On minimal two-letter propositional logic with disjunction

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On minimal two-letter propositional logic with disjunction

Citation for published version (APA):

Bruijn, de, N. G. (1976). On minimal two-letter propositional logic with disjunction. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7612). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1976

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~b59L\

N49 EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

Memorandum 1976-12 Issued October 1976

On minimal two-letter propositional logic with disjunction.

by N.G. de Bruijn. University of Technology Department of Mathematics PO Box 513, Eindhoven The Netherlands

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On minimal two-letter piopositiona1 logic with disjunction.

by

N.G. de Bruijn.

I. Introduction. In this note we shall use the notation of [2J. In partic-ular E*({a,b}; + , A , V ; Q ) denotes the Lindenbaum algebra for

proposi-tional logic with connectives +, A ,v and natural deduction derivation

rules, with alphabet {a,b} and with Q as a set of formulas which are taken as axioms. There is no negation, but if we wish, we can take b to be "fa1sum" and define ~ A as A+b.

With this notation, we can describe intuitionistic one-letter propositional logic by E*({a,bl +, A V ;{b + a}); the axiom b + a is the so-called "falsum rule". (We note that from f- b + a it follows that t- b + P for all p E E({a,b}; +, A , V ) ( this. E stands for the set of all formulas built from

a,b by means of the connectives +, A , V ). The structure of this Linden-baum algebra was described by Nishimura [41 in the form of a lattice. This description can be simplified a little by describing Nishimura's lattice as

the lattice of all closed sets in a particular topological space. In this note we shall expose the material necessary for this purpose; our verification will be independent of Nishimura's paper.

We shall also try to get some further results of the same type. These might possibly help to get insight into the unsolved problem of finding the structure of the Lindenbaum algebra for the case without falsum rule, i.e.

*

E ({a,b}; +, A , V ) .

2. Topological description of Nishimura's lattice. By N we shall denote the following topological space. The point set is the set N = {1,2,3, .•• }, and

the closure cl(S) of a set S(with S eN) is simply the union of all cl({i}) for all 1. t S, and cl({i}) is the set of all j E iN with j = i or j > i+l.

The closed setB of N are (i) the empty set, (ii) the sets cl( {i}) =

{i, i+2,i+3, ... } (i c iN), (iii) the unions cl({i}) u cl({i+I}) ={i,i+l,i+2, ••. } (i E ti). The lattice of these closed sets (with set inclusion as lattice

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We draw two pictures of this lattice. The first one is the one given by Nishimura (D.Scott referred to it as the IIgarden lattice"), the second one

looks less attractive, but is better in showing the connection with

N.

In the pictures, we write i and i,j instead of c1({i}) and c1({i}) u c1({j}).

1

3 s

2

4

3. Exact valuations. In [lJ,[2J the notions of valuations and exact valuations

*

of E (A; ~,A, V ; Q) are explained. In particular we mention that if there is an exact valuation by means of the finitely generated closed sets of a topo-logical space, then the points of that space represent the primes of the Linden-baum algebra.

(a

c E* is called a prime i f (~ is non-derivable and if for all

dis-sectIons (X = fJ, Ay we hav(' (l "" r~ or l~ =y).

We shal I not use any results on exactness of valuations in this note. We do use valuations, however, for showing non-derivability of formulas.

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4.

Abbreviations. The following abbreviations will be used, where A is an alphabet and ~ a set of axioms.

(i) E E(A - + , A,V), E

*

"" E (A

*

*, A, V ; n ).

(ii) I f p,q E E, and i f ~t

t-

p ~,. q and ~ t-- q -+ p, we write p:: q.

(iii) I f p,q E E, and i f both p -+ q == q and q -+ p::: p we write p J. q.

5. Some lemmas.

Lemma I. Assume pEE, q E E, r E E, with r pqq and r qrr. Then, putting

s

=

r -+ (p v q) we have

r

rss,

r

sqs and

r

srr.

Proof. For typographical reasons the proof 1S given 1n Appendix I.

Lemma 2. Assume f t: E, geE, h t: E, k E E, with r gff, t-- fk, r gk. Then t-- kfk.

Proof. Appendix 2.

Lemma 3. The binary relation ~ yxy is transitive. 1.e. if x E E, Y E E, Z E E, ~ yxy and ~ zyz then t-- zxz.

Proof. Appendix 3.

Remark. Another non-trivial binary relation is

r

yxxy, which, however, plays no role 1n this note.

Lemma 4. Assume u

l,u2,u3 E E, and U \ l u2' U21 u3' t-- u3u1u3. We define u4,u5' ... recursively by u. 3

=

u. -+ (u. v u. I) (i ~ 1,2, ••• ). P is the set

1+ J_+2 1 1+

{u.

Ii

E IN} u {u. A U. \

I

i

1 1 1+ IN }. Then we have

(i) u. 1 u. 1

1 1+ (i

=

1,2,3, •.. ),

(ii) I-- u .1.1, U • (whence ~ u. u . )

J J.. J 1 J (i = 1,2, ... ,j > i+l),

(iii) for all p,q E P there exist r,s,tc P such that p -). q :: r, p A q == s,

p v q :: t.

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deduce, by induction, that ~ u. 2u. 3u, 3 for i = 1,2 •... , and that, moreover 1+ 1+ 1+

~ U. ₁₊3u,+Zu. 2 and ₁ ₁₊ ~ u. 3u . ₁₊ ₁₊IU' ₁₊3 for these values of i.

(ii) _{By lemma} ₂ _{(with f}

₌

_{u., g}₁

₌

_u.₁₊_l'_h= u'+₁₂' k

=

u. ₁₊3) we have

~ ui+3uiui+3 (i

=

1,2, ... ). Together with ~ u₃u_lu₃and the ~ ui+3ui+lui+3 obtained under (i), this leads to

r

u.u.u. for all 1,J with j-i

=

2 or 3.

J 1 J

Now by lemma 3 we get it for all i,j with j > i + 1.

(iii) It suffices to take p U

i' q

=

uj ' the other cases then follow by

elementary relations between the connectives.

If j - 1. :: 0 or J - i > 2 we have u.u. _ T, and otherwise we get u.u. u.

1 J 1 J J

(if i - j = by (i), if 1 - j > (by ii); note that ~ pqp leads to pq q).

If u.u. :: T we have u. A U. u. and u. v u _ U .•

1 J 1 J 1 1 j 1

The cases with i = j being trivial, it remains to investigate j

=

i + 1. Trivially u

i A ui+1 E P. Finally we prove ui v ui+J

=

ui+3 A Ui+4 E P. We have

~ U_i (U_i+₃A U_i+₄) and ~ u

i+l(ui+3 A ui+4)(since i+3 - (i+l) > 1), so

~ (u. v u. I)(u, 3 A U

1'+4)' On the other hand, if we start from ~ u. 3 A U. 4'

. 1 1+ 1+ 1+ 1+

we derive ~ u

i+1 v ui+2 (by the definition of ui+4). Since both ~ ui+I(ui v ui+l: and ~ u_i+₂(u_i v u

i+l) (note that we have ~ ui+3' and apply the definition of u

i+3), so by v-elimination we get ~ (ui v ui+I).

Lemma 6. Let p,q,r E E, and p 1 q, q 1 r,

r

rpr. Then abbreviating x

=

q A r, y = P A q we have

!- yx, p :: xy, q xyy, r xyyx.

Proof. For proofs of!- yx, !- p(xy), ~ xyp,

r

q(xyy) we refer to appendix 4. That settles p :: xy. Next we have xyy :: py

=

pep A q)

=

pq; by p 1 q we infer

!- (xyy)q. That settles q :: xyy. Finally q 1 r leads to r :: q(q A r), whence r :: xyyx.

Lemma 7. Let x,y E E, and assume!- yx. Then, abbreviating p = xy, q

=

xyy, r = xyyx, we have p 1 q, q 1 r,

r

rpr.

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6. The Lindenbaum algebra E*({a,b} ; + , A ,

v

;{ba} ).

Theorem. Define u_p u2,u

3 by u l = ab, u 2

=

P as in lemma

4.

Then every e E

E

is (under the axiom ba) equivalent to exactly

one pEP.

Proof. By lemma 7 we have u_l ~ u

2' u2 ~ u3' ~u3ulu3' whence lemma 4 can be applied. Furthermore we have (cL lemma 6) a ::: u

2 A u3' b ::: ul A u2• The

closure of P with respect to + , A , V , as expressed by Lemma 4(iii), now shows that every e E E is equivalent to some PEP.

The fact that the u. and the u. A U. I are pairwise inequiv<llent, is shown

:1 :1 1+

by the valuation v of E into N, generated by veal = {2,3 .... 1, v(b) = {I ,2,3, .•. }.

We obtain v(ba) =

0

as it should for the value of an axiom, and v(u.)

=

d( {i 1),

:1

v(u

i A ui+l )

=

cl({i}) u cl({i+1 }). These values being all different, we observe

the inequivalence of the ui's and u

i A ui+l's.

7. Semi orthogonal systems. In [1], [3J the following situation plays a vital role. It involves a topological space (X,cl) (cl is the closure operator) and a mapping f of X into E such that

(i) if x EX, y lOY, x ~ d({y}), y ~ ({x}) then f(x) ~f(y),

(ii) i f x E d( {y}), y

1

cl( {x}) then f(y)f(x) == f(x), and I- f(x)f(y),

(iii) if both x l cl( {yl) and y c d( [xl) then f(x)- fey).

Let us call such an f semiorthogonal.

We observe that this situation holds for our space N (section 2) and the

mapping f defined by f(i)

=

u. (i

=

1,2, •.. ) where the u. are taken from section 6.

:1 :1

Remark. From lemma I we can derive a more general construction of such a system. We start from p,q,r with I- pqq and q ~ r. Then with u

l = q, u2 = r, u3

=

r + (p A q)

we get u

1,u2,u3 with ul .1. u2' u2 .1. u3' ~- u3u1u3' and lemma 4 can be applied.

Semi orthogonal fts are particularly useful for showing that the structure of the Lindenbaum algebra is the lattice of finitely generated closed sets if we have

(i) For every u 10 A we have a finite set xl"" ,x

n with u ::: f(xl) A

Af(x n),

(ii) I f x,y 10 X and x

:f

y then f(x)

f

f(y).

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8. The Lindenbaum algebra E*({a,b}; -+ , 1\ , v ;{abb}).

This one can be shown to have the same structure as the one of section 6. We take u

J = ab, u2 = ba, u3 = baa, whence lemma 4 can be applied. Furthermore (cf, (i) of section 7) a

=

u

2 1\ u3, b

=

u1' Finally, the valuation v, gene-rated by

v(a)

=

cl({2}) u cl({3}), v(b)

=

cl({I})

has the property that v(u.)

=

cl({i}) (i

=

1,2, •.• ) and v(abb)

=

0.

These

~

things guarantee, by the argument of section 6, that the Lindenbaum algebra has the same structure as the one of section 6.

Remark. Construction$ of systems like the one of section 8 help

to throw l1ght on the unsolved problem of the structure of the Lindenbaum

*

algebra E ({a,b}; -+, 1\ , V ).

The following two lemmas have the same relation to the aXIom abb as lemmas 6 and 7 have to the axiom abo Together they show how the Lindenbaum algebras of sections 6 and 8 can be mapped onto each other. That mapping follows from lemma 10.

Lemma 8. Let p,q,r E E, P ~ q, q ~ r,

r

rpr. Then abbreviating u

=

q 1\ r, v

=

p we have

~ uvv, p _ uv, q

=

vu, r

=

vUU.

Lemma 9. Let u,v E E and assume r uVV. Th~n,abbreviating p

r

=

vuu we have p ~ q, q ~ r, r rpr. Lemma 10. Let x,y,u,v L E. Then we have

r yx, u _ x, v xy

if and only

~ uvv, X _ u, y _ UV 1\ vu

We omit the proofs.

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. Appendix I. Proof of lemma 1. pqq 2 qrr 3 s = r ~ (pvq) 4 !~ 5

r

6 s 7 pvq 8 s 9 rss 10 sq 1 I r 12 pvql 13 r ~ (pvq) 14 q 15 (pvq)q 16

pl

17 pvq 18 q 19 pq 20 q 21 pvq 22 s 23 sqs 24 sr 25

_~

26 sq 27 s 28 r 29 qr 30 r 31 srr (Assumption) (Assumption) (Abbreviation) MP(4,5) MP(6,5) Discharge 5 from 7 Discharge 4 from 8 (from 12) MP(10,13) Discharge 12 from 14 (from 16) MP(15,17) Discharge MP(!,19) (from 20) Discharge Discharge (from 25) MP(23,26) MP(24,27) 16 from 18 1 1 from 21 10 from 22 Discharge 25 from 28 MP{2,29) Discharge 24 from 30

Note: MP(i,j) means that Modus Ponens is applied to lines i and j ; if

i reads x ~ y and J reads x then

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Appendix 2. Proof of Lemma 2. gff (Assumption) 2 fk (Assumption) 3 gk (Assumption) 4 kf 5

~

6 MP(3,5) 7 MP(4,6) 8 gf Discharge 5 from 7 9 f MP( 1,8) 10 k MP(2,9) I I kfk Discharge 4 from 10

Appendix 3. Proof of lemma 3.

yxy (Assumption) 2 zyz (Assumption) 3 4 5 x MP(3,4) 6 yx from 5 7 y MP(I,6) 8 zy Discharge 4 from 7 9 z MP(2,8) 10 zxz Discharge 3 from 9

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Appendix 4. Details of proof of lemma 6. 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 pqq qpp qrr rqq rpr p A q P rp r q A r yx

~

I

p A q xy 16 x 17 q 18 19 20 21 22 23 24

25

26 27

r

q r qp 28 xy 29 30 31 32 33

~

Ip

p y xyy x y p rp r p Assumption

"

from 6 from 7 MP(S,8) from 6,9 Discharge 6 from 10 from 12,13 Discharge 13 from 14 from 17,18 MP(l6,19) from 20 Dische.rge 18 from 21 MP(5,22) MP(22,23) Discharge 17 from 24 MP(2,2S) same derivation as 21 same derivation as 24 from 27,31 Discharge 28 from 32. (x

=

q A r, y

=

p A q).

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• =

=

composite MP's in an obvious manner .

2 3 4 5 yx (xy)(xyy)

r;

xyy 6 (xyy) (xy) 7 8 9 10 I 1 12 13 14 15 16 17

!l

I

XY1 Y xyy Y xy (xyy! (xyyx)1

F

xyyx (xyyx5 (xyYTJ· ~---~~--18 xy 19 20 21 22 23 24 25 26 27 28 29 30 31 xyyx y xyy (xyyx) (xy21 yy

~YX

xy x xyyx Assumption MP(MP(2,3),3) Discharge 2 from 4 MP(8,7) Discharge 8 from 9 MP(MP(6,10),7) Discharge 7 from II MP(MP(13,14),14) Discharge

14

from 15 MP(I,MP(l9,18») Discharge 19 from 20 MP( 18,MP(21 ,MP( 17 ,21») Discharge 18 from 23 from 26 MP(MP(24,27),26) Discharge 26 from 28 MP(I,MP(25,29)) Discharge 25 from 30

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References.

1. N.G. de Bruijn. Exact finite models for minimal propositional calculus over a finite alphabet. Technological University Eindhoven, T.H.-Report 75-WSK-02, 1975.

2. N.G. de Bruijn. The use of partially ordered sets for the study of non-classical propositional logics. Colloque international C.N.R.S., Problemes combinatoires et theorie des graphes, 9-13 Juillet 1976.

3. N.G. de Bruijn. Exact valuations for systems of non-classical propositional logic. To appear in Nederl.Akad.Wetensch.Proceedings Ser A (= Indagationes Math. ).

4. I.Nishimura. On formulas of one variable in intuitionistic propositional calculus. Journal Symbolic Logic 25 (-1960), p. 327-331.