extensional interpretations of modal logics

Extensional Interpretations of Modal LogicsAuthor(s): M. H. LöbSource: The Journal of Symbolic Logic, Vol. 31, No. 1 (Mar., 1966), pp. 23-45Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2270618 .

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rHE JOURNAL OF SYMBOLIC LOGIC

Volume 31, Number 1, March 1966

EXTENSIONAL INTERPRETATIONS OF MODAL LOGICS

M. H. LOB

Introduction. By PL we shall mean the first order predicate logic based on S4. More explicitly: Let Po stand for the first order predicate calculus. The formalisation of Po used in the present paper will be given later. PL is obtained from P0 by adding the rules the propositional constant E- and

E0c, P -*L- F+, r Af

(El. F) -aLisk ,F-(, I") F-10

where (LO. I') stands for the sequence E1fq, . .., L0n, IP being the sequence ?>1, * * *, O>n-'

In seeking a formal interpretation of PL we shall be guided by an intuitive understanding of F1 as a necessity operator, and correspondingly we read nX as 'necessarily b' or '0 is true in all possible worlds'. Our task of interpreting PL will then consist in explicating the notion of a possible world.2

In relation to a given formal system, S say, the notion of a world may be satisfactorily identified with that of a model. Then we are led to define Fnlk, where q is a P-formula, as 'b is true in all models of P'. This approach is, however, not available when b itself contains occurrences of the square. Consider, for instance, the formula Dl Ln0f where 0 is a P0-formula. For the inner formula L1s, as rendered by our interpretation, is not a Po- formula, but belongs to some metalanguage, P1 say. Therefore the notion 'model of P0' is not applicable to it, thus preventing us from interpreting the outer square as we did the inner one. It is natural, however, to attempt to interpret the outer square analogously to the inner one by rendering it, for instance, as true in all of an appropriate class of models of PI. Similarly consideration of formulae such as El D C1i, etc. would lead us to repeat the previous observation and to talk about models of, P2 say, and so on.

Received November 16, 1964. 1 In [8] it is shown that the cut (compare footnote 8)) holds as a derived rule of PL. 2 There are various previous approaches towards this end in the literature, e.g.

[1], [3], [4], [6], [9]. However, to the author's knowledge the present approach is the first which succeeds in making the modal notions of necessity and possibility fully explicit in terms of extensional ones. This point may be illustrated, for example, in relation to the interpretation given in [6]. Here the concept of a model structure has been introduced as a triple [G, K, R], K being a set of models of Po. A world (model) H1 is then said to be possible (with respect to another world H) if the condition H1 RH holds. In this definition R figures as a primitive notion, and we must therefore conclude that possibility has been "explicated" only in the sense that it has been reduced to (defined in terms of) another primitive.

23



24 M. H. LOB

We therefore wish to have at our disposal an infinite hierarchy of meta- languages PO, P1, P2, ... such that the notion of a model of Pi can be defined in Pi+1.

More specifically, we shall choose (? 1) Pi (i 0, 1, .. .) to be a weak version of a ramified type theory of order i and rank i. The union of the Pi's will be denoted by T.

In ? 2 we discuss the arithmetisation of Pi and, in particular, we introduce the following notation: For any formula b of Pi, 0 will be the Gddel number of b expressed as a function in terms of the free variables in ?,.

In ? 3 we define a predicate modi such that the formula modi(Ai+?) expresses in Pi+1 that Ai+1 is a model of Pi. N-models are those models, in which the notations for the non negative integers have their usual meanings. This notion will be expressed by the formal predicates Modi. Lastly, we introduce a yet more restricted class of models - the interpretations. The latter will be characterized by the following property:

All models of PO are interpretations of PO. In general, whenever A1 denotes (under the standard model of PI) an interpretation of PI-,, then any interpretation of P1 assigns an interpretation of P1_j to Ai. The predicate Inti will express the property of being an interpretation of Pi.

We are now ready (? 4) to introduce a translation a of the formulae of PL into those of T. Under a the classical logical constants of PL are trans- lated by the corresponding ones of T. But the translation of O is as follows: If b is a formula of PL, whose degree (i.e. the number of subordinate occurrences of E in b) is i, then the translation of no is

(Ai+s)(Inti(Ai+s) :D Ai+s(oa())).

The main result of the present paper is the theorem: b is provable in PL if and only if C(+) is provable in T. The proof depends on a number of lemmas concerning certain auxiliary systems.

In ? 5 we introduce systems Pi+ similar to Pi, but in Pi+ the predicate variables (with individuals as arguments) will be shown to range over interpretations.

In ? 6 systems Hi are introduced, which are even more restrictive than Pi+, in as much as predicate quantifiers will only be allowed to appear immediately to the left of their respective variables, i.e. (Ai) Ai(#) is a formula of Hi+, but, for instance, (Ai) (?y v Ai(0)) is not. Nevertheless any formula of Hi provable in Pi+ is seen to be provable in Hi. The usefulness of the Hi's or rather that of their union H consists in the following: the translation under a of any formula of PL corresponds naturally to formulae of H. Thus our main theorem is eventually established by showing that any provable formula of H has a proof, in which the universal predicate quantifiers are introduced in conformity with the conditions regarding their use imposed by a.



INTERPRETATIONS OF MODAL LOGICS 25

In ? 7 we give a brief indication of how the methods of this paper may be adapted to provide an interpretation for the first order predicate calculus based on M [12].

?1

The systems Pi. Types: The natural number n is a type of rank n (we shall also refer to n as the rank n). If oa is a type of rank < n, then n + 1 (a) is a type of rank n + 1. (Compare [11] p. 245)

Italicized subscripts range over ranks. Greek subscripts range over arbitrary types. x, y, z, ... are individual variables. Ai, By,..., are predicate variables of ranks i, j, . . .,respectively. Ci(a), Di(#), . . . are predicate variables of types i(0c), j(3), . . and ranks

i, j, . . . respectively. b is said to be a PO-predicate expression of n arguments if the expression

b (xi, . . ., xn) stands for a PO-formula (defined below) containing the free variables xl, . . ., x,.

Prime formulae. PO-prime formulae are expressions of the form Ao(x, y), Bo(z, u),.

Pi-prime formulae are (i) PO-prime formulae, or (ii) expressions of the form Aj(x), Bk(y), ..., Vz(z) (O < j, k, .. .,1 < i), or (iii) expressions of the form Fj(x) (Ca, x) where j(a) is a type such that

j < i. (Thus, Schichten < co of [11] exceed our ranks by 1.) Pi-formulae. b is a Pi-formula if (i) b is a Pi-prime formula, or

b is of the form (ii) -,y, or (iii) py v 0, or (iv) (x)?y, or (v) (Ap)y, the rank of oa being less than i, where my and 0 are Pi-formulae. Expressions of the form b & VI, b v VI, b =p, (EX) will be used to

abbreviate -(-(no v -Iy), -iX v VI, -.1(-I v -IV) v -1 (0 v ?y), -1(X)-- respectively. b is a formula of rank j, if b is a Pj-formula, but not a Pi-,-formula. If F and A are finite sequences of Pi-formulae, then the expression

F -r- A is a Pi-sequence formula. Axioms of Pi are all expressions of the form b - , b being a Pi-prime

formula. Rules of inference of Pi

A Thin ,, A S Thin A



26 M. H. LOB

I k, 0b, F- A Cont ?,

? Perm E,0 0, y, r,

NA

OA

AA (x) b(x),F-A

~(X+X), Fr?

APA (X), (Aj) 0(Y), rP A

Case (i) j = 0; here X stands for a predicate expression of- n arguments and Y for Q(Ao), where Q(Ao) is a fixed predicate expression with n free variables and Ao being the only free predicate variable. An explicit form of Q(Ao) may be obtained from [10].

Case (ii) j> 0, X stands for a predicate variable Bh and Y stands for A1, h ? j.

AQA O(Bh(a)), r -- A A (Bj(x))b(Bj(x)), P -h A

4 ? j

rP a/\+Al S Cont -

S Perm r ' '"ix3 FP E) , A

NS '

-- -A, V, y FP -?> A, 0 vY

AS r A, +(Y) F -? A (x)+b(x) provided y does not occur free in the conclusion.

APS r -?A, b(B1) r -\ A, (A1) b(Aj)

By does not occur free in the conclusion and for j = 0 the number of arguments of By is equal to that of A>.

AQS r -> A, O(Bj(a)) 4 r -> A, (Aj(x))h(Aj(a))

Bj(x) does not occur free in the conclusion.5

?2

Arithmetisation of Pi. If e is an expression not containing free variables e stands for the Gbdel number (henceforth: G.N.) of e with respect to a fixed Gddel numbering of Pi (i 0, 1, 2, . . .). If e contains

3 The rules A-Thin, S-Thin, A-Cont, S-Cont, A-Perm, S-Perm will be said to represent structural inferences.

4 In AS, APS and AQS y, Bj and Bj(,) respectively, will be said to be proper variables.

5 The formulae explicitly displayed in the rules will be called principal formulae.




the free variables $1, ..., an (only) of arbitrary types, then e stands for the function g($i, ..., n), which is such that if e' is obtained from e by replacing tI, . n.., A by closed formulae $j, ., o Of appropriate types then e' g($i, .., $1)

For each closed Pi-formula i, we can find, a Po-formula ipp(u) being the arithmetized version of the description of the syntactic structure of b. Then let Aj(q) stand for (x)(ipp(x) D Aj(x))

In particular, if q is the formula with G.N. k Vp(u) will be taken as the formula Nk(u) standing for

(SO) (A(So) D (Evo) ... (Evk-1) (w) {So (w, VO) & So (vo, vI) & ... & So (Vk-1, u)})

where A(So) stands for Peano's axioms, restricting So to the successor relation between non negative integers.

If b is an open formula with G.N. k containing the free variables ... Xam, xi, . . ., xn only whose G.N.'s are gi, . . ., gm, fi, ., fn,

respectively, then A1($) may be expanded into a formula of the form

(z){(Eu) 6fi & Vyp(u) v A1(z)}, where Vp(u) is given as before and E stands for

(EFjl(l)) ... (EFjm(m))(EFm +1) ... (EF +n))(Es,) ... (Esm)(Eti).** (Etn) (Eyi) ... (Eym) (Eu1) ... (Eum+n-i) (Evi) ... (Evn) {Ng1(sl) & sub(u, sI, yl, ul) & ...

& Ngm(sm) & sub(umi1, Sm, ym, um)

& Nf1(ti) & sub(um, ti, VI, Um+i) & ...

& Nf.(tn) & sub(um+n-i, tn, Vn, Z)

& Fjf(rC1)(X0CJ1 yi) & ... & Fjm(,xm)(Xm, yin)

& G[;(l,)(Fl(al)) & ... & GU(m)](Fm(am))

& Fm '(xi, vi) & ... & F +n (xn, Vn) & G[0](Fm+1) & ... & G[n](Fo +")}.

Here sub(u, s, y, z) expresses that z is the G.N. of the expression obtained by replacing the variable with G.N. s by the expression with G.N. y through- out the expression with G.N. u.

GU(,] (F1(,)) expresses that for any XO; and y Fj(o;)(Xx,, y) is the relation holding between the object XO; and the G.N. of rFXIt, i.e. the formal expression denoting X,.

?3

Models. A predicate A1+1 is a model of Pi if it satisfies the following conditions:

ml. (Bo)(ECo)(x)(y){Aj+i(Bo(x, y)) - Co(x, y)},



28 M. H. LOB

m2. (Bh+1) (ECh+l) (x) {Aj+1 (Bh+l (x)) Ch+1 (x)}, m3,; (Fh+,(ox)) (EGh+,(ox)) (XO;) (x){Aj+l (Fh+,(;) (Xoax)) =:Gh+ (ox) (Xx;, x)}.

for each h < i. If S and ip are Pi-formulae

m4. Aj+1 (-oh) =nAj+1 (), m5. A1+1 (S v tp) A1+1 (I) v Aj+1 m6. Aj+i((x)b) = (x)Aj+i(q), m7,,. Aj+l ((Boa)+) =_(Bx)Aj+1(0),

modi(Aj+i) stands for the universal closure with respect to all free variables except A1+1 of the conjunction of ml.-m7,.

N-models. Let Num(x) be a Pi-formula expressing that x is a non- negative integer. A model A1+1 will be said to be an N-model if it satisfies the following conditions :6

n1. (So)(A(So) D) (u)fAj+,((x) -,So(x, u)) (X) -,So(x, u)}), n2. (So)(A(So) z) (x)JNum (x) D) (y)(Aj+1(SO(x, y)) ==SO(x, y))}).

Modi(Aj+i) stands for the conjunction of the universal closures with respect to all free variables except A1+1 of n 1 and n2 together with modi(Aj+i).

Standard model. V1+1 is a standard model of Pi if it satisfies the following condition:

vi. Modi(Vj+z) v2. Vj+i(Ao(x, y)) Ao(x, y), v3. VJ+1 (Bh+l (x)) Bh+-(x), v4. Vj+1 (Fh+l (a) (XO, x)) Fh^+ (ox) (X., x),

where v3. and v4. hold for each h < i. St Modi(V1+?) stands for the conjunction of the universal closures with

respect to all free variables except V1+1 of vl-v4. M-terms. Predicate variables of rank i > 0 are M-terms of level i and rank i.

If V is an M-term of level i and rank < / and % is an M-term of level j> i and rank k, then WD1P is an M-term of level i and rank k.

If no variable in the M-term Di of level i occurs bound in the formula

(fi~'), then this formula stands for

(ECi) {[(x) (9(9(x)) =Ci(x)] & O(Ci)}. Interpretations. Interpretations of Pi form a more restricted class of

N-models. Their essential property is that predicate variables of rank < i, which under the standard model range over interpretations, do so also under all interpretations (of rank > i).

6 The concept of an N-model was introduced by Kreisel in [5].




This concept is formally defined by recursion as follows:

Into(Al) -= Modo(Al),

Inti(Aj+i) _ Modi(A1+?) & (k)?, 2(Bh){Inthl(Bh) D Inth.l(BF3+1)}7

?4

Translation of PL. Let T be the least system containing all Pi's as subsystems. We define a translation of PL into T, i.e. a function a mapping the formulae of PL into those of T, as follows:

Let b and ip be subformulae of a closed formula g- of PL,

a(b) = Df , if b is a Po-prime formula, or(-nb) =Df -r(b), a(b V V) Df O(b) V (Y), ((X)O) Df (x) {(Aj) [Into (Al) D Al (Do (x))] D a(o)} otherwise,

Do being a PO predicate variable which is not otherwise used in PL.

a( LI0) =Df (Ai+,3){Inti(Ai+3) DAi+3(o(0)}, where the degree of b is i.

a(F A) =Df F A,

where Po and AO, are obtained by replacing each component b of F and A by a(+), respectively.

THEOREM. F -> A is provable in PL if and only if a, rF5-> Aa is provable n T. Here as denotes the existence postulates in the definition of A* (p. 32). For the proof we establish several lemmas regarding certain auxiliary

systems.

?5

The systems Pt. The symbols of P+ are the same as those of Pi apart from the following changes:

No predicate variables of composite types occur, i.e. only the symbols Ai, Bj, ... are predicate variables.

V1 (j 1, 2, ..., i) are predicate constants. The definition of M-terms in Pi will run as before except for reading

the first sentence as 'Predicate variables of rank i and Vi are . . .'. Moreover,

the bar (e.g. in 9 or )(0)) will in Pt be used as a primitive notation. Gothic letters stand for M-terms of Pi+ unless otherwise specified.

7 Here (h),2 0 4 stands for b1 v ... v ei.



30 M. H. LOB

The set of formulae of P+, Ki say, form a subset of the set of Pi-formulae defined as follows: All Po-formulae belong to Ki. If a formula b of rank h belongs to Ki, then so does 9R(O), where 9N is an M-term of level j (h < j) and rank I (I ? i). If b and ip are formulae of Ki, then so are n,, b v ip, (x) b, (A1) b (j < i).

Axioms of Pi+ are formulae of the form + b, or of the form VW() 9X(O), where b is a Po-prime formula, and 9N (b) a formula of Ki.

Rules of inference of P+. These are obtained by adding the following rules to those of Pi and omitting APA, APS, AQA, AQS.

MNA r

, MNS 9X(q), F A 9RHA) r A r/ ~+

MOA .N -, A, MOS ____

9N(O (t)), r 9 r (O w(y MAA )(b(t)),Fr A MAS r ?A4 w(y))

9U ((x)q(x)), r A r A, 9N ((x) x))

y does not occur free in the conclusion.

AMA O(S ), r - A, O(Bj+i)

(A1)O(Aj), F +A' AM F ~A, (A1+1)O(Aj+1)'

the level of % being ? j. B1+1 does not occur free in the conclusion.

MAMA 9N (O(),

r -? A MAMS

r -? A, 9R(O(B1+1)

9)((A1)O(A1)), F A F -> A, 9R((A1+1)O(A1+1))

the level of 1 being ? j. B1+1 does not occur free in the conclusion.

VA Vsq)F~'F +,1~) VAi, r A, V r A\, VJA'

MVA W F0 ) MVS qX(V1(O)), r \A F A (Vi(o))

in the last four rules the rank of S is less than 1;

MA 29(q),Fr?A MS F-?A,2()

MMA M(S)), F - A F A, 1(2?)) (2(A)) I -

A FMS A, % (21 (0))




LEMMA 1. For any N-model A1 of Pi

Aj(Bk(b)) =

the rank of b being less than j. The formal proof though rather laborious is routine. We shall indicate

it only for the case when S is a closed formula. Then a more expanded version of AJ(Bk(q)) is

Aj((u)(Vo(u) z) Bk(U))).

In view of modi(A1) this is equivalent with

(a) (u) JAj (Vp(u)) D) Aj (Bk (U))} . But Vp (u) is of the form N.(u). Thus in view of Modi(Aj) we have

Aj(Vp(u)) = Aj(Ns(u)) - Ns(u) = (u). Hence (a) is equivalent with

(u) (Qy(u) D A (B k

which is an expanded version of F3 A -

LEMMA 2.

(Ah+l)(EBh+2){JIith(Ah+1) D Inth+i(Bh+2) & ($) (Ah+,($) Bh+2($))}

where $ ranges over the g. n. s of Ph-formulae. Informally: For each interpretation Ah+1 of Ph there is an interpretation

Bh+2 of Ph+1, which agrees with Ah+l on all Ph-formulae. Let Ch+2 be an arbitrary predicate variable of rank h+2. We impose

the following conditions on Ch+2:

(a) For any PA-formula b let

Ch+2(q) Ah+l(0), (b) Modh+l (Ch+2) .

Requirements (a) and (b) are clearly consistent. We prove that jointly they imply Inth+l (Ch+4.)

Suppose Mk is an interpretation of Pk-1 (k ? hl-1) then by lemma 1

(C) Ch+2(Mk (f)) =M- + (9 ) for any Pk-1-formula py.

From (a), (c) and observing that Ah+l is an N-model we can by an inductive argument show that Mkh+ 2 is an N-model of Pk-1, i.e. Modk-1

(M5ih+2). Moreover, suppose that Mk is an interpretation of Pk-1. Then from the definition of Int we get

Intk-l(Mk) & Intl-,(DI) & Inth(Ah+l) o Intl-, (Dm k),

o Intl-, (Dm k h

D Intl- (Dkah+2)

D Intl-, (D(MkCh+ 2))



32 M. H. LOB

Therefore

Intk1l(Mk) & Inth(Ah+l) D (l)<k-l(Dt)(Intj-q(Dt) D Intl-j(D(mk^+2))

and in view of Modk-l(M^h+2) we have

Intk-l(Mk) & (Inth(Ah+1) D Jntk-l(Mkh+ 2))

and hence

Inth(Ah+l) D (k) 5--(Mk)(Intk-1(Mk) D Intk-l(Mah+ 2)).

Thus together with Modh+l(Ch+2) we get

Inth(Ah+l) D Inth+l(Ch+2).

Using (a) we obtain

Inth(Ah+l) D Inth+l(Ch+2) & (Ch+2(q) Ah+l(0)),

and hence

(Ah+l)(EBh+2){Inth(Ah+l) D Inth+l(Bh+2) & ($) (Bh+2 ($) Ah+l($))}

This completes the proof of lemma 2. Definitions. We shall call any occurrence of a subformula a of a formula q

an oblique occurrence of oc in any formula of the form 0(q). Moreover, any oblique occurrence of a formula a. in a formula q will also be said to be an oblique occurrence of a. in any formula of the form 0(q).

Let q be a formula of T+ and let q* be obtained from q by replacing each subformula (including all subformulae occurring obliquely in q) of the form

(Ai+1)Ai+1(/) by (A+1) (Inti(Ai+z) o Ai+(0)).

Let r* > A* be obtained from the sequence formula r -> A by replacing each component q of r or A by q*, respectively, and for any predicate variable, B,+j say, free in r or A let r* contain a component of the form Int,(B,+z), and, moreover if V1+j occurs in r or A let r* contain a component of the form St Mod1(Vj+i). Further r* shall contain the formulae

_Q/ (EVk+l) St Modh(Vh+l), Oh, (0 < h < k)

k being the maximal rank of the formulae of r and A, Oh the formula of lemma 2, and corresponding to each individual variables, x say, occurring free in r or A a/ shall contain a formula

(Ai){Into(Aj) D Az(Do(x))}.

LEMMA 3. If r -? A is provable in Pi+, then r* -? A* is provable in Pi. We proceed by induction with respect to the length of a shortest proof

of r -* A, i.e. the least number of applications of the rules of inference of Pi in any proof of rP - A.




For axioms (i.e. formulae of proof length 0) the lemma holds obviously. Let the length of a shortest proof of r -> A be n+ 1 and suppose that

the lemma holds for all sequence formulae the length of whose shortest proofs is n.

Suppose also that the last step in a shortest proof of r -> A is of the form

rl -A1 [F2 -> A2]

r H>A

(Bracketed formulae may occur vacuously). Then by the inductive hypothesis rF* - A*, [rF -> A*] is (are) provable

in Pi. We show for each rule of Pi+ that r* - A* is derivable from

Fl -?AI, [P* -?A*] in Pi. For the non M-rules (by M-rule we shall mean any rule whose first

reference letter is M) other than AMA is trivial. For the rules MNA, MNS, MOA, MOS, MAA, MAS, MAMS, MVA, MVS the verification is routine and we omit it.

The validity for MMA, MMS, MMMA, MMMS follows from lemma 1 by induction over the structure of M-terms.

Next we consider AMA, i.e.

c0(3)), PF -? A (Aj)q!(A,), r -? A

where the level of 3J is h ? j. By the inductive hypothesis

(a) 0 (9) )*, 0 ->A*

is provable in Pi, where q(3))*, 0 stands for (O(3), f)*. Since r* contains a formula Ints(Bs+z) for each variable B8+1 in 3) it follows from the definition of Int1 that we can prove in Pi

() i-\A*, Inth().

From (ac) and (p) we obtain by NA and OA

(y) Inth(V) D f(), 0 -i A*

From MMA, MMS, MMMA, MMMS we see that the operation s is associative. Hence if Ch+j occurs in 3) as the variables of lowest rank, we get from (y)

Inth(Ch+1) D b(Ch"+) 0 - A*.

for a certain M-term R of level h+2, which may be expanded into

(Dh+1ffl(X)(%(Ch+1(X)) =-Dh+l(X)) & Inth(Dh+l) D b(Dh+l)*}, 0 -A A

But the provability of this formula in Pi implies the provability in Pi of

(6) (Dh+l){Inth(Dh+l) D f(Dh+l)*}, 0 -A A*.



34 M. H. LOB

By definition of r* we have Oh in the antecedent, i.e.

(Dh+l)(EAh+2){Inth(Dh+l) D Inth+l(Ah+2) & ($)(Ah+2($) = Dh+l($))}

From this formula we get by a simple inductive argument

(Dh+1)(EAh+2){(Inth(Dh+1) D Inth+l(Ah+2)) & (q(Ah+2)* 5(Dh+l)*)}

and hence

Dh, (Ah+2){Inth+l(Ah+2) D b(Ah+2)4} -} (Dh+l){Inth(Dh+l) D(Dh+l)*}a

From (6) and the last formula we get by using the cut rule8

(Ah+2){Inth+l(Ah+2) D b(Ah+2)*}, 0 A-

Repeating this argument a finite number of times we obtain the required formula

(Aj){Intjl(Aj) D 0 (A) *}, ( . *.

Definition. If r is the sequence of formulae k1, ..., On let (3), r) stand for the sequence 3)(91), ..., 9(+n)

LEMMA 4. If r -* A is provable in Pi+, and 3) is an M-term of rank j and level k (i < k), then (S, r) -? (S., A) is provable in P1.

We prove the lemma by induction with respect to the length of a shortest proof of r -> A.

If the length is 0, i.e. r -> A is an axiom the lemma clearly holds. Suppose the lemma holds for formulae whose shortest proofs are of length

n and let there be a shortest proof of r -? A of length n+ 1. Suppose the last step in such a proof is

r ->i, ()

%HA} r -> By the inductive hypothesis

(a) (91, r) -> (91, 0), 0(1(q)) is provable.

Using the final formula of the following (partial) proof

_P) __ (0 (MMA)

RMw0) >WR (0)

together with (a) we obtain by means of a cut

8 By the cut rule we mean the inference scheme

rP-A,)fr --+ A

r, -A, A By methods similar to those of [2] it may be shown that the cut holds as a derived

rule in Pi.




(b) (9R, r) - (9, ) (0(q) ). From (b) we get by MNA

w19(-i)} f, r) -? (ifT ?), whence we derive the required formula

RM-10i)), PR. r) ->(S., i). Analogous arguments for all other inference rules establish the lemma.

LEMMA 5. If r* > A* is provable in Pi, then r -? A is provable in Pti. It is easy to recognise (though a detailed formal verification would be

laborious) that the M-rules restrict the ranges of the predicate variables of rank j+1 to N-models of P1. Therefore, in particular, the predicate variables of rank 1 are restricted to Into. Suppose the ranges of the predicate variables of rank j+- < h are restricted to Int1.

Now in Pi+ we can prove

(Bad,) (ECj+,) (Ah(Bj+l (0)) =:Cj+l 0)

uniformly in q. From the inductive hypothesis we get

(B1+i){Intj(Bj+1) D (ECj+z)Intj(Cj+1) & (x)(Ah(Bj+l(x)) =Cj+(x))l which is equivalent with

(Bj+,){Intj(Bj+,) D Intj(f3Ah^l)}

whence we have

(Y) 5,(Bj+j) {Intj (Bj+j) D) Intj(f3Ah^l)}.

From this formula together with Modhl(Ah) we obtain Inth-l(Ah) Thus the range of the predicate variables of rank h, and hence inductively

the ranges of all predicate variables of Pi (i - 1, 2, . . .) are restricted to interpretations. But this is what is asserted by the lemma.

?6

The systems 1i. Let Ki be a subclass of formulae of Ki defined as follows: All PO-formulae belong to Ki. If a formula q of rank h belongs to K* so does 9X(q), where 3J is an M-term of level j (h < j) and rank I (I ? i). If k and p are formulae not containing free predicate variables belonging to K7, then

m+, c v A, (x)+ belong to Ki. If A1+z(0) (j+1 < i) belongs to K7, so does (Aj+1)A,+1(i). Let HIb be the system obtained from Pi+ by replacing

AMA by AMS by

AMAn M A j r A

AMS-n r A, Aj(q)

(A,)A,(~), PF -? A F -?,A, (Aj)A,(~)



36 M. H. LOB

j < i and 3) is an M-term of rank j < i and As does not occur free in < i and level <j. the conclusion.

MAMA by

MAMAM a (3(q)),Pr A %((Aj)Aj(b)), F A

The condition on 3) is the same as in AMAa. W is an M-term of level greater than the rank of 9N and its rank ? i. MAMS by

MAM~~z -? a, 3J(AjSqS)) MAMSa __

F- A, U((Aj)Aj(q)) As does not occur free in the conclusion. 9N is an M-term of level > j and rank < i.

LEMMA 6. If f is a K7-forinula provable in Pi+, then f is provable in Hi. This lemma follows easily from an analysis of the formulae of K7.

Let II be the least formal system containing Ho, H1,, ... as subsystems. Let Ka be the least set containing K', K', ... as subsets. Let T be the

mapping of the formulae of PL into Ka, which satisfies the condition

[T(o)]* d(+),

where c is an arbitrary PL-formula. If F -> A is a sequence formula of PL where r and A stand for yl, . . ., ym and 61, . . ., bn respectively, then T(r (-- A) stands for the 11-sequence formula FT -* AT where rT is the sequence T(yj), . . ., T(ym) and AT the sequence T(Q1), . . ., T(Qn).

LEMMA 7. If a PL-sequence formula g is provable in PL, then T(!) is provable in 11.

We proceed by induction with respect to the length n of shortest proofs. When n = 0 the lemma holds.

Let the proof of g be of length n+ 1 and suppose as inductive hypothesis that the lemma holds for all formulae whose shortest proofs are of lengths ? n. We consider two cases: the last inference in the proof of i> is

(a) a PO inference, (b) an instance of I or II. In case (a) the lemma is easily seen to hold for >. Case (b). Suppose the last inference in the proof of t is I and hence

of the form +,r FH-A

Eok, F -Ad




By the inductive hypothesis (i) -{(o), IT -> Ar is provable in II. We attach to the proof of (i) the following ending

(, Ir A'T VA V1(T)), f1T -- Ar M v(T()) of AMAn

(A,)A,(Tz()), rT -AT

Thus we have a proof of T(Ei0), IT -tAr in H. Suppose that the last inference in the proof of g is an instance of II

and hence of the form

F001, * + , O Ms A

By the induction hypothesis

is provable in HIi+,. Whence by lemma 4

(iii) B,&((A11)Aj1(T(ql))), . ., Bh((Aj8)Aj8(T(0s))) -*Bh(())

and hence

(iv) (Bh)Bh((Aj1)Aj1(r(0q))), . * *, (Bh)Bh((A-)8(T(0s))) Bh(r(V))

is provable. Now any well-formed formula of the form

(A1)A1(O) -(Bh)B((A*JAJ(O))

is provable in H, as shown by the following derivation

AJB"(O) -?A.(G) MMS A -(O Bk(AJ(O)) i- MAMSn

(A,)A,(O) B-((A*)A ( M)) AMSn (A,)A,(O) (Bh)Bk((AJ)AJ(O))

Hence by using the cut repeatedly we get from the provable formulae

(Aj,)Aj,(-( -* (Bt)Bt((Ajl)Aj(r(0))

together with (iv) we get

and thus also



38 M. H. LOB

If the rank of -(V) is h-1 (vi) is the required formula. If the rank of ,-(ip) is k-1 < hk1 we note that

(vii) (Bh)Bh(T(V)) -? (Ck)Ck(T(V))

is provable in view of the derivation

Ck(T(y,)) -*Ck(T(?r)) AMAn (Bh)Bh(-r(y)) Ck(T(Yr)) M

(Bh)Bh(r(wy)) (Ck)Ck(TQ(p))

From (vi) and (vii) we then get by means of a cut

(A!JAjl('r(01)), - * * * (A.)Ai.(r(0s)) -(Ck)Ck(r(Y)) .

Definitions. A variable Ai+, will be said to be an Ai+,-term. If W is an

Ai+,-term, then so is any M-term of the form 3'. Formulae of the form R(q), where 3J is an Ai+,-term, will be said to

be basic formulae of kind Ai+,. We shall call formulae of the form (Ai+,)Ai+, (q) Q-formulae.

LEMMA 8. Let rT -+ AT be a formula on the proof-tree for a formula T(,) such that all components of rT and AT are basic formulae or possibly Q-formulae (i.e. Q-formulae need not occur). Then if rT AT is provable in HI without the use of VA and VS, so is the formula obtained from rT AT

by deleting all basic formulae except those of one kind. The lemma holds for axioms. Suppose it holds for all formulae whose

shortest proofs are of length < n. Let the shortest proofs, one of which is D say, of rT --AT be of length n+ 1.

We distinguish two cases: (a) The last formula of D (i.e. rT -? AT) is obtained from If -> A' by

a one premise inference rule. By the inductive hypothesis the lemma holds for Pr -, Ar, and it is easy to see that every one premise inference transmits the property asserted by the lemma from premise to conclusion. Thus in this case the lemma holds.

The last step of D cannot be an OA. For by the conditions of the lemma no formula of form f v iy occurs in Tr.

(b) The last step of D is an MOA. Using again the inductive hypothesis for the premises the validity of the lemma follows easily for the conclusion.

LEMMA 9. Any provable formula P -> A of [H, which is the translation of a sequence formula of Pt, has a proof, in which all instances of rules, any one of whose principal formulae are of the form V(0), where M contains at least one predicate variable (henceforth: M-inferences), precede all non M- inferences.

Let Il and I2 be consecutive inferences in the proof of P -? A, Ii being a




non M-inference, I2 an M-inference, and suppose I, is of the form

Z(ai) [Z(a2)]

Here Z(oc) and C(oc) represent sequence formulae and the ac's stand for the principal formulae. The formula in brackets may be void.

Now a cannot be the principal formula of I2 since it does not contain free predicate variables of rank > 0. If fil is a principal formula of I2 the succession of I, and I2 has the form

Z(oc, Pu) [Z(a2, l)]I

Q, f1) [E(Y(, P2)] '2

The only inferences imposing conditions on g are MAS, AMS3 and MAMS3.

Hence if I2 is not one of these we have that

Q(al, Pu) [(o(al, P2)]

'2 (oY, E) [(a2, 9)] I g- (, A) 1

represents an admissible sequence of inferences and it is easily seen that Z(ocl, fli) and (o(a2, /) are provable if and only if Z(a2, flu) and H(&, P2) are.

Suppose now that I2 is an AMS3. Then the sequence I1, I2 takes the form

Zi(ocl, Aj([)) -[t@(*-, Aj([)) Ii (oc A(0)) I2

(&, (A)A($)) '2 This may be changed into

Z(ocl, Aj([)) '2

Zi(ocl, (Aj)Aj(+)) [Z(a2, (Aj)Aj(+)] QY(oc, (A ) A i(6) )

In the case when I2 is an MAMS3 we argue analogously to the previous case.

Suppose next that I2 is an MAS, i.e.

Z(ai, MOM(y)) [Z (a2, 9N(O (Y))) I

H(0- 9N(O (Y) ) )

H(0-, 9N((x) 0(x)))

If acl does not contain the variable y free, Il and I2 are clearly inter- changeable. If acl contains y free, then as -a cannot contain free occurrences of y by the variable condition on MAS the present scheme Ii, I2 must be



40 M. H. LOB

of the form

(i) V(Y), r ,* , 9N (y)) I1

(ii) (x)p(x), r A, M((y)) 2 (iii) (X)?p(X), r ?A, 9((x)b(x))

Here 9N contains at least one predicate variable. Instead of ip(y) and (x)ip(x) we may also have %(ip(y)) and %((x)ip(x)), where 9 does not contain predicate variables (i.e. 9 = Vq* *Vr) without affecting the subsequent argument.

Since (x)p(x) is the translation of a P+-formula ip(y) is of the form

A1(D0(y)) v y(y). Thus from the provability of (i) follows that of

(a) r -? Al(Do(y)), A, 9D1(0 (y))

In view of the condition on the proper variable of I2, y cannot occur free in (a) except as shown. We may, moreover, assume without loss of generality that the proper variables of any AS or MAS prior to (i) are distinct from y.

Hence the formula

(Al)A1 (Do (y))

can only occur in the antecedent of any sequence formula F belonging to the proof of (a) whenever there are inferences below F, as a result of which this formula becomes the displayed part of formulae of the form

(z){(A,)A,(Do(z)) D) O(z)),

respectively. These inferences must clearly include AS's or MAS's whose proper variable is y. But we have seen above that there are proofs of (a) in which such inferences do not occur.

Hence no formula in such a proof of (a) contains Do(y) as a constituent of the antecedent. Therefore, the occurrence of Do(y) in (i) has been introduced by an S-Thin and so

(b) r ->A, 9N (y)) is provable. From (b) we obtain (iii) as follows

rP-*A 9N w(O(y)) r -\A, 9((x)q0(x))

(x)v(x), r ?A, 9R((x)O(x))

Thus we see that in all cases M-inferences which are preceded by non M- inferences may be moved up in the proof. Hence by repeated performance of this step the type of proof asserted by the lemma to exist may be obtained.




LEMMA 10. If r(g) is provable in II, then t is provable in PL.

Proof by induction with respect to the length of shortest proofs. Suppose r(t) is provable in II. Lemma 9 permits us to assume that in

the proof of --(St) all M-inferences, if any, precede all non M-inferences, if any. If the proof of -r(t) does not contain any M-inferences, then formulae

in -r(t) containing predicate variables of rank > 0 have been introduced by thinnings. Hence the formula r(t'), which is obtained from r(f) by deleting all formulae containing predicate variables of rank > 0, is provable in PO. Thus, a fortiori, S' which is identical with r(,') is provable in PL.

From t' we derive ) in PL by thinnings. If the proof of -r(t) contains M-inferences there are formulae (henceforth:

critical formulae) which are the conclusions of M-inferences but not premises of M-inferences.

Now if for each critical formula T-(O) ? is provable in PL, then ! must be provable in PL. For either S is identical with t or else the proof below -r(0) consists of non M-inferences only.

Any non M-inference is (i) one which is also available in PL or (ii) it is a VA or VS or (iii) an AMAz of the form

9J(r(+b)), P -TAC

(Aj)Aj( ( ])), -T oA

where 93 does not contain any free variables. In other words all constituents of 9R are V1's.

We delete all VA's and VS's from the proof and change all AMAn's of the form (iii) into

0b, PIF

Hence if the translations of critical formulae are provable in PL, then so are the translations of all formulae provable in II.

Suppose then that z-(0) is a critical formula. z-(0) must be of the form

(All)Aj' (-r(0j) ) , . , (Ah) Ajh (-r(Oh) )

(B ;) Be Iv ,*** (B k)B Bt(-(Vk)) v

Since this formula is provable, then so is the formula

(Awl) Awl (Ti) ), * ,(Ash)(Ai(T(oh) ) ->Bjjr(y~) ), .B , k(r(Yk) )

and thus, by lemma 8, we have the provability of the formula

for )A(T(0)),

somr(A?)Ak(()) . r(T(Vr))

for some r k.



42 M. H. LOB

From this we see that

h ) ( 1)),* Vir(> (Ir))

and hence also

(All)All (-r(0j)), ... (Aih,,)Aih.(-r(h) ) -- -(tr)

is provable. Thus by the inductive hypothesis

01k, I I I, ) >h Or0

and from this

F201, * * *, F- h > 14r,

and hence also

F0?1, *** , Fo?h Fall, D*, tk

is provable in PL. But this is the translation of the critical formula T-(w). Our main theorem is an immediate consequence of lemmas 3, 5, 6, 7, 10.

?7

The system PM. Let PM denote the first order predicate calculus based on von Wright's system M. More specifically, PM is obtained from Po by adding the rules I and

II', r - (D' r) -co

We wish here to indicate briefly how the methods of this paper may be adapted to obtain an interpretation of PM.

We introduce a translation X of PM into T, such that X agrees with a

but reading Mod1 for Into and the closure of Am+,(... (Al(Do(x)) ... ) for that of A1(Do(x)), m being the degree of the longest formula containing the respective occurrence of (x)+ not in the scope of O-I.

We now proceed to introduce the system Pi+, which are like Pt except that AMA is replaced by

AMA ~O(Ah),PIF-->A AMA (A1)0(A), I -- A <A'

MAMA is replaced by

MAMA ~19(O(Ah)), IF - A

9R((Aj)0(Aj)), r --A AMS is replaced by




AMS P -> A, O(Ah?1) AMS - A, (A1+1)0(A1?1) h ? i;

and MAMS is replaced by

MAMS P -->A, ' (O (Ah+l)) , h j. M -MS A, - D1((Aj+?)0(A1+?))

Then analogues of the lemmas of ? 5 hold with obvious modifications of the proofs.

In particular, to the modified proof of lemma 3 the following argument is appended:

We clearly have

(A1+1){Modj(Aj+i) D (EAh+l)(Modh(Ah+l) & [Ah+l(0)-Aj(4)])}

if h < j, and hence by induction with respect to the number of logical constants we get

(Aj+i){Modj(Aj+i) D (EAh+l)(Modh(Ah+l) & [O(Ah+l) O(A1+1)])}

provided O(Ah+l) is a formula? of Pt. From this formula we easily get

(Ah+1) (Modh (Ah+1) D 0 (Ah+l)) D) (Aj+,) (Modj (A;+,) D 0(Aj+,)), whence by the following sequence of inferences

Modh(Ah+l), r* -> A*, O(Ah+l)

r* \A , IModh(Ah+l), O(Ah+l)

r -A*, Modh(Ah+l) D O(Ah+l)

r* A- , (Ah+1){Modh(Ah+l) D 0 (A+l)}

and a cut we get

r* A , (Aj+i){Modj(Aj+i) D O(Aj+,)}, which justifies the rule AMS.

Eventually we introduce the systems [If, which are related to the [HI's as the Pi+'s are to the Pj+'s.

The analogue to lemma 6 obviously holds. The last part of the proof of lemma 7 is replaced by the following

argument: Suppose the last inference in the proof of ( is an instance of II' and

thus of the form 01, the i t h

Fo101 . .., )~ By the inductive hypothesis



44 M. H. LOB

is provable in Ili+,. Hence by lemma 4

Bh(r(01) ), . . ., Bh(-r~os)) -- Bh(-r()p))

and thus also

(Bh)Bh(r(0j)), . . *, (Bh)Bh(,(0s)) (Bh)Bh(-(-))

is provable. If the ranks of T(fi) (i 1, . . ., s) and -r(y) are each equal to h-1, then this is the required formula. Otherwise we reduce the rank of the bound variable in the antecedent as in the original proof of the lemma.

We may also reduce the ranks of the displayed bound variables in the antecedent. Since

(A~h j)A~h~j-(0()) :) (Bh)Bh(-r(0))

is provable in II (II being the union of the Hi's). Thus by means of repeated cuts we eventually obtain the required formula

hi hi-I(-r~(0+)), * ,(A' -,)A' (-r(os)) -(Ck)Ck(-r(t)).

The proofs of the analogues of lemmas 8 and 9 again go through with obvious modifications. So does the proof of the analogue of lemma 10 up to the point when the provability of the formula

(a) (Aj1)Aj1('r(#i)), ..., (Ah)A h((T(oh)) - Bi,(-r(pr)) is established.

This is to be followed by an argument, in which we may without loss of generality assume that each of the displayed universal quantifiers has been introduced by an AMA, since if any were the result of an A-Thin we delete the respective formula from the proof of (a), (i.e. we move all A-Thins to the end of the proof of (a)).

The formula in the proof of (a), which is the premise of the last AMA must be of the form

or (is)), a., s-l) s+l, , h B,(

or

(ii) s,,T)), al, . as-1, as+l, . ., Jh r,((0)) --,

or

(iii) Aij-r+s)), a1, * o**,s-1, as+j, * O *ch --,

where Ock stands for (A,,)A k(T(#k)) If in (i) and (ii) A,. is distinct from Bj, then by lemma 8 this case reduces

to (iii) or to

Ocl, Oas-i, Ocs+1. Oh, * . , ad BJr(())-




Then, by a simple inductive argument, we see that (a) is provable from formulae of the form

Cl(-(l)), . .., Cl('r(Om)) * C1('r(0m+l)), .**v Cl('r(Om+n))

But then the formulae obtained from these by replacing C1 by V,, and, therefore, also the formulae of the form

T(01), * . ., r(Om+l) *T(Om+l), .-r(Om+n)

are provable. Hence the formula

41), * , + .,(#n) *Tf(lr)

is provable, and, by the inductive hypothesis, so is

l'k, *. . ,' n - r,

from which we obtain the required formula

Rol. **, Don Li pr

by 11'.

REFERENCES

[1] F. DRAKE, On McKinsey's syntactical characterization of systems of modal logic, this JOURNAL, Vol. 27 (1962), pp. 400-406.

[2] G. GENTZEN, Untersuchungen ilber das logische Schliessen I. Math. Zeitschrift, vol. 39 (1935), pp. 176-210.

[3] JAAKKO HINTIKKA, Modality and quantification. Theoria, vol. 27 (1961), pp. 119-128.

[4] STIG KANGER, Provability in logic, Stockholm studies in Philosophy, vol. 1. Alynquist & Wiksell, 1957.

[5] G. KREISEL, Set theoretic problems suggested by the potential totality. Infinitistic methods, pp. 101-140, Pergamon Press, 1961.

[6] SAUL A. KRIPKE, Semantical analysis of modal logic I. Normal modal calculi. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, vol. 6 (1963), pp. 67-96.

[7] CLARENCE IRVING LEWIS and COOPER HAROLD LANGFORD, Symbolic logic. Century Company, 1932.

[8] SH6JI MAEHARA, Eine Darstellung der intuitionistischen Logik in der klassischen. Nagoya mathematical journal, vol. 7 (1954), pp. 45-64.

[9] J. C. C. MCKINSEY, On the syntactical construction of systems of modal logic, this JOURNAL, vol. 10 (1945), pp. 83-94.

[10] W. V. QUINE, Reduction to a dyadic predicate, this JOURNAL, vol. 19 (1954), pp. 180-182.

[11] KURT SCHtTTE, Beweistheorie, Springer-Verlag, 1960. [12] G. H. VON WRIGHT, An Essay in modal logic. North Holland Publ. Co., 1951.

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