic, then G(Faaj, ••• ,Fa a ) is
I )\ ...
derivable in ML(-),A) from the assumption (4.1).
PROOF. It is well-known that all the classically derivable expressions in a, •••• ,at'\. can be derived in ML( -),1\) if we
just assume the double negation rule, so there is an ML derivation
Replacing in this derivation every a
C by Fa,- a ~ we get a
derivation from (Fa a )FF(Fa al), •••• (Fa a )FF(Fa a ) to
I I I " n ~ ..., G(Fa,a, ••••• F~a~). In order to prove the theorem it now suffices to show that we can derive (Faa)FF(Faa) from aFaa. Such a derivation 1s
aFaa (Faa)FF Faa
Faa aFa (Faa)F a
a F F
(Faa)FF(Faa) a
As a particular case of the theorem we take Pierce's rule abaa which is claSSically derivable. So there exists a derivation of (Faa)(Fbb)(Faa)(Faa) from aFaa and bFbb. Here we show one:
aFaa
bFFb aFa (Faa)(Fbb) bFb a
(Faa)(Fbb)(Faa) a Faa Fbb b F
Faa a b
6. ADDING DISJUNCTION.
On top of ML(-),") we now take the disjunction as a connective, with the usual rules (cf. [2])
a b a -) c b -) c a v' b
a
v
b a v b cThe extended system will be called ML(
->,1\.
v).THEOREM. In ML(->,A,V) the addition of the rule
uFuu (for all propositions u)
is equival~nt to the addition of the excluded third:
w if (-.w) (for all propositions w).
PROOF. We first assume uFuu for all u. We saw in section 3 that this leads to the principle of the superfluous third. We now take any proposition w, and we apply superfluous third to w V (..., w). Assuming w we have w v (-,w) by the first rule
for w, assuming -. w we have w v (..., w) by the second rule. So by the principle of the superfluous third, we have w v (-, w).
Conversely, if w v ( , w) for all w, then the third rule of the disjunction turns into the principle of the superfluous third,
R~FERENCES
[1] N. G. de Bruijn. Exact finite models for minimal propositional calculus over a finite alphabet.
T.H. Report 75-WSK-02 (February 1975), Department of Mathematics, Eindhoven University of Technology, Eindhoven, The Netherlands.
[2] I. Johansson. Der Mlnimalkalkul, ein reduzierter intuitionistischer Formalismus.
February 1985 Dept. of Mathematics and Computing Science Eindhoven University of Technology
PO BOX 513, 5600MB EINDHOVEN The Netherlands
