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Journal ArticleDOI

On two classical results in the first order logic

01 Jan 2004-Publications De L'institut Mathematique (National Library of Serbia)-Vol. 76, Iss: 90, pp 21-24

Abstract: A common core of proofs of the classical consistency theorem of Hilbert and Ackermann and Herbrand's theorem concerning validity of exis- tential formulas is extracted.
Topics: Herbrand's theorem (69%), Ground expression (65%), Second-order logic (61%), First-order logic (59%), Ackermann function (57%)

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PUBLICATIONS DE L’INSTITUT MATH
´
EMATIQUE
Nouvelle s´erie, tome 76(90) (2004), 21–24
ON TWO CLASSICAL RESULTS
IN THE FIRST ORDER LOGIC
Miodrag Kapetanovi´c
Communicated by
ˇ
Zarko Mijajlovi´c
Abstract. A common core of proofs of the classical consistency theorem of
Hilbert and Ackermann and Herbrand’s theorem concerning validity of exis-
tential formulas is extracted.
Two well known results about first order logic appeared in the twenties: a
theorem of Hilbert and Ackermann about consistency of theories with quantifier
free axiomatization and Herbrand’s theorem, characterizing validity of existential
sentences. Both results have finitary character and they share the following fun-
damental idea: a first order problem is solved by reducing it in some sense to the
propositional calculus. The natural question then is whether the proofs of these
results also have something in common. The answer is positive and the core of the
argument is separated as the Main Lemma below. Such kind of analysis is well
known from the work of Gentzen in [3], who showed how several (already known)
important metamathematical results follow from his Hauptsatz. A recent exposi-
tion of the subject, based on the sequent calculus, is in [1] (see also [2]). Our aim
here however is to prove the two above mentioned theorems in the presence of the
cut rule and we envisage a simpler proof.
A Hilbert style presentation is taken from [5] and unexplained notation and
terminology is also therefrom. In order to expound the essence of the argument
the presence of equality is not presupposed. Terms are denoted by s, t, . . . while
ϕ, ψ, . . . are used for formulas and x, y, z, . . . for individual variables. We shall write
s¯x and φ¯x in order to emphasize that all free variables in s¯x and φ¯x are among
¯x = {x
0
, . . . x
n1
}. Also for any set of terms
¯
t = {t
0
, . . . , t
n1
}, by s
¯x
¯
t and φ
¯x
¯
t we
shall denote the term and the formula obtained by the simultaneous substitution
of x
i
by t
i
(i < n ) in s and φ, with the usual constraints on substitutability. The
axiom system for the first order logic, as we said already, comes from [5, p. 21],
but without the identity and equality axiom, which leaves us with propositional
2000 Mathematics Subject Classification: Primary 03F05; Secondary 03F07.
21

22 KAPETANOVI
´
C
and substitution axiom schemes only. The list of five rules of inference remains
unchanged.
Recall that a formula is open if it does not contain quantifiers and a theory
is open if all its nonlogical axioms are open. Here T will always denote an open
theory. Recall also that a formula ϕ
0
is a variant of ϕ if it can be obtained from ϕ
by bound variable renaming (and ϕ is its own variant), so all variants of a formula
share the same free variables.
If a substitution axiom of the form ψ
0
w
t wψ
0
appears in a proof and if
0
is a variant of yψ , then ψ
0
w
t is a ψ -instance. Let ψ
D
denote the disjunction of
all ψ -instances from the proof and ψ itself (thus it is never empty). A formula
of the form yψ is critical relative to a given T -proof if no other formula of that
form, occurring in the proof, has a -rank greater than ²(yψ), where the -
rank function ² is defined inductively: ²(ϕ) = 0 if ϕ is atomic, ²(¬ϕ) = ²(ϕ),
²(φ ψ) = max{²(ϕ), ²(ψ)} and ²() = ²(ϕ) + 1. Let φ
H
denote a formula
obtained from φ by replacing all occurrences of (all variants of) all critical formulas
yψ with the corresponding ψ
D
.
Lemma 1 (Main Lemma). If `
T
φ, then `
T
φ
H
.
Proof. We shall use induction on the length of the proof of φ. First, if φ is
a nonlogical axiom of T , then φ
H
is φ (since it is open) so `
T
φ
H
. Similarly if
φ is a propositional axiom ¬χ χ, then φ
H
is ¬χ
H
χ
H
, hence `
T
φ
H
. Next,
if φ is a substitution axiom χ
w
t wχ there are two possibilities: if wχ is not
critical, then φ
H
is φ, otherwise φ
H
is χ
w
t χ
D
and this is a tautology (χ
w
t is
a disjunct in ψ
D
), therefore provable in T by the tautology theorem [5, p. 27].
If φ is inferred by a rule of inference, notice first that by the remark [5, p. 30]
the proposition holds if only propositional rules are used. There remains the case
when φ is of the form yψ θ and is inferred from ψ θ by the -introduction
rule, (so y , which we can also call critical, is not free in θ ). We have to prove
`
T
(yψ)
H
θ
H
and notice first that `
T
ψ θ
H
by induction hypothesis. We
can rectify the proof so that if yψ and zϕ are critical formulas which are not
variants of each other, then ψ
D
and ϕ
D
have no free variables in common and also
that these variables differ from critical variables in the proof. With this in mind
suppose first that yψ is not critical. Then y is not free in θ
H
and `
T
yψ θ
H
follows by the -introduction rule. But this is just `
T
φ
H
, since (yψ)
H
is yψ .
So the main case is when yψ is critical and several cases arise depending on the
proof of ψ θ .
First if ψ θ is a propositional axiom, then θ is ψ and y is not free in ψ , so
ψ
D
is ψ and θ
H
is θ , hence ψ
D
θ
H
is ψ ψ . Next if ψ θ is a nonlogical
axiom of T , then again θ
H
is θ and `
T
ψ
D
θ follows from `
T
ψ
D
yψ and
`
T
φ by transitivity. Finally if ψ θ is a substitution axiom, then either θ is a
variant of yψ , in which case θ
H
is ψ
D
, or θ
H
is θ (yψ is critical) and in both
cases `
T
ψ
D
θ
H
.
Taking up rules of inference, suppose first that ψ θ is inferred from θ by
the expansion rule. Then `
T
θ
H
by induction hypothesis and `
T
ψ
D
θ
H
by
the same rule.

ON TWO CLASSICAL RESULTS IN THE FIRST ORDER LOGIC 23
If ψ θ is inferred by the cut rule using a cut formula ϕ, then `
T
ψ
ϕ and `
T
ϕ θ . Infer `
T
yϕ θ by the -introduction rule and observe
three cases. First if ²(ϕ) < ²(ψ), then (yϕ)
H
is yϕ , so `
T
yϕ θ
H
by
induction hypothesis. Infer `
T
ψ
D
yϕ from `
T
ψ ϕ , then `
T
ψ
D
θ
H
by transitivity. Next if ²(ϕ) > ²(ψ), then (yϕ)
H
is yϕ
H
, so `
T
yϕ
H
θ
H
,
again by induction hypothesis. Now `
T
ψ ϕ
H
also by induction hypothesis,
hence `
T
ψ
D
yϕ
H
, then `
T
ψ
D
θ
H
by transitivity again. Finally if yϕ
is critical, then `
T
ϕ
D
θ
H
by induction hypothesis. For every ψ -instance ψ
y
t
from the proof we may add a substitution axiom of the form ϕ
y
t yϕ and from
`
T
ψ ϕ we have `
T
ψ
y
t ϕ
y
t, so `
T
ψ
D
ϕ
D
, hence `
T
ψ
D
θ
H
by
transitivity.
Next if ψ is of the form zχ and ψ θ was inferred from χ θ by the
-introduction rule, then z is not free in θ . We have `
T
yχ θ by the -
introduction rule and `
T
yχ θ
H
by induction hypothesis. By a preliminary
variable renaming we can arrange that all disjuncts in ψ
D
begin with z , thus
z is not free in θ
H
either, then apply the -introduction rule again to infer `
T
zyχ θ
H
. But ` zyχ yψ , so `
T
yψ θ
H
, hence `
T
ψ
D
θ
H
.
Finally, supp ose that ψ θ was inferred from `
T
(ψ θ) (ψ θ) by the
contraction rule. We can use the additional induction on the number of applications
of the contraction rule and the fact that `
T
(¬ψ ¬ψ) (θ θ) follows from the
above formula without use of contraction, to get `
T
y¬(¬ψ ¬ψ) θ θ by the
-introduction rule. Hence `
T
[y¬(¬ψ ¬ψ) θ θ]
H
by induction hypothesis,
i.e., `
T
[¬(¬ψ ¬ψ)]
D
θ
H
θ
H
(y¬(¬ψ ¬ψ) is critical), which is equivalent
to `
T
ψ
D
θ
H
. ¤
The two classical metatheorems mentioned at the beginning come out now as
simple corollaries
1
. In both of them the theory T is assumed to be open.
Corollary 1 (Hilbert–Ackermann). If `
T
φ and φ is open, then it can be
derived using only propositional rules.
Proof. If the -introduction rule was not applied there is nothing to prove.
Otherwise let yψ θ be any formula inferred by the rule and we may suppose
that y ψ is critical. An application of the lemma eliminates the application of the
rule and φ
H
is φ since it is open. After finitely many such steps only propositional
rules remain. ¤
Corollary 2 (Herbrand). If `
T
x
1
. . . x
n
χ, where χ is open, then for some
terms t
ij
, 1 6 i 6 m, 1 6 j 6 n also `
T
W
16i6m
χ
x
1
...x
n
t
i1
. . . t
in
.
Proof. Again we may suppose that x
1
χ
0
, where χ
0
x
2
. . . x
n
χ, is critical.
Then by the lemma `
T
W
16i6k
χ
0
x
1
s
i
for some terms s
1
, . . . , s
k
. For the same
reason each of χ
0
x
1
s
i
is, after finitely many steps, also replaced by an appropriate
disjunction and so on, until we are left with a disjunction of χ-instances. ¤
1
A classical source to consult is [4]

24 KAPETANOVI
´
C
References
[1] S. R. Buss, On Herbrand’s Theorem, In: Logic and Computational Complexity (D. Leivant,
ed.), LNCS 960, Springer-Verlag, 1995, pp. 195–209
[2] S. R. Buss, An Introduction to Proof Theory, In: Handbook of Proof Theory (S. R. Buss, ed.),
Elsevier, 1998. pp. 2–78
[3] G. Gentzen, Untersuchungen ¨uber das logische Schliessen, Math. Zeitschrift 39 (1934/5), pp.
176–210, 405–431
[4] D. Hilbert, P. Bernays, Grundlagen der Mathematik II, Springer-Verlag, 1970
[5] J. R. Shoenfield, Mathematical Logic, Addison-Wesley, Reading, 1967
Matematiˇcki institut SANU (Received 14 11 2003)
Knez Mihailova 35/I (Revised 01 03 2004)
11000 Beograd
Serbia and Montenegro
kapi@mi.sanu.ac.yu
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