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

n−1

}. Also for any set of terms

¯

t = {t

0

, . . . , t

n−1

}, 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

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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 ∃wψ

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(∃xϕ) = 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

∃z∃yχ → θ

H

. But ` ∃z∃yχ ↔ ∃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

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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