algebraic semantics for modal logics ii

Algebraic Semantics for Modal Logics IIAuthor(s): E. J. LemmonSource: The Journal of Symbolic Logic, Vol. 31, No. 2 (Jun., 1966), pp. 191-218Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2269810 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 31, Number 2, June 1966

ALGEBRAIC SEMANTICS FOR MODAL LOGICS II

E. J. LEMMON

This paper is a sequel to [7], and the terminology of [7] is largely pre- supposed here. In [7], the algebraic methods of McKinsey-Tarski were employed and extended to yield semantic results of a Kripke kind for a class of relatively weak modal logics, the strongest of which was the Feys- von Wright system T. Deontic versions of both T and E2, called T(D) and D2, and even weaker systems, were handled. The main aim of the present paper is to extend these results to stronger systems of modal logic. Thus the Lewis systems S2-S5, the Brouwersche system B of Kripke [4], the systems E3-E5 of [5], and Lukasiewicz's modal logic, as well as certain new systems, are considered.

Certain modifications of the method of [7] have proved convenient. Thus in Section I, some further results concerning model structures are proved in order that the relationship between S2 and E2, S3 and E3, can be properly stated; in particular, the notions of a refined and connected model structure play a pervasive role throughout. Also, for a simpler treatment of the strong systems of Section IV, a general representation theorem, due in essentials to Dana Scott, for modal algebras is given: the effect of this is to transfer the point at which the decision problem for each system is solved from the purely algebraic stage, as in [7], to the model structure stage, where it proves easier to effect it. As the paper proceeds, it is increasingly left to the reader to draw the obvious semantic conclusions from the algebraic results. Thus we are content with the conclusion that A is a theorem of a certain system iff A is satisfied by R+ for all model structures 9 of a certain kind. Theorem 21 of [7] can then automatically be applied to show that A is a theorem of a system iff A is valid in all * of that certain kind, which gives semantic completeness. Thus Theorem 21 of [7] continues to play a fundamental, if tacit, part in our results.

Much remains to be done. For example, there are various deontic counter- parts to these systems to be considered, such as D3-D5 of [5], as well as the systems which result from adding to either T or S4 reduction theses of various kinds. However, it may be expected that many of the methods of this paper carry over to these other systems. Throughout, I am very greatly indebted to the ideas and stimulus of Dana Scott, though I alone am responsible for the ugly algebraic form into which I have cast some of his elegant semantics.

Received April 4th, 1965.

191



192 E. J. LEMMON

I

The systems E2, E3, ET, E4. The system E2 is thoroughly discussed in [7]. Our main purpose in this

section is to establish a certain derived rule for E2, which will be of importance later, and to sketch completeness results and decision procedures for three extensions of E2.

As in [7], we consider the schemata:

Al: A (B -A); A2: (A (B -C))? ((A -B) -(A C)); A3: (-A -B)-?(B-A); A4: D(A-B) (DA -+ B); A6: L A -A;

and rules:

Ay* A NIB . 2 A SIB RI: A,-B R2: A+

B ' C A -B

Then E2 has A 1-A 4, A6 as axiom-schemata and R I, R2 as rules. In addition to the definitions of [7], we put T = p -*p, F = -(p -*p).

By an e-algebra (epistemic algebra), we understand a structure =

<M, U, n, -, P>, where M is a set of elements closed under U, rf, -, and P such that:

(i) M is a Boolean algebra with respect to u, A, Y (ii) for x,yEMP(xUy) =PxuPy; (iii) for xE MY x < Px.

We know from [7], Theorem 14, Corollary 2, that FE2 A iff A is satisfied by all finite e-algebras.

By a model structure, we understand a structure 9 = <K, Q, U>, where K is a non-empty set of elements, Q a K, and U is a relation defined on K. A model structure is epistemic iff U is reflexive in K - Q. By 9+, the algebra on = <K, Q, U>, we understand the structure <M, U, rA, -, P>, where:

(i) M =SK; (ii) U, n, - are the set-theoretic operations of union, intersection,

and complementation restricted to M; (iii) for A E M, PA = {x : (3y) (y E A A Uxy) V X E Q}.

By [7], Theorem 18, the algebra R+ on an epistemic m. s. ! is an e-algebra; and by [7], Theorem 19, any finite e-algebra is isomorphic to the algebra on some finite epistemic m. s. Thus ([7], Theorem 20), FE2 A iff A is satisfied by V+ for all finite epistemic m. s.'s R. We note also that, for A E M, NA (- P -A){x: (Vy)(Uxy -*yEA) A xQ}.

We wish first to sharpen this result by restricting attention to a special



SEMANTICS FOR MODAL LOGICS 193

kind of m. s. Let us say that a model structure ! = <K, Q, U> is refined iff:

(p) for x,yEK, if Uxy then xEK -Q.

Intuitively, <K, Q, U> is refined if it has no U-leads within Q or out of Q, i.e. Uxy never holds if x E Q. Given any model structure R = <K, Q, U>, we may put:

U1 {<x, y> : xE K - Q A UXY}. Then the m. s. =1 <K, Q, U1> will evidently be refined. We prove:

THEOREM 1. R+ .

PROOF. Put = <K, Q, U> and 91 = <K, Q, Ul>, where Ui is as defined above. The cases of U, n, - are trivial, and we need only show that, for A a KY PA = P1A, where:

P1A = {x: (3y)(yE A A UiXy) V X e Q}. That P1A a PA is immediate, since if Ujxy then Uxy. Conversely, suppose x E PA, so that either (3y) (yE A A UXy) or xE Q. If xE Q. then xE P1A of course. Suppose then that x ? Q. Then (3y)(y EA A Uxy), whence (3y) (y E A A Uixy) and x E P1A. This completes the proof.

This means that, in the completeness results of [7], we may if we choose restrict attention to the algebras on refined m. s.'s. In the case of E2, we have:

THEOREM 2. FE2 A iff A is satisfied by A+ for all finite refined epistemic m. s.'s R.

A further restriction may also be made, as in effect is shown by Kripke [4], 2.2. Let us say that a model structure 9 = <K, Q, U> is connected iff there is an x E K such that, for all y E K(y =A x), U*xy, where U* is the ancestral of U.1 Given 9 = <K, Q, U> and xE K, by x,, the connected m. s. generated by x from 9, we mean <Km, QX, U,>, where:

(i) Kx {y : U*xy} u {x}; (ii) Qx Kx r) Q;

(iii) UX {<y, Z> : Uyz A y, Z E Kx}. THEOREM 3. If A is falsified by R+ for 9 = <K, Q, U>, then there is a

connected model structure fx generated by an element x E K such that A is falsified by kx+.

PROOF. Suppose A is falsified by R+ for an assignment W = <A l, ..., An> (Ai a K) to its variables vj, ..., vn. As in [7], we write Va(B) for the value assigned to the subformula B of A by W. We know that Va(A) =# K, hence we may select x E K - Va(A), and put Ax <Kx, QxY Ux> as the connected model structure generated by x from R. For As 5 K, put A4 Ai r) Kx, and consider the assignment W' (from R+) <A;, ..., A n> to the variables of A. We shall show first that, for any subformula B of A, for all ke KxE

1 This x is not necessarily unique.



194 E. J. LEMMON

k E Va(B) iff k E Va,(B). We use induction on the length of A. If B is a variable vi, we have, for ke KE:

kEVa(vi) iff kEAi iff k EA' (= Ain Kx) iff k E Va.(vi).

If B has the form C --D, we have, for kE Kx: kE Va(C -=*D) iff k f Va(C) or kE Va(D)

iff k g Va,(C) or k E Va,(D) (by inductive hypothesis) iff kEVa,(C D).

A similar argument holds in case B has the form -C. Finally, then, suppose B has the form ICG. Now, for k E Kx, we have:

kEVa(EZC) iff (VI)(IEKA Ukl lEVa(C)) A k Q; (1) kE Va (EZC) iff (VI)(1E KX A UxkIl E Va(C)) A k Qx. (2)

To see that the right-hand sides of (1) and (2) are equivalent, we note first that, for k E Kx, k E Q iff k E Qx. Now suppose that the right-hand side of (1) holds, and let IE Kx, Uxkl; then IE K, UkI, so that "E Va(C); thus, by the induction hypothesis, given that 1 E Kx, 1 E Va,(c). Conversely, suppose that the right-hand side of (2) holds, and let I E K, UkI. Since k E Kx, we have either U*xk or x = k. Either way, given UkI, it follows that 1 E Kx, whence Ukl, so that 1 E Va,(C). Thus, by the induction hypothesis, 1 e Va(C). This completes the inductive proof.

It follows that, for k E Kx, k E Va(A) iff k E Va,(A). But, since x E Kx and x g Va(A), we have x ' Va,(A), so that Va,(A) #& Kx and A is falsified by R+.

Obviously, as a consequence of this theorem, we may restrict our attention, so far as completeness results go, to connected m. s.'s. Thus, for any non- theorem A of E2, we have a finite refined connected m. s. t such that A is falsified by 9+. The proof of Theorem 3 also reveals that, if A is falsified by an assignment W from R+ for some m. s. = <K, Q, U>, then for any x e K Va(A) A is falsified by R+.

There are two particular m. s.'s with which we shall be concerned in what follows. 9e = <K, Q, U> is defined as follows: K = {a}, Q = {a}, and U = 0, for some element a. R+ has the property that PA = K for all A c KY NA = 0 for all A a K; Ve is (trivially) connected and refined.

=5 <K, Q, U> is defined as follows: K = {a, b}, Q = {a}, and U {<b, b>, <b, a>}, for some elements a and b. 95 is also defined, and its graph given, in [7], where it is pointed out that R5 is isomorphic to Lewis's Group I in [9]; it is evidently epistemic, connected, and refined.

THEOREM 4. If A is falsified by R+, then A is falsified by $.

PROOF. Suppose A is falsified by R+, i.e. for some assignment W <A , ..., An> (Ai c {a}) to the variables vi, ..., vn of A Va(A) / {a}, and




so Va1(A) = 0. Consider the same assignment from A+ to vj, ..., vn. Let Va (B) be the value assigned to a subformula B of A by W in R5. We show by induction that a e Va(B) iff a e Va5(B). If B is a variable v?, the result is trivial, as also are the cases where B has the form C -+ D and -C. Suppose then that B is DIC. Now both in A~e and in 5 a E Q, so that a g Va( DC) and a + Va5( DC), and the result holds here too. Since Va(A) 0, a + Va5(A) and A is falsified by S+.

THEOREM 5. If FE2 DT -O DA then FE2 A. PROOF.2 Suppose FE2 D]T -+ DA, and yet not FE2 A. Then, for any

epistemic m. s. *= <K, Q, U> and any assignment W from A+, Va,(DT -* WA) = K. Now it is easily seen that Va(DT) = K - Q, from which it follows that Va,(DA) = K - Q also, i.e. NVa(A) = K - Q. But, for A c K, NA c A, so that K - Q c VC,(A). In the particular case of ft5, since b E K - Q, b E V,(A); also, since b e NVa,(A) = K -Q, given Uba it follows that a e Va,(A), so that Va(A) = K, and A is satisfied by R+. Now if A is a non-theorem of E2, by Theorem 2 there is some (finite) refined epistemic m. s. R' ( <K', Q', U'> such that A is falsified by R'+, i.e. such that Va4(A) A K' for some assignment 1' from A'+. But we have seen that K' - Q' c V<,(A). By the remarks following Theorem 3, A is also falsified by ft'+ for any x E K' - Va,(A); but any such x e Q. Since S' is refined, $x is evidently isomorphic to St+, so that A is also falsified by t+. This contradiction with Theorem 4 proves the theorem.

We shall make use of this derived rule for E2 in the next section. We now turn to the system E3, defined by the addition to E2 of the axiom- schema:

A7: DA -D(DB -*A). E3 may be alternatively axiomatized by adding to E2 the axiom-schema:

A7': D(A -B) -*D(A -DB).

(This is, in effect, how E3 is defined in [5].) For, if in A 7 we take A as A -- B and B as A, we have:

FE3 D(A -B) -* O(DA - D(A -B));

and, using A 4 and R2, we may easily prove:

FE2 [([A D(A -PB)) -* D(EA ADB). Together, these results yield FE3 A 7'. Conversely, A 7 results from A 7' if we observe that FE2 DA -C D(B -* A).

From A7, using A4, we have: FE30A(CB DA-DDB DDA),

2 An alternative, and neater, proof of this theorem, in terms of normal forms for E2, has been communicated to me by Dana Scott, to whom I am also indebted for seeing the importance of it in connexion with S2 and S3. (Cf. the next section.)



196 E. J. LEMMON

from which, taking A as W2A and B as A, we derive at once:

FE3 LI LIA -* L LILA.

This and similar results enable us to derive reduction-theses for E3, in virtue of which it can be shown that the system has exactly 42 distinct and irreducible modalities (cf. Parry [13] for parallel reductions in the case of S3).

LEMMA. FE3 2 (OA A ALIT) *-?A. PROOF. That FE3 0 (OA A LIT) -> B A follows from A7 by contra-

position and easy modal transformations, if we take A as -A and B as T. Conversely, by propositional calculus (A 1-A3), FE3 OA -* (OA A LIT) v (AA - LIT). But FE3 0 A A [EIT -* )(OA A LIT), by A6. Also FE3OA A - CT -0-T, and FE30 -T -BLOC for any C. Putting these together, we have FE3 OA - (OA A UT).

For the algebraic development, we define an e-algebra as transitive iff, in addition to conditions (i)-(iii) for an e-algebra, it satisfies:

(iv) for x e M, P(Px - PO) Px. Using the definitions and ideas of [7], we now prove:

THEOREM 6. 9) = <M, {d}, u, ), -, P> is a regular E3-matrix iff <M, u, n, -, P> is a transitive e-algebra and d = 1.

PROOF. In addition to the proof of [7] Theorem 11, we need the following. That a regular E3-matrix fulfils condition (iv) is a consequence of the Lemma. Conversely, in any e-algebra PO _ Py (cf. [7], Theorem 6 (ii)), so that Px - Py < Px - PO, whence P(Px - Py) < P(Px - PO) = Px, by (iv). Now Nx -+ N(Ny -> Nx) = -Nx U N(-Ny U Nx) P-x U -P(-P-y fl P-x). But -P-x -P(P-x n -P-y), by the above. Thus Nx -+ N(Ny -+ Nx) = 1 and A7 is satisfied.

This theorem shows that the regular Lindenbaum E3-matrix (for whose existence, compare the proof of [7], Theorem 8) can be viewed as a transitive e-algebra with 1 as designated element. We next establish the finite model property for E3.

THEOREM 7. Let A be a wff with r subformulas. Then FE3 A iff A is satisfied by all transitive e-algebras with not more than 22r+1 elements.

PROOF follows the lines, and the notation, of [7] Theorems 13 and 14. In addition to what is proved there, it remains only to show that Pi(Pix - PiO) = Pix, given that P(Px - PO) = Px, where Plx =

Pyl n ... n Py. for yl, ..., yn covering x. In the construction of M1 from M, we no longer need to include PI, since PI 1= 1, but we include instead PO: this entails that M1 has not more than 22r+1 elements. Since PO EM1, P1O = PO by (v) of [7] Theorem 13. Now Plx < Pyj (1 < i ? n), whence Pix - P1O < Pyj - P10 = Pyj - PO. But P(Py - PO) = Pyi E M1, so that Pix - P10 is covered by Pyj - PO. Suppose that Plx - P1O is also covered by zl, ..., Zm. Then Pi(Pix - P1O) = P(Pyl - PO) n ... n




P(Pyn -PO) n Pzil ... n Pzm=Pyl n ... n Pyn n PZ1 i Pzm= Pix n Pzj n ... n Pzm. Thus Pi(Pix - P1O) ? Pix. Conversely, since 9R1 is an e-algebra, P1O ? Pjx, whence Pjx = (Pix - P1O) U P10. But Pjx - P10 ? Pi(Pjx - P1O) and also P1O < Pl(Pix P1O). Hence Pjx ? Pi(Pjx - P10). This proves the theorem.

Turning to model structures, we define an m. s. = <K, Q, U> as transitive iff U is transitive in K. We show:

THEOREM 8. If S is a transitive epistemic m. s., then R+ is a transitive e-algebra.

PROOF. It clearly suffices to show that P(PA - Q) = PA, for A a K, where S == <K, Q, U> is a transitive epistemic m. s. Suppose x E P(PA - Q). Then, if x E Q, x E PA anyway. If x g Q, for some y y E PA - Q and Uxy, whence y + Q and y E PA, whence for some z z E A and Uyz. Thus by transitivity Uxz so that x E PA. Conversely, suppose x E PA. It x E Q, then x E P(PA - Q) anyway. If x g Q, then x E PA - Q, whence x E P(PA - Q) by the assumption that S is epistemic.

THEOREM 9. Any finite transitive e-algebra is isomorphic to the algebra on some finite transitive epistemic m. s.

PROOF. We follow the lines of the proofs of Theorems 17 and 19 of [7]. Given 9R as a finite transitive e-algebra, we know how to find a finite epistemic S = <K, Q, U> such that 9 is isomorphic to R+ under an isomorphism S. By Theorem 1, we also know that R+ = A+, where S1 is the refined m. s. corresponding to R. Hence 9 is isomorphic to R+ under S. Further, by the isomorphism S P(PA - Q) = PA for A c K, since V is transitive. Now suppose, for x, y, z E K, that Ujxy, Ulyz; then x, y g Q since M1 is refined. We have that y E P{z}, y E P{z} - Q, whence X E P(P{z} - Q). It follows that x E P{z} and that Ulxz, so that U1 is transitive.

THEOREM 10. KE3 A (i) iff A is satisfied by R+ for all transitive epistemic m. s.'s M (ii) iff A is satisfied by R+ for all finite transitive epistemic m.s.'s .

PROOF is immediate from Theorems 7, 8, and 9 (compare proof of Theorem 20 in [7]). If, following the lines of [7], we define E3-validity of A to mean that A is valid in all transitive epistemic m. s.'s, by [7] Theorem 21 we have:

THEOREM 1 1. FE3 A iff A is E3-valid. A further result for E3, useful later, is: THEOREM 12. If FE3 LIlT -+ WA then FE3 A. PROOF is parallel to that of Theorem 5, if we note that S5 is a transitive

epistemic m. s. so that R+ satisfies E3. Our third system, ET, results from the addition to E2 of the axiom:

A8: D1T -+D D1-T.

Since the converse follows from A6, we have that FET L- T --* W DT. The



198 E. J. LEMMON

semantics of ET is readily given in the next few theorems. Let an e-algebra be closed iff, in addition to conditions (i)-(iii) for an e-algebra, it satisfies:

(v) PPO = PO.

THEOREM 13. 9 = <M, {d}, U, n, -, P> is a regular ET-matrix iff <M, U, l, -, P> is a closed e-algebra and d = 1.

PROOF. Compare Theorem 6. That a regular ET-matrix satisfied (v) is immediate from the fact that FET OOF O-* OF. Conversely, given -(v), we have NN1 =N1, so that N(x -x) -+NN(x -x) = -NI u NN I 1.

THEOREM 14. Let A be a wff with r subformulas. Then FET A iff A is satisfied by all closed e-algebras with not more than 227+1 elements.

PROOF. Compare Theorem 7. It suffices to show that PlPl0 = PiO, given PPO = P0. As in the proof of Theorem 7, we include P0 in the construction of M1, so that PPO = P0 E M1. Thus by (v) of [7] Theorem 13, PlPO = PP0. By the same result, P0 = P10. Thus PlPl0 = PPO= PPO = PO= P10.

We next define a model structure M - <K, Q, U> as closed iff whenever Uxy then yE K - Q, for x, y E K. Thus a closed model structure has no U-leads into or within Q.

THEOREM 15. If S is a closed epistemic m. s., then A+ is a closed e- algebra.

PROOF. Suppose M == <K, Q, U> is a closed epistemic m. s. It will suffice to show that PQ = Q. Q c PQ is trivial, so suppose x E PQ. Then (3y) (y E Q A Uxyl) v x E Q. But the first disjunct is impossible since t is closed.

THEOREM 16. Any finite closed e-algebra is isomorphic to the algebra on some finite closed epistemic m. s.

PROOF. As in Theorem 9, we know that 9 is isomorphic to R+, and so to R+ where M, is refined. Further, by the isomorphism, since 9R is here closed, PQ = Q. Now suppose that Ulxy and yet y E Q, for x, y E K. Then X E PQ = Q, which contradicts the refinement of S1.

As usual, we have completeness results: THEOREM 17. FET A (i) iff A is satisfied by R+ for all closed epistemic

m. s.'s M (ii) iff A is satisfied by R+ for all finite closed epistemic m. s.'s R. And, defining ET-validity of A to mean that A is valid in all closed

epistemic m. s.'s, we have: THEOREM 18. FET A iff A is ET-valid. A crucial difference between ET and E2, E3 is that the analogue of

Theorems 5 and 12 breaks down. This is mainly because 95 is not closed, since Uba and yet a E Q. Indeed, if it were true that if FET UIT -C EA then FET A, from A8 we could conclude FET UIT, yet, as we shall see, ET, like the other E-systems, has no theorem of the form WA.

Our fourth system, E4, results simply from E2 by the addition of both




A7 and A8. Equivalently, we may add to E2 the axiom-schema: A9: CA --W A.

For FET WA -C WWT since FE2 WA -C LiT, and FE3 WA (Li LOT D WA) by A 7 and A 4. Hence FE4 A 9. Conversely, FE2 EZWCA F-> (W-B WA), so that A7 is an immediate consequence of A9; and so is A8. (It is in this alternative way that E4 is defined in [5].)

In view of the present axiomatization of E4, completeness results and a decision procedure are forthcoming for it by a simple combination of the results for E3 and ET, and do not require separate treatment. The relevant algebras are closed transitive e-algebras,3 and the relevant model structures are closed transitive epistemic model structures. If we say that A is E4-valid iff A is valid in all closed transitive epistemic m. s.'s, we have:

THEOREM 19. FE4 A iff A is E4-valid. Thus algebraic semantics have been given for the four systems of the

present section.

II

The systems S2, S3, T, S4. The systems to be treated in this section, S2, S3, T, and S4, unlike the

systems of the last, are well-known in the literature (see e.g. Prior [14]). We shall see that appropriate semantics for them can be readily developed from those for the previous systems, in a quite natural way.

We need no new axiom-schemata, except that we shall write WAi for the schema resulting from schema Ai by prefixing the necessity symbol (1 ? i < 9). However, we shall use two new rules:

R lW(A -B) R WA R2 W 1(WFA-F1WB)'

R. A

Our four new systems are defined as follows:

S2 {A6, WA1-WA4, WA6; R1, R2'}; S3 ={A6, WA 1-WA4, WA6, WA7; R1}; T = {A1-A4, A6; R1, R3}; S4 = {A1-A4, A6, A9; RI, R3}.

These formulations of T and S4 are virtually standard. As for S2 (S3), we note first that A 1-A4 (A 1-A 4, A 7) follow as theorem-schemata by A6; second that, by a simple induction, if A is provable from A 1-A 4, A6 (A 1-A 4, A6, A7) by RI alone, then FS2 (S3) WA. (This holds for A 1-A 4, A6

3 These algebras can be seen to be alternatively definable as e-algebras satisfying the condition that PPx = Px.



200 E. J. LEMMON

(A 1-A4, A6, A7) themselves; and a step of RI to B from A and A -*B can be paralleled by steps from EIA and EZ(A -+ B) to EIB by A4 and RI). Since all tautologies are derivable from A 1-A 3 by R1 alone, we have that, if A is a tautology, then FS2 (S3) WHA. The equivalence of the above formulation of S2 to that in [5] is then immediate. For S3, we additionally note that, using A7, we can easily derive R2' as a rule (take A as A -- B and B as T. and note that FS3 LiT); this enables us to prove LIA7' from DA7 in a way parallel to the E3-derivation of A7' from A7 (see last section), and shows equivalence to the formulation of S3 in [5].4

THEOREM 20. (i) FE2 A iff FS2 WA; (ii) FS2 A iff FE2 iT --* A; (iii) FE3 A iff PS3 GA; (iv) Fs3 A iff FE3 LiT -*A.

PROOF. The proof that, if PE2 A, then PS2 FIA is by induction, as mentioned in [5]. It is also obvious that, if PE2 LT -* A, then FS2 A, since FS2 L-IT. Conversely, suppose FS2 A: we show by induction on the length of proof that FE2 LIT -* A. The result is immediate for the axioms, since FE2 LCT -> DAi (i = 1, 2, 3, 4, 6) by R2. A step of RI from B and B -+ C to C can be paralleled in E2 by steps from LIT -+ B and LIT -+ (B -+ C) to LT -- C using A2 and RI. Suppose, finally, that LI(LIB -+ DC) follows from D(B C) by R2', and that FE2 [:lT -+ D(B -+ C). Then by Theorem 5 FE2 B C, whence PE2 LiB -+ DC by R2, whence clearly FE2 T -+ (LIB -+ DC), whence FE2 LIT -+ LI(LIB -? DC) by R2 again, as was to be shown. This proves (ii). For the other half of (i), suppose FS2 LIA. Then, by the result just obtained, PE2 FIT -? WA, whence FE2 A by Theorem 5. This gives (i).

For (iii) and (iv), we have that, if PE3 A, then PS3 LOlA by induction (a step of R2 from A B to LIA -O LIB can be paralleled by steps from LI(A -- B) to LI(LIA O-+ B) using A7'). The proof that, if PE3 LIT -+ A, then 1S3 A is immediate, given 1S3 7T, and the converse is also trivial, since we do not have R2' to worry about. This proves (iv). For the other half of (iii), suppose 1S3 LiA. Then, by the result just obtained, FE3 EiT

C-WA, whence FE3 A by Theorem 12. This gives (iii). THEOREM 21. (i) FT A iff FET FIT -+ A; (ii) PS4 A iff PE4 LOT -- A. PROOF. That, if PET (E4) LOlT -- A, then FT (S4) A is immediate, given

FT(S4) LT (and the obvious inclusion of the theorems of ET (E4) in those of T (S4)). The converse is proved by a simple induction: a step of R3 from B to DB is paralleled by steps from E]T -+ B to LT -+ LIIB, using R2 and A8.

It is a consequence of Theorem 20 (ii) and (iv) and Theorem 21 that the decision problem for the four new systems reduces to that for the previous

4 It is perhaps worth remarking that the above gives a finite axiomatization of S3, along the lines of Simons [15]; such axiomatizations can also be given for S4, E3, E4, but not for E2, S2, ET, or T, as is shown in [8].




four, which is of course already solved. (Conversely, Theorem 20 (i) and (iii) reduces the decision problem for E2 and E3 to that for S2 and S3.) Further, these results enable us to derive completeness results for the new systems from those for the old.

At the algebraic level, let us say that a wff A with n variables is weakly satisfied by an e-algebra 9J iff f(A), the n-adic matrix function corresponding to A, takes values > NI for every n-tuple of elements of WN. This is equivalent, obviously, to a rethinking of 9J as a matrix; instead of taking I as sole designated value, we are now viewing all elements > Ni as designated, i.e. D, the set of designated elements, has become the principal filter (additive ideal) generated by Ni; then weak satisfaction is equivalent to satisfaction by the new matrix.5

THEOREM 22. For any e-algebra 9), UiT -+ A is satisfied by ? iff A is weakly satisfied by 9W.

PROOF. f(OT--A) - I iff Ni I_ f(A) - 1 iff Ni ? f(A) Algebraic completeness results for all four systems follow, by Theorems

20-22, from the corresponding results for the earlier four. THEOREM 23. PS2 (S3, T, S4) A iff A is weakly satisfied by all finite (tran-

sitive, closed, closed transitive) e-algebras. Actually, better results for T and S4 can be obtained, in terms of satis-

faction in normal e-algebras (where a normal algebra, as in [7], is one in which the condition P0 = 0 is met) rather than weak satisfaction in closed e-algebras. The relevant results are, actually, forthcoming from the study of normal algebras and model structures in [7]; but it will be more convenient, and in some ways more illuminating, to derive them indepen- dently here. For this will throw more light on the importance of the closure condition and the axiom A8, and show why similar results are not obtainable for S2 and S3, whose algebras fail to meet this condition.

We note first that, if A is weakly satisfied by all (finite) e-algebras, then a fortiori A is satisfied by all (finite) normal e-algebras, since, if PO = 0 and so Ni = 1, weak satisfaction simply is satisfaction. For the converse relationship, it is clearer to work at the level of model structures. If A is weakly satisfied by $+ for m. s. = t'<K, Q, U>, then for any assignment W K - Q c V,(A). In terms of Kripke models 0(v, k) (for idea, see [4]; for notation, see [7]), weak satisfaction of A in $+ is equivalent to truth of A for all models P(v, k) at all 1 e K - Q in $: what we may call the weak validity of A in R.

Let us now consider a model structure R = <K, Q, U> which is closed.

5 Note that this new matrix is not a regular matrix, in the sense of [7]. For it is easy to see that condition (iii) of [7] Definition 5 will not in general be met (e.g. if x = 1, y = NI, we have x m y NI E D, but not always x = y). We incidentally see here why finite matrices with more than one designated value play a role in the study of the weaker Lewis systems.



202 E. J. LEMMON

Then by the closure condition there are no U-links into Q. If we select x E K - Q, it follows that in xS <Kx, Q, Ux>, the connected m. s. generated by x from St, Qx = 0, so that R+ is normal. We use this fact in the proof of the following theorem. We say that- A is weakly falsified by SV+ iff A is not weakly satisfied by V+.

THEOREM 24. If A is weakly falsified by V+ for closed epistemic =

<K, Q, U>, then for some x E K - Q A is falsified by R+, where S+ is a normal e-algebra.

PROOF. If A is weakly falsified by SV+, then for some assignment W (K - Q) - Va(A) # 0. Select x E (K - Q) - Va(A). Then S + falsifies A by the remark following Theorem 3, and R+ is normal by the remarks preceding this theorem.

If, in addition to being closed, St = <K, Q, U> is also transitive, then, for x E K - Q, + will not only be normal but transitive also. Hence, if A is a non-theorem of T (S4), there is a finite (transitive) e-algebra, and so a finite (transitive) V+ for epistemic m. s. Rt, in which it is weakly falsified. Thus by Theorem 24 and remark following, there is a finite (transitive) normal e-algebra V in which it is falsified. We therefore obtain all the usual completeness results for T and S4. Note in particular that a normal epistemic algebra is simply an extension algebra in the sense of [6], and that a normal transitive epistemic algebra is simply a closure algebra in the sense of McKinsey-Tarski [12]. Hence Theorem 23 yields:

THEOREM 25. FT(S4) A iff A is satisfied by all finite extension (closure) algebras.

T-validity is defined in [7] as validity in all normal epistemic m. s.'s. We may define S2-validity as weak validity in all epistemic m. s.'s, S3- validity as weak validity in all transitive epistemic m. s.'s, and S4-validity as validity in all normal transitive epistemic m. s.'s. We then have:

THEOREM 26. kS2 (S3, T, S4) A iff A is S2- (S3-, T-, S4-) valid. The relationship between E2 and S2 should already be fairly clear, in

view of Theorem 20. It can be made clearer, however, in the following way. We define a hierarchy of systems E2n such that E20 = E2 and, for n > 1, E2n has as axioms all A such that FE2 A together with LiIT, and sole rule of inference R1. We discuss first E21.

THEOREM 27. E21 _ S2. PROOF. The inclusion of (the theorems) of E21 in (those of) S2 is obvious.

Conversely, suppose kS2 A. Then FE2 UT -* A by Theorem 20, whence kE21 A. Thus the class of S2-theorems is exactly the class of E2-theorems together with LIlT, closed under detachment.

THEOREM 28. kE2n A iff hFE2 LOlnT -* A. PROOF by induction. Thus the decision problem for all systems E2n is solved via that for E2.

Appropriate semantics for all systems may be given by analogy with those




for S2. Thus, let us say that A is n-weakly satisfied by 9R iff for all assign- ments from 9N f(A) > Nn1. By analogy with Theorem 23, we have:

THEOREM 29. 'EE2n A iff A is n-weakly satisfied by all finite e-algebra. We may say that A is n-weakly valid in 9 iff A is true for all models

0(v, k) at all 1 E (K - Pn-1Q) in R. Then A is E2n-valid iff A is n-weakly valid in all epistemic m. s.'s. We easily show:

THEOREM 30. FE2n A iff A is E2n-valid. The systems E2n form an increasing series, in the sense that the theorems

of E2n are properly included in those of E2fn+l; for E2n has no theorems of the form LIJn+lA, as e.g. the matrices 9Rn of [8] show. Further, the totality of all theorems belonging to any E2n is just the class of T-theorems, as is also shown in [8]. The argument of Theorem 5 can be appropriately generalized to show that, if FE2 DTT -* W8A, then FE2 A, whence it can be shown, along the lines of Theorem 20, that FE2 A iff hE2n OnA. This entails that a generalized version of R2' can be derived for each E2n:

A =>n B R2,n: A =>n OB

Thus substitutability of 'n-strict' equivalents, i.e. A and B such that FE2nA >n B, is available for E2n, but not of rn-strict equivalents for m < n. This generalizes known results for S2.

We might, finally, try defining a similar hierarchy based on E3, such that E30 = E3 and E3n, for n > 1, has as axioms A for FE3 A together with FLjnT, and R 1 as sole rule of inference. We find, however:

THEOREM 31. E31 - S3; E3n = S4 for n ? 2. PROOF. That E31 S3 follows from Theorem 20 (iv). Evidently if

FE3n A then ks4 A, and if FE3n A then kE3n+lA. Hence it suffices to show that if Fs4 A then FE32 A. Now FE3 W LiT -* (WA -O W WA), by A7 and A4. Thus FE32 A9. Suppose FE32 A. We show by induction that FE32 WA, and thus that R3 is a derived rule of E32. Given FE3 A, we have FE3 DT -* OA by R2, whence FE32 WA. Also FE32 03T by A9 and the axiom O2T. Thus if A is an E32-axiom FE32 WA. Obviously, when C follows from B and B -* C by R1, OC follows from OB and O(B -* C) by A4 and R1. This completes the proof.

Thus, whilst S2 may be thought of as the first step in hierarchy of systems leading from E2 to T, and whilst S3 bears the same relationship to E3 as S2 does to E2 in most respects, the analogous hierarchy between E3 and S4 collapses into S4 itself at the second stage. This fact is closely connected with the finite axiomatizability of S4 - see [8], especially Theorem 1.



204 E. J. LEMMON

III

A general representation theorem6. With respect to their various inclusions, the eight systems so far considered

form a cube in a natural way. Let us imagine the E-systems at the vertices of its base: then E3 and ET form extensions of E2 in which A7 and A8 are added respectively, and E4 forms an extension in which both are added. We may picture the four systems of the last section placed vertically above the previous four at the vertices of the top face: as Theorems 20 and 21 show, each top system has as theorems exactly the sentences A where UT -+ A is a theorem of the bottom system. (See the chart in Section VI.) We now wish to consider eight further systems, obtained from the given eight as it were along a new dimension. This new dimension may be viewed as the addition of the schema:

A 10: UT -(A -*LYA);

except that, when extending S2 and S3, we add CA 10 instead (from which A 10 follows by A6). As we shall see, many of the resulting systems are familiar in the literature.

Our completeness results so far have hinged essentially on first showing that a non-theorem of a given system was falsified in a suitable finite algebra and then using representation theorems for finite algebras in terms of the algebras on certain finite model structures. These methods appear not to be straightforwardly available for the new systems. We, therefore, reverse the two steps. We prove first a representation theorem applying to all algebras, whether finite or infinite, and then establish the finite model property at the model structure stage rather than at the purely algebraic. (In fact, this order might have been followed throughout.) The present section yields this general representation theorem.

We go back to the notion of a modal algebra in [7], essentially an algebra like an e-algebra except that condition (iii), that x < Px, is not demanded. If 9 = <M, U, n, -, P> is a modal algebra such that M is denumerable, then of course there can be no model structure 9 = <K, Q, U> such that 9 is isomorphic to A+, since the algebra V+ will have either finitely many or non-denumerably many elements, depending on whether K has finitely many element or not. Hence, as with Stone's general representation theorem for Boolean algebras, our best result will be to show that 9 is isomorphic to a suxbalgebra of S+ for some R. In fact, our proof will follow closely, and be a generalization of, Stone's.

For an arbitrary modal algebra 9 = <M, U, n, -, P>, we shall use

6 For the central results of this and the following section, I owe much to ideas communicated to me in correspondence with Dana Scott.




m, n as variables ranging over M. We put F as the set of all proper maximal filters of the Boolean algebra <M, U, n, ->, and let x, y, z be variables ranging over F. (A proper maximal filter x E F is a subset of M which is a filter (additive ideal) such that 0 O x and for all m E M either m E x or - m E x.) Now let A be any subset of M. All finite intersections of members of A are in M, of course; let A' be the set of them, i.e. A' = {n (3m1) ... (3mp) (n ml r ... i Mnp A M1, ..., np E A} (p ? 1). Clearly A s A'. We put As {m: m ? n for n E A'}. (In case A = 0, put As = {1}.) Evidently, A' As, so that A a As. Intuitively, if 9) is a Lindenbaum algebra, we may think of As as the set of 'theorems' generated by the members of A taken as 'axioms'; compare Stoll [17], Chapter 6, Sections 6-9. Indeed A s is always a filter, as we now show:

LEMMA 1. For A a M, AS is a filter of <M, U, n, >.

PROOF. Suppose mE As, n E AS. Then there are finite intersections of members of A, mi' and n' E A', such that m > m', n > n'. But obviously m' n n' is a finite intersection of members of A, and m n n > m' n A', so that m n n E As. Now suppose m E As, n E M. Then m > m' for some m' E A'. But m U n ? m, so that m U n > m' and m U n E As.

Let us say that A a M is consistent iff 0 f As, otherwise inconsistent. If A is consistent, As is a proper filter by Lemma 1 and so contained in a proper maximal filter (by Zorn's Lemma - see Stoll [17], Chapter 6, Theorem 5.2 and Section 9 (3)), i.e. (3x) (x E F A A : x). Conversely, suppose (3x)(x F A A s x), and select ml, ..., mi eA. Then ml, ..., mpE x so that nm1 E x since x is a filter; thus As c x so that 0 g As since x is proper, and A is consistent. Thus A's consistency is equivalent to A's containment in a proper maximal filter of F.

LEMMA 2. For A c M and mE M, if A U {-m} is inconsistent, m E As. PROOF. Suppose A U {- m} is inconsistent. Then 0 E (A U {- m})s,

i.e. there are members nl, ..., np E A u {- m} such that 0 = ni n ... n np. Either -m = n1 for some (1 < j ?p) or not. Suppose -m = n1 for some j. Then, unless 0 = m (i.e. -m = n1 for all j), in which case m = 1 E As anyway, evidently 0 = -m n n' for n' E A', whence m ? n' and miEAs. If -m is not one of the n1, then nfln ... n ni= 0EA' and, since 0 ? m, m again E As.

LEMMA 3. For A c M, (Vx)(xEF AA s xTimex) iff mEAs. PROOF. For A : M, suppose m E AS and A c x for a proper maximal

filter x. Then, as we reasoned above, As c x so that mE x. Conversely, suppose m g As. Then, by Lemma 2, A U {- m} is consistent and so contained in some proper maximal filter y E F. But in that case A c y also, and yet, since -m ey, m g y.

(Lemma 3 is intuitively equivalent to the assertion that a 'sentence' m belongs to all complete and consistent extensions x of A iff m is a 'consequence' of A.)



206 E. J. LEMMON

THEOREM 32. Let 9) =<M, U, n, -, P> be a modal algebra. Then there is a model structure M* <K, Q, U> such that 9R is isomorphic to a subalgebra of 9+.

PROOF. Let K simply be F, the set of all proper maximal filters of <M,u,n, ->. For mEM, put +(m) = {xEF: mEx}. Put Q = {XeF: PO E x}, so that x E Q iff POE x. Finally, for x, y E F, put Uxy iff (Vm)(mE y - Pm E x). We write 9 - <K, Q, U> and use P* for the possibility operation in f+, i.e. for C a K, P*C = {x: (3y)(y E C A Uxy) V x E Q}. We shall show that 9 is isomorphic to a subalgebra of A+ under b. Indeed, this will be a simple consequence of the four results:

(i) for m, nE M, b(mn U ) =+b(m) u +(n); (ii) for m, n eM, 0b(m n n) = +(m) n +(n);

(iii) for m EM, k(-m) = (m) (iv) for m E M, b(Pm) = P*b(m).

Of these, the first three, as in Stone's representation theorem, are simple consequences of the definition of b and properties of proper maximal filters. For (iv), suppose first that x E P*b(m), i.e. either for some y such that y E +(m) Uxy or xE Q. If y E #(m), m E y, and, if Uxy, (Vm)(m E y - Pme x), so that PmE x and xE e/(Pm). If xE Q, POE x. But Pm > PO for any modal algebra, so that PmaE x since x is a filter, and again xE O(Pm). Thus P*O(m) a O(Pm).

The converse alone presents problems. Suppose x t P*b(m), i.e. (V/y)(Uxy -b y # +b(M)) A x g Q. Now it is easy to verify that Uxy iff (Vn) (Nne x -*n E y). Hence (Vy) ((Vn) (NnE x -nE y) - -mE y), since y is maximal. Put A = {n : NnE x}, so that n e A iff NnE x. Then (Vy)(A a y -*- mE y). Hence by Lemma 3 -amE As. This means that there is some n' l nj ... n n. for nf eA (1 _ j <?p) such that -am n'. But then Pm= N - m Nn' =Nni n ... n Nnv, by properties of modal algebras (see [7] Theorem 6). However, Nn1jE x, by definition of A, so that Nn' E x and - Pm E x. Thus Pm g x, i.e. x f O(Pm). This shows that b(Pm) a P*b(m), and completes the proof of (iv).

If we take the set of subsets of K of the form +(m) for some m E M, it is clear from (i)-(iv) that this set is closed with respect to U, I, -, and P*;

thus this set forms a subalgebra of R+ to which 91 is evidently isomorphic under b. Hence our proof is complete.

We next wish to show that, if restrictions of various kinds are imposed on the algebra 9, the representation theorem still holds with appropriate restrictions imposed on the corresponding model structure R. We recall that in [7] a deontic algebra is a modal algebra satisfying the condition that PI = 1 and a deontic model structure is a model structure for which for all x either (3y) Uxy or x E Q.

THEOREM 33. Let 9 be a deontic (epistemic, normal, closed, transitive) algebra. Then there is a deontic (epistemic, normal, closed, transitive)




model structure R such that 91 is isomorphic to a subalgebra of R+. PROOF follows in terminology and structure that of Theorem 32. Suppose

9N deontic, i.e. PI 1. Now, for all xEF, 1 Ex, so that +b(1) F ==K, and P*K =P*0(1) = (P 1)-= (1) =)K, or {x: (3y) Uxy v xE Q} K, and 9 is deontic. Suppose 9) epistemic, i.e. m < Pm, and m E x. Then Pm E x since x is a filter, and so Uxx. Thus At is epistemic. Suppose 91 is normal, i.e. PO = 0. Now, for all xEF, 0~x, so that PO~x and Q = 0. Thus 9 is normal. Suppose 9 is closed, i.e. PPO = PO. By Theorem 1, we know that f+ = f+, where 91 is refined. Suppose Uixy and yet y E Q. Thus xEK - Q, (Vm)(mEy -?PmEx), POEy. Then PPO = POEx and xEQ, a contradiction. Hence 91 is closed, and 91 is isomorphic to a subalgebra of 9+. Finally, suppose 9) is transitive, i.e. P(Pm - PO) = Pm. Again by Theorem 1 we have h+ = + for refined ki. Suppose now Uixy, Ulyz. Then x, yE K - Q, so that - POE y since y is maximal; also (Vm) (m E y Pm E x), (Vm)(m E z -* Pm E y). Suppose m E z. Then Pm e y. Thus Pm - P0 Ey since y is a filter. Hence P(Pm - PO) = PmE x. Since x E K - Q, Ujxz follows, and fl is transitive.

It is clear from the nature of this proof that combinations of restrictions on algebras give no special problems: for example, closed (transitive, closed transitive) e-algebras are isomorphic to subalgebras of the algebras on some closed (transitive, closed transitive) epistemic model structure. In view of this, we can give, via the representation theorems and the various regular Lindenbaum matrices, new characteristic matrices for all systems so far considered, either in [7] or in the present paper: whilst the Lindenbaum matrices contain only denumerably many elements, the new matrices contain non-denumerably many, of course. We illustrate in the case of E2. We know this system has a regular characteristic matrix which is an e-algebra, by [7], Theorems 8 and 11. By Theorem 33, this e- algebra is isomorphic to a subalgebra of f+ for some epistemic R. By [7] Theorem 18, R+ is itself an e-algebra, and so an E2-matrix. On the other hand, for any non-theorem A of E2, A is falsified by a subalgebra of R+ by the isomorphism, and so clearly by A+ itself. Thus R+ is characteristic for E2. And so for the other systems - with the exception of S2 and S3, where special qualifications need to be made in the light of the results of the last section. These facts may be summed up by giving a list of appropriate model structures for each system:

System Corresponding appropriate m. s. C2 Any m. s. D2 Deontic m. s. E2 Epistemic m. s. T(C) Normal m. s. T(D) Normal deontic m. s.



208 E. J. LEMMON

T Normal epistemic m. s. E3 Transitive epistemic m. s. ET Closed epistemic m. s. E4 Closed transitive epistemic m. s. S4 Normal transitive epistemic m. s.

THEOREM 34. For each of the systems C2, etc., there is a corresponding appropriate m. s. f such that f+ is a characteristic matrix for the system.

IV

Eight further systems. Of the eight systems adumbrated at the beginning of the last section,

we begin with an uninteresting one: though once its problems are solved, those of the remaining seven are easy. The system E2(S) shall be the system resulting from the addition to E2 of A10. I know of no earlier mention of this system in the literature.

THEOREM 35. (1) FE2(S) LIT (FA -G A); (2) F E 2(S) F-1(0A B) (A -*F1B); (3) FE2(S) V(A -* LIB) (OA -- B).

PROOF of (1) is trivial. For (2), we note that FE2(S) EZB -* (A -* LI<>A), since clearly FE2(S) EIIB -E LIT. By A4, FE2(S) LI((OA -* B) -* (LIK>A EZB), and (2) now follows. (3) is proved similarly, using (1).

We call an e-algebra 9R symmetric iff it satisfies the condition: (vi) for xE M, x -PO < NPx. THEOREM 36. 9) == <M, {d}, U, a, -, P> is a regular E2(S)-matrix iff

<M, U, n, -, P> is a symmetric e-algebra and d = 1. PROOF. That a regular E2(S)-matrix satisfies condition (vi) is immediate

from A 10. Conversely, NI -* (x -* NPx) = x - PO -- NPx = 1, by (vi). There is no difficulty in showing that E2(S) has a characteristic regular

Lindenbaum matrix, so that by this theorem there is a symmetric e-algebra which characterizes it. Let us say that S = <K, Q, U> is symmnetric iff U is symmetric in K - Q. We next need:

THEOREM 37. Let 9) be a symmetric e-algebra. Then there is a symmetric model structure At such that 9 is isomorphic to a subalgebra of 9+.

PROOF follows the lines of Theorems 32 and 33. If 9 is symmetric, m - PO ? NPm. Suppose Uxy, i.e. (Vm) (m E y - Pm E x), for x, y e K-Q. Then it can quickly be shown that (Vm) (Nm E x -i mE y); also, given x E K - Q) - PO E x since x is maximal. Now, given m E x, we have M - PO E x, whence NPm e x, so that PmE E y. This shows that Uyx, i.e. that S is symmetric.

THEOREM 38. The algebra on a symmetric model structure is symmetric. PROOF. Let A = <K, Q, U> be a symmetric m. s. We require to prove




that, for A a K, A - Q a NPA. Suppose then that xE A - Q, so that XE A, x g Q. Now x E NPA iff x Q A (Vy)(Uxy -> (3z)(z E A A Uyz) V yE Q)). Suppose, then, that Uxy. If y E Q, the result is trivial. If y f Q, then Uyx by the symmetry of U in K - Q. But then (3z)(zE A A Uyz).

We can now prove: THEOREM 39. There is a model structure f such that $+ is characteristic

for E2(S). PROOF. Let f be the model structure given by Theorem 37 from the

Lindenbaum algebra V for E2(S). Then any non-theorem of E2(S) is falsified by $+ by Theorem 37. Conversely, any theorem of E2(S) is satisfied by f+ by Theorems 36 and 38.

We are now ready to establish the finite model property for E2(S). We establish first, however, a rather general result which can be used as a basis for proving the finite model property for a wide variety of systems.

THEOREM 40. Let $ = <K, Q, U> be any model structure, and Al, ..., A,, subsets of K. Then there is a finite model structure 3 = <K, Q, U> with at most 2r+l elements with subsets A1, ..., Ar G K such that:

(i) As KiffA=K-; (ii) -Ai = A1 iff -A = A1; (iii) Ai U AJ=Ak iff AJ u A= Ak; (iv) Ai rn AIAk iff A nj =Ak; (v) if PA == A1, then 'PAi = A1;

where P and P are, respectively, the possibility operations in R+ and 9+. PROOF. Put S = {Al, ..., Ar, Q}, and, for x, yE K, put x _ y iff

(VA)(A E S -* (xE A M-+ y E A)). Then - is an equivalence relation in K, which in fact partitions K into not more than 2r+l equivalence classes. For xE K, put -{y: y = x}. For A a K, put A= {x: xE A}. Put K { : x E K}, the set of --equivalence classes of K, with a cardinality not greater than 2r+l. For x E K, A a K) it is easy to verify that (i) if xEA then XEA and (ii) if XEA and A ES then xEA. Thus we have already defined Q, Ai, ..., Ar. We finally put, for x, 9 E K, UNg iff (VA)(A E S A PA E S -> (y E A -? x E PA)). This definition is proper, i.e. independent of the choice of x and y in x and 9, since, for y, y' E 9, y E A iff y' E A given A E S, and for x, x' E i, xE PA iff x' E PA given PA ES, by the definition of

Of the five conditions of the theorem, (i)-(iv) are quite trivial. For (v), we show first that PA i PAL, given PAi = A1E S. Suppose then that x E PAL. Then, since PAj E S, x E PAL, so that either (3y)(y E Ai A Uxy)

or xEQ. Suppose that yEAi and Uxy. Now if yEA, for any A a K, we have x E PA given Uxy; hence certainly Uxy. But also y E Ai. Thus x e PAL. Alternatively, if x E Q, x E Q, whence x E PAL. Thus PAi a PAL.



210 E. J. LEMMON

Conversely, suppose x e PAL, so that either (35) (A i A Ux') or X e Q. Now, if 3 E At and Ux, we have yE Ai (since At eS) and (VA)(A E S A

PA ES-* (yEA -* xE PA)). But Ai, PAj ES; hence xEPAi and so

e PAR. Alternatively, if x E Q, X e Q (since Q E S), so that also x E PAL, x E PA . Thus PAt a PA i.

Now suppose PAi = A1. Then PAi = PAi = AJ. This proves (v) and the theorem.

This theorem can be used, in place of [7] Theorem 13, to give the finite model property for C2 and other systems; in fact, the proofs are related, the finite algebra R+ being equivalent in a sense to the algebra 911 of that theorem, and the relation U replacing McKinsey's notion of one element of T11 covering another. For present purposes, however, we need a variant covering the case of symmetry.

THEOREM 41. If $ = <K, Q, U> is a symmetric model structure, and A1, ..., Ar are subsets of K, then there is a finite symmetric model structure

= <K, Q, U> with at most 2r+l elements with subsets A1, ..., Hr a K satisfying the five conditions of Theorem 40.

PROOF follows the outlines of that for Theorem 40. S, K are as before; however, for x, y E K, we now define Uxy iff (VA)(A E S A PA E S -*

(yEA -*xEPA)A (x 'QAy 'Q-* (xEA -* yEPA))); the propriety of this definition hinges on the presence in S of Q. As before, (i)-(iv) are immediate. Suppose PAi- A1 E S, and consider x E PAL. Then either (3y)(yE Ai A Uxy) or xE Q. If xE Q, X E PAi of course. For A a K, if yE A then by Uxy xE PA. Further, if x ? Q, y ? Q, then from Uxy we deduce Uyx (U is symmetric on K - Q), so that if x E A then y E PA. Thus UNx; but 9 e As; hence xe PEAi. Conversely, if xc E PAL, either

(35l)(5 E Ai A UN-) or x E Q. If XF e Q, then anyway x~ E PAi. We have Af e S, PAi E S, hence, given UN- and y E Ai, we have x E PAi, xc E PAi. This gives (v). It remains to show that U is symmetric in K - Q. Suppose then Usxyq, x ? Q, 5 ? Q, so that x ? Q, y ? Q. It is easily verified that by the definition of U Ujx.

A clear consequence (compare McKinsey [11], Theorems 5 and 6) is that a sentence falsified by $ is falsified by a suitable $+: we let Al, ..., Ar be the values taken by the r subformulas of A for a falsifying assignment in $+; then Al, ..., Ar will be the values taken by the same subformulas for an appropriate falsifying assignment in $+. Thus Theorem 41 established the finite model property for E2(S), and we have:

THEOREM 42. FE2(s) A (i) iff A is satisfied by $+ for all symmetric epistemic m. s.'s $ (ii) iff A is satisfied by $+ for all finite symmetric epistemic m. S.'s r.

PROOF simply requires an extension of Theorem 41 to efiistemic m. s.'s.




However, it is immediate that, given Uxx for x g Q, we have Ux for xc t Q.

The suitable validity definition and theorem for E2(S) is obvious. We finally note:

THEOREM 43. If FE2(S) EIT -- [IA then FE2(S) A. PROOF is parallel to that of Theorem 5 (cf. Theorem 12), if we note that

$5 is a symmetric epistemnic m. s., so that S+ satisfies E2(S). The second system, to be called E3(S), results from adding A 10 to E3.

We note first: THEOREM 44. hE3(S) EIO< WA -G WA. PROOF. We have FE2 GO) WA -* < WA, FE2 DO WA -E LT (by R2),

hence hE2 GO) WA -E T A >LO A. Taking B as T in A7 and using R2, we have FE3 ODA-O)LI(LT - WA). Thus FE3 LITAO<)LA - LITA <>Li(OT -G 0A). Now, using A10, we have FE2(S) WTAO)W(WT WA) -G WA. Thus FE3(S) WO WA -E WA.

As a consequence of Theorems 36 and 6: THEOREM 45. 9N = <M, {d}, u, n, -, P> is a regular E3(S)-matrix iff

<M, u, n, -, P> is a symmetric transitive e-algebra and d = 1. Next: THEOREM 46. There is a model structure R such that $ is characteristic

for E3(S). PROOF. $ is given by Theorem 37 from the Lindenbaum algebra 9)N

for E3(S). Let $1 be the refined model structure corresponding to $, and suppose Ujxy, Ulyz, whence xE K - Q, y E K - Q. Then (Vm)(mE y -> Pm E x), (Vm) (m E z - Pm E y). Now suppose m E z. Then Pm E y. Also, since y E K - Q, - PO E y (for y is maximal). Therefore Pm - POE y, whence P(Pm - PO) E x. But Pm = P(Pm - PO) since V is transitive. Thus Pm E x and Ulxz, so that $1 is transitive. The result follows from Theorems 1, 8, 36, 38, and 45.

THEOREM 47. If $ = <K, Q, U> is a symmetric transitive model structure, and Al, ..., Ar subsets of K, then there is a finite symmetric transitive model structure $ = <K, Q, U> with at most 22(r+ 1) elements with subsets A1, ..., Ar C K satisfying the five conditions of Theorem 40.

PROOF. We modify the proof of Theorem 40 (cf. Theorem 41): now we put S = {Ai, ..., Ar, Q, A- Q, ..., Ar- Q, 0, so that i contains not more than 22(r+? ) equivalence classes, and, for x, 5EK, we define Uxy iff x ' Q A (VA)(A E S A PA E S- (y eA -xE PA) A (X ' Q A Y ' Q (x E A --*y E PA))). (Compare the U of Theorem 41.) As before, (i)-(iv) are immediate. For (v), suppose PAf = AX E S, and consider Xc E PAi. Then either (3y)(yE Ai A Uxy) or xE Q. If xE Q, sx E PAj as before. But if x g Q, we establish UNg as in the proof of Theorem 41, so that X~ E PAi. That if X E PAj then X~ E PAj is also proved in the same way. This gives (v), and it remains to show that $ is symmetric and transitive. That U



212 E. J. LEMMON

is symmetric in K - Q is immediate from the proof of Theorem 41. For transitivity, assume UNg and U5z, i.e. x Q, y ' Q, (VA)(A ESA PA ES (yeA XEPA) A (X QA Y1Q (XEA yE PA))), (VA)(A ESA PA ES (zEA yEPA) A (Y Q A Z Q y (yEA zE PA))). Now suppose A E S, PA E S, and z A A Then yE PA, whence yE PA - Q. Now PA - Q E S by the definition of S (indeed, for all A E S, A - Q e S), and, since P(PA - Q) _ PA by the transitivity of f, P(PA - Q) E S also. Hence xE P(PA - Q), i.e. x E PA. Finally, suppose x, z f Q, and xE A. Since y f Q, yE PA - Q, so that z eP(PA - Q) = PA. Thus Uxz, as was to be shown.

THEOREM 48. FE3(s) A (i) iff A is satisfied by R+ for all symmetric transitive epistemic m. s.'s A (ii) iff A is satisfied by A+ for all finite symmetric transitive epistemic m. s.'s R.

PROOF as the proof of Theorem 42. Finally we note that, since 5 is also transitive, we have the analogue

of Theorem 43: THEOREM 49. If FE3(S) EZT -+ WA then FE3(S) A. In view of Theorem 44 and bearing in mind the known modalities struc-

ture for E3 (the same as for S3 - see Parry [13]), we can show that E3(S) has exactly 26 distinct and irreducible modalities. They are in fact identical .with the 26 modalities of the system described in Parry [13], p. 151, and the same matrices show that further reductions are impossible.

The next two systems, EB and E5, result from adding A8 to E2(S) and E3(S) respectively; or, equivalently, from adding A 10 to ET and E4 respectively; and simpler axiomatic formulations of both can be given. E5 was mentioned, but wrongly formulated, in [5]; the proper correction, due to Kripke, is given in [18]. Whilst E4 has the same 14 irreducible modalities as S4, E5 has 10 as compared with S5's 6. Completeness and decision results are readily forthcoming on the basis of earlier theorems.

THEOREM 50. 9N <M, {d}, u, n, -, P> is a regular EB- (E5-) matrix iff <M, u, n, -, P> is a closed symmetric (transitive) e-algebra and d = 1.

THEOREM 51. There is a model structure 9 such that A+ is characteristic for EB (E5).

PROOF requires only, in addition to Theorems 37 and 46, that we show that the SV+ given by Theorem 32 is closed if 9N is, i.e. if PO PPO. But, using the terminology of the proof of that theorem, we know that P*0 (PO), so that P*P*0 = P*O(PO) = O(PPO) - (PO) = P*0, and R+ is closed.

THEOREM 52. If 9 = <K, Q, U> is a closed symmetric (transitive) m. s., and A1, ..., Ar are subsets of K, then there is a finite closed symmetric (transitive) m. s. = <K, Q, U> with at most 2r+1 (22(r+ 1)) elements with subsets A1, ..., Ar c K satisfying the five conditions of Theorem 40.




PROOF requires only additions to those of Theorems 41 and 47. On the assumption that $ is closed, we have Q PQ. But Qe SE so that PQ E S. By condition (v), therefore, PQ PQ = Q. and $ is closed also.

THEOREM 53. FEB (E5) A (i) iff A is satisfied by A+ for all closed symmetric (transitive) epistemic m. s.'s S (ii) iff A is satisfied by $+ for all finite closed symmetric (transitive) epistemic m. s.'s R.

The analogue of Theorems 43 and 49 does not, of course, hold for EB and E5, any more than it holds for ET or E4.

The remaining four systems can be dealt with together, and bear the same relationship to E2(S)-E5 that S2-S4 bear to E2-E4. S2(S) results from adding WA 10 to S2; S3(S) from adding WA 10 to S3; B from adding A 10 to T; and S5 from adding A 10 to S4. Of these systems, S2(S) and S3(S) would appear to be new; B is the Brouwersche system of Kripke [4]; and S5 is the familiar Lewis system. Of course, in the presence of R3, B and S5 may be more easily axiomatized by adding A -* W<A to T and S4; and even simpler axiomatizations of S5 are known (see Prior [14]). The main theorem connecting the new four with the old four is the following.

THEOREM 54. (i) FE2(S) fE3(S)) A iff lS2(S) (S3(S)) EA; (ii) Fs2(s) (S3(S)) A iff FE2(S) (E3(S)) WT -* A; (iii) lB (S5) A iff FEB(E5) WT -> A.

PROOF is exactly parallel to that of Theorems 20 and 21, using Theorems 43 and 49 for (i) and (ii).

This theorem reduces the decision problem for the new systems to that for the previous four, already solved. The main results of Section II may now be virtually transcribed with the obvious appropriate modifications. In addition to giving suitable algebraic semantics for the new systems S2(S) and S3(S), which resemble S2 and S3 in having no theorem of the form Wg W2A, this yields an independent proof of Kripke's semantic results for B and S5. We summarize the main features in the following theorem:

THEOREM 55. (a) HS2(S) (S3(S)) A (i) iff A is weakly satisfied by all [finite] symmetric (transitive) e-algebras (ii) iff A is weakly satisfied by A+ for all [finite] symmetric (transitive) epistemic m. s.'s 9;

(b) 1B(S5) A iff A is satisfied by A+ for all [finite] normal symmetric (transitive) epistemic m. s.'s A.

A hierarchy of systems E2(S)n can be defined by analogy with the systems E2n of Section II. Then S2(S) = E2(S)l. Each E2(S)n is decidable, and suitable semantics can be given in terms of n-weak satisfaction. The existence of this hierarchy can be used to show that B is not finitely axiomatizable, in the way in which the E2n-hierarchy is used in [8] to show that T is not finitely axiomatizable.

It is easy to design a model structure which reveals that B has infinitely many distinct and irreducible modalities; hence so do S2(S), E2(S), and EB.



214 E. J. LEMMON

However, like E3(S), S3(S) has just 26 irreducible modalities. In view of Theorems 44 and 54 (i), it contains the system of Parry [13], p. 151, which results from adding Ll([1O WA -* WA) to S3, and has exactly the same modalities structure. I can find, however, no proof that WA 10 is provable in Parry's system, so that it appears to be an open question whether the two systems are identical. As is shown in [2], the addition of WK1 WA -- WA to S4 yields S5; it can similarly be shown that its addition to E4 yields E5. It would be nice to show that its addition to E3 yields E3(S), and thus that S3(S) is the same as Parry's system.

V

Three degenerate systems; intersection results. We consider next three additional axiom-schemata: All: A -L-A; A 12: OA; A 13: CMB -*(A --- A).

By PC, we understand the system {A 1-A 4, A6, A 1 1; R 1}; by E, the system {A 1-A4, A6, A12; R1}; and by L, the system {A1-A4, A6, A13; R1}. For PC, the rule R3, and so R2, is at once forthcoming from A 1. Indeed, by A 11 and A 6 Fpc A - MA, and the system collapses very obviously into the classical propositional calculus. As such, it contains S5, and therefore all the systems we have hitherto considered. Put K {b} for some element b, Q = 0, and U = {<b, b>}; then, for Rpc = <K, Q, U>, Rc is clearly a characteristic matrix for PC, since as a Boolean algebra it is isomorphic to the familiar two-valued one, and satisfies the condition Px = x. The system E is similarly degenerate, for we have FE CA * F; R2 may be obtained as a derived rule for E, since FE WA -- B for any B (in fact, this shows A4 and A6 to be redundant axiom-schemata). E contains E5, and so all the E-systems which we have considered. It is characterized by the matrix R+ of Section I, since R+ is also isomorphic as a Boolean algebra to the two-valued one, and satisfies the condition Px = 1.

A 13 is provable in both PC and E, so that L is contained in both PC and E. We shall see that L, which is in fact a version of Lukasiewicz's modal logic,7 bears an even closer relationship to PC and E. We first note that FL(A--NB) -i(LA-- LZB).ForbyA13FL(A--NB) -i( A-* LZ(A- NB)); the result follows by A 4. Hence R2 is obtainable as a derived rule for L. It is now easy to see that L contains E5, and so all E-systems other than E itself.

7 See Lukasiewicz [10] and Smiley [16]; here, however, we understand that modal system defined by Lukasiewicz's matrix but lacking functorial variables in the for- mation rules.




For the algebraic treatment of L, let us say that an e-algebra is discrete iff it satisfies the condition:

(vii) Px = X U PO. It is easy to verify:

THEOREM 56. 9 = <M, {d}, u, n, -, P> is a regular L-matrix iff <M, u, n, -, P> is a discrete e-algebra and d = 1.

Let us say that a model structure A == <K, Q, U> is discrete if f A satisfies the condition that if Uxy then x = y.

THEOREM 57. Let 9N be a discrete e-algebra. Then there is a discrete m. s. A such that 9 is isomorphic to a subalgebra of 9+.

PROOF follows the lines of Theorems 32 and 33 (cf. Theorem 37). Suppose 9) satisfies (vii), i.e. for m E M, Pm - m u PO. For the A given by Theorem 33 (epistemic case), consider 91, the refined counterpart of A, and suppose Ulxy, i.e. (Vm)(mE y -- PmE x), where x Q, i.e. POG x. For mE y, we have Pm E x, and so m U PO e x, and so either m E x or PO E x by properties of maximal filters, whence m E x; thus y c x. For the converse, we note that Nm = m - PO by the discreteness of 9N and, since Ulxy, (Vm) (Nm E x mE y). Assume mE x; - POE x since x is maximal, so that m - PO Nm E x, whence m E y. Thus x c y, and so x = y. Hence Al is discrete, and, by Theorem 1, 9) is isomorphic to a subalgebra of A.

THEOREM 58. The algebra on a discrete epistemic model structure is discrete epistemic.

PROOF. Let A = <K, Q, U> be a discrete epistemic m. s., so that if Uxy then x=y and Uxx for X E K - Q. We show, for A c K, that PA =A u Q. Since A c PA and Q c PA, A u Q c PA. Conversely, suppose x E PA. Then, if x ' Q, (3y)(Uxy A yE A). But then x = y and X E A, so that PA c A U Q.

This gives at once (cf. Theorem 39): THEOREM 59. There is a model structure A such that A+ is characteristic

for L. We now remind the reader of A of [7], namely 3 = <{a, b}, {a}, {<b, b>}>.

From Smiley [16], it can be seen that A+ is Lukasiewicz's matrix for the L-modal logic. We shall show that A+ is characteristic for L; this will of course solve the decision problem for L since R3 is finite. It is clear that R3 is discrete epistemic, so that it is certainly a matrix for L, by Theorems 56 and 58. Now consider a non-theorem A of L. A is falsified by the R+ of Theorem 59, say by an assignment Wf. Then for any x E K - Va,(A) A is falsified by the algebra A+ on the connected model structure Ax (see remark following Theorem 3). Now, if x E K - Q, R+ will be isomorphic to PC, for it is a consequence of the discreteness of s that Kx = {x} simply. Thus A is a non-theorem of PC. Similarly, if x E Q, R+ will be isomorphic to S+, and A is a non-theorem of E. Bearing in mind that any theorem of L is both a theorem of E and a theorem of PC, we have:



216 E. J. LEMMON

THEOREM 60. FL A iff both Fpc A and FE A. It is well-known (see e.g. Smiley [16], p. 149) that A+ can be viewed

as the cross-product of the two matrices A+ and l+. Hence, if A is falsified by either A+ or St+, then A is falsified by Al+. Thus:

THEOREM 61. R+ is a finite characteristic matrix for L. A final result relates L and PC, proved in a way parallel to the proof of

Theorems 20, 21, and 54: THEOREM 62. Fpc A iff FL DT -+ A. We next note a useful result for what follows: THEOREM 63. FE A iff FE2 OF --* A. PROOF. Since FE OF by A 12, given FE2 OF -+ A, it follows that FE A.

The converse is by a simple induction on the length of the E-proof of A, noting that FE2 OF -* OA by R2.

Let us, for notational simplicity, use the names of systems as names for the classes of their theorems. Then Theorem 60 can be more briefly ex- pressed in the form: L = PC n E. The next theorem shows that similar intersection results are available for all sixteen systems discussed earlier.

THEOREM 64. (i) E2 = S2 r) E; (ii) E3 -a S3 n E; (iii) ET = T n E; (iv) E4 - S4 n E; (v) E2(S) = S2(S) rn E; (vi) E3(S) = S3(S) fl E; (vii) EB=B n E; (viii) E5 S5 n E.

PROOF. That E2 c S2 n E is trivial, and so for the other seven cases. For the converse, suppose FS2 A and FE A. Then FE2 LiT -* A by Theorem 20 (ii) and FE2 OF -+ A by Theorem 63. But FE2 FiT v OF. Therefore FE2 A. The other seven cases employ Theorems 20, 21, and 54; we observe that E2 is contained in all other E-systems.

A proof in terms of model structures for all these results is also forthcoming. We use two cases for illustration. Suppose A is a non-theorem of ET. Then there is a closed epistemic m. s. ! such that *+ falsifies A. Select x E K - Va(A), where W is an assignment falsifying A. It is easy to see that, depending on whether x E K - Q or x E Q, A is either falsified by a normal epistemic m. s. *+ or by A+ (we assume $ chosen so as to be refined), and so is either a non-theorem of T or a non-theorem of E. Again, suppose A is a non-theorem of E3. Then there is a transitive epistemic m. s. 2 such that R+ falsifies A. Choosing x as before, we see in this case that either A is weakly falsified by R+ and so is a non-theorem of S3 or A is falsified by ,+ and so a non-theorem of E. The other six cases are similar to one or other of these.8

8 These intersection results are in the style of Halld6n [3], who showed that S3 = S7nS4. However, no attempt is made in the present series of papers to handle such oddities as S6-S8; no doubt they are related in some comparatively simple algebraic way to the system E.




VI

Appendix on systems and matrices. The systems considered in Sections T, II, and IV can be represented in

the following trim containment chart:

B S5[6]

T~~~~ -

4= [14

/~~~~~S v==S()21

E 2 ,,3 (s)[26]

E2 E3[42]

Here, the top 4 systems possess the rule R3 and are characterized by normal matrices; the next 4 have theorems of the form WA but none of the form W WA; the next 4 are characterized by closed matrices; and the bottom 4 have the property that if F WT -O WA then F A. The right-hand cube is a 'symmetric' counterpart to the left-hand cube. A number following a system in square brackets is the number of distinct irreducible modalities in the system; where no number appears, the system has infinitely many such.

To show independence of unlinked systems and that no further inclusions are possible, it suffices to do this for the top eight. For example, if we have shown that S3 is not included in T, it follows by Theorems 20 and 21 that E3 is not included in ET. Further, we know that all E-systems, being contained in E, have no theorems of the form WA and so are properly contained in the systems vertically above them.

That the systems S2, S3, S2(S), and S3(S) are properly contained in T, S4, B, and S5 follows from A+, which weakly satisfies them all yet weakly falsifies W CZIT. To show T's proper containment in S4 and S2's in S3, we need clearly the algebra on a non-transitive model structure, which must therefore have at least 3 elements. The simplest is the following: =

<{a, b, c}, 0, {<a, a>, <b, b>, <c, c>, <a, b>, <b, c>}>; then A+ satisfies T and S2, but falsifies A7. This also shows, together with $+, the independence of S3 and T. Next, consider S+ (Lewis's Group II: see [7], Section V). ftl4 is normal epistemic transitive, but not symmetric, and so satisfies S4 but not A 10. Thus S2, S3, T, S4 are properly contained in S2(S), S3(S), B, S5. We finally need a normal epistemic symmetric m. s. which is non-transitive; again 3 elements are required; we simply extend the U of ! above to in-



218 SEMANTICS FOR MODAL LOGICS

clude <b, a> and <c, b>. This matrix satisfies B but falsifies A 7, thus showing the proper containment of S2(S) in S3(S) and B in S5, as well as the independence of S3 and S2(S) and S4 and B.

A final observation. Dugundji [1] has shown that no system between S1 and S5 possesses a finite characteristic matrix. This result, therefore, applies to all the top 8 systems in the above chart. It is a consequence that no E-system belonging to the lower 8 has a finite characteristic matrix either; for any such matrix would yield, by the methods of this paper, one for the system vertically above the system in question. Of course, the degenerate systems PC, E, and L all do have finite characteristic matrices, namely PC) A+ , and A+.

REFERENCES

[1] J. DUGUNDJI, Note on a property of matrices for Lewis and Langford's calculi of propositions, this JOURNAL, vol. 5 (1940), pp. 150-151.

[2] M. A. E. DUMMETT and E. J. LEMMON, Modal logics between S4 and S5, Zeit- schrift fur mathematische Logik und Grundlagen der Mathematik, vol. 5 (1959), pp. 250-264.

[3] S6REN HALLDEN, Results concerning the decision problem of Lewis's calculi S3 and S6, this JOURNAL, vol. 14 (1950), pp. 230-236.

[4] SAUL A. KRIPKE, Semantical analysis of modal logic I, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 67-96.

[5] E. J. LEMMON, New foundations for Lewis ntodal systems, this JOURNAL, vol. 22 (1957), pp. 176-186.

[6] E. J. LEMMON, Extension algebras and the modal system T, Notre Dame journal of formal logic, vol. 1 (1960), pp. 3-12.

[7] E. J. LEMMON, Algebraic semantics for modal logics I, forthcoming in this JOURNAL.

[8] E. J. LEMMON, Some results on finite axiontatizability in modal logic, forthcoming in Notre Dame journal of formal logic.

[9] C. I. LEwIs and C. H. LANGFORD, Symbolic logic, New York, 1932. [10] J. LUKASIEWICZ, A system of modal logic, Journal of computing systems,

vol. 1 (1953), pp. 111-149. [11] J. C. C. MCKINSEY, A solution of the decision problem for the Lewis systems

S2 and S4, with an application to topology, this JOURNAL, vol. 6 (1941), pp. 117-134. [12] J. C. C. MCKINSEY and ALFRED TARSKI, The algebra of topology, Annals

of mathematics, vol. 45 (1944), pp. 141-191. [13] WILLIAM TUTHILL PARRY, Modalities in the Survey system of strict implication,

this JOURNAL, vol. 4 (1939), pp. 137-154. [14] A. N. PRIOR, Formal logic (2nd edition), Oxford, 1962. [15] L. SIMONS, New axiontatizations of S3 and S4, this JOURNAL, vol. 18 (1953),

pp. 309-316. [16] TIMOTHY SMILEY, On Lukasiewicz's L-modal system, Notre Dame journal

of formal logic, vol. 2 (1961), pp. 149-153. [17] ROBERT R. STOLL, Set theory and logic, San Francisco, 1961. [18] NAOTO YONEMITSU, Review of [5], this JOURNAL, vol. 23 (1958), pp. 346-347.

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