Modal Operators, Equivalence Relations, and Projective Algebras

Modal Operators, Equivalence Relations, and Projective AlgebrasAuthor(s): Chandler DavisSource: American Journal of Mathematics, Vol. 76, No. 4 (Oct., 1954), pp. 747-762Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2372649 .

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MODAL OPERATORS, EQUIVALENCE RELATIONS, AND PROJECTIVE ALGEBRAS.*l

By CHANDLER DAVIS.

1. Introduction. In modal logic, as initiated by C. I. Lewis in 1918, one applies to propositions a, b, ... not only the ordinary Boolean operations n, U, and ', but also a modal operator which for present purposes will be written C. This is a constant symbol with the following interpretation: If a is thought of as assertion the truth of a proposition, then Ca is thought of as asserting its possibility; similarly, (Ca')' asserts its necessary truth.

Depending on the sense put on the words " possible " and " necessary," different properties will be assumed for C applied to propositional variables [12]. These properties may be expressed as identities, such as those below. Waiving the logical questions, some of these sets of identities may be studied as definitions of kinds of operators on Boolean algebras. ParticLilarly interesting are S4 operators, satisfying

(1.1) CaaD a,

(1.2) CO -O,

(1.3) CCa=Ca,

(1.4) C(aU b) -CaU Cb.

Here (1. 1) - (1. 3) are, with the condition of " isotoneness "

(1. 4') C(a U b) D Ca,

just the assumptions made on "closure operators " on an arbitrary lattice [1, 19]; while all of (1. 1)-(1. 4) are satisfied if C is the operation of topological closure in a topological space, a and b being subsets and the Boolean operations being set operations. Boolean algebras with an S4 operator have been studied as generalizations of topological spaces [16].

* Received September 12, 1953. 1 This material is essentially excerpted from the author's thesis [5]. The work was

done without knowledge of the earlier work of Chin and Tarski (see [4]), which it presumably overlaps a good deal. It is unfortunate that their investigation has never been published.

747



748 CHANDLER DAVIS.

S5 operators satisfy, beside (1. 1)-(1. 4),

(1. 5) a n Cb = O implies Ca n Cb=O.

In the case of a topological space, the condition (1. 5) on the closure operator implies that every open set is closed; in fact, (1. 5) with (1. 1) implies in general (1. 5') C(Ca)'= (Ca)'.

Boolean algebras with an S5 operator do not therefore generalize any top- ologically interesting spaces, but they do provide generalizations of the simpler and " more set-theoretical " notion of equivalence relation.

This will be discussed more fully, together with other preliminary remarks on S5 operators, in Section 2. In Section 3 is a discussion of the reason for considering Boolean algebras with several S5 operators. The remainder of the paper gives theorems on the structure of these systems. They are shown to be in a rough sense " coordinatizable " by deriving them from the projective algebras of Everett and Ulam (Sections 4-5) ; only the two-operator case is treated here. The relatioii of these systems to the first- order functional calculus of logic is stated in Section 6.

The terminology to be used for Boolean algebras is as follows. "Meet" means " n," " join " means" U." The relation a = a U b, i. e., a D b, will be called " a is above b " or "b is below a." I and 0 are the " unit" and " null " elements respectively. A prime denotes complement if it follows an element of a Boolean algebra, though it may have other meanings in other connections. If a is a set of elements of a Boolean algebra, U C will be the join of all elements in the set.

2. Properties of S5 operators. First of all, the following more eco- nomical set of conditions will be convenient:

LEMMA 2. 1. If 03 is a Boolean algebra, any function C on #3 into 2 is an S5 operator provided it satisfies

(1. 1) Ca D a,

(1. 4') C(a U b) D Ca,

1. 5) C(Ca)' = (Ca)'.

Proof. (1. 4') is a consequence of (1. 4) alone. As already mentioned, (1. 1) and (1. 5) give (1. 5'): Ca n (Ca)' =- 0, hence by (1. 5), Ca n C(Ca)' = 0, that is, C(Ca)' C (Ca)', and the inequality cannot be strict because of (I. 1).



MODAL OPERATORS. 749

The other half is to show that (1. 1), (1. 4'), (1. 5') imply (1. 1)-(1. 5). Now (1. 1) implies CI I, so, by (1. 5'), CO =CG(CI)' = (CI)'= 0; (1. 2) is proved. Next, (1. 3) results from (1. ') alone: CCa = C(Ca)" = C(C(Ca)')' = (C(Ca)')' = (Ca)" Ca. The proof of (1. 5) is also easy: Suppose a n Cb = 0. This is equivalent to a C (Cb)', so that by (1. 4') and (1. 5'), Ca C C(Cb)'= (Cb)', that is, Ca n Cb =-0. (1. 5) is proved. Only (1. 4) remains.

In the usual terminology, call any fixpoint of C "closed"; the closed elements are those of the form Ca, by (1. 3). Well-known arguments show that (2.1) a C Cb implies Ca C Cb,

(2. 2) a and b closed implies a n b closed.

Indeed, a C Cb gives Ca C CCb =- Cb by (1. 4') and (1. 3). And for a and b closed, (1. 4') gives C(a n b) C Ca n Cb - a n b; the inequality cannot be strict because of (1. 1), so (2. 2) is also proved.

Complements and meets of closed elements are closed, by (1. 5') and (2. 2) respectively, so C`3 is a subalgebra and

(2. 3) a and b closed implies a U b closed.

The proof of (1. 4) is now easy. By (1. 1), a U b C Ca U Cb; by (2. 3), Ca U Cb is closed; by (2. 1), C(a U b) C Ca U Cb. But the reverse inequalitv is obvious from (1. 4'). The proof is complete.

It is easily shown that (1. 1), (1. 4'), (1. 5') are independent conditions if t3 has at least eight elements.

The following theorems make precise the sense in which S5 operators generalize equivalence relations.

First the relation to equivalence relations is stated in the case where 8 is a full Boolean algebra, that is, isomorphic to the Boolean algebra of all subsets of some set:

THEOREM 2. 1. Let #3 be a full Boolean algebra. For each equivalence relation cj on the points of 3, let Co be the operator defined as follows: for- a e 03, Cpa is the join of all points of 03 which are c-equivalent to any point below a. Then Co is S5; and the correspondence ( -> Co is a lattice isomorphism of the partition lattice P(03) of all ( onto the lattice of all S5 operators on S3. Also CG3 is always a subalgebra of #3 complete in 03 and a full Boolean algebra; and the correspondence 4p -> C0G3 is a lattice dual isomorphism of P(03) onto the lattice of all such subalgebras.

2



750 CHANDLER DAV1S.

Explanations. 55 operators are ordered as functions into a poset [1], namely, C1 D C2 means that, for all a ?, X3, C1a D C2a. Equivalence relations on a set of points p, q, * * - are ordered by defining c D at to mean that pallq implies poq. Subalgebras of 3 are ordered as usual: by inclusion. A point of a Boolean algebra is a minimal element strictly above 0. The join referred to in the second sentence of the theorem is of course the join operation of X3, not set union.

This theorem is not really new (see [16, 17] ; also [18, 19]) ; also the ideas involved are perhaps made sufficiently clear by the remarks in Section 1. Accordingly, the proof is omitted; it is given in [5].

Next, a representation theorem relates the general case to the special case above.

THEOREM 2. 2. If C is an S5 operator on the Boolean algebra X, and if W' is the full Boolean algebra in which 03 is imbedded by the Stone representation theorem, then there is an S5 operator C' on 63' which is an extension of C. If several S5 operators are extended to S3' in this way, the correspondence is order-preserving in both directions.

This is a special case of a representation theorem of J6nsson and Tarski [10], but the proof is particularly simple in this case.2

For p a point of D', define C'p == n {a; a D p, a ? CB3}; define further CO = 0. For other b ? D', define C'b as U {C'p; p C b, p a point}. Then C' may be proved to be an S5 operator by Lemma 2.1, or as follows.

Define pcpq, for points p and q of p3', to mean that p C Ca implies q C Ca for a ? D3. The relation p is obviously reflexive and transitive. To verify that it is symmetric, take pcq and p ?X Ca; then, by (1. 5'), p C (Ca)' = C(Ca)', so q C (Ca)', q T Ca. Hence 4 is an equivalence relation. The S5 operator

CO corresponding to it according to Theorem 2. 1 is identical with C', by definitions.

C' must be shown to agree with C on B3. The definition shows that if a is closed under C it is closed under Ct. From this it follows that, for all b ?D 6, C'b C Cb. The reverse inequality is more difficult.

Take any be&3; C'b=-U {C'q;qCb,q a point}; and each such C'q is the join of an equivalence class under c. Hence to show that Cb C C'b it is enough to show, for any point p C Cb, the existence of a point q C b with ppq.

Consider the set a = {b n d; d D p, d ? C#3}. First, a contains with

2 The referee contributed to its simplicity.




any two elements their meet: if b n d 6 a, b n d 6 a, then, because d n d D p and dnd0 C, bn dn bnd e a. Next, a does not contain O: for bnd = O and d ? CD3 would imply b C d' ? C0, d' D Cb D p, which would forbid d D p. A standard argument shows there is some point q of D3' which is below every element of a; in particular, q C b. Also poq, by the definition of a. There- fore Cb C C'b.

This proves the first statement of the theorem. Now let C,' and C2' be the S5 operators on D3' obtained in this way from

the S5 operators C, and 02 on S. That C0' C C2' implies C0 C C2 is obvious. The converse follows almost as easily: If C, C C2, then C26 is contained in C,3, so by definition Cl/p C C2'p for p a point of p3', so C1'b C C2'b for all b ? p3', which was to be shown.

3. Discussion of S5 operators. All the considerations of this paper were motivated by an interpretation of the notion of possibility, therefore of modal operators, which will now be set forth.3 It is related to the ideas of Wajsberg [22], Carnap [2, 3], and McKinsey [14].

Let us consider a logical system as a way of representing the relationships between statements which may correctly or incorrectly be made about the universe. (The " universe " here may be any " isolated " physical system or conceptual system.) No two such statements need be considered as distinct unless it is believed that one of them might hold without the other's doing so. In other words, let us consider a statement to assert something only insofar as it distinguishes a state of affairs which it asserts from an otherwise con- ceivable state of affairs which it denies. This point of view leads naturallv to limiting oneself to propositions " in extension ": To begin with, one imagines a set of " possible " alternate universes. Thereafter, a proposition a is considered " in extension," that is, it is identified with the set of alternate universes in which it holds. (Each proposition considered is to be one whicb asserts something about the universe: one, therefore, which can assert something about each alternate universe, and can hold or not hold in each alternate universe. It is not to be a proposition about sets of alternate universes; nor yet a proposition about propositions.) Now if the proposition asserted by a holds in some alternate universe one can say that the proposition asserted by Ca, " it is possible that a," holds in all the alternate universes.

Now in the formal system constructed following these ideas, logical

3 The discussion of logic in this section, as in ? 1, is informal. Nowhere in this paper is there a formal discussion of logical systems generalizing Lewis's S5, though the mathematical systems treated would be matrices for sujch systems.



75 a2 CHANDLER DAVIS.

connectives between propositions will correspond to operations on the elements of a complemented lattice of subsets of the set of all alternate universes. This lattice might be taken to be the Boolean algebra of all subsets, as is done by Carnap [3], who distingiushes each of his alternate universes initially by the subset of a given set of atomic propositions which hold on it. Here, the lattice will be taken to be some field of sets, therefore a Boolean algebra (although it is not at all clear that other lattices are uninteresting in this connection) ; it will not be assumed full.

Tne operator C on this Boolean algebra which was discussed above gives Co = 0, and for a 0, Ca = I. It is an S5 operator-that which corresponds, if the Boolean algebra is full, to an equivalence relation with only one equivalence class.

Non-trivial S5 operators, or more than one S5 operator, might appear in the same context. An early argument for the introduction of two modal operators [6, 21] need not be discussed here (see [11]). A strong argument, however, is made by Weyl [23] for considering more than one "degree of possibility." He gives the following example. Suppose a train leaves Seattle at 9:00 a. m. Pacific Time. It is physically impossible for it to arrive in Chicago one microsecond later. It is technically impossible, though physically possible, for it to arrive in Chicago at 10 :00 a. m. Pacific Time the same day. Weyl's example illustrates the desirability of a totally ordered set of "degrees of possibility" 4; it can be amplified as follows to illustrate the case for a partly ordered set. It may be technically possible for the train to arrive at 9 :00 a. m. two days later, but impossible because no track is available for it to enter the Chicago station at that time; the technically impossible arrival time of 10:00 on the day it started migbt however be possible by the latter criterion. Thus we have two " degrees of possibility " neither of which implies the other. The example is artificial, and a different one will serve better.

Let the alternate universes be " all " division rings. Assume for simplicity that the Boolean algebra being considered is that of all sets of alternate universes; that is, that there is, among those propositions being considered, a proposition distinguishing each of the division rings. Let C,a assert that the proposition asserted by a is possible as far as commutativity or non-commutativity is concerned. (Equivalently one could say, let C, be Co of Theorem 2. 1, taking for 0 the equivalence relation having two equivalence classes: the commutative division rings, the non-commutative ones.) Thus if a asserts,

4Possibility operators to be ordered as fuinctions into the Boolean algebra of propositions, see ? 2.




"This division ring has two elements," then C0a is true for all fields (all members of the first equivalence class) and for no other division rings. Let C2a assert that the proposition asserted by a is possible as far as finiteness or non-finiteness of the division ring is concerned. If a asserts, " The identity of this division ring has infinite order," then C2a is true for any division ring with an infinite number of elements and for no others; and so forth. 0C and C2 are again S5 operators, and neither C0 C C02 nor C D C02.

Return for a moment to the first, relatively trivial interpretation of possibility above. It would have been equally reasonable to replace the propositions a by propositional functions Fa (-) in the following way. For each proposition a about the universe, let Pa(x) assert that a holds in the alternate universe x. (For fixed x, Pa(X) is not an assertion of the same form as a, so that expressions involving the Fa () cannot meaningfully be combined with expressions involving the a by logical connectives.) The association of a to FPa(-) is of course a Boolean isomorphism. To what does the S5 operator 0 described above correspond? By its definition, Ca = I if (3 y) Pa (y) is true, Ca = 0 otherwise. If one is willing to regard the proposition (3 y) Fa (y) as a propositional function (whose variable is not written and is without effect on the truth-value), then the conclusion may be simply stated: The operator C on propositions corresponds to existential quantification on the associated propositional functions. This correspondence between modal propositional calculus and the ordinary "einstellige Priidikatenkalkul" was mentioned but not discussed by Wajsberg [22].

So much for the case of one trivial S5 operator. The introduction of more than one modal operator naturally suggests the extension from the calculus of propositional functions of one variable to the first-order functional calculus, in which propositional functions may have more than one argument. In fact, there is a close relationship between the two situations; this relationship is the subject of the following sections.

4. S5 operators and projective algebras. The projective algebras of Everett and Ulam [7] were intended as a preliminary attempt at supplying a model for the first-order functional calculus similar to that supplied for propositional calculus by Boolean algebra. The definition is as follows.5

A projective algebra is a Boolean algebra ?2 (whose unit element will be written "ci "), with a distinguished point po; with unary operations (" projections ") a --> a, and a -> a.; and with a binary operation a, b -> a El b defined for a C i, and b C tiy; subject to the following laws:

5 See also [4].



754 CHANDLER DAVIS.

Pi. (a U b)$ a,, U b,,; (a U b), y= ay, U by.

P2. i$y iy$ PO.

P3. a = 0, ax = 0, and ay= 0 are all equivalent.

P4. axx- = ax, ayy = ay.

P5. Let OCaCi,,, OCbCiy. Then (aElb)x=a, (a El b)y= b; also, for any c? 2, cx =a and cy = b imply c C (a O b).

P6. ix D po ix; pO29 M y = y

P7. (a U l) El i (a l i) U (a J iy); i$ E (b U b) = (ix E b) U (ix C1 ).

Dl. aE0 O=Olb=O.

The typical case motivating these conditions is the projective algebra of subsets of a direct product, in which 2 is a Boolean algebra of some subsets of a Cartesian product of a set by itself, and where the projections a -- ax, and a - ay are the projections of a set in 92 onto the respective axes. Every projective algebra is isomorphic to a projective algebra of this sort [7, 15], but not necessarily to a projective algebra of all subsets of a direct product.

Every projective algebra 92 can be made into a Boolean algebra with two S5 operators in a natural way. Namely, S5 operators on 9 as a Boolean algebra are given by defining

(A) C1a = ax Dil,i C2a =il ay.

This result is a special case (where e is taken equal to i) of Theorem 4. 1 which follows.6

THEOREM 4. 1. Let 92 be a projective algebra and e a fixed element of S. On the Boolean algebra of elements below e (complement being defined as relative complement) the following are S5 operators:

(B) C1a = (a. O iy) n e, Ca =(ix O ay) n e.

THEOREM 4. 2. Every Boolean algebra with two S5 operators is obtained in this way.

Proof of Theorem 4. 1. It is enough to give the argument for C,, since the proof for C2 would go just the same. The conditions of Lemma 2. 1 must be verified; accordingly, it will be shown that

/4X 1 (a El i_ ne \

a;

6 Cf. riemark at end of ? 5.




(4.4) ((aU b) El iya) ne = ((a.2Eliy) U (ba,li,y)) ne;

(4. 5') C0((C0a)' n e) (C1a)' n e.

Here (1. 5') takes the form (4. 5') because the prime is still being used to denote the complement in 9 (not the relative complement); (4. 4) is the statement of (1. 4), which is stronger than (1. 4').

These identities can be verified using the representation theorem for projective algebras, or formally, as follows. In the proof, results from [7] will be cited by number, without quoting them.

First, if a 0 O then (4. 1) becomes trivial and (4. 4) follows from P3, Dl. If none of the elements involved is 0, one has, by Cl, C9, and C10, (ax E 'iy) n e D (a. E ay) n e D a, proving (4. 1); while C19 gives (4. 4).

If a = 0, (4. 5') reduces to C1e = e, which follows from (B) and (4. 1). It remains to prove (4. 5') for 0 C a C e. Now

(C0a) f n e (a, El iy)' n e (obvious from (B) - ([(a,' n ia,) l iy] U [i0 Z (i n fi)]) n e (by C22)

((a" n fi) El iy) n e (by Di). Also

Cl ( (C1a) n e) ([(a E Liy) 'f n e]0 EiJy) n e (by (B) C (([ (a. 0 iy)'] Tn ex) El i) n e (by C4, C9)

([(a. E,Liy)] x Liy) n (e E,liy) n e (by Clil)

([(a.,LIiy)']0xLIiy) n e (by P5, C9) ([(a,' n i) i iy] rL iy) n e (see above)

((ax'n i,) El iy) n e (by P5).

Hence " C " holds in (4. 5'). But the reverse inequality is immediate fromt (4. 1). The proof is finished.

5. Continuation. Proof of Theoremr 4. 2. Imbed the given Boolean, algebra t3 in 3' by Theorem 2. 2; let the extensions of the given S5 operators C, and C0 be written simply C1 and C2. The proof is in three parts: construction of a certain full projective algebra 9'; construction of a certain sub-projective-algebra 2 of 9'; and showing 3 can be obtained from 9? by the method of Theorem 4. 1.

Elements of 3' will be writen as before (in particular, "p " and "q will denote points) ; lower-case Greek letters will denote elements of 9i (thus its unit element is written "L").



7 56 CHANDLER DAVIS.

Part 1. The points v- of 9.' are to be as follows:

(5. 1) for every point p of 03', a point [p];

(5. 2) for every point Cip of C0S' and point C2q of C2,', a point [Clp, C2q];

(5. 3) for every point C0p of C123', a point [CGp];

(5. 4) for every point C2q of C2,', a point [C2q];

(5. 5) an additional point 7ro.

ilere it is clear that the same element r of 3' may give rise to more than one point of 9.'-e. g., if r is a point both of 3' and of CU3'. In such a case the notations [r]1, for the point of the form (5. 1), and [r]3, for the point of the form (5. 3), could be used to distinguish the two; however, the arguments which follow will be sufficiently clear without doing this. The point 7ro will be the distinguished point (? 4) of 9'.

Let 9' be the full Boolean algebra generated by the points (5. 1)-(5. 5). Denote by XX the join of all points of the form (5. n), n = 1, 2, 3, 4, 5.

The projections of 9' will be defined first for points 7r.

7rx: [p]x==-[0Cp] 7ry: [PY- [C2P1

[Clp, C2q]x [C0p] [Clp, C2q]y = [C2q]

[C0p] x [CiP] [WYP] 7ro

[C2p] = 7ro [C2P]y = [C2p]

(wo) X = 7ro ; (o ) y 7ro.

Next, for arbitrary a e S', define , = U {r; j7r C a,7r a point}; and a,

similarly. Of the conditions (Pl-P7 and D1) which must be checked to prove 2'

a projective algebra, PI, P3, and P4 are already clear from the definitions. Now , X3 U X5, ty = X4 U X5, in the notations above. Recognizing this,

one verifies P2 immediately. Also for a e 9', a Ct if and only if a is a join of points of the forms

(5. 3) and (5. 5), with a similar condition for /8 C ty. For such a and 8, the operation "fO" must be defined. First, aE 0 0 O C1 d 0 may be assuamed; this is Dl. Next, for 0 C a C Lw and O C /3 C t,, let a fO l] = U {7r; w Ca, ry C, 1,r a point}. This makes P6 and P7 obvious.

P5 must be checked. As a sample, here is the demonstration that a C (af O,8) . Consider any point qX C a. If 7r 7ro, then 7r (7r O p), where p = 7r O p is any point below /8. (Remember /3 + 0.) The other case




iS r = [C,p]: then either X = [C,p, C2q],, with [C2q] -[Cp, C2q]y C ,; or, if /8 =3 one has -r =-_r,, T =7ry. (It might also happen that w- [p],

with [p]y, C ,/, but this can not be guaranteed; this was of course the reason for introducing the points (5. 2) originally.) In either case, 7r C (a El /3) Therefore a C (ac EO,8),.

It has been proved that 2' is a projective algebra.

Part II. Next the subset 2 of 92' will be defined. Here a, a1, a2 represent sets of points of (respectively) p3', C,3', C2,'.

(5. 6) If U a e a, then U{[p]; p c a} e 2; (5.7) if U a,sC1c?, Ua2 CC23,

then U {[1Cp,C2q];cip8a1,C2q8a2} C2;

(5.8) if U a,FC 3s, then U {[Cip];cipeGa} 2; (5.9) if U G2 eC2 B, then U {[C2p];C2peG2}c9;

(5.10) 7roC2.

Let 2 be the sub-Boolean-algebra of 2' generated by these prescribed elements.

This ensures that 2 is closed under the Boolean operations. The proof that it is closed under the other operations follows.

Any az 92 may be written as the join of the a n X", n =1 , 2, 3, 4, 5. Also a nfX 2, and, by P1, (a n x) 2 for all n implies a,, c 2. So the next task is to prove, for a C X, that a 8 2 implies a,, ? 2.

For n = 5, that is, a 5r0, this is automatic. For n = 1, 3, 4, it is almost as obvious. Take for example a C Xi, a ? 2.

It is easy to see that a is of the form (5.6): a= U {[p];pCa,ac e}. Then a2, = U ([Cp] ; CGp C C6a}; since C1a ? C,B. this is of the form (5. 8), hence in 2.

Finally, consider a C X2, a e 2. Call an element of the form (5. 7) a "2 -rectangle." Now a is a combination of a finite number of 2-rectangles, together with elements disjoint from X2, under Boolean operations. Note that, for any 2-rectangles /3 and y, A n fy and /1' n X2 are joins of 2-rectangles. Therefore a is a join of 2-rectangles; and therefore, by P1, only the case where a is a single 2-rectangle needs to be considered. But this case is clear.

Thus 2 is closed under x-projection; similarly for y-projection. There remains the operation "C E."

Assume a = U { [COp]; Cip C a ? C13}, 3= U { [C2p]; C2p C b C26}; so that a, e8 2. In order for a point r of 2' to be below aE [1, it must



758 CHAN'DLER DAVIS.

satisfy wT0, C a and vy C /3. This is equivalent to saying that either r- [p], with p C a n b, or wr [C1p, C2q], with Cip C a, C2q C b. The join of all 7r of the first kind is an element of the form (5. 6), because a and b, hence a n b, are in 2; the join of all 7r of the second kind is an element of the form (5. 7). Therefore a E /,, the join of all 7r of either kind, is in Q.

This was not quite general enough, it assumed 7ro ?1 a, 7ro L ,B. However the remaining cases involve no difficulty, and one concludes that _2 is closed under the operation c"EL

It is now easy to see that a is a projective algebra.

Part III. The first two parts of the proof were preliminary: the supplying of a projective algebra a from which an isomorphic image of 2 could be obtained by the construction of Theorem 4. 1. There remains the proof proper: showing that this can in fact be done using the Q prescribed.

Define the function 4? on 23' by cDa = U { [p] ; p C a, p a point>. Then 42B' consists of those elements of 9' below bcDI, which is the element called X1- above. Any a e4 S' is in (DS if and only if it is in 9; and 4? is a Boolean isomorphism of 2 on 42S (in fact of 2' on 4?'), if complementation in '1)2' is understood as relative to cP1.

So much is obvious. Now by Theorem 4. 1, 01 and 02 defined as follows are So operators on 4?2: O1a = (a., o if) n iil, C2a = (tx El ay) n 4?I. It must be shown that 4?Cla l 014a, 4?C2a = 2iba.

Assume [p] C O,a, a = 4?a, a ? 2. Since [p] C a0 O Ly, one has [p]$ C a; and every point below a, is [0Cq] for some [q] C a, that is, for q C a. Hence, for such a q, p C Cip - 1 Cq C C1a. This proves 40CLa D 0d14a.

Let p be a point of 2' below C1a, a e 2; then [p] C 4?C1a. By the proof of Theorem 2. 2, there is some point q below a such that p C C1q. For such a q, [p]$ [C,p] = [C,q] = [q]x C ((Da)0,. Thus p, being a point below (DI whose x-projection is below ('4a).,, is below (4ba O ty) n [11 014?a. This proves 4?C1a C 014?a.

An identical argument relating C2 with 0, completes the proof of the theorem.

It should be remarked that in case e of Theorem 4. 1 is i, then for any a # 0, C2UCa 02 (ax0, iy) ix El iy= i (see [7, C10]). But there exist 2 such that C2C1a assumes other values than 0 and I. Therefore if Theorem 4. 1 had asserted only the special case e = i, the converse Theorem 4. 2 would have been false.




6. Relation to the first-order functional calculus. This relation is most easily expressed by dealing with Mautner's Boolean tensors [13] instead of with the logical theory itself. The logical coordinate system of Mautner's theory will be left fixed,7 so that a Boolean r-tensor may be treated as simply a function on the set S X S X * X S (r factors) to the two-element Boolean algebra.

The Boolean r-tensors f _ f (xi, , xr), Xk S S, form a system 03r closed under the Boolean operations. However D3r is not closed under the following operations:

(6. 1) outer sum formation

f(xi, *, Xr) U g(yi, *nYr) = h(xi, * Y . ,. . Yr) s 62r;

(6. 2) outer product formation, dual to (6. 1);

(6. 3) setting arguments equal

f(. . . X?n *i . xj, ) f( *) -- f xin * xi ) (6. 4) sum contraction (i. e., existential quantification)8

U {f (xi, * * * , Xr) ; Xk S} = g (X1, Xk-, Xk+i, *, Xr) eS r-1

(6. 5) product contraction, dual to (6. 4).

All of (6. 1) -(6. 5) correspond to admissible processes in forming formulas of the functional calculus.9

If all of (6. 1) -(6. 5) are forbidden, B3r is merely a Boolean algebra. This corresponds to the situation of the " alternate universes " of Section 3 before SS operators were introduced. Corresponding to the introduction of S5 operators there, a limited class of Boolean tensor operators will be specified which will take 03r into itself. (Note that (6. 1)- (6. 3) will still be forbidden, and will have no analogue.) Namely, for any y e S let

(6. 6) U {f(x, ' *, Xr) ; Xk c S} = (Ckf) (X1, . . , Xkl, y, Xk+1. , Xr)

Clearly Ckf is in Or, but its k-th argument is without effect. Conversely,

7 It will be clear that Theorem 6. 1 has invariant significance in Mautner's sense and Theorem 6. 2 does not. Note that my notation for Boolean tensors in this section differs from Mautner's.

8 The notation in the left-hand member implies that the k-th variable is bound by the summation.

9 It is unconventional to include setting arguments equal (6. 3) as a process analogous to the others; that is, " P (a, x) " is not usually regarded as having been formed from " F (x, y)." But cf. [9; Kap. 5a].



760 CHANDLER DAVIS.

a function in 03r whose value is independent of its k-th argument is of the form Ckf. This characterization of Ck3r makes it clear that it is a complete sub-Boolean-algebra of 3r*. Also Ck can be seen to take any f e 6, to the meet of those elements of Ck3r above f. Therefore (Theorem 2. 1) Ck is an S5 operator.

Furthermore, if t is any set {k1,** , kt} of the indices 1,* , r, then the operator (6.7) C CkCk, Ck,

clearly depends only on g (not on the order of the Ck) and is an S5 operator. Some further properties of the CQ may be pointed out. It is natural to

order them as functions into 3r, in fact to consider them as a subset of the lattice of all S5 operators on Br (Section 2 and [1]). Now if one defines an operator corresponding to the null set of indices by

(6. 8) Cof f,

then one concludes that CO.u , Ce U C6. It is not hard to show also that CG ?, CQ n c6. (The proof, based on Theorem 2. 1, is given in [5].) Finally, if t = {1, * *, r}, then COf(xi, * * *, xr) = I for all xl, x, provided there is any set of arguments for which f (xi, * *, xr) =.

Collecting these facts gives the following.

THEOREM 6. 1.7 The operators CQ on 3r defined by (6. 6)-(6. 8) are S5 operators. They form a sublattice of the lattice of all S5 operators (containing its null and unit elements), and C- is a lattice-isomorphism onto the Boolean algebra of all sets of indices 1, r.

Thus Boolean algebras with S5 operators generalize Boolean tensors only if the operations allowed on the latter are restricted; they provide a Boolean model for the functional calculus only in a limited sense. Quantification is included (notice that the dual of (6. 6) is ( Ckif)') but (6. 1) - (6. 3) are not.

Projective algebras are evidently in the same situation. In fact, identify every f e 932 with the set of all points of S X S which it maps into 1. Fix a point zo of S. Define 10 fZ as the set-intersection of {zo} X S with Cif (as defined by (6. 6)); similarly fy = (S X {zo}) n C2f. Also define, for subsets T and U of S.

({zo} X T) O (U X {zo}) =U X T C S X S.

lo [7, C28].




THEOREM 6. 2.7 With respect to the distinguished point {z0} X {z0} and the operations just defined, 32 becomes the projective algebra of all subsets of the direct product S X S. Every projective algebra of all subsets of a direct product of a set with itself is obtained in this way.

This is clear without giving details. For the first sentence, compare [7]. For the second sentence, simply construct D32 over the set in question.

ANN ARBOR, MICHIGAN.

BIBLIOGRAPHY.

[1] G. Birkhoff, Lattice theory, American Mathematical Society Colloquium Publica- tions, vol. 25 (1948), 2nd ed., N.Y.

[2] R. Carnap, "Modalities and quantification," Jourtal of Symbolic Logic, vol. 11 (1946), pp. 33-64.

[3] , Meaning and necessity, Chicago, 1947. [4] L. H. Chin and A. Tarski, "Remarks on projective algebras," Bulletin of the

American Mathematical Society, Abstract 54-1-90 (1948). [5] C. Davis, Lattices and modal operators, Harvard Doctoral Thesis, 1950. [61 A. F. Emch, "Implication and deducibility," Journal of Symbolic Logic, vol. 1

(1936), pp. 26-35; also ibid., p. 58. [7] C. J. Everett and S. Ulam, "Projective algebra I," American Journal of Mathe-

matics, vol. 68 (1946), pp. 77-88. [8] D. Hilbert and W. Ackermann, Grundzilge der theoretischen Logik, 2nd ed., Berlin,

1938. [9] D. Hilbert and P. Bernays, Grundlagen der Mathematik. I Band, Berlin, 1934.

[10] B. Jonsson and A. Tarski, "Boolean algebras with operators," Bulletin of the American Mathematical Society, Abstract 54-1-88 (1948).

[11] C. I. Lewis, " Emch's calculus and strict implication," Journal of Symbolic Logic, vol. 1 (1936), pp. 77-86.

[12] C. I. Lewis and C. H. Langford, Symbolic logic, New York, 1932. [13] F. I. Mautner, "An extension of Klein's Erlanger Program: logic as invariant

theory," American Journal of Mathematics, vol. 68 (1946), pp. 345-384. [14] J. C. C. McKinsey, " On the syntactical construction of systems of modal logic,"

Jou6rnal of Symbolic Logic, vol. 10 (1945), pp. 83-94. [15] J. C. C. McKinsey, " On the representation of projective algebras," American

Journal of Mathematics, vol. 70 (1948), pp. 375-384. [16] J. C. C. McKinsey and A. Tarski, " The algebra of topology," Annals of Mathe-

matics, vol.. 45 (1944), pp. 141-191.



762 CHANDLER DAVIS.

[17] J. C. C. McKinsey and A. Tarski, " Some theorems about the sentential calculi of Lewis and Heyting," Journal of Symbolic Logic, vol. 13 (1948), pp. 1-15.

[18] 0. Ore, " Theory of equivalence relations," Duke Mathematical Journal, vol. 9 (1942), pp. 573-627.

[19] 0. Ore, " Some studies of closure relations," Duke Mathematical Journal, vol. 10 (1943), pp. 761-785.

[20] Tang Tsao-Chen, "Algebraic postulates and a geometrical interpretation for the Lewis calculus of strict implication," Bulletin of the American Mathe- matical Society, vol. 44 (1938), pp. 737-744.

[21] P. G. J. Vredenduin, "A system of strict implication," Journal of Symbolic Logic, vol. 4 (1939), pp. 73-75.

[22] M., Wajsberg, "Ein erweiterter Klassenkalktil," Monatshefte filr Mathematik und Physik, vol. 40 (1932), pp. 113-126; also ibid., vol. 42 (1935), p. 242.

12:3 1 If. Weyl, " The ghost of modality," in Philosophical essays in memory of Edmund fiusserl. ed. Marvin Farber, Cambridge (Mass ). 1!)40, pp. 278-303.



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Modal Operators, Equivalence Relations, and Projective Algebras

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