A formal definition of derivation trees in systems of natural deduction

A formal definition of derivation trees in systems of natural

deduction

Citation for published version (APA):

Balsters, H. (1982). A formal definition of derivation trees in systems of natural deduction. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8220). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1982

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum 82-20 November 1982

A formal definition of derivation trees in systems of natural deduction

P.O. Box 513

5600 ~m Eindhoven

The Netherlands

by

Introduction

A FORMAL DEFINITION OF DERIVATION TREES IN SYSTEMS OF NATURAL DEDUCTION

by

H. Balsters

The motivation behind this note was a certain dissatisfaction with regard to the lack of formal rigour in familiar natural deduction presentations of first order logic. Usually such presentations are only in part formal and often use is made of intuitive pictures of so-called "derivation trees", without actually giving precise rules of their construction. The

tech-nique of "cancelling of hypotheses", for example, ususally lacks a proper formal treatment. Which hypotheses are to be cancelled and what cancelling formally amounts to "is not made clear. Sometimes one has the impression that the cancelling technique is governed onl'y by some pragmatic rule of thumb. In this paper a formal recursive definition will be given of the set of derivation trees in a classical first order system of predicate logic, based on natural deduction. For an exposition of the intuitive background of our subject, see [1].

•

2

-I. Preliminaries

O. Language

0.0. Alphabet

Our alphabet consists of the following symbols:

(i) k-ary predicate symbols _PO' k

P.,

k _P2' k

...

,

_Pn' k

(H) k-ary function symbols _{fa, f} k k _{1 ' £2'} k

...

,

fk _n'

(iii) constant symbols _{cO' c}

t ' c2,

...

,

c n'

(iv) variables _x

_o'

_{xl' x2'}

...

,

_{xn '}

(v) connectives A, .. , .L,

V

(vi) the cancellation symbol $ (from cancelled $entence)

(vii) auxiliary sYmbols (, )

,

where k, n E :N and. • {O,lt2, ••• }.

The set of variables is denoted by

VAR.

Furthermore, we shall use meta-variables c (for constant symbols) and x,y (for meta-variables).

0.1. We introduce three syntactical categories: TERM (the set of t~), FORM (the set of

6oromula4)

and CANCEL (the set of

cancelled

6or0muia4).

0.1.0. Definition

TERM is the smallest set X such that

(i) VAR C X (H) c E X n (1'1'1') ₁ ' f _{tl, ••• ,t} X h fk( ) X k E t en n t1, ••• ,tk E where k,n E: :N.

3

-0.1.1. Definition

FORM is the smallest set X such that (i) .L E X

(ii) if t1 ••••• t k E TERM then

P~(tl

••••• tk) E X

(iii) if ~.~ E X and 0 E {A. ~ } then (~ 0 ~) E X

(iv) if ~ E X then Vx ~ E X

n

where k.n Eli.

The formulas in (i) and (ii) are called ato~. In what follows we shall use I~ as an abbreviation for (~ ~ .L).

0.1.2. Definition

CANCEL:- {$(~)I~ E FORM}.

Our language now consists of FORM U CANCEL.

0.1.3. Remark. From now on we shall omit indices and just write an f or p

in-d f f k k 'd .

stea 0 or p to aVOl messy notatlon.

n n

I. Free variables

1.0. Definition

Let t E TERM. then the set FV(t) of 6~ee v~bleh

06

t is defined by

(i) FV(x):= {x} (ii) FV(c ):= f/J n k (iii) FV(f(t1 ••••• t k

»:=

_j>= U FV(t.) _{1 J} where k,n E :IN.

4

-1. -1. Definition

Let ~ € FORM, then the set FV(~) of 6~ee v~ble4 06 , is defined by

(i) FV(.L):= f/J k (ii) FV(p(t1, ••• ,tk)}:a U FV(t.) j= 1 J (iii) FV(~

0

$):= FV(~) U FV(~) (iv) FV(Vx ~):= FV(~) - {x} where k € ::N and 0 E {J\, -+ }.

2. The substitution operator 2.0. Definition

Let 8,t E TERM, then [x:= t]s is defined by

t]y:- { t

,

if x.y (i) [x:= y t otherwise (ii) [x:= tJc:= c (iii) [x:: tJ f(t1, ••• ,t_{k) "" f([x:= tJt1, ••• ,[x:= t]tk)} where k E :N. By i! we denote syntactical equality.

2.1. Definition

Let t E TERM and ~ € FORM, then [x:- tJ~ is defined by

(i) [x:= t].L := .L

(ii) [x:= t] p(t1, ••• ,t_k):- p([x:- tJt1, ••• ,[x:- tJt_k) (iii) [x:- tJ (,

0

~):= ([x:= t](jl

0

[x:= tJ~)

(iv) [x:= tJ Vy':=

{Vy

(jl

Vy[x:= tJ!p where k € ::N •

,ifxllY

2.2. Definition

Let t € TERM and ~ € FORM.

t -i.6 6lt.ee

60lt.

x in ftI iff (i) , is an atom

5

-(ii) ql a (,) 0 <IlZ) and t is free for x in both ') and ~Z (0 € {A, -+})

(iii) 1p .. Vy ljI and a) x .. y, or b) x

t

FV(ljI),

or c) y

i

FV(t) and t is free for x in IP.

II. Derivation trees

In this section we define the set of

deJt.ivation

~ee4 in a system of natural deduction in a completely formal way, without resorting to

two-dimensional diagrams. Ordered pairs will be denoted by <a,e> and ordered triples by <a.a,y>.

O. Defini tion

The set of

deJt.ivation

tJt.ee4~ T is the smallest set X such that (i) FORM u CANCEL c X

(ii) if T E X and IP € FORM then <T,1p> € X

(iii) if T₁,T₂ € X and ~ € FORM then <T1,TZ'IP> € X •

1. Defini don

Let T €

T,

then

the It.oot

R(T)

06

~ee T is defined by

(i) if q € FORM u CANCEL then R(q):- q

(ii) R«T,1p»:= IP (iii) R«T

6

-2. Definition (cancelling of leaves of a tree) C is the function defined by C € TTxFORM and

(i) i f q E FORM u CANCEL then C(q,'/'):= { $(q) , if q • !JJ

't' q otherwise

(iii) C«TI,T2'~>''''):= <C(TI,W) , C(T₂,!JJ),,> C is called the

cancel 6unetion.

3. Definition

Let T E

T,

then L(T) the set of

uncancelted

lean tabel4 06

T is defined

by

(i) if q € FORM then L(q):- {q}

if q E CANCEL tpen L(q):=

¢

(ii) L«T,~»:= L(T)

(iii) L«TI,T2'~»:= _{L(T I)} u L(T₂) 4. Definition

The set of d~v~n tftee6

VeT

is the smallest set X such that

(i) FORM c X

(ii) if D

1,D2 E X then <DI,D2,(R(D1) A R(D2

»>

E X (A-introduction)

(iii) if D E X and R(D)

= (,

A W) then <D,~> E X and <D,W> E X

(A-elimination) (iv) if D_I,D₂ E X, R(D

I) = , and R(DZ) = (~ ~ !JJ) then <D1,D2,!JJ> c X ( ~ -elimination)

(v) if D E X, R(D) = '" and ~ E FORM then <C(D,,), (, ~ !JJ» E X ( ~ -introduction)

(vi) i f D E X, R(D) "" Tlcp then < D , q»' E X

·

. "

.

.,

.

..

7

-(vii) if D € X, R(D)

=

~ and x E VAR - u {FV($) 1$ E L(D)}

then <D,VxQ;I> E X (V-introduction)

(viii) if D E X, R(D)

=

Vx~, t E TERM and t is free for x in ~ then

<D,[x:- t } p E X (V-elimination) •

What we have presented here is a simple set-theoretic definition of the concept of derivation tree which is completely formal; it does not de-pend on any extra information or suggestive visual aids.

References

[lJ Van Dalen, D., : Logic and Structure Springer Verlag, Berlin 1980.