Zdaehr . f. moth. Lagik und arundlagen d . Math. Bd. 24, S. 405-408 (1918)

ON THE ADDITION OF WEAKENED L-REDUCTION AXIOMS TO THE BROUWER SYSTEM

by MICHAEL BYRD in Madison, Wisconsin (U.S.A.)

0. The system T: is the normal modal logic obtained by adding the axiom:

s4, , L"p 2 Ln+lp ,

to the Brouwer system B. These systems were first investigated by IVO THOMAS, and, in [l], he showed that all these systems are distinct. Since T: = 55, a natural ques- tion to pursue is the extent to which the other Ti-systems resemble 55. In this paper, I exhibit an important point of similarity and an important point of contrast. Spe- cifically, like 55, each Tf-system has a finite number of modalities. Indeed, my proof permits the prediction of an upper bound on modalities for each system. On the other hand, it is well-known that in 55 there are only finitely many distinct propositions expressible in any finite number of variables. In contrast, I show that this is not true for T,+ if n > 1.

Before turning to the proof of these claims, I want to suggest a few reasons why the T+-systems are worthy objects of study. First, of course, these systems constitute an interesting way of approaching 55, a way dissimilar to the better known terrain between 54 and S5. From a semantic point of view, the way the T+-systems approach S5 is intriguing and suggestive. Note that model structures for S5 have the property t,hat every point is accessible to every other point. The T+-systems represent a sys- tematic weakening of this universal immediate accessibility. For in the system T,+ it may not be possible to get immediately from point x to point y. However, it will always be possible to get from x to y in n - 1 steps. That is, there will be points x,, . . ., xn-, such that xRx,, x,Rx,, . . ., and x{,-,Ry. An examination of just what difference passing through these intermediaries makes seems likely to enhance con- siderably our understanding of the topology of modal logics.

B second point of interest concerns the system T,+. In [2], RESCHER and URQUHART suggest that this system correctly represents the logic of the Megarian conception of necessity in certain kinds of temporal structures. On this conception of necessity, a statement is now necessary just in case it is true now and a t all earlier and later times. If time is treated as linear, necessity so conceived is exactly S5. An alternative, and appealing, view of time is that it may branch in the future, although it is linear toward the past. If this view of time is adopted, it is less clear what the logic of Megarian necessity is. RESCHER and URQUHART conjecture that it is T2f. (See [2], p. 129.) One consequence of the present study is that this claim is false, although it is true that the structure of modalities in T,f is the same as that for Megariari necessity in forward- branching, backward-linear time.

1. The fact that the T+-systems have a finite number of modalities follows as in Theorem 1 of [3]. Let M"oL"1M"z. . . (a , 2 0 ) represent the general form of a modality. For the sequence a,, a,, . . . , a,, to be irreducible in T,f , several conditions must be met. First, since Lnp 3 Ln+lp holds in T: , no a , can be greater than 12. Second, since T,' is an extension of B, it follows that, for all j , LjMjLjp is equivalent to L'p, and dually. This means that for a sequence to be irreducible, it must be that if

406 MICHAEL BYRD

ai 2 ai+,, then uci+l > ai+2. If this condition is not met, then an L M L or M L M se- quence can be reduced to an L or M Sequelice. Only three kinds of sequences meet these two conditions :

1. a strictly ascending sequence with no element greater than n. 2. a strictly decreasing sequence (except possibly a, = a,) with no element greater

than n. 3. a sequence strictly ascending to a number m, less than n + 1, followed by a

strictly decreasing sequence whose initial element is no greater than m. Since there are only finitely many such sequences for each n, all T+-systems have a finite number of modalities.

One nice consequence of this general result is that it is a simple matter to determine the structure of modalities for Megarian necessity in f orward-branching, backward- linear time. It turns out to be the same as T,+, as the RESCHER-URQUHART conjecture suggests. To see this, note that the general result implies that T,+ has a t most 20 distinct positive modalities, and examinabion of cases shows tjhet 8 (or 9) are irreducible. They are:

LLP -1 LP I

M L p I I Figure 1

I

Since all theorems of T,+ are valid for Megarien necessity in forward-branching, backward-linear time, it follows that this concept of modality can distinguish no more than the 8 modalities of Figure 1 . It is then straightforward to find models that make all the required distinct,ions.

2. In 55, it is possible to reduce any sentreme to a sentence with no nesting of modal operators. As a consequence, only a finite number of distinct propositions are express- ible in any finite number of variables. This is not true for any other T+-system. To prove this claim, I exhibit an infinite sequence of sentences in two variables which are pairwise distinct in T,+. Since the Tf-systems form a decreasing chain, it follows that the same is true for each weaker system.

The relevant sequence is defined, as follows :

ADDITION O F WEAKENED L-REDUCTION AXIOMS TO THE BROWWER SYSTEM 407

To prove the Ai's distinct, consider the structure ( W , R), where W consists of three disjoint denumerable sets: {a,, a,, . . .}, {b,, b,, . . .}, (c,, c,, . . .}. The relation R is defined by cases :

1. a,Raj, for any i , j . 2. biRbj, for any i , j . 3. ciRcj, for any i , j . 4. ci+,Rai, for any i. 5. aiRbj, if i 2 j. 6. biRcj, if i 2 i. 7 . if xRy by cases 1.-6. , then yRx.

The structure ( W , R) validates T$. It is clearly reflexive and symmetric. To show that ( W , R) validates LLp 3 LLLp, it suffices to prove that for every x and y in W , there is a x in W such that XRZ and zRy. If x and y are both a-worlds, or both b-worlds, or both c-worlds, the result is guaranteed. Suppose x is a,; and y is bl,. Then ajRa,? and akRbl,. . If x is bj and y is c k , then bjRbk and bkRck. Finally, assume x is aj and y is ck . If k > 0 , then ajRa,<_, and ak-,Rck. If k = 0 , then ajRbj and bjRc,, since j 2 0. Con- verse cases hold by symmetry of R.

Now, consider the valuation V on which (1) V ( p , ai) = V ( q , ai ) = T, for all i. (2) V ( p , bi) = T, V(q , bi) = F , for all i. (3) V(p , ci) = F , V(q , ci) = T, for all i.

I show t,hat for all n, V(Ai, a,) = T iff i 5 n. Basis. Clearly, A , is true on V at a,. For A , to be true at a,, V ( M ( ( p & w q ) &

& N ( ( - p & q) & MA,) ) , a,) = T. Now, p & - q is true on V only a t b-worlds. But b, is the only b-world accessible from a,. So, M ( ( - p & q) & X A , ) must be true a t b, on V . Again, only c-worlds make - p & q true. Consequently M A , must be true on V at, c, since co is the only c-world accessible to b,. But, for M A , to be true a t c,, there must be a world accessible to co which makes p & q true. This cannot be true since only b-worlds and c-worlds are accessible to c o . It therefore follows that A , is false a t a,. Further, each A i , i 2 1, is false there, since A i implies A j , when i 2 j .

Induc t ion . Suppose that for j < m, V ( A k , a j ) = T iff k 5 j . I show first that V(A,,, , unl) = T. Now A , = ( p & q) & M ( ( p & -q ) & M ( ( w p & q ) & MA,-,) . Since a,,!Rbn1, b,,Rc,,, c,nRa,n-l, and, by inductive hypothesis, V(A,-, , am-,) = T, the required result holds. The next problem is to prove that V(A,+, , a,,) = F . Suppose the contrary. Since A,,,+, = ( p & q ) & M ( ( p & -4) & M ( ( - p & q ) & MA,), A,+1 is true a t a, only if p & - q & M ( ( w p & q ) & MA,,,) is true at some world accessible t,o a,,. This world must be a b-world bk and it must be that m 2 k. Of course, M ( ( - p & q) & MA,,,) must be true a t bk. This requires a world accessible to bi, where - p & q is true. This must be a world c j , with k 2 j . The sentence MA,,, must be true at cj with m >= i. If j = 0, this is clearly impossible. If j > 0 , there must be a world accessible to c j , where A , is true. For A , to be true, p & q must be true; so, the required world must be an a-world. But the only a-world accessible to cj is aj-,, where j - 1 < nL. However, A,,, is false a t aj-, by inductive hypothesis, thereby yielding the required contradiction, and concluding the proof.

I remarked earlier that this proof enables us to show the falsity of the RESCHER- URQUHART conjecture about the logic of Megarian necessity in forward-branching, ba,ckward-linear time. The sentence A, 3 A , is valid in this sort of temporal structure, whereas my proof establishes that this sentence is not a theorem of T,f. To see that A , 3 A , is valid, suppose that A , is true at time t . This entails that there is moment before or after t a t which ( p & w q ) & M ( ( - p & q) & MA,) is true. Call this moment t'

408 MICHAEL BYRD

and suppose t < t’. Since M ( ( - p & q) & MA,) holds a t t’, there must be a moiiient earlier or later than t’ where ( - p & q) & M A , is true. Whether t” < t‘ or t‘ < t“, it follows, by backwards linearity, that t < t’ < t” or t < t” < t’ or t” < t < t‘. I n any of these cases, A , is true at t as the reader may verify. So, suppose t’ < t . Now if t” is comparable to t (i.e. t < t” or t” < t ) , the situation is as before, and A , is true at t . Assume then that t and t” are incomparable. If so, by transitivity, it cannot be that t“ < t’. This yields the following situation, where t‘ < t ” :

The sentence MA,, is true a t t‘. There must therefore be a time t”’ earlier or later than t“ where ( p & q) & M ( ( p & - q ) & M ( - p & a ) ) is true. If t”‘ < t” , then either t’ < t”’ or t”’ < t‘ by backwards linearity. In either case, A , is true a t t , since we can go from t to t‘ to t” to t’” to t’ to t” to t”’ to t‘ to t“. On the other hand, if t“ < t“ ’ , then t’ < t” < t”’, and we can argue similarly trhat A , is true at t .

So, the logic of Megarian necessity for this case is not T,+ . What is the correct logic? The appropriate logic is the system T,f (Meg), obtained by adding the axiom:

60 T,f . (See [5].) Is T,f appropriate for Megarian necessity in any interesting temporal structure?

Let me close by mentioning several other significant features of the T+-systems. Each system T,f can be proved sound and complete for the class of frames <W, R), where R is reflexive, symmetric and meets the condition: C,t . The SCOTT-LEMMON method of filtrations can be applied to the canonical model of a T+-logic, yielding a proof that every non-theorem of that system is falsified in a finite model of the sort just described. Consequently, all T+-logics are decidable.

Finally, I should note some similarities between the present work and MAKINSOK ’S

proof (in [4]) that there are infinitely many Diodorean modal functions in S4.3. The resemblance between the sequences used is obvious. But, interestingly, MAKINSOX’S sequence employs only a single variable, thereby showing the existence of infinitely many distinct propositions expressible in one variable. My const,ruction uses two variables. Is this essential? I do not know, but I believe that it is.

Meg. (Lp & -Lp) ZI L(Lq ZI Lp) )

(x) (y) (35,). . . (3~,-~) (zRx, and xlRx2 and . . . and x,-,Ry).

References [l] THOMAS, I., Modal systems in the neighborhood of T. Notre Dame J. Formal Logic 5 (1964),

[ Z ] RESCHER, N., and A. URQUHART, Temporal logic. Springer-Verlag, Wien- New York 1971. [3] ULLRICH, D., and M. BYRD, The extensions of BAM,. J. Philosophical Logic 1977, 109-117. [4] MAKINSON, D., There are infinitely many diodorean modal functions. J. Symb. Logic 31 (1966),

[5] BYRD, M., Megarian necessity in forward-branching, backward-linear time (Forthcoming).

59-61.

406 - 408.

(Eingegangen am 9. November 1976)