algebraic semantics for modal logics i

Algebraic Semantics for Modal Logics IAuthor(s): E. J. LemmonSource: The Journal of Symbolic Logic, Vol. 31, No. 1 (Mar., 1966), pp. 46-65Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2270619 .

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THE JOURNAL OF SYMBOLIC LoGic Volume 31, Number 1, June 1966

ALGEBRAIC SEMANTICS FOR MODAL LOGICS I

E. J. LEMMON

Modal logic received its modern impetus from the work of Lewis and Langford [10]. In recent years, however, their axiomatic approach, aided by somewhat ad hoc matrices for distinguishing different modal systems, has been supplemented by other techniques. Two of the most profound of these were, first, the algebraic methods employed by McKinsey and Tarski (see [11] and [12]) and, second, the semantic method of Kripke (see [5] and [6]); and there have been others. The aim of the present series of papers is to afford a synthesis of these methods. Thus, though new results are given, the interest lies rather in revealing interconnexions between familiar results and in providing a general framework for future research. In general, we show that semantic completeness results of the Kripke kind can be deduced from the algebraic results by means of one central theorem (Theorem 21).

In this first paper, the bulk of the work is devoted to showing that the McKinsey-Tarski algebraic method, used so successfully in connexion with S4, can be extended to a group of six modal systems the strongest of which is T. A second paper will deal with stronger systems, and it is hoped that a third will deal with the addition of quantifiers to all these systems.

The method for handling all these systems is identical. We first establish a correlation between regular matrices for each system and a certain kind of algebra (Theorems 9 and 11). Second, using the Lindenbaum matrix (Theorems 7 and 8), we prove that each system has the finite model property and so is decidable (Theorem 14). A consequence is that we may restrict attention to linite algebras, so that, third, we establish representation theorems for such algebras in terms of an algebra based on the set of all subsets of a given set (Theorems 17 and 18). These representations yield the connexion between the algebraic approach and Kripke's, so that semantic completeness results are immediately forthcoming.

I

The weak modal systems We shall in the present paper consider modal systems defined in terms

of the following axiom-schemata and rules:1

Received January 11th, 1965. l All systems are to be thought of as having -> (material implication), - (negation),

and n (necessity) as undefined connectives, together with some denumerable list p, q, r, ... of propositional variables. Formation-rules are as usual.

46



SEMANTICS FOR MODAL LOGICS 47

Al: A-(B-A); A2: (A (B ARC)) -- ((A SU>-B) (A AR); A3: (-Ad-B) >(B A); A4: n (A B)-- (LA ? B); A5: AY- -W-A; A6: WA A;

R A)A-B A B A RI: ~~R2: R3:- B n~LA-WB' WA'

As definitions we throughout adopt the following:

DI: AvB=-AR-B; D2: AAB= -(-Av -B); D3: A *-B (A-B)A (BO-A); D4: OA = - A; D5: A= B =l(A-?B); D6: A B (At =B) A (B => A); D7'n: nnA= C[n..A;

n D8,n: O'nA = ... OA;

n .D9,f: A o,.n B =n S(A -- B); DlO'n: A o.-n B =(A a:n B) /v (B o:-n A).

In terms of the axiom-schemata and rules, we define six modal systems, as follows:

C2 ={A 1-A4 ; RI, R2}; D2 ={A l-A5 ; RI, R2}; E2 ={A l-A4, A6; RI, R2}; T(C) ={A l-A4 ; RI, R3}; T(D)={A1-A5 ; R1,R3}; T ={A1-A4, A6; R1, R3}.

The use of axiom-schemata rather than axioms in the definition of the systems avoids the need for a separate rule of substitution. The presence of A l-A3, RI among the postulates for each system, together with DI -D3, provides it with the full classical propositional calculus as a substructure,2 and we shall take this for granted in what follows. It should be remembered that in both systems containing A6 (E2 and T) A5 is immediately forthcoming as a theorem-schema. Also, for the T-systems, R2 may be obtained as a derived rule via R3, A4, and RI. These facts, taken in conjunction with the definitions of the systems, at once justify the following inclusion

2 See e.g. Church [1].



48 E. J. LEMMON

table3 for the systems: E2 - T

D2 T(D)

C2 T(C)

(Here a line from left to right marks the inclusion of the left-hand system in the right-hand system.)

Of these systems, T is equivalent to the well-known system M (T) of Feys-von Wright (see [3] and [17]), and E2 is presented, in a slightly different formulation, in [71. T(D) and D2 are, respectively, deontic counter- parts to them, in which A6 is replaced by the weaker 'deontic' A5. T(C) and C2 are the corresponding weaker systems in which not even A5 is assumed. The distinctive mark of C2, D2, E2 is that the strong rule R3 of the T-systems is replaced by the weaker R2. It is worth recalling that, as is shown in [7], Lewis's S2 is intermediate between E2 and T.

Our main concern in the sequel will be with C2. Formally, this concern is motivated by the fact that results for C2 can rapidly be extended to corresponding results for the other systems. But in the second place this system would appear to be a kind of minimal modal logic; though weaker systems can, and have, been devised,4 they would appear to have little interpretational interest; a possible interpretation of C2 is given in [9]. The following theorem gives some of the main theorem-schemata of C2; their proofs are straightforward, and not given here.

THEOREM 1.

(1) FC2 E(AA B) AAA OB; (2) FC2 O(A v B) OA v OB; (3) Fc20(AA B) OAA OB; (4) FC2 LA v OB - (A v B); (5) FC2 RnA -(B >,nA); (6) FC2 n-A -*(A =n B); (7) FC2 (A =n+m B) --(FnA =>m onB); (8) FC2 (A =>n+m B) -(OnA =>. ong); (9) FC2 FnA (-A on A);

3 That all inclusions are proper, and that no other inclusions hold, are consequences of matrices given in the last section of this paper.

4 For example, the systems El, Dl, SO.5, and SO.9 of [7], though not properly contained in C2, are weaker in certain respects than C2. And the same, of course, applies to Lewis's SI and Feys's Si0.




(10) FC2 DTnA+-+((B F->B) >nA); (I11) FC2 (A =>nB) +- > On(A A -B); (12) FC2 (A on+m B) ((B =>n C) =>m (A => C)).

Some main theorem-schemata of D2 and E2 which are not in C2 are given by the next theorem:

THEOREM 2.

(1) FD2 K(A A); (2) FD2 (A A) *- (A -A); (3) FE2 A -OA.

A notable feature of the T-systems is the strengthening of theorems of the corresponding weaker systems by prefixed D-I's, in accordance with R3:

THEOREM 3. If FC2(D2,E2) A, then FT(C)(T(D),T) C] A.

A special feature, which we shall need later, is:

THEOREM 4. FT(c) I(A -- A) * (A -* A).5

One feature shared by all six systems, though not possessed, for example, by S2, is the substitutivity of material equivalents in the sense of the next theorem. We shall make much use of this in what follows.

THEOREM 5. For S = C2, D2, E2, T(C), T(D), T, if Fs A * B then Fs ... A... .-.. B... (where ... B ... results from .A. ... by replacing zero or more occurrences of A in ... A... by B).

PROOF by induction on the length of ... A..., making essential use of R2 or R3.

II

Algebras and matrices We know from the work of McKinsey-Tarski [12] that there is a close

connexion between Lewis's S4 and closure algebras. A similar connexion is established in [8] between T and a generalization of closure algebras called extension algebras. We consider here a further generalization, called simply a modal algebra, to be used in our study of C2.

DEFINITION 1. A structure 9 = <M, '., n, -, P> is a modal algebra iff M is a set of elements closed under operations a, a, , and P such that:

(i) M is a Boolean algebra with respect to a, r,

(ii) for x,yEM, P(x %y) = Pxr Py.

DEFINITION 2. Nx = -P--x.

5 In fact, more generally we have FT(c) (A =>n A) +-+ (A =>-m A) for all m, n.



50 E. J. LEMMON

THEOREM 6. In any modal algebra E = <M, P>, ',-, :

(i) for x,yEM, N(xny) NxrNy; (ii) for x, yE M, if x ? y then Px ? Py and Nx ? Ny (where x < y

iff X J y-y iff X y =x). PROOF. (i) N(x c y) = -P- (x ny) = -P(-x %j -y) (P-x %P-y) - P-x n-P-y Nx n Ny. (ii) Suppose x _ y, i.e. x %jy y. Then Px % Py P(x % y) Py,

whence Px ? Py. Similarly, using (i), we show that Nx ? Ny.

We shall be interested in using modal algebras as (logical) matrices.

In general, an algebra may be said to become a matrix when some subset D of its set of elements is selected as the set of designated elements (thus to each algebra with a set of elements of cardinality N there correspond 2N

distinct matrices). A given propositional logic can be interpreted in terms of a matrix as follows: we take the propositional variables of a wff of the logic to range over the elements of the matrix, and interpret the connectives of the logic as operations in (or definable in) the matrix; in this way with each wff A containing n propositional variables is associated a (unique) matrix-function f(A) of n variables. We then say that A is satisfied by a matrix iff, under the given interpretation, the value of f(A) for every n- tuple of elements of the matrix lies in D; otherwise we say that A is /alsijied by the matrix. We also say that a system S is satisfied by a matrix iff all the theorems of S are satisfied by the matrix. And a system is characterized

by a matrix (or a matrix is characteristic for a system) iff the wffs of the system satisfied by the matrix are all and only the theorems of the system.

Conversely, given a suitable formalism (i.e. a set of connectives) and matrix interpretation, any matrix determines a propositional logic: namely the logic whose theorems are exactly the wffs of the formalism satisfied by the matrix under the given interpretation. Obviously the matrix will be characteristic for its corresponding system.

For any propositional logic L, let WL be the set of its wffs (in terms of connectives c1, . . ., cs), and TL be the subset of its theorems. Then we have the following quite general fundamental theorem (due to Linden- baum6):

THEOREM 7. Let L be a propositional logic such that TL is closed under substitution on propositional variables. Then there exists a characteristic matrix DIL for L.

PROOF. Suppose that L has connectives ci', . can, such that cr" is ai-adic (1 ? i ? n). The elements of our matrix will be the members of WL and the designated elements the members of TL. Corresponding to each c~a we define an ai-adic matrix function c** as follows: for wi,

6 See e.g. McKinsey [11].




waZ E WL we put c*(wi, ..., wad) equal to the wff which results from applying the connective cr" to the wffs wi, . ., was Put 9RL = <WL, TL, cl, ... ., c>. Interpreting cr" as c*, we see at once that all theorems of L are satisfied by 9L. For suppose t E TL: then any assignment of arguments from WL to f(t) yields as value simply a substitution-instance of t, which by hypothesis is in TL. Conversely, if w g TL, then the value of f(w) lies outside TL for just that assignment to its variables which consists of the appropriate propositional variables of w. Thus 9JL is characteristic for L.

COROLLARY. For each of the systems C2, D2, etc., there exists a characteristic matrix.

Although we shall make use of the Lindenbaum matrix, whose existence is established by this theorem, in what follows, it is worth bearing in mind that this matrix is quite trivial and not necessarily the most interesting characteristic matrix for a given system. A system may have many non- isomorphic characteristic matrices. Thus for the classical propositional calculus clearly the most interesting characteristic matrix is the one based on a two-element Boolean algebra (i.e. the usual truth-table matrix), whilst the Lindenbaum matrix has denumerably many elements and is not even Boolean.

The matrices in which we are here interested are structures 9X <M, D, a', n, -, P> such that D c M and 'a, n are dyadic, -, P monadic operations on M class-closing on M. A matrix is fprofper iff D C M. In connexion with these matrices, we shall use the following definitions for particular matrix-functions:

DEFINITION 3. x y y -x %- Y.

DEFINITION 4. x y (x > y) n (y -A-x). We further define a regular matrix as follows: DEFINITION 5. A matrix = <M, D, a, n , -, P> is regular iff (i) 9X is proper; (ii) D is an additive ideal of M; (iii) if x +- yED, then x =y.

(A subset D of M is an additive ideal iff for x E D, y E D x n y E D and for X E D, y E M x % y E D.)

From now on, we have in mind a uniform interpretation of the connectives of modal systems in terms of matrix-functions, namely: '-a-' is interpreted as -*, '-' as -, and 'LI' as N. This in fact means that A v B is interpreted as - -x '. y rather than x '. y, in view of D1; but these functions will be identical in all interesting cases (though they are not identical in the Lindenbaum matrix). The same applies to A A B and other defined connectives.

We next show how, from the characteristic Lindenbaum matrices already



52 E. J. LEMMON

shown to exist for the six modal systems, regular characteristic matrices can be constructed. This is done, essentially, by forming a new matrix whose elements are the equivalence classes (under A--) of the elements of the old.

Let W be the set of wffs of the six modal systems. Let T be the set of theorems of C2 (D2, etc.). For A e W, put E(A) {B: B <-+ A e T}. Let V, A, -, 0 be those functions on W which, for wffs A, B eW, yield respectively the wffs A v B, A A B, -A, and OA. We define new functions Vi, Al, 1, 01 on the set of all sets E(A) as follows:

E(A) vi E(B) = E(A v B); E(A) Al E(B) = E(A A B);

-E(A) E(= A); 01E(A) E(OA).

Finally, we call Wl the set of all sets E(A) for A e W, and T1 the set of all sets E(A) for A e T. In fact, it is obvious that T1 has only one member, since, for A, B e T, A +-? B e T.

If we put A __ B iff A +-? B E T, it is easy to verify that __ is a congruence relation over W. (This is a consequence of the fact that all six systems contain the classical propositional calculus, together with Theorem 5.) Hence we are justified in treating the structure <Wi, Ti, vl, Al, -1, 0> as a matrix7; we name it 9X1. We shall show that 91 is a characteristic matrix for C2 (D2, etc.).

THEOREM 8. There exist characteristic regular matrices for C2, D2, etc., such that only one element is designated.

PROOF. We treat only the case of C2; the cases of the other systems only differ in that T is a different set of wffs. That 91N is a characteristic matrix is an easy consequence of the fact that <W, T, v, A, -, <> is a (variant of) the Lindenbaum matrix. Since C2 is consistent, T C W, so that T1 C WI and 9Ji is proper. For E(A), E(B) e T,, A, B e T, whence (by propositional calculus) A A B e T, so that E(A A B) = E(A) Al E(B) e T,. Similarly, we can show that, if E(A) e T1, E(B) e Wl, then E(A) vi E(B) e T,. Hence T, is an additive ideal of W,. Now suppose that E(A) +-+i E(B) e Ti, i.e. that (E(A) --l E(B)) Al (E(B) -*1 E(A)) E T1, where E(A) -*1 E(B) 1E=(A) vi E(B). Then E(A +-+ B) E T1, so that A " B e T, i.e. A __ B. It follows that E(A) = E(B). Hence 91 is regular. That T1 is a unit set has already been noted above.

We next wish to establish a relationship between regular matrices satisfying C2 (C2-matrices) and modal algebras. The next theorem states

7 Unless is a congruence relation, we shall not in general have the substitutivity of identity, which, of course, we are assuming for matrices as for algebras.




this relationship; in it we use '1' for the universe element of the Boolean algebra within the modal algebra.

THEOREM 9. 1 - <M, {d}, v, n, -, P> is a regular C2-matrix iff <M, a, ^,-, P> is a modal algebra and d = 1.

PROOF. Let 9N = <M, {d}, V, n, -, P> be a regular C2-matrix. To prove that <M, v, n, -> is a Boolean algebra, it suffices to take some set of postulates for such algebras and establish these: we use Rosenbloom [15], p.9. Of his seven postulates, Al, A6, A7 are trivial consequences of the fact that 9N is a matrix. For A2, we prove x n y = y n x as follows: A A B +-* B A A is a theorem of C2, so that x n y +-* y n xE {d}, since 9N is a C2-matrix. It follows by (iii) of Definition 5 that x n y = y n x. A3, A4, A5 are similarly consequences of the fact that C2 contains the classical propositional calculus, together with (iii) of Definition 5. That P(x v y) - Px '- Py is a consequence of Theorem 1 (2), together with (iii) of Definition 5. Thus <M, vI, , -, P> is a modal algebra. Since <M, a.', ', ->

is a Boolean algebra, we may put 1 = x -x. Given FC2 A v -A, we have X %J -X E {d}, i.e. x v -X d, whence d= 1.

Conversely, let 9N =<M,-,, n, -, P> be a modal algebra. Since <M, a-, n, -> is a Boolean algebra, it is obvious that the schemata A 1-A3 are satisfied by <M, {1}, vI n, -, P>. The matrix-function corresponding to A4 is -N(-x y) ; (-Nx ; Ny) P(x -y) ; (P-x - P-y)= P((x n -y) V -X) '; -P-y =P(-x '-y) -P-y = (P-X '.; P-y)

-P-y = P-x '- 1 = 1, so that the schema A4 is satisfied. For RI, suppose x 1 and x --y 1. Then y (x c-x) '; y (x - y) n (-x vy) =(1 (I-'y) n 1 = 1. For R2, suppose x -d-y 1. Then -x .-y = 1 and x _ y, so that Nx 5 Ny by Theorem 6 (ii). Thus Nx -- Ny -Nx '- Ny = 1. Thus <M, {1}, a;, n, -, P> is a C2-matrix. That this matrix is proper follows from the fact that <M, a-, n, -> is a Boolean algebra and so contains at least 2 elements. It is immediate that {1} is an additive ideal of M. Finally, given x +-* y = 1, x = y by Boolean algebra, so that <M, {1}, a;, n, -, P> is also regular.

In view of this theorem, we need not distinguish between a modal algebra and the corresponding matrix in which 1 is taken as sole designated value; and we shall not do so in the sequel. Thus any modal algebra provides a matrix satisfying C2 in a natural way. Conversely, the C2 characteristic matrix of Theorem 8 is shown to be a modal algebra.

In order to set up similar correlations between the other five systems and appropriate algebras, it is necessary to define suitable algebras. Hence we say:

DEFINITION 6. An algebra is deontic iff, in addition to being a modal algebra, it satisfies the postulate:

(iv) P1 I 1.



54 E. J. LEMMON

DEFINITION 7. An algebra is epistemic iff, in addition to being a modal algebra, it satisfies the postulate:

(v) x < Px.

DEFINITION 8. An algebra is normal iff, in addition to being a modal algebra, it satisfies the postulate:

(vi) PO - 0.

THEOREM 10. All epistemic algebras are deontic. PROOF. Given (v), we conclude 1 < PI. But by Boolean algebra

PI < 1, whence P 1 I., which is (iv).

THEOREM 11. 9 =<M, {d}, V, -, , P> is a regular D2- (E2-, T(C)-, T(D)-, T-)matrix iff <M, v, n,-, P> is a deontic (epistemic, normal, normal deontic, normal epistemic) algebra and d = 1.

PROOFS in general follow that of Theorem 9, if we note in addition the following. That a regular D2-matrix fulfils condition (iv) follows from Theorem 2 (2). That a regular E2-matrix fulfils condition (v) follows from Theorem 2 (3). That a regular T(C)-matrix fulfils condition (vi) follows from Theorem 4. Conversely, given (iv), Nx --- Px - P-x v Px P (-x x) = PI 1, so that A5 is satisfied. Given (v), Nx --x P-x x = (-x P-x) v x (sinceP-x -x v P-x) 1 P-x 1, so that A6 is satisfied. Given (vi), suppose x 1. Then Nx =-P-x -PO = -0 = 1, and R3 is satisfied.

A normal epistemic algebra is an extension algebra in the sense of [8]; thus Theorem 11 for T is virtually equivalent to Theorem 4 of [8]. However, in [8] the dual interpretation of the modal logic T was employed.

III

The finite model property The results of the last section already provide completeness results for

the six modal systems of a rather uninteresting kind:

THEOREM 12. FC2(D2,E2,T(C),T(D),T) A iff A is satisfied by all modal (deontic, epistemic, normal, normal deontic, normal epistemic) algebras.

PROOF. Any theorem of C2 is satisfied by any modal algebra in virtue of Theorem 9; conversely any non-theorem of C2 is falsified by the regular characteristic C2-matrix of Theorem 8, and so by some modal algebra by Theorem 9. The cases of the other systems similarly uses Theorems 11 and 8.

In this section, we are concerned to show that Theorem 12 still holds if a restriction to finite algebras is made. We shall in fact show that each system possesses the finite model property, i.e. that for each non-theorem A of each system there is a finite matrix satisfying the system but falsifying A. Thus, in addition to sharpening Theorem 12 and paving the way for the




semantics of the next section, the result shows that all six systems are decidable. Our method of proof, as before, is an imitation of McKinsey- Tarski (see [11] and [12]).

THEOREM 13. Let KM, < , ,-, P> be a modal (deontic, etc.) algebra, and let ai, . . ., a, be a finite sequence of elements of M. Then there is a finite modal (deontic, etc.) algebra El = <M,, %l, nM, -1, Pi> with at most 22r+1 elements such that:

(i) for 1 <i <r, aE Ml; (ii) for x, yE Mi, x i y x % y; (iii) for x,yEMl,xniy x ny; (iv) for x E Ml, -lx -x; (v) for x E Ml such that Px E Ml, Plx = Px. PROOF. Let M, be the set of elements of M obtained from P1, al, ...,ar

by any finite number of applications of a, n, and -; by Boolean algebra, there are not more than 22r+1 such elements. We put Lii, 1i, -1 equal to au, r, - restricted to Ml. It is immediate that (i)-(iv) of the theorem are satisfied. For x E Ml, we say that x is covered by y iff y E Ml and Py E Mi and x < y. Since PI E Ml as well as 1, it follows that every element is covered by some element, if only by 1. Where x is covered by yl, . . ., yn. we put Plx = Pyl n ... n Pyn (so that Plx E Ml). If x is covered by

, ..., Yn., then x < yj (1 < i < n), so that Px < Pyi by Theorem 6 (ii) and Px < Plx. Conversely, if x E Ml and Px E Ml, then x is covered by itself, so that Pix = Px n Pyl ... n Pyn, where yl, . . ., yA are the other elements that cover x. Thus Plx < Px. Putting these results together, we see that (v) is satisfied.

It remains to show that 91 = <Mi, ii, i, -1, Pi> is a modal (deontic, etc.) algebra, given that V is. For modal algebras, the proof that Pl(x 'jl y) = Pix Ljl Ply is identical with that given in McKinsey [11], p. 125, and quite straightforward. For deontic algebras, we need to show that, given PI = 1, then Plu 1. But, since both 1 and PI are in Mi, (v) applies and Pi 1 - PI = 1.8 For epistemic algebras, we need to show that, given x < Px, then x < Plx. But we have already shown that in general Px < Pix. For normal algebras, we need to show that, given P0 0, then PiO = 0. But if P0 = 0 then P0 E Mi, so that by (v) again P10 P0 = 0. The cases of normal deontic and normal epistemic algebras are simply combinations of the cases already considered.

By a sub!ormula of a wff A we mean a wff which occurs as a part (proper or improper) of A. We now prove:

8 For deontic algebras, and so for epistemic algebras (Theorem 10), the present theorem can be strengthened slightly by replacing 22r+1 by 221; for it is no longer necessary to stipulate that PI be used in the construction of M1.



56 E. J. LEMMON

THEOREM 14. Let A be a wff with r subformulas. Then FC2(D2,etc.) A iff A is satisfied by all modal (deontic, etc.) algebras with not more than 22r+1 elements.

PROOF. If FC2(D2,etc.) A, then, by Theorem 12, A is satisfied by all modal (deontic, etc.) algebras. Conversely, suppose that A is a non-theorem of C2 (D2, etc.) with r subformulas. Then A is falsified by the appropriate regular matrix of Theorem 8, say E <M, {1}, aJ, n, -, P>. Let the propositional variables in A be vi, . . ., vn, and let al, . . ., an be elements of M which form an assignment to vi, . . ., vn which falsifies A. Suppose that, for this assignment, the values of the subformulas of A other than vi, . . ., vn are an+1, ..., ar. We may assume that the last such subformula is A itself, so that ar # 1. Now <M, v-, n, -, P> is a modal (deontic, etc.) algebra (Theorems 9 and 1 1), so that by Theorem 13 there is a finite modal (deontic, etc.) algebra 9R with at most 22r+1 elements satisfying the five conditions of that theorem. By condition (i), we can consider the same assignment al, . . ., an to vi, . . ., vn in the matrix V1. By conditions (ii)-(v), it is clear that this assignment assigns the same value to A in 91 as it assigned in 9, namely ar # 1. Thus A is falsified by 91.

COROLLARY 1. The systems C2, D2, E2, T(C), T(D), T all have the finite model property, and so are decidable.

COROLLARY 2. FC2(D2etc.) A iff A is satisfied by all finite modal (deontic, etc.) algebras.

IV

Algebras and model structures In [6], Kripke has shown how normal model structures can be used to prove

completeness and other results for the system T, and how particular kinds of normal model structures can be similarly used in connexion with stronger systems such as S4 and S5.9 In what follows, we generalize Kripke's notion, and show that, in terms of different structures, natural representation theorems can be obtained for the algebras introduced earlier. Given these theorems, Kripke-type completeness results are very easy consequences of the completeness results of the last section.

A model structure (m. s.) is an ordered triple s = <K, Q, U>, where K is a non-empty set of elements, Q c K, and U a relation defined on K. Intuitively, we may think of K as a set of 'possible universes', and interpret

9 The origin of these structures in the recent history of modal logic is a little hard to determine. Hintikka has used similar semantical methods, and so has Prior, who seems to have derived the idea from some unpublished notes of Meredith ('the calculus of properties', see Thomas [16]). The view of these structures as telling us about the interconnexions between 'possible worlds' seems due to Geach. Their connexion with the algebraic treatment of S4 is implicit in Dummett-Lemmon [2].




Uxy (for x, y e K) as meaning that y is 'accessible' from x or that y is possible relative to x.10 To understand the role of the subset Q, it is'necessary to recall that the weak systems C2, D2, E2 contain no theorem of the form WJA, so that they are consistent with the addition of OA as an axiom- schema: from an interpretational standpoint, this means that we must allow for the possibility of universes in which anything (including a contra- diction) is possible, and Q is intuitively the set of such universes.11

Given a model structure S = <K, Q, U>, we define f+, the algebra on Mi, as <M, v , -, - P>, where:

(i) M $K; (ii) ,, - are the set-theoretic operations of union, intersection, and

complementation restricted to M; (iii) for A E M, PA - {x: (3y) (yEA A Uxy) V XEQ}

THEOREM 15. If A is a model structure, then A+ is a modal algebra. PROOF. That <M, v-1 n, -> is a Boolean algebra is immediate from

condition (ii) above. Further, for A, B e 3K, we have: P(A v B) ={x: (3y)(y e A vJ BA UXY) V XE Q}

- {x: (3y)(y EAA Uxy) v (3y)(y E BA UXy) V XE Q} - {x : (3y)(ye AA Uxy) V x EQ} % {x : (3y) (y E BA Uxy) V XE Q} -PA 'vPB,

so that R+ is a modal algebra. Let us write 'L" for the null set. Then by (iii) above we have at once: THEOREM 16. In the algebra on any model structure: (i) PN= Q; (ii) Q a PA, for any A.

The main representation theorem of this section is the following: THEOREM 17. Any finite modal algebra is isomorphic to the algebra on

some finite model structure. PROOF. Let 9) =<M, a., n, -, P> be a finite modal algebra. Then, by

Stone's representation theorem, <M, v., n, -> is isomorphic to the algebra of all subsets of a given set K. Let b be the isomorphism. For A c K, put P'A = OPf-l A, and put Q = b(PO). Finally, for x, y e K, put Uxy iff xE P'{y}. Let ft = <K, Q, U>, which is evidently a model structure. We show that 9R is isomorphic to 5+ under b. Let the possibility operation in ft+ be P*, i.e. P*A = {x : (3y)(y E A A Uxy) V X e Q}. b is evidently an

10 See e.g. Kripke [6], p. 70, or Prior [14] for a more informal account. 11 This device was suggested to me by a conversation with S. Kripke; in [4], Kripke

announces semantic completeness results for at least E2 and D2 of the systems considered here, and presumably these were obtained by considerations akin to the present ones.



58 E. J. LEMMON

isomorphism with respect to v, r, and -, so that it remains only to show that O(Px) = P*(Ox)

We first prove that for x E M O(Px) vi Q = O(Px). By Theorem 6 (ii), PO ? Px so that Px v PO = Px. Therefore b(Px) '(Px vPO) 1(Px) v O(PO) = O(Px) v Q, by definition of Q. Now consider an atom a E M. Clearly Oa is a unit set {u} for some u E K, and we have:

P*(Oa) ={x: (3y)(yE {U} A UXy) V XE Q} - {x: UxU V XE Q} - {x: XE P'{U} V XE Q} - {x: x E (Pa) v xE Q} - O(Pa) v Q - ((Pa).

Any element x E M may be represented in the form ai v ... .v am for atoms al, .. ., am. Thus for x eM:

?,(Px) = O(P(al ... v am)) - (Paj v ... v Pam) - (Paj) v ... v b(Pam)

P*(Oal) VP*(am) P*(Oal v ... v bam)

P*(O(al .. am))

P*(Ox) Thus b is also an isomorphism with respect to P.

For the other five kinds of algebra, we need suitable definitions of corresponding model structures. Let us define a deontic m. s. as an m. s. <K, Q, U> in which U and Q satisfy the condition:

(6) for x E K, either (3y) Uxy or x E Q. Let us define an epistemic m. s. as an m. s. <K, Q, U> in which U and Q satisfy the condition:

(e) for x E K, either Uxx or x E Q. ((s) is clearly equivalent to the condition that U be reflexive in K - Q.) Finally, let us define a normal m. s. as an m. s. <K, Q, U> in which Q = (D. We may conveniently think, in fact, of a normal m. s. as a structure <K, U> rather than a structure <K, Q, U>: there are no 'queer' universes. For the algebra on a normal m. s., condition (iii) on A E $K may be written:

(iii)' PA = {x : (3Y)(y E A A UXy)}.

Similarly, for normal deontic m. s.'s, condition (6) can appear in the form:

(6)' for xE K, (3y)Uxy;




and for normal epistemic m. s.'s, (e) becomes the condition that U be reflexive in K. Thus normal epistemic m. s.'s are identical with what Kripke calls normal m. S.'S.12

THEOREM 18. If SR is a deontic (epistemic, normal, etc.) model structure, then R+ is a deontic (epistemic, normal, etc.) algebra.

PROOF. Let St = <K, Q, U> be a deontic m. s. To prove that A+ is a deontic algebra, it suffices, in view of Theorem 15, to prove that PK - K. But PK {x: (3y)(yE K A Uxy) v xE Q} = K, by condition (6). Let

= <K, Q, U> be an epistemic m. s. To prove that R+ is an epistemic algebra, it suffices, in view of Theorem 15, to prove that, for A S K, A c PA. But by (e), for XE A, Uxx v XE Q, whence (3y)(yE A A UXy) V

X E Q, so that X E PA. Finally, let - = <K, Q, U> be a normal m. s. To prove that S+ is a normal algebra, it suffices, in view of Theorem 15, to prove that PD = D. However, by Theorem 16 (i), PD = Q, and we have Q = D.

Representation theorems may now be given.

THEOREM 19. Any finite deontic (epistemic, normal, etc.) algebra is isomorphic to the algebra on some finite deontic (epistemic, normal, etc.) model structure.

PROOF. We employ the same terminology and definitions as in the proof 'of Theorem 17. The proof of isomorphism is as before, and it only remains to show that A is a deontic (epistemic, etc.) m. s. First, then, suppose that 9J is a deontic algebra. Since P1 = 1, by the isomorphism b P*K = K, or {x: (3y) Uxy v x E Q} - K. Condition (6) follows at once, and A is a deontic m. s. Second, suppose that 9 is an epistemic algebra. Since for X E M X < Px, by the isomorphism b for any A : K A c P*A. Hence in particular for x E K {x} s P*{x}, so that x E P*{x}. It follows that (3y) (y E {X} A UXy) V X E Q, from which condition (8) follows at once, and SR is an epistemic m. s. Third, suppose 9N is a normal algebra. Since PO = 0, by the isomorphism b P* = (D. But by definition Q = O(PO) = P*(O) =

P*(D. so that Q = D and A is a normal m. s.

THEOREM 20. FC2(D2,etc.) A (i) iff A is satisfied by A+ for all model structures A (deontic model structures 9, etc.) (ii) iff A is satisfied by A+ for all finite model structures A (deontic model structures 9, etc.).

PROOF. If FC2 A, then A is satisfied by all modal algebras by Theorem 12, and so by S+ for all model structures S, by Theorem 15. Conversely, if A is a non-theorem of C2, there is a finite modal algebra falsifying it by

12 This terminological departure is, perhaps, unfortunate; yet I prefer to think of the normalcy of Kripke's structures as residing in the fact that Q = D rather than in the fact that U is reflexive, in which case we need a special term for the latter feature. Kripke also selects for attention a special element G of K (the 'actual' universe); but there is no point in this for present purposes.



60 E. J. LEMMON

Theorem 14, so that A is falsified by R+ for some finite model structure A, by Theorem 17. The other cases employ Theorems 18 and 19.

Theorem 20 adds to our completeness results (compare Theorem 12 and Corollary 2 to Theorem 14). It only remains to be seen that these last results are equivalent to semantic completeness. This will be a direct consequence of the fact that satisfiability by A+ for an m. s. A is equivalent to validity in A, in Kripke's sense. (Actually, we shall employ a slight variation on Kripke's terminology.)

First, we need, following Kripke, to define the appropriate semantic notions. By a model /or a wff A in a model structure <K, Q, U>, we understand a binary function PD(v, k), where v ranges over the propositional variables of A and k over the elements of K, whose values lie in the set {T, F}. Thus in effect <>(v, k) consists of an assignment of truth-values T, F to each propositional variable of A in each 'world' of K. Thus by (D(v, k) each variable is associated with a set of worlds of K, those in which it is true (so that the variable may be thought of as representing a function from worlds to truth-values), and each world is associated with a set of variables, those which are true in it (so that the world may be thought of as a function from variables to truth-values). In a natural way, PD(v, k) associates with a variable v a set A 5 K which can be construed as the value of that variable in an assignment from the algebra S+.

We next define a unique extension of d>, VP'(B, k), where B is any subformula of A, as follows:

(i) if B is atomic (i.e. a variable), VP'(B, k) (D(B, k); (ii) F'(-C, k) = T iff V'(C, k) = F; (iii) d'(C -- D, k) = T iff either d'(C, k) = F or VI'(D, k) -T; (iv) P'(LIC, k) = T iff both P'(C, 1) = T for all 1 E K such that Ukl and

k f Q. As a consequence of this definition, together with D1, D2, and D4:

(v) P'(C v D, k) T iff either VI'(C, k) = T or P'(D, k) T; (vi) P'(C A D, k) T iff both P'(C, k) = T and P'(D, k) T; (vii) P'(OC, k) = T iff either V'(C, 1) = T for some l e K such that Uki

or k E Q.

This interpretation needs little comment, except to observe that in virtue of (vii) OC is true in world k iff either there is some world accessible from k (possible relative to k) in which C is true or k is one of the 'queer' worlds in which anything is possible. SC is interpreted in parallel by clause (iv).

We say that A is true lor model (D(v, k) at 1 E K in a model structure <K, Q, U> iff (D'(A, 1) = T. We say that A is valid in <K, Q, U> iff A is true for all models (D(v, k) at all 1 E K in <K, Q, U>. We say that A is valid iff A is valid in all model structures. We further say that A is




D2- (E2-, T(C)-, T(D)-, T-) valid iff A is valid in all deontic (epistemic, normal, normal deontic, normal epistemic) model structures.

Given any wff A and model (D(v, k) for A in a model structure =

<K, Q, U>, in terms of D we may define an assignment %(QI) to the variables vj, . . ., vn of A from A+, by putting V(vi) = {x: x E K A

(D(vi, x) = T} (1 < i < n), and %C(QD) = <V(vi), . . ., V(v.)>. Conversely, given an assignment W = <A1, . . ., An> (As i K) from A+ to the n variables of A, we may define a model 4)(W) (v, k) for A in A, by putting @((W) (vi, x) = T iff x E Ai. For any assignment SC from A+ to the variables of A and subformula B of A, let V%(B) be the value assigned to B in A+ for the assignment W.

LEMMA. (i) Let A be a wff and ID(v, k) a model for A in 9 - <K, Q, U>. Then for all x E K .D'(A, x) = T iff xE V%(,,)(A).

(ii) Let A be a wff and W an assignment to the variables of A from Al+ for some Al <K, Q, U>. Then for all x E K 4I(D)'(A, x) = T iff x E VA(A).

PROOF of (i), by induction on length of A. If A is one of the variables vi, ..., vn, the result holds by the definition of A(<)). For the inductive step, suppose the result holds for B and C. We note first that by quantifier logic:

(Vy) (Uxy ->A)'(B, y) = T) *-> (Vy) (Uxy -?y E Vw(() (B)) (1) Now suppose A has the form B -> C. Then, for all x E K:

D'(B -+C,x) = T<-('(B,x) = Fv V'(C,x) = T x N Vw((D,(B) v x e VW(O) (C)

4>x e ( VSA(1D(B) V VW(@,D(Q) *- x 1e Vw((D,(B -->C).

A similar proof obtains if A has the form -B. Suppose finally that A has the form LIB. Then, for all x E K:

V)( DB, x) T (Vy) (Uxy V'(B, y) T) A X Q (Vy)(Uxy y E Vs(D)(B)) A X Q (by (1)) x E NV(,D)(B)

4>X E VW(()( LI]3). PROOF of (ii) is similar: we note that the result holds for the case when

A is one of the variables vj, . . ., vn by the definition of 4>(W). Our main theorem relating the algebraic and the semantic approach

may now be stated as follows:

THEOREM 21. Let Al <K, Q, U> be any model structure, and A any wff. Then A is satisfied by l+ iff A is valid in Al.

PROOF. Let A be a wff satisfied by V, and consider a model 1D(v, k) for A in A. Then Vw(()(A) = K, whence by the Lemma (i) (D'(A, x) = T for all x E K. Thus A is valid in Al. Conversely, suppose that A is valid



62 E. J. LEMMON

in A, and consider an assignment W to its variables. Then, for all x E K, 4I(D)'(A, x) = T, whence by the Lemma (ii) for all xE K xE V%(A), so that VA(A) = K and A is satisfied by R+.

COROLLARY. FC2(D2,E2,etc.) A iff A is valid (D2-valid, E2-valid, etc.). PROOF by Theorem 20 (i), Theorem 21, and the various definitions of

validity.

The completeness result for T in this corollary is equivalent to that for T in Kripke [6], p. 86. The completeness results for the other five systems are new, though in some cases announced by Kripke in [4]. However, the results are here obtained independently of the considerations pertaining to semantic tableaux employed by Kripke, and in such a way as to relate them to comparable algebraic theorems.

V

Four-valued matrices Wide use has been made in the literature on modal logic of 4-valued

matrices for determining independence, etc. The representation theorems of the last section provide a ready way of determining all regular 4-valued matrices which satisfy C2; for any such matrix is isomorphic to the algebra on a model structure sf = <K, Q, U> where K contains 2 elements, say a and b. Elementary computation shows that there are 64 such model structures (there are 4 possible choices of Q and 16 of U). However, of the 64 corresponding algebras, many are isomorphic; there turn out to be only 15 essentially distinct ones, as follows:13

=I <K, {a, b}, (D>; =2 <K, {a}, eF>; =3 <K, {a}, {<b, b>}>; =4 <K, {a}, {<b, a>}>; =5 <K, {a}, {<b, b>, <b, a>}>;

R6 <K, 4W, D>; =7 <K, (D, {<b, b>}>; =8 <K, ID, {<b, a>}>; =9 <K, (D, {<a, a>, <b, b>}>;

Rio = <K, (D, {<a, b>, <b, a>}>; =1 = <K, (D, {<b, b>, <b, a>}>;

13 This is a result of the obvious symmetry between a and b, together with the fact that U-leads within and out of Q make no difference to the construction of t+; this latter fact will be exploited in the sequel to this paper.




912 = <K, F, {<b, b>, <a, b>}>; 13 <K, 0, {<b, b>, <a, b>, <b, a>}>; =14 c <K, (, {<a, a>, <b, b>, <b, a>}>;

915 = <K, (, {<a, a>, <b, b>, <a, b>, <b, a>}>.

These 15 model structures can be more perspicuously represented as directed graphs (see [13]), as follows:

a b a b a b 0 P 0.

'R (Q= a,b0) A2 (Q }a;) A3 (Q= at)

a b a b a b 0 n No ___ 0 0

A4 (Q= ag) A5 (Q= at) A6 (Q= ()

a b a b a b O 0? of - - q .0

A7 (Q=() A8 (Q= () A9 (Q= ()

a b a b a b - N o O 0 M O go __ P 0

(OQ= (D) fi1 (Q =(D) 912. (Q= (D)

a b a b a b

A13 (Q=D) A14 (Q4D) A15 (Q=F)

To facilitate comparison with the usual presentations, we may agree to denote the elements of ft+ (1 _ i < 15) as follows: 1 = {a, b}, 2 = {b}, 3 = {a}, 4 = (. The connectives -- and - have the same interpretation in all matrices:

q p q 1j1 2 3 4 p I-P

I 1 2 3 4 1 4 2 1 1 3 3 2 3 3 1 2 1 2 3 2 4 1 1 1 1 4 1



64 E. J. LEMMON

The fifteen matrices can be distinguished by the column assigned to [1, as given in the next table.

1 l* 2 3 4 + 26 27 + *9 1+0 +1 12 $+3 1+4 15

1 4 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 4 2 2 4 4 1 1 3 2 3 3 1 3 4 4 3 4 2 4 2 4 1 3 1 3 2 3 4 4 3 4 4 4 2 4 4 4 1 3 3 4 4 3 4 4 4 4

Of Lewis and Langford's five groups (see [10]), Group I is identical with A+, Group II with A+, Group III with A+, and Group IV with A. Group V does not form a regular matrix; although it satisfies A4, it does not satify R2, and for it we have P(x v y) # Px v Py (P(4 v 2) = P2= 2, but P4 v P2 = 3 v 2 1). It should be remembered that, as regular matrices, all fifteen of the present ones have 1 as sole designated value; it may well be, however, that the same matrices with both 1 and 2 designated have their use in other connexions.

A2 falsifies A5 and also -ilA, thus showing that C2 is properly contained in T(C) and D2. A4 satisfies A5 but falsifies A6 and WA, showing that D2 is properly contained in E2 and T(D). 6 satisfies the rule R3 but falsifies A5, showing that T(C) is properly contained in T(D) and independent of D2. %3 satisfies A6 but falsifies WA, showing that E2 is properly contained in T. Finally, Al satisfies A5 and R3 but falsifies A6, showing that T(D) is properly contained in T and independent of E2.

In fact, what schemata are satisfied by which matrix can be read off at a glance from the associated graphs, if we bear in mind the relevant model conditions. Thus the matrices which satisfy T are the ones based on reflexive graphs in which Q = 4F, namely R9, +4, A+. Some of the other matrices in this set will be used in the sequel to this paper.

REFERENCES

[1] A. CHURCH, Introduction to mathematical logic I, Princeton, 1956. [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] R. FEYS, Les logiques nouvelles des modalitds, Revue neoscholastique de

philosophie, vol. 40 (1937), pp. 517-533, and vol. 41 (1938), pp. 217-252. [4] SAUL A. KRIPKE, Semantic analysis of modal logic (abstract), this JOURNAL,

vol. 24 (1959), pp. 323-324. [5] SAUL A. KRIPKE, A completeness theorem in modal logic, this JOURNAL, vol. 24

(1959), pp. 1-14. [6] SAUL A. KRIPKE, Semantical analysis of modal logic I, Zeitschriftffur mathe-

matische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 67-96.




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

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

[9] E. J. LEMMON, Deontic logic and the logic of imperatives, Logique et analyse, vol. 8 (1965), pp. 39-7 1.

[10] C. I. LEWIS and C. H. LANGFORD, Symbolic logic, New York, 1932. [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, Some theorems about the sentential

calculi of Lewis and Heyting, this JOURNAL, VOL. 13 (1948), pp. 1-15. [13] OYSTEIN ORE, Graphs and their uses, New York, 1963. [14] A. N. PRIOR, Possible worlds, Philosophical quarterly, vol. 12 (1962),

pp. 36-43. [15] P. ROSENBLOOM, The elements of mathematical logic, New York, 1950. [16] Ivo THOMAS, A final note on SI' and the Brouwerian axioms, Notre Dame

journal of formal logic, vol. 4 (1963), pp. 231-232. [17] G. H. VON WRIGHT, An essay in modal logic, North-Holland, 1951.

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