syntactic aspects of modal incompleteness theorems

Syntactic aspects of modal incompleteness theorems by

J. F. A. K. VAN B E N T H E M (University of Groningen)

1. Introduction

THIS PAPER is concerned with propositional modal logic. Nota- tional conventions will be explained as the need for them arises.

Modal formulas are constructed using proposition letters (p, q, r, ...), Boolean operators (7: not, +: if . . . then ..., A : and, v : or, c*: if and only if) and unary modal operators 0 (necessarily), 0 (possibly). It actually suffices to take 7 , + and 0 as primi- tives, using the well-known definability of the other operators. The minimal modal logic K has a set of propositional axioms complete (for propositional logic) with respect to the rules of detachment (modus ponens) and substitution. Moreover, it has the modal axiom U(p+q)+(Op-+Oq), as well as the modal rule of “necessitation” (to infer Ocp from cp). (Notice that, very often, K is axiomatized without using the rule of substitution, but with axiom schemata.) Deducibility in K may then be defined as follows. Z kKcp if a finite sequence of modal formulas exists with cp at its end, such that each formula in the sequence either belongs to C, or is an axiom of K, or follows from previous formulas by an application of some rule of inference.

This notion of deducibility admits of a semantic characterization through the following concepts. A frame is an ordered couple ( W, R ) consisting of a set W (of so-called “worlds”) with a binary relation R on W (“accessibility”). Frames will be denoted by 8( = ( W, R ) ) . Truth of modal formulas in frames is definable by the intermediary of a valuation V on such a frame 8 which assigns subsets of W to proposition letters. Using the well-known Kripke truth definition, V may be lifted to the set of all modal formulas in a canonical fashion. Now cp is true in 8 (“$k cp”) if V(cp) = W for all valuations V on 8. The following notion of modal consequence then

64 J . F. A. K . VAN BENTHEM

arises naturally: for all sets Z of modal formulas and for all modal formulas cp,

Z /=Y iff, for all f r a m e s 5 , 5 / = c only if9kcp.

(‘$5 k C” means that 8 k r~ for each ~ E Z . )

Now, an easy induction on the length of derivations in K shows that Zbcp only if Zkcp. In fact, many well-known modal completeness results describe sets C for which the equivalence

holds for all modal formulas cp. But, in 1974, K. Fine and S. K. Thomason gave examples of sets Z for which this equivalence fails (cf. [6] and [ 131). More, and simpler, examples were discovered later on. The following one is from [4]:

and

cp=p+Op.

Indeed, the situation for b is hopeless, as was shown in [14]: the notion of semantic consequence in monadic second-order logic (with a binary first-order predicate constant) is effectively reducible to it. Therefore, k cannot be recursively axiomatized, since the latter notion is unaxiomatizable-even hyper-arithmetical.

On the other hand, there is a semantic characterization for deducibility in K , provided that the concept of “frame” is generalized in a suitable manner (cf. [15]). A general frame is an ordered couple (8,933) consisting of a frame 8 together with a set 933 of subsets of W which is closed under the set-theoretic operations of complement (taken with respect to W), intersection and “interior”. (The interior l (X) of a subset X of W is { W E WI for all V E W, Rwv only if V E X } . ) A valuation on a general frame (9,933) is a valuation on 5 taking its values in 933. Truth of modal formulas may then be defined like above. (Note that 933 is rich enough to contain all

SYNTACTIC ASPECTS OF MODAL INCOMPLETENESS THEOREMS 65

images V(cp).) The corresponding notion of semantic consequence is the following:

cp iff, for all general frames (3, %I >, c <~,%)FC only if(B,%I>kcp.

The standard Henkin type completeness proof establishes that the following equivalence holds for all C and cp,

(**) Cbcp iff Ckgcp.

This move corresponds to the one made in [9], where “general models” were introduced (in addition to “standard models”) to save some kind of completeness theorem for higher-order logic.

Typically, a modal incompleteness result of the above-mentioned kind (“C cp, but not C bcp”) is proven as follows. For the negative result, a general frame is exhibited in which C holds, but cp fails. (This suffices to show that cp is not derivable from C in K , by (**).) For the positive result, a direct semantic argument is given showing cp to be true in all frames in which C is true. Such arguments-and this is the point of the present paper-may be regarded as derivations of cp from C in some system of second-order logic (which may be isolated by inspecting the proof). Thus, modal incompleteness theorems become non-conservation results: cp is derivable from C in some second-order extension of K , but not in K itself. In the following section, it will be shown that K is already incomplete in this sense with respect to the weakest possible system of second-order deduction. This settles the question raised in [4].

2. Weak second-order logic

2.1 A simple system of deduction. The above truth definition for modal formulas in frames amounts to a treatment of modal formulas as second-order sentences in the following sense. Fix some correspondence between proposition letters q and unary predicate variables Q. Moreover, fix some individual variable x. Let the standard translation ST(cp) of a modal formula cp be obtained through the recursion 5 - Theoria 2: 1919


ST(q) =Qx

ST(cp-4) = ST((P)+ ST($) SmJcp) = ~y'y(Rxy-,[y/xI ST(cp)),

ST(7cp) =7 ST(cp)

where y is the first individual variable (in some fixed enumeration) which does not occur in ST(cp). Let cp contain the proposition letters p l , . . .,pn. The closed standard translation ~ T ( ( P ) of cp is

VPl . . . VP,VxST(q). Note that frames (and even general frames) may also be regarded as semantic structures for the second-order language L, with unary predicate variables and one binary, first-order predicate constant R. Clearly, E(q) is an L,-sentence. Now, it is easy to see that, for all frames 3,

Accordingly, it follows that

~kcp iff {ST(a)la~~)bST(cp). The notion of semantic consequence for L,-sentences is not axiomatizable, as was remarked in the introduction. (To see this, note, e.g., that second-order Zermelo Fraenkel set theory can be formulated in L2. This theory implies the complete theory of the standard model for arithmetic. Therefore, a routine application of Turskcs unde- finability theorem yields the desired conclusion.) Still, one may consider various reasonable (though incomplete) axiomatic theories of deduction for this language.

Consider some set of axioms complete for first-order predicate logic, say the one in [5 ] with predicate axioms

-Vx( cp --$ $) '(VXCp -Vx$) -Vxcp+[t/x]cp

able for the variable x in the formula cp) -q-'vxcp

in cp).

(provided that the term t be substitut-

(provided that x does not occur free

(Enderton allows universal quantifier prefixes, which enables him to get by with modus ponens as his single rule of inference.) Now, the


most conservative policy would seem to consist in having these same axioms for the case of predicate variables P, Q. But, a decision has to be made as regards the terms t. If these are to be just predicate variables (L , has no unary predicate constants), then the resulting axiom will be very weak, with trivial instances like

VPVx(Vy(Rxy -Py) - Px) -+ Vx(Vy(Rxy - Qy) -+ ex).

Such axioms do not suit our purposes. E.g., note that the antecedent formula is E ( O p + p ) . Now, it is a well-known fact that O p + p is true in exactly those frames whose accessibility relation is reflexive (cf. [l]), and it would be pleasant to be able to derive this in L,.

(1) Rxx+(Vy(Rxy-+Py)- Px) is easily derived in first-order logic, and hence (2) VxRxx-Vx(Vy(Rxy-Py)~Px) is. Next, apply the (derived) rule of generalization to obtain the second-order closure

Using the first and third axioms mentioned above (with “ P for “x”), the following may be derived from (3) :

In fact, one side of this equivalence is obvious:

( 3 ) VP(2) .

(4) VxRxx -+ VPVx(Vy(Rxy - Py) - Px) . (5) VPVx(Vy(Rxy 4 Py) -+ Px) +vx VP( Vy(Rxy - Py) -+ Px)

But, what about the converse implication? The only natural way to proceed is as follows.

is derivable in a routine fashion. Now, one wants to take for P: { W E WIRxw}, or-syntactically-subformulas of the form “Pu” are to be replaced by “Rxu”. This will yield (6 ) from whose consequent Rxx will follow at once (using the provability of Vy(Rxy+Rxy)). Then, obviously, some final touches will yield

VP( Vy(Rxy -+ Py ) + Px) + (Vy( Rxy + Rxy ) - Rxx), (7) v~vx(vy(Rxy-,Py)~Px)-+vxRxx.

Substitutions like the above are studied in [l], being a great help in finding first-order equivalents (in R and =) for those modal formulas which (like U p - p ) have a first-order relational equivalent on the class of all frames. (For the limitations of this substitution method, cf. also [2].)

A good formulation for our second axiom is, then,


-vpcp + [$/PI cp 2

where $ is any L,-formula without occurrences of second-order quantifiers, having some free individual variable x such that [$/P]cp arises from cp by replacing subformulas of the form “Pu” by “[u/x]$” . (Modulo some trivialities concerning freedom and bondage of variables, this formulation works as it should.) The resulting system of deduction for L, will be called weak second-order logic in this paper. Deducibility in it will be denoted by “k,”. Stronger systems may be obtained by adding further axioms, e.g., forms of the axiom of choice.

Deducibility in the minimal modal logic K implies deducibility in weak second-order logic in the following sense:

~bcp only if {ST(~)(~EC) bST(cp). (The proof of this implication consists in a simple checking of cases.) In [4], the question was raised if the converse holds as well. For, if not, then K is incomplete in the strongest sense, viz. with respect to the weakest reasonable second-order logic. This, then, is our main question:

“ls conservative over with respect to modal formulas?’

2.2 Previous examples are too complicated. Now, existing modal incompleteness proofs did not answer our question. E.g., for the example from [4] (which was mentioned in section l), it turns out that

(i) cp is not derivable from C in K , (ii) ??f(cp) is derivable from {=(o)(cr~Z} in weak

second-order logic together with some axiom of choice; but

(iii) =(q) is not derivable from this set in weak second- order logic alone.

This last fact merits special attention, because its proof is not completely obvious. ((i) is in the paper, (ii) follows from a simple inspection of the arguments given in it.)


CLAIM. p + Op does not follow from O p + p ,

Op-, oop in weak second-order logic. O(UP+04)" 0 ( 0 4 + 0 P ) , O(P+OP)+(OP+P) and

Proof. Consider the general frame (8, m ), where = ( W, R ) with - W is the set of all natural numbers, -R consists of all ordered couples ( m , n) such that

--cz13 consists of all finite and all cofinite subsets of W .

In the above-mentioned paper, it was shown that C is true in (5, m ), whereas cp is not. By (**), this established (i) above. But, a little more information may be extracted from this example.

First, note that each set X in crx\ is L,-definable in 8 in the following sense. Lo is the first-order language with identity having the binary R as its only non-logical constant. Now, an L,-formula c p x exists (with free variables x, xi, . . ., x,; say) such that, for some parameters wl , . . ., W,E W,

n>m- 1, and

X={WEW1q=cpx[w, w1, ..., % I > . In fact, the finite and the cofinite sets are the only L,-definable subsets of W! This is a purely model-theoretic observation, which may be proven as follows.

(a) Suppose some infinite set X is L,-definable in 8. whose complement W-X is infinite too. Say, Xis defined by cp(x, xi, . . .,x,) using parameters w l , . . ., w,. It is easy to see that each element of W is L,-definable. (Start with the smallest and work upwards.) There- fore, the parameters w l , . . ., w, may be removed, by means of their definitions. This procedure yields a new formula cp(x) defining X i n s . Then, both Vx3y(Rxy A cp(y)) and Vx3y(Rxy A 7cp(y)) are true in 8 (both X and W - X being cofinal in W).

(b) The frame 5 has an ,!,,-elementary extension consisting of 8 followed by a linear sequence of copies of the frame ( W', R ' ) with

- W' is the set of all integers, and -R' like above.

(This follows from a standard compactness argument.) In fact, 5 with a tail of one copy of ( W', R ' ) is an Lo-elementary extension of 8

70 3. F. A. K. VAN BENTHEM

already. (The laborious proof of this fact is omitted here.) Now, any function which leaves 5 unchanged, but moves all points in the tail over some fixed distance (to the left, or to the right) is an Lo-automorphism of the longer frame. Therefore, all elements of the tail have the same Lo-theory.

(c) Combination of (a) and (b) yields a contradiction. By (a), the tail ( W ‘ , R‘) must contain both points at which cp(x) is true and points at which it is false. But, by (b), this is impossible.

Thus, it has been shown that ca\ consists of exactly all Lo-definable subsets of W.

It follows that % is closed under Lodefinability in the following sense. Let cp be an L,-formula with free individual variables x, xl, ..., x, and free predicate variables X1, ..., X, such that cp contains no second-order quantifiers. Let wl, . . ., W,E W and A , , ..., A,€%. Then the set {we Wl8bcp[w, w, , . .., w,, A , , ..., A d } belongs to %? again (substitute the definitions for A , , . . ., A , in cp).

Here, finally, comes the connection with weak second-order logic. To see this, a suitable notion of “weak second-order consequence” has to be defined as follows.

Let CU(cp> be a set of L,-formulas, possibly containing free individual and (unary) predicate variables.

+,cp iff, for all general frames (5, ca\) satisfying (1) % is closed under Lo-definability, and

where f is an assignment of worlds in W to individual variables and sets of worlds in ca\ to (unary) predicate variables, it holds that (8, ca\ ) cp m.

(2) ($7 B) l=wl,

(Condition (1) ensures that the axiom VPcp-,[$/P]cp in its above formulation will be true in (8, %? ) under any assignment.) An easy induction on the length of derivations shows that

Applying this insight to the above-mentioned modal formulas,

SYNTACTIC ASPECTS OF MODAL INCOMPLETENESS THEOREMS 7 1

or rather, their L,-translations, at once yields the statement made in the claim. Q.E.D.

Thus, the current examples of incomplete modal logics do not answer the question of section 2.1. (Fine’s and Thomason’s examples are of a similar complexity.)

2.3 Another method? Another promising approach tries to take advantage of existing derivations in k, (which, hopefully, cannot be given in K ) . Two examples were mentioned in [4]:

(1) Up+ 0 U p is derivable from O(Op+p)+ Op (Lob’s formula; cf. [12]) by the following substitution for P in ~ ( U ( O p + p ) + O p ) :

substitute Vy(Ruy+Rxy) for Pu.

(2) (Sobocihski’s formula; cf. [ 1 11) by the substitution of

U p + 0 U p is derivable from O( O(p + U p ) + p ) + p

R X U A X # U for Pu. (In this case, one obtains as a “by-product” the formula

VxVy( Rxy+ (Ryx+x = y ) ) , i.e., anti-symmetry.)

These facts were already known to the present author some years ago. Unfortunately, they did not give rise to pleasant incompleteness results. (In fact, no “natural”-already ,existing-modal logic has been shown to be incomplete yet.) In 1974, D.H.J. de Jongh found a deduction for Op+UOp from Lob’s formula in K (it was found independently by G. Sambin) and in 1978 W. J. Blok found one for Op+O U p from Sobociriski’s formula. (The former deduction was given in [4], the latter in an appendix to that paper called “Transitivity follows from Dummett’s axiom”, which has appeared in Theoria, vol. 44 (1978), pp. 117-1 18.)

2.4 The main theorem. Both methods having failed, it remained to wait for a stroke of luck. It came when S. K. Thomason discussed an incompleteness result in [8] with the author. Thomason mentioned a trick for converting non-normal incomplete modal logics into normal ones (cf. [3]) which was, more or less, applicable to a non-normal example which the author had known for some time

72 J . F. A. K. VAN BENTHEM

without seeing how it could be useful. Here it is. Consider the modal formula

0 (0p-p) -+P? which translates into L2 as

VPVx(Vy( Rxy -+ (Vz( Ryz-t Pz) -+ Py)) -+ Px).

Even in its local form, without the quantifier Vx, it implies a contradiction in weak second-order logic, by the substitution of u # x for Pu:

(1) (2) (3) (4) ( 5 ) Rxx A (Rxx-+x#x)

Vy( Rxy -+ (Vz(Ryz+z # x) -+y # x)) -+ x # x -Vy( Rxy -+ (Vz( Ryz-, z # x) -+y # x)) ly(Rxy A Vz(Ryz-+z # X) A y = X) Rxx A Vz( Rxz -+ z # x)

(6) x#x.

But, on the other hand, a general frame (8, %? ) exists such that, for some W E W, W E ~ ( ~ ( ~ p + p ) - + p ) for all valuations v on 5 taking their values in %. It is defined as follows:

- W is the set of all natural numbers together with 00,

-R is the set of all ordered couples (m, n) (m > n) and

- %? consists of all finite subsets of W not containing co (m, W ) ( W E w), and

and all cofinite subsets of W which do contain 00.

An easy calculation shows that ca\ satisfies the required closure conditions. Moreover, co is the above-mentioned point W. For, if WE V(O(Op-+p)), then each W E W belongs to V ( 0 p - p ) . Now, start with 0: it belongs to V ( 0 p ) (having no R-successors) and hence O E V(p). But, then, 1 E V ( 0 p ) (0 being its sole R-successor), etc. Thus, all natural numbers belong to V b ) , so V(p) is cofinite and, therefore, c o ~ V(p) (since V ( ~ ) E % ) . This example shows that no contradiction is derivable from O(Op-p)+p in K minus the rule of necessitation. (The easy proof of this fact is omitted here.) But, in K , one is derivable as follows. Let I be any contradiction.


(1) O(UI-+L)-+J- (by the rule of substitution) ( 2 ) 001 (from (1) in K ) (3) O O O I (from ( 2 ) by necessitation) (4) I (from (2) and (3) in K ) .

Still, by Thomason’s trick, the above example may be exploited to yield a negative answer to our main question. Let T be any tautology.

THEOREM. Foro=OOT- ,O(O(Op~p)+p) and q = O I v 001, it holds that

(9 n c + > t K V !

(ii) {SW)I k2 W V ) .

Proof. To prove (i), consider the following general frame (5, % ). s= ( W , R ) , where -W is the set of all natural numbers together with 00,

-R is the set of all ordered couples (m, n ) O + i ,

(m, n e N ; m>n), (00, w ) (weW, w#co+l) together with (m + 1, co), and

-%is the set of all finite subsets of W not containing co and all cofinite subsets of W which do contain co.

A calculation like above shows that % has the required closure properties. Moreover, co still belongs to all sets V(O(Op+p)+p) (for valuations V on 5 taking their values in %) and hence 00 + 1 belongs to all sets V(c7(O(Op+p)-*p)). Now, since V(O0T) consists of just 00 + 1 (for any V), it follows at once that o is true in the general frame (8, % ). But, clearly, I J l v O O l is not: 00 + 14 V(cp) (for any valuation V on 3 ). This proves (i), by the equivalence (**) given in the introduction.

(ii) is proven by a simple deduction in weak second-order logic, which is almost like the one given above. E ( o ) is

V PVx(Vy( R x y -+ 3z( Ryz A T)) +

Vy(Rxy+(Vz(Ryz-,(Vu(Rzu4Pu)-+ Pz))-+Py))).

One moves the second quantifier Vy to the front (immediately after the first x) and substitutes u # y for Pu. Etc. Q.E.D.

74 J. F. A. K. VAN BENTHEM

2.5 Supplementary remarks. To conclude this section, we make a few more remarks about weak second-order logic.

First, the final observation in the proof of the claim in section 2.2 provides us with a completeness theorem for weak second-order logic:

C k2 cp if and only if C cp; for all C and cp.

(One direction follows by induction on the length of derivations in weak second-order logic, as has been noted before. The other direction follows by a Henkin type completeness proof, like the one described in [5 ] , chapter 5.)

Then, it was noted above that weak second-order logic plays a role in the determination of first-order relational equivalents for modal formulas (which have such an equivalent at all). Some relevant concepts in this connection are the following:

-M1 is the set of modal formulas cp such that, for some first-order sentence c1 in R and = , cp and a are true in exactly the same frames,

-Mpf is the set of modal formulas cp such that, for all general frames (8, %3 ) whose %3 is closed under Lo-definability, if (5,m ) cp, t hen8 cp.

It is shown in [2] that M?'f is contained in M1; by using the fact that, if cp~Mpf, then E(cp) is implied by its first-order substitution instances (with respect to its universal second-order quantifiers). Then, =(q) is already implied by a finite number of such substitution instances (thanks to compactness). Moreover, E(cp) implies all such substitution instances of itself, whence it is actually equivalent to the conjunction of the formulas just mentioned.

Yet M p f does not exhaust M1. E.g., the modal formula (0p+OlJp)~( lJOp+OOp) belongs to the latter set, but not to the former.

is a quite interesting set, because it contains all sets of first-order definable modal formulas which have been described by constructive means in the literature (cf. [lo]). Weak second- order logic provides us with a syntactic characterization of M F f .

Still,


CLAIM. c p ~ M p ~ iff there exists some first-order sentence a (in R and=) such that h F ( c p ) o a .

Proof. The direction from left to right will be obvious from a closer inspection of the above argument to the effect that, if c p ~ M f ~ ~ , then E(q) is equivalent to a conjunction of its first-order substitution instances. For the converse, suppose that h=(cp)ocr. Now, if (8, % ) is a general frame whose % is closed under Lo-definability such that (Bj % ? k cp, then-by previous observations-e(cp)oa is true in (3, %), and hence a is. It follows that a is true in 8, being a first-order sentence unaffected by% . But, then, viewing the frame 8 as the (equivalent) general frame (8, %')-where% ' is the set of all subsets of W, which is obviously closed under Lo- definability-we have that actE(cp) is true in 8 , whence =(q) is true in B as well. Q.E.D.

Unfortunately, no pleasant connection exists between first-order definability of modal formulas cp and the more traditional property of completeness (in the sense that {cp}F,$ iff cpk$, for all modal formulas $). There are complete modal formulas, like Liib's formula, which are not first-order definable (cf. [I]). On the other hand, there are first-order definable modal formulas (in Ml), like the conjunction of the set C presented in the introduction-which is equivalent to VxVy(Rxyox=y)-which are not complete. For a while, the author hoped that, at least, the following would hold:

each formula in M;'" is complete;

M p f being so much better-behaved than M1 in many respects. But, again, the above theorem constitutes a counter-example. For, the modal formula i-JOT+O(O(!Jp+p)-+p) belongs to M p f (as is clear from the above proof, it is provably equivalent to a first-order sentence-viz., the translation of 00 I v 0 i-in weak second-order logic) without being complete.

3. Conclusion

Given the fact that deducibility in K captures so little of the strength of modal consequence (k), it becomes of interest to study modal

76 J. F. A. K . VAN BENTHEM

logics with different underlying systems of deduction. Especially for the case of b, some natural questions arise. Which modal logics are complete with respect to it? There will be more of these than just the ones which were complete already in the sense of section 2, witness the logic mentioned in our theorem. Thus, more logics become complete, and more comprehensive general completeness theorems might become available than the traditional ones like Bull’s theorem on the extensions of S4.3 (cf. [7]). This, then, is the first kind of question one may ask: given a notion of second-order deducibility (stronger than K ) , to study (in) completeness of modal logics with respect to it.

Another approach starts from a given class R of frames and asks if the notion R of ‘%-consequence” defined by

c kacp fl, for ail frames 5 in R , 5 c only if 8 q, is axiomatizable; and, if so, by which L,-theory. No such result holds for the class of all frames, by S. K. Thomason’s theorem referred to above. On the other hand, Bull’s theorem implies that, if R is the class of all reflexive linear orderings, K axiomatizes FS? already. What about, e.g., the class of all partial orderings?

Then, there are more connections between first-order definability and complete axiomatizability. E.g., if cp is a first-order definable modal formula, then {$lcpk $} is recursively axiomatizable. For, cpk$ iff m(cp)/=D($) iff ak??($) (where a is any first-order equivalent of cp) iff a FST($) (because no second-order variables occur in a), and the latter notion is recursively axiomatizable (being just ordinary first-order consequence). But, as we have seen above, {$Icpk$} need not be axiomatized by cp (and K ) . Is there any system of deduction X such that, if cp~M1, then {$lcpk$}=

These are just a few examples of the host of new problems which arise when modal logic is viewed from the perspective of arbitrary L,-theories of deduction, instead of the old (and weak) minimal modal logic K.

= {$lcpt-X$l?


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syntactic aspects of modal incompleteness theorems

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