On Possible Worlds in Propositional Calculi

On Possible Worlds in Propositional Calculi by

R. A. BULL

(University of Leeds)

Studying the All Possible Worlds interpretation of the modal calculus S5, in [4], Prior uses a propositional operator Q, with the intended interpretation that Qa is true everywhere if a is true in exactly one Possible World, and is false everywhere otherwise. The point of this operator is that it enables individ- uals-the Possible Worlds-to be discussed within a purely propositional calculus. (See also Prior’s use of a similar operator in the context of tense-logics and instants, in [3].) Prior works with two extensions of S5: he either defines Q using propositional quantifiers, taking

Qa = dl KMalTpALCapLCaNp;

or alternatively takes Q as a new primitive, with the axioms:

Q1 CQPMP 4 2 CQpAWqLCpNq 4 3 CQPLQP 44 CLCpqCQpCQpQq.

The purpose of this paper is to give completeness proofs for these systems with respect to models in which Q has the intended interpretation. In the case of Q being defined in terms of propositional quantifiers, it proves necessary to add a further axiom:

Q* ZPKPQP

(Prior’s Q6 in [4]). I should point out that the models I need when working with propositional quantifiers are secondary, in 12 - Theoria, 3: 1968

172 R. A. BULL

the sense that not all mappings from the set of Possible Worlds to {T, F} are permitted. However there is a characteristic value for each Possible World in the models: it is t o ensure this property that the axiom Q’ must be added. The system S5 plus Q1-Q4, which I refer t o as SSQ, turns out to have the finite model property, so that it is decidable.

§ 1. It is well-known that S5 is characterised by models on ordered sets, where the ordering is RST’ and holds on all pairs of members of the set. In what follows I shall use q as the algebraic operator corresponding to Q. What I want t o show is that S5Q is characterised by models of the form described, extended with the operator q given by:

qx = 1 if x = {a} for some a in the set, = 0 otherwise.

In fact I shall characterise S5Q with finite models of this form, although it is immediate that all models of this form verify S5Q.

It is well-known that S5 has a characteristic Lindenbaum model of congruence classes of formulas, which is a closure algebra with the unit designated, L being interpreted by J. With SSQ since we have 44 , there is again a characteristic Lindenbaum model of congruence classes of formulas, which is a closure algebra with the unit designated, together with the further operator q.

[The concept of homomorphism can be naturally extended to models with one designated element. A model C with one designated element is sub-directly reducible t o models Xi, i in I, if and only if:

(1) Each Ci is a proper homomorphic image of C, under a mapping Oi say.

(2) Given a pair of elements x and y of C, there is a mapping Oi such that

Oix # Oiy.

(This is a handier alternative to sub-direct reduction.) A model

Reflexive, symmetric and transitive.

Birkhoff’s original definition of with one designated element

ON POSSIBLE WORLDS IN PROPOSITIONAL CALCULI 173

which is not sub-directly reducible is said to be sub-directly irreducible. As a nominal extension of a standard result for algebras, every model with one designated element is sub-directly reducible to sub-directly irreducible models with one designated element. Further, the sub-directly irreducible models of this reduction characterise the system characterised by the given model. On the one hand, each Xi is a homomorphic image of 2, and therefore verifies the system. On the other hand, a formula rejected by C is rejected by any Ci which does not map the designated element and a non-designated value of the formula together, and there must be a t least one such Xi by the second condition for sub-direct reduction.]

I now sub-directly reduce the Lindenbaum model for SSQ t o sub-directly irreducible models, which again characterise SSQ. Let (M, {I}, n , U ,‘, C, q) be such a sub-directly irreducible model. I claim that (M, n, U ,’, C) is a sub-directly irreducible closure algebra. Suppose the contrary. Then there are proper homomorphisms Oi, i in I, of (M, n , U ,’, C ) which satisfy condition (2) for sub-direct reduction. If I can define an operator qi on the image of M under Oi in such a way that

Oiqx = qiOix,

then I will have sub-directly reduced (M, {I}, n, U,’, C, q), contrary to hypothesis, giving the required result.

To do this I define qi on the image of M under Oi by the above equation. I t remains to show that this definition is unique, i.e. that

But

implies

implies

implies

since

Oix= Oiy implies Oiqx= Oiqy.

Oix=Oiy

Oi( (x’ U y) n (x U y’)) = 1

OiJ((x’U y) n (x U y’)) = 1

Oi((qx)’u qy) = 1

174 R. A. BULL

(J((X’U Y> n (x ” Y’>))’U ((qx)’ u qv) = 1

in (M, { 1}, n , U ,’, C, q) by the verification of Q4.

Similarly

Therefore

which implies

as required.

@(qx u (qy)’) = 1.

oiC(Cqx>’U qY> n Iqx u IqY)’)) = 1

oiqx = oiqy,

We see then, that SSQ is characterised by models (M, { l}, n , U,’, C, q) where (M, n, U,’, C) is a sub-directly irreducible closure algebra. Now Birkhoff has shown that in any sub-directly irreducible closure algebra,

J x U J y = l iff x = l or y = l

We also have that each (M, { l}, n , U ,’, C, q) verifies ALNLpLp. I t follows that in these models

(Jx)’= 1 or x = 1,

Jx=O for x # 1. so that

By the verification of 43 , each element qx is an open element, so that

q x = 1 or qx=O.

By the verification of Ql,

so that Cx = 0 implies qx = 0,

x = 0 implies qx = 0.

An element x of a Boolean algebra is said to be an atom if and only if

x z o A (y) n = o v x n y’=o).


Let x be a non-zero element of M which is not an atom of (M, n , U ,’). Then there is an element y of M such that

x n y z 0 a n d xny’zO, :. x’ U y‘ + 1 and x’ U y # 1, :. J(x’ U y’) = 0 and J(x’ U y) = 0, :. J(x’ u y) u J(x’ u y’) = 0, :. qx = 0

by the verification of Q2. To summarise these results: we see that S5Q is characterised

by models (M, {1}, n, U,’, C, q) where (M, n, U,’) is a Boolean algebra;

Cx=O if x=O, = 1 otherwise;

and qx is 1 or 0, and

qx = 1 only if x is an atom.

Note that all models of this form verify S 5 Q , whether or not qx= 1 for every atom x. I can use these properties to show, first that S5Q has the finite model property, and then that it is characterised by its intended models.

Let us suppose that a formula a is rejected by a model (M, { l}, n , U ,’, C, q) of the form described above. I must now find a finite model of the same form which also rejects a. Let V be an evaluation in (M, { l}, n , U ,’, C, q) which rejects a, and let X be the set of the values of the parts of a under V. Take the finite subalgebra (Mx, n , U,’, C, q) generated in (M, n , U ,’, C, q) by X. To see that this subalgebra is finite, note that the Boolean algebra generated in (M, 0, U,’) by X is finite, and that it is closed under C and q, since Cx and qx are always 1 or 0. Every atom of (M, n , U ,’) must also be an atom of (Mx, n , U ,’) if it is in Mx, so in the finite model we have

qx= 1 only if x is an atom.

Thus (Mx, {l}, n, U,’, C, q) is a finite model of the form described above.

176 R. A . BULL

To see that u is rejected in this finite model, define an evaluation V‘ in it by allocating:

V ’ ( y ) = V ( y ) for every variable y in a, V ’ ( y ) = 1 for all variables y not occurring in (1.

Since (Mx, n, U ,’, C, q) is a subalgebra of (M, n, U,’, C, q),

V ’ ( U ) = V ( U ) f 1, as required.

The intended finite models for SSQ are given by taking each closure model on a set of positive integers { 1, 2, . . ., n} ordered by i 5 j for 1 I i, j 5 n, and defining:

q x = l i f x = { r } f o r s o m e r w i t h l < r I n , = 0 otherwise.

Note that this model is of the form described above. I shall prove that SSQ is characterised by this set of models, by showing that every finite model of the form described above can be embedded in such a model on an RST ordered set. (Embedding preserves rejection, so that the new set of finite models rejects all non-theses of S5Q; while it still verifies all the theses of SSQ.)

The given finite models can be represented as models on ordered sets, where the ordering is RST and holds on all pairs of members of the set, using Stone’s Theorem. (The finite Boolean algebra is isomorphic to a field of the subsets of a set; since a 5 b for each a and b in the set,

Cx=O for x=O, = 1 otherwise,

in the model on the RST ordered set; as required.) With this representation, the atoms are the subsets of the form {a}, so that

qx = 1 only if x = {a} for some a in the set.

However we may have q{a} = 0 for some a in the set.

{ail 1 I i I m + n} ordered by ai I aj for 1 I i, j I m + n, and that Let us suppose then, that we have a closure model on a set

q{ai} = 1 for 1 I i I m, q(am+i}=O for 1 I i l n .


Take the intended model on {iI 1 < i < m + 2n} ordered by i 5 j for 1 < i, j < m + 2n. Define a mapping O of the elements of the first model into the elements of the second by:

(I) O{ai}=(i} for 1 < i s m ,

(2) @(ail, . . ., aili}=O(ail} U . . . U O(aili}. O{am+i} = (m + Zi - 1, m + Zi} for 1 5 i 5 n.

This is immediately an embedding for the closure algebras. Also 0 is a homomorphism with respect t o q. For in the given model,

q x = 1 iff x={ai} for some 1 < i < m ;

while in the intended model,

q O x = l iff Ox={i} for some 1 < i < m + Z n ,

which requires

q O x = l iff x={ai} for some I < i < m ;

so that the conditions on q x = 1 and qOx= 1 are the same. I t follows that 0 embeds the first model into the second, as required.

Thus S5Q is characterised by its finite intended models. § 2. Let us consider the system S5n obtained by extending S5

(I assume the Godel axiomatisation) with propositional quantifiers, given by the quantifier introduction rules:

k cap - k CITpaP kCup*kCuITpB, if p does not occur free in u.

Of course substitution must be suitably restricted. Prior [2] shows that the Barcan schema

CITpLaLl7pu

can be derived in this system. In [l] I show that S5IIis characterised by secondary ‘algebraic’ models on ordered sets, where the ordering is RST and holds on all pairs of members of the set. In these secondary models, the values of propositions range over some field of subsets of the set, the whole set being the designated element. The classical operators and L are given as usual in models on ordered sets. For the propositional quantifiers I have:

178 R. A . BULL

x E V(npa) iff x E Vi(a) for all i in I, where {Vili E I} is the set of evaluations in which Vi(Y) = V ( y ) for all variables y except perhaps p.

Within S517, Q can be defined by

QLI = dr KManpALCapLCaNp,

where p is a variable which does not occur free in a. Now in the models for S5II which I have described,

V(Qa) = 1 if V(a) is an atom of the field of values, = 0 otherwise.

Let the RST ordered set be (K, I), and let the field of values be 132 with elements {Kj / j E J}. Let N be the set of atoms of 132, and let H be the set of members of K which are not members of atoms of 132. Set up a new RST ordered set { K’, 2 } by taking K’= N U H; and set up a new set of values {Kijj E J} by taking

This gives us a sound new model with field of values 132‘ say, with the property that if the evaluation V’ is given by the alloca- tions:

then

K ’ = { x / ( x E N A x G K j ) V ( x E H A x E K j ) } .

V’(y) = K’ iff V(y) = Kj, for all variables;

V’(a) = KS iff V(a) = Kj, for all formulas.

(It is a straightforward task to establish this property by induction on the operators, and the soundness of the new model follows from it. The point is that the new closure algebra is a homomorphic image of the given one, formed in such a way that (m‘, C ) is an isomorphic image of (132, C).) Because of the use of N in the definitions of K’ and Kj’] only one-member subsets of K’ are atoms of 132’. Thus S5II is characterised by secondary ‘algebraic’ models on RST ordered sets in which

V(Qa) = 1 only if V ( a ) ={a} for some a in the set.

However, in contrast to S5Q, it is not the case that S5I7 is characterised by secondary ‘algebraic’ models on RST ordered sets in which


V(Qu) = 1 iff V(u) = {a} for some a in the set.

To see this, consider the formula

CMpzqKQqLCqp.

This formula is verified by all models with the property I have just stated. But it is rejected by the following sound model for S5n. Take a complete non-atomic Boolean algebra Nz, and embed it in the field of subsets of a set K, using Stone’s Theorem. Let the image of Nz in K be m’. Take an RST ordering < on all pairs of members of K. Now we have a secondary ‘algebraic’ model for S5I7 on (K, 5 ) with field of values m’; it is sound because m‘ is complete. Let A be a non-zero member of Nz’ which does not contain any atom of m‘, and take an evaluation V in the model for which V(p) = A . We now have

q{x}=O for x E A :. qB =O for BGA :. (B) (qB = 0 V J(B’ U A) = 0)

:. 9 (qB n JIB’ U A)) = 0 :. VCCqKQqLCqp) = 0 :. V(Mp) = CA = 1 and VIZqKQqLCqp) = 0 :. V(CMpZqKQqLCqp) = 0,

:. (B) (qB n J(B’ U A) = 0)

and the formula is rejected by V as required.

adding the further axiom To avoid this non-standard interpretation, I strengthen S5n,

Q* ZPKPQP.

This new system, which I call S5II*, seems to be the one needed for Prior’s purposes.

To find the models for this new system, I need the Deduction Theorem in S5II. In other words, I need the result that B is derivable in the extension of S5II with a if and only if CLDp, . . . 17pnu/3 is derivable in S5n, where p,, . . ., p,, are the free variables of a. The first follows from the second immediately. The proof that the second follows from the first is straightforward, an

180 R. A. BULL

induction on the length of proof trees. The induction basis is the fact that the result holds if p is u or an axiom of S5. The induction steps are on the five rules: substitution, detachment, k u * FLa, and the rules for introducing quantifiers.

The induction step on substitution is immediate, since LUpl . . . Up,a has no free variables.

The induction step on detachment holds by:

c a p , c a c p y *Cap, CCapCay * c u y .

The induction step on Fri * tLcr holds by:

CLaB *LCLU/? *LCLap, CLCLapCLuLp 3 C L a L p .

The induction steps on the rules for introducing quantifiers hold by:

cG% * CKupy *CUpKu/?y * CKITpaITpB y * CKuUppy

(since p is not free in u)

and *CaCUppy;

*CaC/$ * CKapy *CKapITpy, if p does not occur free in /3

(since p is not free in a)

=Gc/9Upy.

We have then, that p is derivable in S5n* if and only if CLZpKpQpp is derivable in S5n. But CLZpKpQpp is derivable


in S5II if and only if it is verified by every secondary ‘algebraic’ model on an RST ordered set in which

V(Qa) = 1 only if V(a) ={a} for some a in the set. This holds if and only if is verified by all models of this form

which verify LZpKpQp. For in each model, either VCLZpKpQp) = 1 for all V, and VCCLZpKpQpa) = V(a); or V(LZpKpQp) = 0 for all V, and VCCLZpKpQpa) = 1. To obtain the condition for a model of the form under consideration to verify LZpKpQp, I argue:

a E VLZPKPQP) iff x E VCZpKpQp) for each x in K iff x E Vi(KpQp) for some i in I, for each x in K iff xEVi(p) and xEVi(Qp) for some i in I, for

iff Vi(Qp) = 1 for the Vi given by Vi(p) = {x}, each x in K

for each x in K,

(since Vi(Qp)#O only if Vi(Qp)= 1, and Vi(Qp)= 1 only if Vi(p) = {b} for some b in K)

iff V(Qa) = 1 if V(u) = {x}, for each x in K

(since V’(Qa) is a function of the value V’(a) for each evaluation V’, in a given model).

Thus S5II’ is characterised by secondary ‘algebraic’ models on RST ordered sets in which

V(Qa)= 1 if V(a)={a} for some a in the set, = 0 otherwise,

as required. Note that this entails every subset {ala E K} being a member of the field of values of the model.

182 R. A. BULL

R E F E R E N C E S [ I ] R. A. BULL, O n modal logics with propositional quantifiers. Forthcoming in

[2] A. N. PRIOR, Modality and quantification in S5. The Journal of Symbolic

[3] A. N . PRIOR, Past, Present, and Future. Oxford, 1967. [4] A. N. PRIOR, Egocentric logic. Forthcoming in Nous.

The Journal of Symbolic Logic.

Logic, 21 (1956), pp. 6 0 6 2 .

Received on July 16, 1968

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On Possible Worlds in Propositional Calculi