horacio costa - first order classical modal logic

8/12/2019 Horacio Costa - First Order Classical Modal Logic

1/39

Horacio Arlo-Costa

Eric Pacuit

First-Order Classical

Modal Logic

Abstract. The paper focuses on extending to the first order case the semantical pro-

gram for modalities first introduced by Dana Scott and Richard Montague. We focus on

the study of neighborhood frames with constant domains and we offer a series of new

completeness results for salient classical systems of first order modal logic. Among other

results we show that it is possible to prove strong completeness results for normal sys-

tems without the Barcan Formula (like FOL + K) in terms of neighborhood frames with

constantdomains. The first order models we present permit the study of many epistemic

modalities recently proposed in computer science as well as the development of adequate

models for monadic operators of high probability. Models of this type are either difficult

of impossible to build in terms of relational Kripkean semantics.

We conclude by introducing generalfirst order neighborhood frames and we offer a

general completeness result in terms of them which circumvents some well-known prob-lems of propositional and first order neighborhood semantics (mainly the fact that many

classical modal logics are incomplete with respect to an unmodified version of neighbor-

hood frames). We argue that the semantical program that thus arises surpasses b oth in

expressivity and adequacy the standard Kripkean approach, even when it comes to the

study of first order normal systems.

Keywords: First-Order Modal Logic, Neighborhood Semantics, General Frames

1. Introduction

Dana Scott and Richard Montague proposed in 1970 (independently, in [45]

and [42] respectively) a new semantic framework for the study of modalities,which today tends to be known as neighborhood semantics.

A neighborhood frame is a pair W, N, where W is a set of states,

or worlds, and N : W 22W

is a neighborhood function which associatesa set of neighborhoods with each world. The tuple F, V, where F is aneighborhood frame and V a valuation is a neighborhood model. A modalnecessity operator is interpreted in this context as follows: M, w |= iff()M N(w), where ()M is the truth-set corresponding to in the givenmodel.

Without imposing specific restrictions on the neighborhood function itis clear that many important principles of normal or Kripkean modal logics

Presented by Name of Editor; Received

Studia Logica (2001) 68: 136 cSpringer 2001


2/39

2 Horacio Arlo-Costa and Eric Pacuit

will not hold in a neighborhood model. At the same time it is possible toshow that there is a class of neighborhood models, the so-called augmentedmodels (see the definition below), which are elementary equivalent to the

relational models for normal modal systems of propositional modal logic.So, the program of neighborhood semantics has normally been consideredas a generalization of Kripke semantics, which permits the study of classicalsystems that fail to be normal.

Early on (in 1971) Krister Segerbeg wrote an essay [47] presenting somebasic results about neighborhood models and the classical systems that cor-respond to them and later on Brian Chellas incorporated these and othersalient results in part III of his textbook [16]. Nevertheless for more than 15years or so after 1971, in the apparent absence of applications or in the ab-sence of guiding intuitions concerning the role of neighborhoods, non-normalclassical modal logics were studied mainly in view of their intrinsic mathe-matical interest. This situation has changed in important ways during thelast 18 or so years. In fact, many of the normal axioms, like the additiv-ity principle, establishing the distribution of the box over conjunction, havebeen found problematic in many applications. As a result many formalismsproposed to retain:

(M) ( ) ( )

(N)

while abandoning:

(C) ( ) ( )

Many recently explored, and independently motivated formalisms, haveabandoned the full force of additivity while retaining monotony (M). Exam-ples are Concurrent Propositional Dynamic Logic [30], Parikhs Game Logic[43], Paulys Coalition Logic [44] and Alternating-Time Temporal Logic [1].Moreover recent research [6] has shown that a large family of Non-Adjunctivelogics, previously studied only syntactically or via a variety of idiosyncraticextensions of Kripke semantics can b e parametrically classified neatly asmembers of a hierarchy of monotonic classical logics, all of which admit

Of course there is a relatively long history of arguments favoring the abandonment

of (C). For example Ruth Barcan Marcus proposed the abandonment of (C) which shecalled the factoring principle both for deontic and for (some) epistemic interpretationsof the box.


3/39

First-Order Classical Modal Logic 3

clear and simple neighborhood models. A salient member of this family isthe logic of monadic operators of high probability studied via neighborhoodsemantics in [36] and [5]. This is a clear case where the intended interpre-

tation of neighborhoods is quite intuitive: the neighborhoods of a point arethe propositions receiving probability higher than a fixed threshold.

More generally one can see the neighborhoods as having an epistemicrole (as suggested in [5]), namely representing the knowledge, belief or cer-tain other attitude of an agent at a certain point. This strategy permits thedevelopment of elegant and economic models of attitudes that can only bemodeled alternatively either via the abandonment of the axiom of foundationin set theory (or via co-algebras) or by postulating a large array of primi-tive epistemic states (devoid of propositional representation). The comingsubsection elaborates on this issue, which ultimately is concerned with theway in which possible worlds are conceptualized and concretely encoded inmodels of modalities.

Possible worlds and modalities

Moshe Vardi considered in [49] the use of neighborhood models in order torepresent failures of logical omniscience, high probability operators as well aslogics of knowledge, time and computation. Nevertheless, after consideringthe wider class of classical modal systems(encompassing both normal andnon-normal modal systems) and their neighborhood models, Vardi discardedthem without exploring them logically. Vardi gave two reasons for not uti-lizing neighborhood models for studying classical modalities (which he dubsintensional logic following Montagues terminology). The central reason is

that this approach leaves the notion of a possible world as a primitive notion[...]. While this might be seen as an advantage by the logician whose interestis in epistemic logic, it is a disadvantage for the userof epistemic logic whoseinterest is mostly in using the framework to model belief states (page 297).Vardi proceeds instead to establish that: a world consists of a truth assign-ment to the atomic propositions and a collection of sets of worlds. This is, ofcourse, a circular definition.... Barwise and Moss [12] showed how to makethis strategy coherent by abandoning the axiom of foundation in set theory.

Most of the mainstream work in models of modalities has been usuallydone by utilizing a space of possible worlds, which are, in turn, understoodas unstructured primitives. This strategy is abandoned in Vardis proposal.

Vardi seems to appreciate that the contents of a neighborhood N(w) canbe seen pre-systematically (for applications in epistemic logic, for example)as the propositional representation of the epistemic state of a given agent


4/39


at world w. But Vardi wants to have as well a notion of p ossible worldincluding as part of it the representation of the epistemic state of the agent(or interacting agents). This can only be done by abandoning the idea of

w as an unstructured primitive point. Therefore he proposes seeing w as astructured entity with an epistemic component N(w). As Vardi explains thisstrategy leads to circularity, which only ceases to be vicious in the contextof an underlying set theory without the axiom of foundation.

There is yet a different way of facing this problem [21]. The idea is toassume thatW O S1 ... Sn whereO is the set ofobjectivestates andSi is a set of subjectivestates for agent i. Therefore worlds have the form(o, s1,...,sn). In multi-agent systemso is called the environment stateandeach si is called a localstate for the agent in question.

Halpern [31] characterizes an agents subjective state s i by saying that itrepresents is perception of the world and everything else about the agentsmakeup that determines the agents reports. To avoid circularity it is quitecrucial that both environment states and the local states of agents are nowunstructured primitives. It is unclear the extent to which this strategy isconducive to concrete representational or logical advantages. It should b eclear, on the other hand, that neighborhoods have equal representationalpower, while reducing the set of needed primitives. In fact, one can havefor each worldw a set of neighborhood functions Ni(w) (with i ranging be-tween 1 and n), where each neighborhood mimics the semantic behavior ofthe local state si. In fact, for each modality Mi and each structured point(o, s1,...,sn), such that MiA is satisfied at w, we can have a correspond-ing unstructured pointw in a corresponding neighborhood model where theproposition expressed by A is in Ni(w). This is so even if there are non-

propositional elements constitutive of relevant aspects of the agents makeup determining the agents reports. As long as these reports are proposi-tional the neighborhoods can encode the information needed to representthe reports in question.

A concrete application where worlds or statesare understood along thethick lines just sketched is concerned with the pioneer work of J. Harsanyidevoted to model games of incomplete information played by Bayesian play-ers [33]. In this case we have a set ofexternal statesSand astate of the worldin a type spaceT ofSis an (n+1)-tuple (s, t1,...,tn) ST1 ...Tn, wherefor each individuali, Ti is a finite set of types. Intuitively the (n + 1)-tuple(s, t1,...,tn) specifies the relevant external state and the epistemic types for

each agent, where each epistemic type corresponds, in turn, with an infi-nite hierarchy of (probabilistic) b eliefs. Recent work has appealed to modallogic in order to formalize (hierarchies of) probabilistic beliefs of this kind


5/39


via modal operators with the intended interpretation: iassigns probabilityat least p to ... (see, for example, the review of recent work presented in[13]). As we will make clear below (via the consideration of various prob-

abilistic applications) the use of neighborhood models off

ers a perfect toolfor the study of type spaces of this kind where external and epistemic statescan be neatly separated. Although applications of this sort already exist(see, for example, the relevant chapters in [7]) the first order case is seldomstudied. Extensions of these type in models with constant domains (whichare the ones considered and tacitly recommended in [21]) are neverthelesshard or impossible to study by appealing the relational models where worldsencoded as (n + 1) tuples. We will show below that models of this type are,in contrast, rather natural if one uses neighborhoods.

Models for first order modalities

Unfortunately the recent interest in articulating applications for neighbor-hood semantics has not motivated yet the systematic study of first orderclassical logics and first order neighborhood models. One of the first sourcesof insight in this area can be found in the book published by Dov Gabbay in1976 where a variation of neighborhood semantics is systematically used [23].More recently one of us (Arlo-Costa) presented in [5] preliminary results inthis area showing that the role of the Barcan schemas in this context isquite different from the corresponding role of these schemas in the relationalcase. In fact, the use of neighborhood semantics permits the developmentof models with constant domains where neither the Barcan (BF) nor theConverse Barcan formulas (CBF) are valid. Moreover [5] provides necessaryand sufficient conditions for the validity of BF and CBF.

The recent foundational debates in the area of quantified modal logicoppose, on the one hand, the so called possibilists who advocate the use ofquantifiers ranging over a fixed domain of possible individuals, and on theother hand, the actualists who prefer models where the assumption of theconstancy of domains is abandoned. A salient feature of the standard firstorder models of modalities is that for those models the constancy of domainsrequires the validity of both the BF and the CBF (see [22] for a nice proof ofthis fact). Many philosophers have seen the possibilist approach as the onlyone tenable (see for example, [18], [41], [50]), and as a matter of fact the

These schemas are presented below in Definition 3.1. The schemas are named after the

logician and philosopher Ruth Barcan Marcus who proposed them in [8] and [9]. Paperswritten before 1950 are usually referenced under Barcan, while papers after 1950 underMarcus.


6/39


possibilist approach is the one that seems natural in many of the epistemicand computational applications that characterize the wave of recent researchin modal logic (see, for example, the brief section devoted to this issue in

[21]). Nevertheless, while the possibilist approach seems reasonable on itsown, the logical systems that adopt the Barcan Formulas and predicate logicrules for the quantifiers might be seen as too strong for many applications.The problem is that in relational models one cannot have one without theother. This has motivated some authors to adopt more radical approachesand to construct, for example, models with individual concepts (functionsfrom possible worlds to the domain of objects). The approach providedby Garson [25] in particular is quite ingenious although it seems limited tofirst order extensions of K = EMNC (and the use of individual conceptsdoes not seem immediately motivated in some simple applications consideredhere, like the logic of high probability).

Notice, for example, that (as indicated in [5]) if the box operator is un-derstood as a monadic operator of high probability the BF can be interpretedas saying that if each individual ticket of a lottery is a loser then all ticketsare losers. While the CBF seems to make sense as a constraint on an op-erator of high probability the BF seems unreasonably strong. At the sametime nothing indicates that a possibilitic interpretation of the quantifiersshould be abandoned for representing a monadic operator of high probabil-ity. On the contrary the possibilistic approach seems rather natural for thisapplication. It is therefore comforting that one can easily develop first or-der neighborhood semantics with constant domains where this asymmetry isneatly captured (i.e. where the CBF is validated but the BF is not). In par-ticular we argue that the non-nested fragment of the system FOL+EMNisadequate for representing first order monadic operators of high probability.

In this article we present a series of representation results that intendto give a preliminary insight on the scope and interest of first order neigh-borhood semantics. We will not limit our study to the analysis of non-normal systems. On the contrary one of our main results shows that astrong completeness result in terms of first order neighborhood models withconstant domainscan be offered for the normal system FOL+K. Relationalmodels (with constant domains) cannot characterize syntactic derivability inFOL +K. The problem motivating this gap is that the CBF is syntacti-cally derivable from FOL+ K but the BF is not. So, one needs to haverelational models with varying domains in order to characterize the system

E is the weakest system of classical logic.


7/39


in question. But, again, there are many interesting applications, rangingfrom the modeling of contextual modals in linguistics [38] to the logic offinitely additive conditional probability where the use of varying domains is

not immediately motivated and where the asymmetry between the CBF (asvalid) and the BF (as invalid) holds.

Neighborhood semantics is usually considered as a mild extension ofstandard Kripke semantics in part because it is usually assumed that theaugmented neighborhood models are mappable to the standard models ofnormal systems, and because it is usually assumed as well that these stan-dard models provide the models one needs in order to characterize the normalsystems. We claim here that the latter assumption is dubious at best. Infact, we show that there are indeed Scott-Montague models of importantnormal systems constructible within a possibilistic approach and withoutstandard counterparts.

Martin Gerson [26] has argued convincingly that an unmodified versionof the neighborhood approach has some of the same problems regardingincompleteness than the standard approach. He showed this by proving in-completeness with respect to neighborhood semantics of two normal systems,one between T and S4 and another which is an extension of S4. We adopt asimilar remedy concerning these problems than the one adopted in [15] and[32], namely the adoption ofgeneral first order neighborhood frames. There-fore we introduce general frames for first order modalities (not previouslystudied in the literature) by utilizing algebraic techniques reminiscent to theones employed in the algebraic study of first order logic with operators. Wethen present a general completeness result covering the entire family of firstorder classical modal logics. We conclude by discussing some examples and

suggesting topics for future research.

2. Classical systems of propositional modal logic

This section reviews some basic results about classical systems of proposi-tional modal logic. The reader is referred to the textbook [16] for a completediscussion.

Let 0 be a countable set of propositional variables. LetL(0), be thestandard propositional modal language. That is L(0) iff has one ofthe following syntactic form,

One needs to use, for example, the models with undefined formulas presented in [34],page 277-280; which are equivalent to the models TK-models used by Giovanna Corsi in[17], page 1478.


8/39


:= p | | |

wherep 0. Use the standard definitions for the propositional connectives

, and and the modal operator . The standard propositional languagemay be denoted L when 0 is understood.

Definition 2.1. A neighborhood frame is a pair W, N, where W is a

set of states, or worlds, and N :W 22W

is a function.

Given a neighborhood frame, F = W, N, the function N is called aneighborhood function. The intuition is that at each state w W,N(w)is the set of propositions, i.e. set of sets of states, that are either necessaryor known or believed, etc. at state w.

Definition 2.2. Given a neighborhood frame F = W, N, a classicalmodel based on F is a tuple F, V, where V : 0 2

W is a valuation

function.

Given a classical model M = W,N,V, truth is defined as follows, letw Wbe any state:

1. M, w|=p iffw V(p) where p 0

2. M, w|= iffM, w|=

3. M, w|= iffM, w|= andM, w|=

4. M, w|= iff()M N(w)

where ()M W is the set of all states in which is true. The dual of the

modal operator , denoted , will be treated as a primitive symbol. Thedefinition of truth for is

M, w|= iffW ()M N(w)

It is easy to see that given this definition of truth, the axiom scheme is valid in any neighborhood frame. Thus, in the presence ofthe Eaxiom scheme (see below) and a rule allowing substitution of equiv-alent formulas (which can be proven using the RE rule given below), wecan treat as a defined symbol. As a consequence, in what follows we willnot provide separate definitions for the operator since they can be easilyderived. We say is valid in M iffM, w|= for each w W. We say that

is valid in a neighborhood frame F iff is valid in all models based onF.

The following axiom schemes and rules have been widely discussed.


9/39


P C Any axiomatization of propositional calculus

E

M ( ) ( )

C ( ) ( )

N

RE

M P

Let E be the smallest set of formulas closed under instances ofPC ,E and

the rulesREand M P. Eis the smallest classical modal logic. The logicECextends E by adding the axiom scheme C. Similarly for EM, EN, ECM,and EMCN. It is well known that the logic EMCN is equivalent to thenormal modal logic K(see [16] page 237). Let S be any of the above logics,we write S if S.

Let N be a neighborhood function, w W be an arbitrary state, andX, Y W be arbitrary subsets.

(m) IfX Y N(w), then XN(w) and Y N(w)

(c) IfXN(w) and Y N(w), then X Y N(w)

(n) W N(w)

It is easy to show (see [16] page. 215) that (m) is equivalent to

(m) IfX N(w) and XY, then Y N(w)

We say that a neighborhood function N is supplemented, closed un-der intersection, or contains the unit if it satisfies (m) (equivalently ifit satisfies (m)), (c) and (n) respectively.

Definition 2.3. A frame W, N is augmented ifN is supplemented andfor each w W,

N(w) N(w)


10/39


Call a frame supplemented if its neighborhood function is supplemented,similarly for the other semantic properties above. It is well-known that thelogic Eis sound and complete with respect to the class of all neighborhood

frames. The other semantic conditions correspond to the obvious syntacticcounterparts. For example, the logic EMC is sound and complete withrespect to the class of all frames that are supplemented and closed underintersection. The completeness proofs are straightforward and are discussedin [16]. One final note about the propositional case will be important for thispaper. The class of augmented frames is equivalent to the class of Kripkeframes in the following sense.

Theorem 2.4 ([16] page 221). For every Kripke model W,R,V, there isan pointwise equivalent classical modelW,N,V, and vice versa

The proof can be found in [16] page 221. We only sketch the main points.

LetMK

=W,R,Vbe any Kripke model. To define a pointwise equivalentclassical modelM = W,N,V, we stipulate that for each X Wand eachw W,

XN(w) iff{w |wRw } X

It is easy to see thatN is augmented and that M is pointwise equivalent toMK. To define MK from a classical model M = W,R,V, stipulate thatfor each w W,

wRw iffw

N(w)

Again, it is easy to see that M is pointwise equivalent to MK.

3. Classical systems of first order modal logic

The language of first order modal logic is defined as follows. Let V be acountable collection of individual variables. For each natural numbern 1,there is a (countable) set ofn-place predicate symbols. These will be denotedby F, G, . . .. In general, we will not write the arity of a predicate F. Aformula offirst order modal logicwill have the following syntactic form

:= F(x1, . . . , xn)| | | | x

LetL1 be the set of well-formed first order modal formulas. The other stan-dard Boolean connectives, the diamond modal operator and the existentialquantifier are defined as usual. The usual rules about free variables apply.


11/39


We write (x) when x (possibly) occurs free in . Denote by [y/x], inwhich free variable x is replaced with free variable y. The following axiomsare taken from [34]. Let S be any classical propositional modal logic, let

FOL + S be the set of formulas closed under the following rules and axiomschemes:

S All axiom schemes and rules from S.

x(x) [y/x] is an axiom scheme.

Gen x

, where x is not free in .

For example, FOL+ E contains the axiom scheme PC,E, and therules Gen,MP. Given any classical propositional modal logic S, we writeFOL+S if FOL + S (equivalently can be derived using the above

axiom schemes and rules).Notice that in the above axiom system there is no essential interaction

between the modal operators and the first-order quantifiers. Two of the mostwidely discussed axiom schemes that allow interaction between the modaloperators and the first-order quantifiers are the so-called Barcan formulaand the converse Barcan formula.

Definition 3.1. Any formula of the form

x(x) x(x)

will be called a Barcan formula (BF). The converse Barcan formula

(CB F) will be any formula of the form

x(x) x(x)

Technically, the Barcan and converse Barcan formulas are schemes not for-mulas, but we will follow standard terminology. For simplicity, we will writeS + BFfor the logic that includes all axiom schemes and rules ofFOL + Splus the Barcan formula B F. Similarly for S + CBF.

Definition 3.2. A constant domain neighborhood frame for classicalfirst-order modal logic is a tuple W,N,D, where W is a set of possibleworlds, N is a neighborhood function and D is any non-empty set, called

the domain.According to the notation used in [34], which we are following here, this axiom followsfrom two additional principles called the principles ofreplacementand agreement. See [34]page 241. These principles guarantee that y is free for x occurring in (x).


12/39


Definition 3.3. A constant domain neighborhood model based on aframe F=W,N,D is a tupleM= W,N,D,I, whereIis a classical first-order interpretation function. Formally, for each n-ary predicate symbol F,

I(F, w) Dn

.

A substitution is any function : V D. A substitution is said tobe an x-variant of if (y) = (y) for all variable y except possibly x,this will be denoted by x

. Truth is defined at a state relative to asubstitution. LetM= W,N,D,I be any constant domain neighborhoodmodel and any substitution.

1. M, w|= F(x1, . . . , xn) iff(x1), . . . , (xn) I(F, w) for each n-placepredicate symbol F.

2. M, w|= iffM, w|=

3. M, w|= iffM, w|= and M, w|=

4. M, w|= iff()M, N(w)

5. M, w|= x(x) ifffor each x-variant ,M, w|= (x)

where ()M, W is the set of states w W such thatM, w|= .

Before looking at completeness for classical systems of first-order modallogic, we survey the situation with respect to first order relational structures

(deriving from seminal work by Saul Kripke). A relational frame is a tupleW, R, where Wis a set of states and R WWis an accessibility relation.A (constant domain) first order relational model based on a relational frameF = W, R is a tuple W,R,D,I where D is a set and I is a first-orderclassical interpretation (defined above).

Truth is defined as above except for the modal case:

M, w|= ifffor each w W, ifwRw thenM, w |=

The following observations are well-known and easily checked (see [34] page245).

Observation 3.4 ([34]). The converse Barcan formula is provable in thelogicFOL + K


13/39


Observation 3.5 ([34]). The Barcan formula is valid in all first order re-lational models with constant domains.

Since the weakest propositional modal logic sound and complete for allrelational frames is K, we will focus on the logic FOL + K. Given Obser-vation 3.5, if we want a completeness theorem with respect to all constantdomain relational structures, we need to consider the logic K + BF. Sound-ness is shown via the following theorem (Corollary 13.3, page 249 in [34]).Given any Kripke frame F, we say that F is a frame for a logic S iffeverytheorem ofS is valid on F.

In order to make this paper self-contained, we now review some well-known techniques and results concerning first-order normal modal logics.The reader already familiar with such results may want to skip to Section3. Recall, that a normal modal logic is a propositional logic that containsat least the axiom scheme Kand the rule N ec.

Theorem 3.6 ([34] page 249). Let S be any propositional normal modallogic, then a Kripke frameF is a frame for S iffF is a frame forS + BF.

The proof of the only if direction is a straightforward induction on aderivation in the logic S + BF. As for the if direction, the main idea is thatgiven Fwhich is not a frame for S, one can construct a frame F that isnot a frame for S + BF.

Essentially this theorem shows that the notion of a frame for first orderKripke models is independentof the domain D. That is, the proof of theabove theorem goes through whateverD may be.

We need some more definitions before proving a completeness theorem.Let of formulas of first order modal logic.

Definition 3.7. A set has the property ifffor each formula and each variable x, there is some variable y, called the witness, such that[y/x] x(x) .

The proof of the following Lindenbaum-like Lemma is a straightforwardand can be found in [34] page 258

There are, of course, many different ways in which a relational semantics can b epresented and not all are inspired by Kripkes early work. For example, Barcan Marcusoffered in [10] a model theoretic semantics with constant domains where b oth Barcan

schemas are valid.The terminology follows the one used in [34]. The property is also known as Henkins

property.


14/39


Lemma 3.8. If X is a consistent set of formulas of L1, then there is aconsistent set of formulas Y of L+1 with the -property such that X Y,whereL+1 is the languageL1 with countably manynew variables.

We can now define the canonical model for first order Kripke structures.The canonical model,MC=WC, RC, DC, IC, is defined as follows. Giventhe first- order modal language L1, let L

+1 be the extension ofL1 used in

Lemma 3.8 and V+ the variables in this extended language. Finally letMAX() indicate that the set is a maximally consistent set of formulasofL+1.

WC={| M AX() and has the -property}

RC iff{ | }

DC=V+

x1, . . . , xn IC(, ) iff(x1, . . . , xn)

Let be the canonical substitution: for each x V+, (x) = x. Thetruth Lemma is

Lemma 3.9. For any WC, and any formula L+,

iffMC, |=

Of course, the proof is by induction on the formula . The base case

and propositional connectives are as usual. The if direction for the modaland quantifier case is straightforward. We will discuss two cases. Supposethat x(x) . Then since is maximal, x(x) , and so by the-property, there is some variable y V+ such that [y/x] , and so[y/x] . Thus by the induction hypothesis, MC, |= [y/x], where is the x-variant of in which (x) = y. Hence, MC, |= x(x).The modal case relies on the following Lemma, which requires the Barcanformula in its proof.

Lemma3.10. If is a maximally consistent set of formulas (ofL +) with the-property and is a formula such that , then there is a consistentset of formulas ofL+ with the-property such that{| }{} .

Given this Lemma, the proof of the truth Lemma above is straightfor-ward.


15/39


3.1. First-order neighborhood frames

As with Kripke frames, validity of certain axiom schemes in a frame cor-responds to properties on that frame. In this section, we discuss the con-

nections between validity of formulas and properties of the correspondingframe. We first need some notation. Recall that a neighborhood frame is atuple W, N where Wis a set of states and N :W 22

W

is the neighbor-hood function. A first-order constant domain neighborhood frame is a tripleW,N,D where W and N are as above and D is an arbitrary non-emptyset called the domain. By a frame, we mean either a neighborhood frame ora first-order neighborhood frame.

A filter is any collection of sets that is closed under finite intersectionsand supersets. A filter is non-trivial if it does not contain the empty set.We say that a frame F is a filter if for each w W, N(w) is a filter. Aframe F is closed under infinite intersections if for each w W, N(w) is

closed under infinite intersections. Finally, a frame is augmented (recallDefinition 2.3) ifN is supplemented and for each w W,

N(w) N(w).

Obviously, every augmented frame is closed under infinite intersections, butthere are supplemented frames closed under infinite intersections that arenot augmented (provided that W is infinite).

In [5] it is shown that the presence of the Barcan and the ConverseBarcan formulas implies interesting properties on the corresponding frame.Before reporting these results, we need some definitions.

Definition 3.11. A frame F is consistent iff for each w W, N(w)=and{}N(w).

Definition

3.12.

A first-order neighborhood frameF=W,N,Dis non-trivial iff|D|> 1

First of all, a couple of facts about frames which are trivial or not con-sistent. First of all, if F is a trivial first-order neighborhood model withconstant domain, then x(x) x(x) is valid. Thus both the Barcanand converse Barcan formulas are valid on trivial domains. Secondly ifF isnot consistent, then for trivial reasons both the Barcan and converse Barcanare valid.

Observation 3.13 ([5]). LetF be a consistent constant domain neighbor-hood frame. The converse Barcan formula is valid onF iffeitherFis trivialorF is supplemented.

Proof. Suppose that Fis a consistent constant domain first-order neigh-borhood model. IfF is trivial, then, as noted above, the converse Barcan


16/39


formula is valid. Suppose that F is supplemented and that M is an arbi-trary model based on F. Let w be an arbitrary state, we will show thatM, w|=x(x) x(x). Suppose thatM, w|= x(x). Therefore

(x(x))M,

N(w).Since (x(x))M, =

x((x))M,

and N(w) is supplemented, for

each which is an x-variant of , ((x))M,

N(w). But this impliesM, w|= x(x). Hence the right to left implication is proven.

For the other direction, we must show that if F is a consistent, non-trivial and not supplemented, then the converse Barcan formula is not valid.SinceFis not supplemented, there is some state w and sets X and Y suchthat X Y N(w) but either X N(w) or Y N(w). We need onlyconstruct a model M in which an instance of the converse Barcan formulais not true. Let Fbe a unary predicate symbol. We will construct a modelM based on F where M, w |= xF(x) but M, w |= xF(x). WLOG

assume thatXN(w).Then we have two cases: 1. Y Xand 2. Y X. Suppose we are inthe first case. That is Y X, then Y = X Y N(w). Suppose that 1.for each v X Y and for all d D , d I(F, v) and 2. I(F, v ) = forallv W X.

Then for any substitution, (xF(x))M, =

x(F(x))M,

=XY

N(w). Hence M, w|= xF(x).Now to complete the description of the model, choose an element a

D and set a I(F, y) for each y X(XY) and such that for noother b D, b I(F, y). Then if (x) = a, (F(x))M, = X N(w).Hence, M, w|=xF(x). The non-triviality ofFis needed to ensure that(F(x))M, = (xF(x))M,. As for case 2 (Y X), we can require thatI(F, v) = for all v Y (X Y), reducing to the first case.

Observation 3.14. FOL + EM C BF

We now turn our attention to BF. In [5] it is shown that the Barcanformula corresponds to interesting properties on first-order constant domainneighborhood frames. We first need some notation. Let be a cardinal. Wesay that a frame closed under intersections if for each state w and eachcollection of sets {Xi | i I} where|I| ,iIXi N(w).

Definition 3.15. A consistent first-order neighborhood frame W,N,D ismonotoniciff the number of objects in its domain is as large as the numberof sets in the neighborhood with the largest number of sets in the frame.


17/39


[5] shows the following result about monotonic frames: Let Fbe a con-stant, consistent and monotonic domain neighborhood frame.

x(x) x(x)

is valid on every model based on Fif and only ifFis either trivial or closedunder infinite intersections. Monotony cannot be dropped from an unrevisedformulation of this result. In fact, consider the following counterexample:

Counterexample Let the neighborhoods ofF contain exactly aninfinite (but countable) family of sets X = Xi with i= 1, 2,.... Thedomain of worlds is also infinite (but countable). Let F be closedunder finite intersections but not under infinite intersections.

Let now the domain of objects contain exactly two objects a, b. Itis clear that F is non-trivial (its cardinality is strictly greater than1). Therefore if the revised formulation of the result from [5] (where

monotony is dropped) were true this entails that there is a modelM based on F, world w and substitution , such that: M, w |=x(x); butM, w|= x(x).

Since we have M, w|=x(x) we know that for each x-variant

of ,

M, w|= (x);

Given the constitution of the domain for each substitution there isonly onex-variant of it, which we will cal . Therefore we have thatboth (x(x))M, and (x(x))M,

are inN(w). SinceM based on

F, and Fin closed under finite intersections then we have that:

x((x))M, N(w).

contradicting the assumption that M, w|= x(x)

Nevertheless the example shows that there are non-monotonic frames (notclosed under infinite intersections) where the Barcan schema is validated.Nevertheless one can adapt the method of proof used in [5] to prove the

following result giving necessary and sufficient conditions for the frame va-lidity of the Barcan formula (where the cardinality restriction imposed bymonotony is dropped):


18/39


Observation 3.16. Let F be a consistent constant domain neighborhoodframe. The Barcan formula is valid onF iffeither 1. Fis trivial or 2. ifDis finite, thenFis closed under finite intersections and ifD is infinite and

of cardinality , thenFis closed under intersections.

Proof. Suppose that Fis a consistent constant domain first-order neigh-borhood model. If F is trivial, then, as noted above, the Barcan for-mula is valid. Suppose condition 2. holds and let M = W,N,D,Ibe any model based on F. Given any state w W and substitution, we must show that M, w |= x(x) x(x). Suppose thatM, w |= x(x). IfD is finite, then {((x))

M, | x } is finite andsince ((x))M,

N(w) for each x and N is closed under finite in-

tersections

{((x)M,

| x } N(w). Therefore,M, w|= x(x).The proof is similar ifD is is infinite.

For the other direction suppose that F=W,N,Dis non-trivial. Since

D is nontrivial it contains at least two distinct elements, say d, c D suchthatd=c. Suppose thatD is finite andFis not closed under finite intersec-tions. SinceFis not closed under finite interesections there is a state w suchthatN(w) is not closed under finite intersections. This means that there aretwo sets X, Y such that X, Y N(w) butX Y N(w). To see this, let nbe the size of the smallest collection of sets C such that each element ofCis in N(w) butCN(w). Ifn= 2 we are done. Otherwise, suppose thatn >2. In this case we can partition C into two subclasses, C1 and C2 suchthat each element ofCi is inN(w) andCi N(w) (fori = 1, 2). Note thatC1 C2 N(w) by assumption. But then{C1, C2} is a collection ofsize two both of whose elements are in N(w) and whose intersection is not

in N(w). This contradicts the assumption thatn > 2. Thus there are twosetsX and Y such that X Y N(w).

Given X, Y N(w) such that XY N(w) we construct a modelbased onFthat invalidates the Barcan formula. LetFbe a unary predicatesymbol. The idea is to define I such that (F(x))M, = X if (x) = cand (F(x))M, = Y if (x) = d. Hence for each v X, let c I(F, v).Then if (x) = c, (F(x))M, = X. For each v Y, let d I(F, v). Thenif (x) = d, (F(x))M, = Y. For the other elements c of the domain (ifthey exist), fix a set, say Y, and let c I(F, v) for each v Y. Let be any arbitrary assignment. We have M, w |= xF(x). However,(xF(x))M, = XY N(w). Hence, M, w |= xF(x) and so the

Barcan formula is not valid.Suppose that D is infinite and of cardinality . Since F is not closed

under intersections there is a statew and a collection {Xi| i I}where


19/39


|I| such that

iIXi N(w). Since |I| |D| there is a 1-1 functionf :I D . Thus we can find for eachXi a unique c D , call it cXi . Theargument is similar to the above argument. Let Fbe a unary predicate. For

each Xi, defineF(x) so thatF(x)M,

=Xiprovided(x) =cXi

. That is foreach Xi, for each v Xi, let cXi I(F, v). For the other elements c of the

domain (if they exist), fix any set, sayXj, and letc I(F, v) for eachv Xj.Now for any , F(x)M, = Xi for some i I, hence (F(x))

M, N(w).However (xF(x))M, =

x

F(x)M,

=

iIXi N(w). Hence theBarcan formula is not valid.

3.2. Completeness of classical systems of first-order modal logic

In this section we discuss the completeness of various classical systems offirst-order modal logic. We start by defining the smallest canonical model

for classical first-order modal logic. Let be any first-order classical modallogic. Define M = W, N, D, I as follows. LetM AX() indicatethat the set is a -maximally consistent set of formulas ofL +1.

W= {| M AX() and has the -property}

XN() ifffor some , X={ W | }

D = V+

x1, . . . , xn I(, ) iff(x1, . . . , xn)

For every variable x V+, (x) =x

whereV+ is the extended set of variables used in Lemma 3.8. The definitionof the neighborhood function N essentially says that a set of states of thecanonical model is necessary at a world precisely when claims that itshould. For any formula L1, let|| be the proof set of in the logic ,that is,

||={| W and }

The fact that N is a well-defined function follows from the fact that contains the rule RE.


20/39


Definition3.17. LetM = W,N,D,Ibe any first-order constant domainneighborhood model. M is said to be canonical for a first-order classicalsystem provided W=W, D= D, I=I and

|| N() iff

Thus the model M is the smallest canonical model for a logic . Itis shown in Chellas ([16]) that ifM = W,N,D,I is a canonical model,then so isM =W, N, D , I ,where for each W,N() =N() {XW | X = || for any L1}. That is N

is Nwith all of the non-proofsets.

Lemma 3.18 (Truth Lemma). For each W and formula L1,

iffM, |=

Proof.

The proof is by induction on

. The base case and propositionalconnectives are as usual. The quantifier case is exactly as in the Kripkemodel case (refer to Lemma 3.9). We need only check the modal cases. Theproof proceeds easily by construction ofN and definition of truth: iff(by construction) || N() iff(by definition of truth) M, |= .

Notice that in the above proof, as opposed to the analogous result for re-lational models, we can construct a constant domain model without makinguse of the Barcan formula. The following corollary follows from the truthLemma via a standard argument.

Theorem3.19. For any canonical modelM for a classical first-order modal

logic , is valid in the canonical modelM iff

Corollary 3.20. The class of all first-order neighborhood constant domainframes is sound and complete forFOL + E.

Notice that in the canonical model M constructed above, for any state, the set N() contains only proof sets, i.e., sets of the form { | } for some formula L1. For this reason, even if contains the Maxiom scheme, N may not be supplemented. Essentially the reason is ifX N(), andXY,Ymay not be a proof set, so cannot possibly be inN(). The supplementation of a frame F=W, N (W,N,D), denotedF+, is a tuple W, N+ (W, N+, D) where for each w W, N+(w) is the

smallest collection of sets containing N(w) that is closed under superset. Itcan be shown that the supplementation M +

is a canonical for FOL + EM

(by adapting to the first order case the proof offered in [16] page 257).


21/39


Theorem3.21. FOL + EM is sound and complete with respect to the classof supplemented first-order constant domain neighborhood frames.

Example: Qualitative probability defined over rich languagesThe systemEMN and first order extensions of it seem to play a central rolein characterizing monadic operators of high probability. These operatorshave been studied both in [36] and in [6]. Roughly the idea goes as follows:letWbe a set of states and W a-algebra generated byW. LetP : W [0, 1] be a probability measure and t (0.5, 1). Let Ht Wbe the set ofevents with high probability with respect to t, that is H t = {X | P(X)>t}. It is easy to see that Ht is closed under superset (M) and contains theunit (N). A similar construction is offered in [36], where the authors claimthat the resulting propositional logic is EMN.

The results offered in [36] show that the approach in terms of neighbor-

hoods has interesting (and potentially rich) applications. These applicationsgo beyond the study of monadic operators of high probability. In [6] Arlo-Costa considers the more general case of non-adjunctive modalities and hedevelops a measure of the level of coherence of neighborhoods. These modelscan, in turn, be generalized in order to classify parametrically other familiesof paraconsistent logics, aside from the non-adjunctive ones.

The probabilistic applications considered in [36] can also be generalized.In fact, they cover only the case of propositional languages. But, as HenryKyburg has pointed out in [35] researchers in various communities (most re-cently in Artificial Intelligence) have been interested in having a qualitativenotion of probability defined over a language at least as reach as first orderlogic. As Kyburg correctly points out in [35] one of the (misguided) reasonsadvanced by the researchers in the logical branch of AI for notusing prob-ability in knowledge representation was related to the alleged difficulties indefining a notion of probability over a rich language. Kyburg reminds thereader of the seminal work already done in this area by Gaifman and Snir[24] and by Scott and Krauss [46]. Nevertheless Kyburg and Teng do not ap-peal to these accounts of probability over first order languages in their ownmodel of qualitative (monadic) probability. This might have been caused bythe fact that first order classical modal logics have not been studied carefullyin the previous literature.

The model offered in [36] differs from the one sketched here in various importantmanners. First it attributes probability to sentences, not to events in field. Second it

works with a notion of primitive conditional probability which is finitely additive. Thirdprobability in Kyburg and Tengs model is not personal probability but objective chance,which is interval-valued.


22/39


We claim that the accounts of monadic operators of high probability pre-sented in [35] and [5] can be generalized by appealing to the tools presentedhere. We will sketch in this example how this can be done by appealing to

the account off

ered in [24]. This is one the most detailed models of how to at-tribute probability to first order sentences in the literature. There are, never-theless, various discontinuities between this account and Kyburg and Tengs.For example Gaifmans account intends to axiomatize a notion of probabil-ity which does not conflict with the standard Kolmogorovian account, whileKyburg and Teng work with finitely additive primitive conditional proba-bility. These foundational issues might have some logical repercussions butthe semantic framework utilized here is largely neutral with respect to thoseissues. So, the proposal below can be seen as a model of high probabilityoperators for a Kolmogorovian notion of probability defined over a first orderlanguage.

LetL0 be a first order language for arithmetic. So L 0 has names for themembers ofN, aside from symbols for addition and multiplications, variablesthat take values onNand quantifiers, etc. Notice that this language is richerthan the one used above. For example, it has constants, which in this caseare numeralsn1, . . .. We use n ambiguously for natural numbers and theirnumerals. Let in additionL be an extension ofL0containing a finite amountof atomic formulas of the form R(t0, . . . , tk) where ti is either a variable ora numeral. Let P r be a nonnegative real-valued function defined for thesentences ofL and such that the following conditions hold:

1 If|= then P r() = P r()

2 If|=, then P r() = 1.

3 If|=( ), then P r( ) = P r() + P r()

4 P r(x(x)) =S up{P r((n1) . . . (nk)) :n1 . . . nk N,k = 1, 2, . . .}

Condition (4) is the substantive condition, which in [46] is called Gaif-mans condition. It is clear that we can use this notion of probability inorder to define a modality is judged as highly probable by individual imodulo a thresholdt. So, for example, for (a finite stock of) atomic formulaswe will have

M, w|= R(t1, . . . , tn) if and only if (R(t1, . . . , tn))M, N(w) if and

only ifP rw(R((t1), . . . , (tn)) > t.

And in general neighborhoods contain the propositions expressed by sen-tences ofLto which an agent of reference assigns high probability. Of course,


23/39


other interesting modalities, like sequence s is random or is judged asmore probable than , etc. can also be defined. In view of the previousresults we conjecture that the logic that thus arises should be an extension

of thenon-nested fragmentofFOL + EMN we refer to the logic encodingthe valid formulas determined by the aforementioned high probability neigh-borhoods. Of course, BF should fail in this case, given that it instantiatescases of what is usually known as the lottery paradox

The situation is more complicated in the case ofFOL + E + CB F. Weneed some definitions in order to consider this case.

Definition3.22. A classical system of first order modal logic iscanonicalif and only if the frame of at least one of its canonical models is a frame for.

Now we can establish a result showing that FOL+ E + C BF is notcanonical (in the strong sense we just defined). That is we will show thatthe frame of the smallest canonical model for FOL + E + CB Fnot a framefor FOL + E + CB F.

Observation 3.23. FOL + E + CBF is not canonical.

Proof. Assume by contradiction that FOL+ E + C BF is canonical. Ifthis were so, given that (by construction) the frame of any canonical modelfor FOL +E+ C BF is non-trivial (see Definition 3.12 for non-triviality),there is at least a frame of one of the canonical models ofFOL + E + CB F(which is also a frame for FOL + E + CBF) and that frame should also besupplemented (by Observation 3.13) as C BFis obviously valid. And if thiswere the case then every first order instance of Mshould also be valid inthe canonical model. But this implies thatMFOL + E + CB F, which isa contradiction.

It is easy to see that the previous argument can be extended to show thatFOL+ E+ C BF is not complete with respect to any class of non-trivialframes (i.e. a class where each frame is non-trivial). However, it is notdifficult to see that if we take the class of non-trivial supplemented first-orderneighborhood frames and we add to it a trivial and non-supplemented frame(of the sort used in Observation 3.17) this class fully characterizes FOL +E + CBF. This is so given that C BFcontinues to be valid with respect to

the widened class, but the addition of the trivial frame guarantees that Mis no longer valid in the widened class of frames. We first need a definition.The augmentation of a frame F = W, N (W,N,D), denoted aug(F),


24/39


is the supplementation of F, where F is the tuple W, N! (W, N!, D)and for each w W, N!(w) =N(w) {

N(w)}. The supplementation of

a modelM based on a frame F, denoted M+, is the corresponding model

based on F+

. Similarly for augmentation.Observation 3.24. The augmentation of the smallest canonical model forFOL + K is not a canonical model forFOL + K. In fact, the closure underinfinite intersection of the minimal canonical model for FOL + K is not acanonical model forFOL + K.

Proof. LetM where =FOL + K be the smallest canonical model forFOL+Kand aug(M) =W, N!, D, I its augmentation. Suppose thatx(x) . Then for each y V+, (y) , and so for each y V+,|(y)| N!(). Hence

xV+|(x)| N

!

(). This follows if we assumeN is closed under infinite intersections. Since |x(x)| =

xV+ |(x)|

N!

(

), we have x(x)

. Therefore, for each

, B F

. But this is acontradiction since BF is not provable in FOL + K.

Lemma3.25. The augmentation of the smallest canonical model forFOL +K + BFis a canonical forFOL + K + BF.

Proof. Suppose that =FOL+K+BFandMis the smallest canonicalmodel for . Letaug(M) =W, N!, D, I be the augmentation ofM.We must show, for any formula L1 and any state W,

iff|| N!()

The proof is by induction on the complexity of.

The boolean connectives and the base cases are straightforward.

(Quantifier Case) Suppose that is x(x). Then ifx(x) , us-ingC BF, x(x) . Since is a maximally consistent set, for eachy V+, (y) . By the induction hypothesis, |(y)| N!() foreach y V+. By the construction of N!

(),

yV+|(y)| N

!

(),

hence |x(x)| N!() as desired. Suppose that x(x) .Then x(x) and using BF, we have x(x) . Thenif |x(x)| N!() we have for each y V

+, |(y)| N!()

since N!() is supplemented. By the induction hypothesis, for each

y V+, (y) . Hence x(x) . But this is a contradiction, so|x(x)|N!(), as desired.


25/39


(Modal Case) Suppose that is . For the left to right direction, if , then|| N() and so|| N!() (sinceN() N

!

()).For the other direction, assume that || = || N!(). Hence

N(

)|

|. We must show

. Now, it is easy to see thatfor each maximally consistent set ,

N() iff{| }

Thus for each with { | } , . Then {| }. Hence by compactness there are 1, . . . , n in { | } with(1 n) . Hence , as desired.

Theorem

3.26. FOL

+K

is sound and complete with respect to the classof filters.

Example: Finitely additive conditional probabilityWhen neigh-borhoods encode qualitative expectations for finitely additive measures (incountable spaces) they form non-augmented filters validating FOL + KbutnotB F.

Some distinguished decision theorists (like Bruno De Finetti and LeonardJ. Savage) as well as some philosophers (Isaac Levi) have advocated the useof finitely additive conditional probability in the decision sciences. Lester E.Dubins offers an axiomatic characterization of finitely additive conditionalprobability in [20]. As De Finetti suggested in [19] it is possible to extract

a superiorityordering from finitely additive probability, which has, in turn,been used more recently in order to define full belief and expectations fromprimitively given conditional probability in a paradox-free manner ([48], [3]).Moreover finitely additive conditional probability has also been used in orderto define non-monotonic notions of consequence ([3], [29]). We will showhere that the resulting modalities can be represented in salient cases byneighborhood frames forming non-augmented filters.

We will proceed as follows: we will first define a qualitative structurein neighborhoods (basically a non-augmented filter). Then we will definea conditional measure satisfying Dubins axioms by utilizing this qualita-tive structure. Finally we will show that the structure in question offers a

characterization of belief and expectation corresponding to the measure.Let the space be the (power set of the) positive integers and consider

E = {2n : n = 1, . . .} be the even integers in the space, and let O =


26/39


{2n 1 : n= 1, . . .} be the odd integers in the space. LetP(i) = {1/2n:ifi = 2n} and let P(i) = 0, otherwise. So the unconditional P is countablyadditive, whose support is the even integersE. ButP(.|O) might be uniform

on the odd integers. This can be reflected by the fact that there is a coresystemover O defined as follows: Let the outermost core C1 =O . ThenC2= {1}c, C3 = {1, 3}

c, etc. For any n: 1, ...; define a rank system rn for Cnas follows:

(ranks)rn = {w : w Cn Cn+1}

Notice that each rank contains exactly one odd number with r1 = {1}.Now we can define conditional probability as follows. If there is a largestinteger isuch thatA Ci =, defineQ(B|A) (forA, B O) as: pi(A B)/pi(A). Otherwise set Q(B|A) to 1 if there is r i such that A Ci = , andA Ci B and Q(B

c|A) to 0 for Bs satisfying the given conditions. For

the remaining infinite sets such that both B and B c are infinite, arbitrarilyset one of them to 1 and the complement to 0. Finally set Q(B|A) to 0 forevery other event in the space. According to this definition each co-finiteset in O has measure 1 (because it is entailed by at least a core) and eachnumber in O carries zero probability. Of two infinite but not co-finite sets,say S = {1, 5, 9,...} and H = {3, 7, 11, ...}, we can assign 1 to the set oflower rank (so S carries measure 1 and H zero).

We shall now formally introduce the notion ofprobability core. We followhere ideas presented in [3], which, in turn, slightly modify the schema firstproposed in [48]. A notion that plays an important role in those works isthe notion ofnormality. The basic idea is that an event A is normal for

Q as long as Q(|A) = 0. Conditioning on abnormal events would lead toincoherence.

A probabilitycorefor Q is an event Kwhich is normal and satisfies thestrong superiority condition (SSC) i.e. ifA is a nonempty subset ofK andB is disjoint from K, thenQ(B|A B) = 0 (and so Q(A|A B) = 1). Thusany non-empty subset ofKis more believable than any set disjoint fromK.

Now it is easy to see that the system of cores Cn constructed aboveconstitutes a system of cores for Q as defined in the previous paragraphs.Cores can be used in order to characterize qualitative expectations relativetoQ: an event Eis expected relative to Q as long as it is entailed by some

core for Q. So, given Q we can construct all neighborhoods of a frame asthe set of expectations for Q. A binary modality can also be intuitivelycharacterized, namely conditional expectation: relative to any finite subset


27/39


ofO its largest element will always be expected. It is easy to see that theseneighborhoods form non-augmented filters and that the first order logic ofqualitative expectations should at least obey the axioms of the non-nested

fragment ofFOL + K.Expectations have been utilized in [3] in order to define non-monotonicconsequence: B is non-monotonically entailed by A relative to Q if andonly if B is expected relative to Q(.|A). An important argument in [3]shows that in infinite spaces and for logically infinite languages (equippedwith at least a denumerable set of atoms) the definition of non-monotonicconsequence sketched above obeys standard axioms of rational consequenceproposed by Lehman and Magidor [39] only if the underlying measure isnot countable additive, so the filter structure presented above is essentiallyneeded in order to characterize probabilistically both qualitative expectationand conditionals (relative to a conditional measure Q)

Theorem 3.27. FOL + K + BFis sound and complete with respect to theclass of augmented first-order neighborhood frames.

ExamplePerhaps the simplest example is constituted by the expecta-tions induced by a measure Q defined over a finite space. In this case coresystems always have an innermost core and the corresponding neighborhoodsare augmented. In general [2] shows that the core systems of countable ad-ditive measures always have an innermost core, also inducing augmentedneighborhoods.

4. General Frames

The argument presented to this point shows that although central normalsystems like FOL+ K cannot be fully characterized via constant domainrelational models, there are many modalities obeying this logic which seem torequire interpretations in terms of constant domains. Moreover the analysisof operators of high probability provides neither examples of modalities notobeying neither the Barcan Formula nor the full strength ofK, but which arebest interpreted in terms of frames with constant domains. The adoption ofneighborhood semantics in the tradition suggested by Scott and Montaguesolves this and many other related problems (pointed out in the introduction)rather well.

Other notions of expectation proposed in the literature, like the one presented in [40],do not seem to exhibit the logical structure encoded in FOL+K; but they can also beanalyzed with the tools offered here.


28/39


But an unmodified version of the program of neighborhood semantics(even at the propositional level) suffers also from certain important (andindependent) inadequacies. The problem in question is that there are many

classical modal logics which are not complete with respect to any class ofneighborhood frames. The problem was first suggested by M. Gerson [26]who showed the existence of two incomplete logics with respect to neighbor-hood frames: one is the logic L (between T and S4) and the other the logicL (an extension of S4 independently defined by Kit Fine).

The logics L and L are also incomplete with respect to relational se-mantics a la Kripke, so Gersons result shows that neighborhood semanticsinherits other types of important inadequacies of the Kripkean program. Inaddition it is also well known that the first order logics strictly between S4.4andS5withoutthe Barcan Formula are also incomplete (in relational seman-tics) [28]. We conjecture that the latter incompleteness can be removed byusing first order neighborhood frames. But the first kind of incompletenesscannot be eradicated by utilizing first order neighborhood frames.

The solution for this kind of incompleteness in relational semantics con-sists in adopting the so-calledgeneral framesconstituted by a frame togetherwith a restricted but suitably well-behaved set ofadmissible valuations see[15] for a textbook presentation of general frames. The terminology generalframes can be traced back to van Benthems paper [14]. More completehistorical references can be found in footnote 6 in chapter 9 of [34]. An iden-tical maneuver is feasible in neighborhood semantics. [32] presents a generalresult of this type circumventing incompleteness results for the family ofmonotonic classical logics.

Hansens general monotonic frames are nevertheless not ideal for our

own purposes. First Hansens main goal is to prove duality results betweenneighborhood frames and algebras with operators and second she only fo-cuses on propositional systems. So, we will define here certain general firstorder neighborhood frames which we will call general first order frames.

Let F = W,N,D be a first-order neighborhood frame with constantdomain. The neighborhood function induces a function N : 2

W 2W

defined as follows. Let X W, then N(X) = {w W | X N(w)}.Intuitively,N(X) is the set of states where the proposition X is necessary.

Let W,N,D be a first-order neighborhood frame with constant domain.Let D denote the set of all functions from to D. For i , s D andd D, let sid denote the function which is exactly the same as s except

for the ith component which is assigned d. For s, s

D, we say that sand s are i-equivalent if s = sid for some d D. When convenient, wewill think of a function s D as an infinite sequence of elements from


29/39


D. Intuitively, these sequences represent a substitution : V D. Tomake this representation concrete, we need to fix an ordering on the set ofvariables V, i.e., assume that V = {v1, . . . , vn, . . .}. We fix this ordering

on the set of variables for the rest of this section. Then we can be moreprecise about the correspondence between sequences and substitutions. Forany substitution, : V D, there is a unique sequence, denoted by s,such that s(i) =(vi).

Similarly, for each sequence s D, there is a unique substitution, de-noted by s, such that s(vi) =si. Obviously, we have that vi

impliessis (and vice versa).

Consider a function f : D 2W. Intuitively the functionf should bethought of as representing the equivalence class of formulas logically equiv-alent to a relation (vi1, . . . , vin), in the sense that for each substitution ,f(s) = ((vi1, . . . , vin))

M,. Given a first-order neighborhood frame withconstant domainF=W,N,D, we say that a collection of functions {fi}iSis appropriate for F if,

1. For eachi S, there is aj Ssuch that for alls D,fj(s) =Wfi(s).We will denote fj byfi.

2. For each i, j S, there is a k S such that for all s D, fk(s) =fi(s) fj(s). We will denote fk byfi fj.

3. For each i S, there is a j S such that for all s D, fj(s) =N(fi(s)). We will denote fj byfi.

Definition 4.1. W,N,D,A, {fi}iSis a general neighborhood frame

with constant domain if W,N,D is a first-order neighborhood framewith constant domain, A 2W is a collection of sets closed under comple-ment, finite intersection and theN operator and{fi}iSis a set of functionsf : D 2W appropriate for W,N,D, where for each i and each s D,fi(s) A and each fi satisfies the following condition:

(C) For each vi V,

sis

f(s) A

An interpretation I is A-admissible for {fi}iS in a general first-order neighborhood frameF=W,N,D,A, {fi}iS provided for eachn-aryatomic formulaF and each substitution ,

1. ((F(x1, . . . , xn))M, A, where M is the first-order neighborhood model

based onFwith interpretation I, and


30/39


2. For each n-ary atomic formula F, there is a function fi such that foreach s D, w fi(s) iffs(vi1), . . . , s(vin) I(w, F), for any set ofvariables, vi1 , . . . , vin .

No further restrictions in the cardinality of the index set S are imposed.A general first-order neighborhood model with constant domain

is a structure Mg =W,N,D,A, {fi}iS, I, where W,N,D,A, {fi}iS isa general first-order neighborhood frame with constant domain and I is anA-admissible interpretation for {fi}iS. Truth and validity are defined asusual. It is easy to see that the following is true in any such model:

() for any scheme with n-free variables vi1 , . . . , vin , there is afunctionfi that corresponds to the scheme (v1, . . . , vn) in the sensethat for each substitution , ((v1 . . . , vn))

M, =f(s).

Lemma 4.2. For each formula L1 and any general first-order neighbor-hood model with constant domainMg = W,N,D,A, {fi}iS, I, ()

M, A for all substitutions .

Proof. Let Mg = W,N,D,A, {fi}iS, I be a general first-order neigh-borhood model with constant domain and any substitution. The proofis by induction on . The atomic case is by definition. The Booleancases are obvious as is the modal case. Let be vi(vi). We must show((x(x))M, A. Now, (vi(vi))

M, =

vi((vi))

M, .

By (), there is a function fj such that fj(s) = ((vi1 , . . . , vin))M,s for

all sequences s and variables vi1, . . . , vin . Hence, it is easy to show that

vi((vi))

M,

=

sis fj(s). By property C,

sis fj(s

) A.Therefore, (vi(vi))

M, A.

In order to define the general first-order canonical neighborhood model,we only need to define the canonical set of admissible sets and a set ofcanonical functions. The definitions of the domain D, the interpretationI and are as in the definitions for the canonical frame for . Theadmissible sets are defined as follows.

A = {|| | L1}

As for the canonical functions, enumerate all n-ary predicate symbols inL1. Then define for eachn-ary predicate symbolPwith variablesv i1 , . . . , vina functionfP :

D 2W as follows, for each s D,


31/39


fP(s) =|P(si1, . . . , sin)|

Let the set of canonical functions{fi }iSbe the smallest set of functions

containing the set{fP |P an n-ary predicate symbol}and closed under theconditions 1, 2, and 3 above. For every formula L 1, there is a functionf such that f

(s) = |(si1, . . . , sin)| for every sequence s D. It is

easy to see that the set {fi }iS satisfies condition C. Fix an arbitraryfunctionfi {f

i }iS. Since the set of functions is the smallest set satisfying

the closure conditions, there must be some expression such that fi(s) =|(s1, . . . , sn)|. Then

sis

f(s) =|x(x)|, wherex = si. Furthermore,

it is easy to check that I is an admissible interpretation for {f

i }iS. Thus,Mg

= W, N, D, A, {f

i }iS, I is a general first-order neighborhoodmodel with constant domain.

Theorem4.3

.Let

be any classical first-order modal logic.

is sound andstrongly complete with respect to the class of general first-order neighborhood

frames with constant domains for .

Proof. Let W, N, A, {fi }iS, D be the minimal general canonicalframe Fg

. Strong completeness requires showing that any -consistent set

of formulas is satisfiable in a modal based on Fg

. Since I is an admissibleinterpretation, the truth Lemma will hold forMg

. Hence, it is easy to show

that for any consistent set of formulas , Mg

|= for each . Thus weneed only show that Fg

|= for each . That is, letIbe anA-admissible

interpretation for {fi}iSand suppose M =W, N, A, {f

i }iS, D, I

is a model based on Fg

. We must show for any substitution : V+ D ,

M |= for each .Let Ibe an admissible interpretation for {fi }iSand therefore an ad-

missible interpretation ofFg

and let : V+ D be any substitution. Fixa formula . Then for each atomic formula Fi(vi1 , . . . , vin) in, there issome functiong {fi }iSsuch thatg(s) = (F(vi1, . . . , vin))

M,s for all se-quencess. By construction of the{fi }iS, there is a formula (vj1, . . . , vjm)such that g(s) = |(sj1, . . . , sjm)|. Note that in general, be have aritym=n (different from F). The goal is to construct a new formula whereeach Fi is replaced by the correpsonding . For a given atomic formulaF(vi1, . . . , vin) and a corresponding formula (vj1, . . . , vjm), the main casethat we need to consider is the one when m n and there is a subset

{vj1, . . . , vjk} {vj1 , . . . , vjm}, such that {vj1, . . . , vjk} {vi1 , . . . , vin}. Inthis case, replace F(vi1 , . . . , vin) with ((vi1), . . . , (vik), vjk+1, vjm).


32/39


Claim: For each W,Mg

, |= iffM, |= .

The proof is by induction on . If = F(vi1 , . . . , vin), then there are

two cases to consider.Suppose that =((vi1 ), . . . , (vik ), vjk+1, . . . , vjm). Then let D

be any sequence where 1= (vi1), . . . , n=(vin), n+1= vjn+1, . . . , m=vjm . Let

1 = (vj1), . . . ,

k = (vjk), for the variables {vj1, . . . , vjk}

{vi1, . . . , vin} see above where the formulas are introduced.

We have,

M, |= F(vi1, . . . , vin) iff (vi1), . . . , (vin) I(F, )

iff 1, . . . , n I(F, )

iff g() =|(1, . . . , k, k+1, . . . , m)|

iff (1, . . . ,

k, k+1, . . . , m)

iff ((vj1), . . . , (vjk), vjk+1, . . . , vjm)

iff Mg

, |= ((vj1), . . . , (vjk), vjk+1, . . . , vjm)

The case when =((vi1 , . . . , vim)) is similar to the above except usethe sequence 1= (vi1), 2=(vi2), . . . , n=(vin).

The Boolean cases are straightforward. Suppose that = . By theinduction hypothesis,

()Mg

, = ()M

,

Thus Mg

, |= iff()M

g, N() iff()M

, N() iffM,

|= .Suppose that = x(x). ThenM, |=x(x) iff

x

((x))M , =

(x)

(((x)))M

g,

I

iffMg

, , I|= (x)((x))

(of Claim)

Then for any , since is closed under universal substitution.By the truth Lemma,Mg

, |=

. Hence by the above claim, M, |=

where M

is any model based on Fg

and any substituion. Hence, isvalid on Fg

.


33/39


5. Conclusion and Further Work

Relational semantics for first order classical modalities suffers at least fromthree major inadequacies. First, the semantic study of non-normal classicalsystems is either impossible or indirect and unnecessarily complicated (somemonotonic non-normal systems can be simulated in terms of polymodal nor-mal logics see [37]). This is particularly important given the fact that manyinteresting modal logics of interest in computer science and philosophy arenon-normal.

Second, the standard relational semantics with constant domains imposesthe validity of both the Barcan and the Converse Barcan schemas. This isunduly restrictive given that there are many interesting modalities which arebetter represented with constant domains and for which the Barcan schema(or the Converse Barcan) fails to be intuitively valid. Monadic operators ofhigh probability as well as many epistemic modalities are salient examples.

Third, even when systems like FOL + K can be characterized in termsof relational semantics utilizing expanding domains there are many otherincompleteness results for systems strictly between S4.4 andS5 without theBarcan schema. The origin of these incompleteness results seems differentlymotivated than in the case of other incompleteness results for purely propo-sitional systems.

Some of these inadequacies can be removed via the adoption of generalrelational frames, but not all of them. For example, the adoption of generalrelational frames cannot give the Kripkean program the ability of character-izing certain normal systems without the Barcan schema in terms of class ofrelational frames with constant domains.

In contrast, the adoption of general first order neighborhood frames re-moves al lthese problems at once and delivers a very intuitive and appealingalternative semantical framework. Our results show that the adoption ofvarying domains in modal logic remains optional but it is not mandatory inorder to characterize normal systems like FOL + K. A general completenessresult can be proved for the entire first order classical family of modal logicsin terms of (general) constant domain frames.

We have tackled here some problems that seem central but much remainsto be done in this area. Concerning applications many notions of interestin distributed Artificial Intelligence and Game Theory can be studied withthe tools we offer. Some examples were provided in the introduction. As

an important additional example we conjecture that a strong completenessresult for the type spaces in the sense of Harsanyi is provable for first orderoperators of the type individual i assigns probability at least a. Concerning


34/39


extensions an obvious further step is to consider the case of varying domains.We expect that also in this area there are significant gains by applying theneighborhood approach (particularly in the case of transition logics and in

the case multi modal logics combining operators of tense and belief). Thereis also a promissory area of investigation related to the logic of conditionalsand non-monotonic inference. Dyadic modal classical modal operators wereconsidered in passing in [16] but this area of research remains practicallyunexplored.

Duality results for general first order neighborhood frames in terms ofcylindric algebras with operators remain also unknown. And although [32]contains some preliminary results concerning recent work in co-algebras, thisis also a topic that remains open. One should expect interesting connectionswith co-algebras given that knowledge operators treated in terms of neigh-borhoods (as is explained in passing in the introduction) can be alternativelycharacterized in terms of non-well founded models for modalities of the kindstudied in [12].

As Fitting and Mendelsohn remarked in [22] (page 134) the lack of acompleteness proof that can cover constant domains, varying domains, andmodels meeting other conditions... has often been felt. Garson suggests aswell that ideally, we would like to find a completely general completenessproof. Our focus in the last sections of this paper was to offer such a generalcompleteness proof for the entire class of first order classical modal systems.

Some authors have recently published results that aim at meeting Gar-sons challenge. The results offered in [17] are perhaps an example. In thiswork the author explains that the production of such a proof [a general com-pleteness proof] is the aim of this paper ([17], page 1483). The paper then

focuses on considering ...all free and classical quantified extensions of thepropositional modal logic K obtained by adding either the axioms of iden-tity or the Converse Barcan Formula or the Extended Barcan Rule ([17],page 1483). A general strategy for proving completeness results for thesequantified logics is then provided. All this is done by utilizing variationsof well-known relational semantics. Original Kripke semantics, Kripkesemantics and what the author calls Tarski-Kripke semantics are utilizedamong others and compared. The appeal to this semantic framework haswell-known limitations that seem to hamper the generality of the resultspresented in the paper.

First the presented results are confined to the class of classical normal

first order modal systems that admit a characterization via this type ofrelational semantics. As a matter of fact the article includes some incom-pleteness results as well, marking the limits of this semantic approach. So,


35/39


[17] clearly does not present a general completeness result even if one con-fines attention exclusively to the subset of classical first order modal logicsthat are normal.

Second, as a consequence of the previous remarks, [17] does not considerfirst order classical non-normal systems at all.Third the appeal in [17] to relational semantics forces the consideration

of models with varying domains. The lack of serious study of alternativesto relational semantics at the first order level has perhaps created the im-pression that the recourse to these kinds of models is obligatory in order toconsider general completeness results. We conclude here that this is not thecase by offering a general completeness result for the entire class of classicalfirst order modal systems in terms of general frames with constant domains.As we remarked above nothing precludes, nevertheless, the study of firstorder neighborhood frames with varying domains. A more detailed com-parison of the behavior of these kind of models with the relational models(and unified strategies for completeness) studied in [17] could be thereforeof interest as a topic for further study.

To a large extent the many philosophical discussions that followed Quineskeptical comments about quantified modal logic, have been based on theobservation that quantified modal logic seems to lead to the toleration andeventual acceptance ofpossibilia. Ruth Barcan Marcus puts the problem ina clear way:

Since we have the option of coextensive domains, QML is not com-mittedto possibilia. Yet admission of p ossibilia would seemto be anatural extension, for informally, the notion of possible worlds lends

itself to framing counterfactuals not merely about the properties ac-tual objects might have and the relations into which they might haveentered but about alternative worlds that might have individuals thatfail actually to exist, or fail to have individuals that do actually ex-ist. The [Kripke-style] semantics accommodated such interpretations([11], 195).

Barcan Marcus has always been skeptical about semantics that exhibitthis tolerance:

[...] modal discourse need not and should not admit possibilia despitethe elegance of the generalization ([11], 213).

She defended this position mainly in philosophical terms, and she, ofcourse, noticed that this philosophical attitude is consistent with the adop-tion of the so-called Barcan schema.


36/39


Our paper can be seen as providing further formal support for BarcanMarcuss philosophical view. Moreover we show as well that a modal se-mantics which rejects possibilia although consistent with the adoption of

the Barcan schemas, does not require the adoption of either of them for itscoherence. Modal discourse in its most general possible expression (the clas-sical family of first order modal systems) can be coherently, completely andparametrically interpreted without any recourse to possibilia of any sort.

References

[1] Alur, R.,. Henzinger, T. A and Kupferman. O. Alternating-time temporal logic, In

Compositionality: The Significant Difference, LNCS 1536, pages 23-60. Springer,1998.

[2] Arlo Costa, H. Qualitative and Probabilistic Models of Full Belief, Proceedings of

Logic Colloquim98, Lecture Notes on Logic13, S. Buss, P.Hajek, P. Pudlak (eds.),

ASL, A. K. Peters, 1999.

[3] Arlo Costa, H. and Parikh R.Conditional Probability and Defeasible Infer-ence,Technical Report No. CMU-PHIL-151, November 24, 2003, forthcoming in the

Journal of Philosophical Logic.

[4] Arlo Costa, H. Trade-Offs between Inductive Power and Logical Omniscience in

Modeling Context, V.Akman et al.(eds.):CONTEXT 2001,Lecture Notes in Artificial

Intelligence2116, Springer-Verlag Berlin Heidelberg, 1 -14, 2001.

[5] Arlo Costa, H. First order extensions of classical systems of modal logic, Studia

Logica, 71, 2002, p. 87 118.

[6] Arlo Costa, H. Non-Adjunctive Inference and Classical Modalities, Technical Report,

Carnegie Mellon University, CMU-PHIL-150, 2003 (forthcoming in the Journal of

Philosophical Logic).

[7] Bacharach, M.O.L., Gerard Varet, L.-A., Mongin, P. and Shin, H.S. (eds.)EpistemicLogic and the Theory of Games and Decisions, Theory and Decision Library, Vol. 20,

Kluwer Academic Publishers, 1997.

[8] Barcan (Marcus), R.C. A functional calculus of First Order based on strict implica-

tion, Journal of Symbolic Logic, XI, 116, 1946.

[9] Barcan (Marcus), R.C. The identity of individuals in a strict functional calculus of

First Order, Journal of Symbolic Logic, XII, 1215, 1947.

[10] Barcan Marcus, R.C. Modalities and intensional languages, SyntheseXIII, 4, 303-

322, 1961. Reprinted in Modalities: Philosophical Essays, Oxford University Press, 3

36, 1993.

[11] Barcan Marcus, R.C. Possibilia and Possible Worlds, in Modalities: Philosophical

Essays, Oxford University Press, 189 213, 1993.[12] Barwise, J. and Moss, L. Vicious Circles: On the Mathematics of Non-Wellfounded

Phenomena, C S L I Publications, February 1996


37/39


[13] Battigalli, P. and Bonanno, G. Recent results on belief, knowledge and the epistemic

foundations of game theory, Proceedings of Interactive Epistemology in Dynamic

Games of Incomplete Information, Venice, 1998.

[14] Benthem, J.F.A.K. van, Two simple incomplete logics, Theoria, 44, 25-37, 1978.

[15] Blackburn, P, de Rijke, M and Yde Venema, Modal Logic, Cambridge Tracts in

Theoretical Computer Science, 58, Cambridge University Press, 2001.

[16] Chellas, B. Modal logic an introduction, Cambridge University Press, 1980.

[17] Corsi, G. A Unified Completeness Theorem for Quantified Modal Logics, J. Symb.

Log. 67(4): 1483-1510, 2002.

[18] Cresswell, M. J. In Defense of the Barcan Formula, Logique et Analyse, 135-6,

271-282, 1991.

[19] De Finetti, B. Theory of Probability, Vol I, Wiley Classics Library, John Wiley and

Sons, New York, 1990.

[20] Dubins, L.E. Finitely additive conditional probabilities, conglomerability, and disin-tegrations,Ann. Prob. 3:89-99, 1975.

[21] Fagin, R., Halpern, J. Y., Moses, Y. and Vardi, M Y. Reasoning about knowledge,

MIT Press, Cambridge, Massachusetts, 1995.

[22] M. Fitting, M. and Mendelsohn, R. First Order Modal Logic, Kluwer, Dordrecht,

1998.

[23] Gabbay, D. Investigations in Modal an

horacio costa - first order classical modal logic

Documents