Modality and Quasimodality in English

D. Terence Langendoen

Department of Linguistics, University of Arizona

Presented at University of Leiden, 1 October 2002

This paper explores the application of principles of modal logic to certain expressions in English, some of which are identified by their syntactic properties as 'modal', including must, may, can, and need, and others which are not, such as be certain, and the universal and existential quantifiers. It investigates the interaction of modals, including modal quantifiers. It defines the related notion 'quasimodal', and shows that many familiar expressions (including at least one syntactic modal) are quasimodal, including be likely, should, seem, many, most, and the plural numerals. It details the interactions of modals with quasimodals, as well as the interactions among various quasimodals. [Note: the terms 'quasimodal' and 'quasimodality' have replaced the original 'semimodal' and 'semimodality', because the latter terms are used in the literature in a different sense.]

Modals

There are two types of modal operators on a set S over which an entailment relation Þ is defined: necessity (□ 'box') and possibility (à 'diamond'), which is the 'dual' of □. In general, □ preserves entailment (e.g. if P, Q Þ R, then □P, □Q ÞR) but fails to distribute over disjunction, whereas à distributes over disjunction but fails to preserve entailment; see A. Koslow (1992) A Structuralist Theory of Logic, Cambridge University Press. Another way to describe entailment preservation is to say that □ 'collects' over conjunction, as in (□1). These 'modal laws' are listed in (□1), (□2), (à1), and (à2).

(□1) For every P, Q in S: □ (P) Ù □ (Q) Þ □ (P Ù Q).

(□2) There are P, Q in S such that: □ (P Ú Q) Þ □ (P) Ú □ (Q) fails.

(à1) For every P, Q in S: à (P Ú Q) Þ à (P) Ú à (Q).

(à2) There are P, Q in S such that: à (P) Ù à (Q) Þ à (P Ù Q) fails.

Some Common Modal Properties

Interdefinability of Necessity and Possibility Modals

In most cases, □ and à are interdefinable using the 'not-not' equivalences in (ØØ) and (ØàØ).

(ØØ) For every P in S: Ø (□ (Ø (P))) Û à (P).

(ØàØ) For every P in S: Ø (à (Ø (P))) Û □ (P).

Necessity Entails Possibility

In many cases, necessity entails possibility, as expressed in (□Þà):

(□Þà) For every P in S: □ (P) Þ à (P)

Some Varieties of Modals

Logical Modals

One familiar pair of modals is logical necessity and possibility defined in (□l) and (àl). It is easy to determine that □l and àl satisfy the modal laws (□1), (□2), (à1), and (à2), are interdefinable using (ØØ) and (ØàØ), and satisfy (□Þà).

(□l) □l (P) iff P is a theorem (tautology) in S.

(àl) àl (P) iff P is not an anti-theorem (contradiction) in S.

Epistemic Modals

A pair of modals that is expressed by common lexical items in English and many other natural languages is epistemic necessity and possibility defined in (□e) and (àe). It is also easy to determine that □e and àe satisfy the modal laws (□1), (□2), (à1), and (à2), are interdefinable using (ØØ) and (ØàØ), and satisfy (□Þ à).

(□e) □e (P) iff P is certain in S.

(àe) àe (P) iff P is not epistemically impossible in S.

Since every logical necessity is also an epistemic necessity but not conversely, and every epistemic possibility is also a logical possibility but not conversely, the relations in (□lÞ e) and (àeÞ à l) also hold.

(□lÞe) For every P in S: □l (P) Þe (P), but not conversely.

(àeÞàl) For every P in S: àe (P) Þ àl (P) , but not conversely.

Expression of Epistemic Modals

Epistemic necessity is expressed by the modal auxiliary must, a 'raising' operator, and the predicate adjective construction be certain, which allows but does not require raising. Epistemic possibility is expressed by the modal auxiliary may, also a raising operator, and the predicate adjective construction be possible, a non-raising operator.

e (P) there must be a screw loose; it is certain that there is a screw loose; there is certain to be a screw loose; Irene must have practiced; it is certain that Irene practiced; Irene is certain to have practiced

àe(P) there may be a screw loose; it is possible that there is a screw loose; Irene may have practiced; it is possible that Irene practiced

Deontic Modals

Another pair of modals that is expressed by common lexical items in English and many other natural languages are deontic necessity and possibility defined in (□d) and (àd).

(□d) □d (P (i)) iff the individual i (the 'obligatee') is obligated to undertake P in S.

(àd) àd (P (i)) iff the individual i (the 'permittee') is permitted to undertake P in S.

It is disputed whether what I am calling deontic necessity and possibility are modal operators, and if so whether the not-not equivalences hold for them, and whether deontic necessity entails deontic possibility. I hold that □ d and à d are modal operators, that the not-not equivalences hold, but that deontic necessity does not entail deontic possibility, i.e. that □d (P (i)) entails neither P (i) nor àd (P (i)). It is possible that one does not do or is permitted to do, what one must do. Nor does P (i) entail àd (P (i)). It is possible that one is not permitted to do what one does.

Expression of Deontic Modals

Deontic necessity is expressed by the modal auxiliary must and the constructions have to and be required to. Deontic possibility is expressed by the modal auxiliaries may and can, and the constructions be allowed to and be permitted to. The raised subject is the obligatee or permittee, unless (1) the complement is passive, in which case the agent of the complement is the obligatee, or (2) the raised subject is not something that can be an obligatee or permittee, in which case an appropriate obligatee or permittee is supplied.

d(P(i)) Irene {must, has to, is required to} practice; it is required that Irene practice

Irene {must, has to, is required to} push Harry; it is required that Irene push Harry

d(P*(h, i)) Harry {must, has to} be pushed by Irene; it is required that Harry be pushed by Irene

d($x(P*(h,x))) Harry {must, has to} be pushed; it is required that Harry be pushed

d("x(P*(h,x))) Harry {must, has to} be left alone; it is required that Harry be left alone

d($x(make(x,P))) this rock {must, has to} move; it is required that this rock move there {must, has to} be someone or something that makes this rock move; it is required that there be someone or something that makes this rock move

àd(P(i)) Irene {may, can, is allowed to, is permitted to} practice

Special Property of Deontic Possibility

In addition, àd obeys the stronger version of (à1) given in (àd1).

(àd1) For every P, Q in S: àd (P (i) Ú Q (i)) Þ àd (P (i)) Ù àd (Q (i)).

Irene {may, can, is allowed to, is permitted to} sing or dance Þ Irene {may, can, is allowed to, is permitted to} sing and Irene {may, can, is allowed to, is permitted to} dance

'Binary' deontic modals

By definition, modal operators are unary, so that a binary relation like require (h, P (i)), where h is the 'obligator', the individual who obligates i to do P, has to be modified so as to be analyzed as a deontic necessity modal. The modification is to 'incorporate' the obligator into the operator, as in require-h (P (i)). With such modifications, it can be shown that require-h (P (i)) is a deontic necessity modal □d , and allow-h (P (i)) and permit-h (P (i)) represent a deontic possibility modal à d .

require-h(P(i)) = □d(P(i)) Harry requires Irene to practice

allow-h(P (i)) = àd (P(i)) Harry {allows, permits} Irene to practice

A 'Need' Modal

The verb need, as in Harry needs Irene to push him, is very close in meaning to the deontic necessity operator require as in Harry requires Irene to push him; it is, in fact, synonymous with the non-deontic interpretation of the latter. Moreover, like binary require, binary need can be analyzed as a necessity modal by incorporation of the 'needer' into the operator, as in need-h (P (i)), analyzed as □n (P (i)), where i represents the 'needee', the individual that the needer needs to carry out P.

need-h( (i)) = □n (P (i)) Harry needs Irene to push him

In addition, like must, but unlike require, need occurs as a unary raising necessity operator □n with a clause as its complement, in which the subject occurs as the needee, and the needer is not indicated. (Cf. Irene must push Harry, representing □d (P (i)), in which Irene is the obligatee, and the obligator is not indicated.)

n(P(i)) Irene needs to push Harry

Careful consideration of examples like Irene needs to push Harry reveals that they are systematically ambiguous. The subject may also be understood as both needer and needee, indicating that on that interpretation, need occurs as a 'control' verb.

need-i (P(i)) = □n (P (i)) Irene needs to push Harry

This ambiguity is present even if the predicate P is intransitive.

n(P(i)) Irene needs to practice ('raising' interpretation: Irene is needee, needer is not indicated)

n(P(i)) Irene needs to practice ('control' interpretation: Irene is both needer and needee, as in Irene feels the need to practice)

When the complement of need is passive, the needee is the unspecified agent of the complement.

need-i ($ x (P* (h, x))) = □n ($ x (P* (h, x))) Irene needs Harry to be pushed

However, if the subject of the complement is not overt, only the raising interpretations is possible.

n ($ x (P* (i, x))) Irene needs to be pushed (raising interpretation: needee is unspecified agent of push and needer is not indicated)

The control interpretation need-i ($ x (P* (i, x))) = □n ($ x (P* (i, x))) in which the needee is the unspecified agent of push and Irene is needer (as in Irene feels the need to be pushed) is not available, I presume because the control interpretation requires identity between the needer and needee. Moreover, if the passive complement is expressed as a gerund, the same raising interpretation is obtained.

n($x(P*(i,x))) Irene needs pushing

Finally if the complement is active and the raised subject is not an appropriate needee, then one must be supplied.

n($x(make(x,P))) this rock needs to move there needs to be someone or something that makes this rock move

Is There a 'Need' Possibility Modal?

I am not aware of the existence of a need possibility modal.

Abilitative Modals

Another possibility modal that is expressed by common lexical items in English and other languages is abilitative possibility defined in (à a).

(àa) à a (P (i)) iff the individual i (the 'abler') is capable of undertaking P in S.

To establish (à1), it suffices to note that it is no easier for the abler to do P (i) Ú Q (i) than to do the easier of P (i) and Q (i). That is, if à a (P (i)) Ú à a (Q (i)) is false, then so must à a (P (i) Ú Q (i)). For (à2), it suffices to note that for certain P (i) and Q (i), P (i) Ù Q (i) is harder for the abler to do than either of P (i) and Q (i) alone, so that à a (P (i)) Ù à a (Q (i)) is true but à a ((i) Ù Q (i)) is false. Abilitative possibility is expressed in English by the modal auxiliary can and the construction be able to.

àa(P(i)) Irene {can, is able to} practice

Abilitative necessity is expressed in English by the locution can't help.

a(P(i)) Irene can't help practicing

Can Be Proved and Be Provable

The combination of the abilitative possibility modal can with the operator be proved, and its synonym be provable, express a special necessity modal □p. Since □p clearly satisfies (□1), to establish that it is a necessity operator, we need only consider the case of a proposition P such that both it and its negation are not provable. Then, assuming the law of the excluded middle holds for P, □p (P Ú Ø (P)) is true but □p (P) Úp (Ø (P)) is false, satisfying (□2).

Quantificational Modals

The universal quantifier "x obeys (□1) and (□2), and the universal quantifier $x obeys (à1) and (à2). They are also interdefinable using the not-not equivalences, and "x (P (x)) Þ $x (P (x)) but not conversely for any domain with at least two individuals. Hence they are paired necessity and possibility modals that we can represent □qx and àqx, as in the following examples. However, for convenience, I continue to use the standard notation for representing these quantifiers.

qx (P (x)) = "x (P (x)) everyone practiced

àqx (P (x)) = $x (P (x)) someone practiced

Modal Interactions

Necessity and Possibility Interactions

If à (P) Þ □ (à (P)), an axiom of the modal system S5, holds, then the modal law (à□) follows from the general principle à (□ (P)) Þ à (P) and the transitivity of 'Þ'.

(à□) à (□ (P)) Þ □ (à (P))

This law has the following special cases, where " and $ have their usual meanings, and □ means any nonquantificational necessity operator and à any nonquantificational possibility operator.

($" ) $x (" y (P (x, y))) Þ " y ($x (P (x, y))).

$y (" x (P (x, y))) Þ " x ($y (P (x, y))).

($□) $x (□ (P (x))) Þ □ ($x (P (x))).

(à") à ("x (P (x))) Þ "x (à (P (x))).

Necessity-Necessity and Possibility-Possibility Interactions

("" ) "x (" y (P (x, y))) Û "y (" x (P (x, y))).

("□) "x (□ (P (x))) Þ □ ("x (P (x))). ('Barcan formula')

(□") □ ("x (P (x))) Þ "x (□ (P (x))). ('converse Barcan formula')

($$) $ x ($ y (P (x, y))) Û $ y ($ x (P (x, y))).

() $x (à (P (x))) Þ à ($x (P (x))). ('dual of Barcan formula')

(à$) à ($x (P (x))) Þ $x (à (P (x))). ('dual of converse Barcan formula')

Quantifier and Other Modal Interactions

Existential Quantifier and Necessity Modal

($e) there is someone who is certain to have practiced Þ there is certain to be someone who has practiced, but not conversely.

Note that both of the sentences someone is certain to have practiced and it is certain that someone has practiced are ambiguous, expressing either proposition in the illustration of ($□). In addition, the sentence someone must have practiced is ambiguous in the same way.

($d) there is someone who must practice Þ there must be someone who practices (understood deontically), but not conversely.

Note that the sentence someone must practice is ambiguous; its meaning may be schematized as $x (□d (P (x))) or as □d ($x (P (x))).

Universal Quantifier and Possibility Modal

(àe") it is possible that everyone has practiced Þ for every person, it is possible that he or she has practiced, but not conversely.

Note that the sentence everyone may have practiced is ambiguous, expressing either proposition in the illustration of (à"), whereas the sentence anyone may have practiced expresses only the weaker one.

(àd") everyone may practice Þ anyone may practice, but not conversely.

Universal Quantifier and Necessity Modal

("e) and (□e") everyone is certain to have practiced Û it is certain that everyone has practiced.

("e) and (□e") everyone is required to practice Û it is required that everyone practice.

Note that everyone must practice is technically ambiguous, but the two interpretations are logically equivalent.

Existential Quantifier and Possibility Modal

(e) and (àe$) there may be someone who has practiced Û there is someone who may have practiced.

(d) and (àd$) someone is permitted to practice Û it is permitted for someone to practice.

Note that someone may practice is technically ambiguous in a similar way to everyone must practice.

Non-Quantifier Modal Interactions

Since à (P) Þ □ (à (P)) does not hold for the epistemic and deontic modals, these modal systems do not obey (à□). In particular it is certain that it is possible that it is raining neither entails nor is entailed by it is possible that it is certain that it is raining, and Irene must be permitted to practice neither entails nor is entailed by Irene may be required to practice.

Quasimodals

Certain operators fail the modal laws in the way described in (D1) and (D2). I refer to such an operator as a quasimodal (D). Quasimodals sometimes 'fall between' necessity and possibility modals logically, i.e. we sometimes find □ (P) Þ D (P) Þ à (P).

(D1) For every P, Q in S: D (P Ù Q) Þ D (P) Ù D (Q), but not conversely. (D fails □1, à 2.)

(D2) For every P, Q in S: D (P) Ú D (Q) Þ D (P Ú Q), but not conversely. (D fails □2, à 1.)

Quantifier Quasimodals

The quantifiers many, most, and the plural numerals acting as quantifiers fall between the universal (necessity) quantifier and the existential (possibility) quantifier. For example, for a sufficiently large domain of entities quantified over, and letting mx stand for the quantifier many, " x (P (x)) Þ mx (P (x)) Þ $ x ((x)). mx is a quasimodal Dmx since:

(m1a) For every P, Q in S: mx (P (x) Ù Q (x)) Þ mx (P (x)) Ù mx (Q (x)).

(m1b) There are P, Q in S such that: mx (P (x)) Ù mx (Q (x)) Þ mx (P (x) Ù Q (x)) fails.

(m2a) For every P, Q in S: mx (P (x)) Ú mx (Q (x)) Þ mx (P (x) Ú Q (x)).

(m1b) There are P, Q in S such that: mx (P (x) Ú Q (x)) Þ mx (P (x)) Ú mx (Q (x)) fails.

Similar observations establish that most and the plural numerals are also quantifier quasimodals.

Epistemic Quasimodals

The epistemic operator be likely falls between the epistemic necessity modal □d and the epistemic possibility modal à d: □d (P) Þ be likely (P) Þ à d (P)), and so is a candidate quasimodal De. Assuming, in accordance with the laws of probability, that De (P Ù Q) entails De (P) Ù De (Q) but not conversely (i.e. that the likelihood of the conjunction of two events is not greater than the likelihood of the two events separately), and that De (P) Ú De (Q) entails De (P Ú Q) but not conversely (i.e. that the likelihood of either of two events is not greater than the likelihood of their disjunction), De satisfies the quasimodal laws (D1) and (D1), and so is an epistemic quasimodal.

Deontic Quasimodals

The deontic operator should does not (in fact cannot) fall between the deontic necessity modal □d and the deontic possibility modal à d, but arguably satisfies (D1) and (D2) and if so is a quasimodal Dd. This would account for the perception that deontic should is weaker than deontic must and stronger than deontic may.

Modal and Quasimodal Interactions

Modal and Quasimodal Interactions Involving at Least One Quantifier

($D) $ x (D (P (x))) Þ D ($ x (P (x))) but not conversely.

(àP) à (Px (P (x))) Þ Px (à (P (x))) but not conversely.

(D") D (" x (P (x))) Þ " x (D (P (x))) but not conversely.

(P□) P x (□ (P (x))) Þ □ (P x (P (x))) but not conversely.

I consider three types of cases, depending on whether the modal or the quasimodal is a quantifier, or both.

Modal Quantifiers and Quasimodal Operators

($De) there is someone who is likely to be practicing entails there is likely to be someone who is practicing, but not conversely.

For example, suppose that there are three people each of whose likelihood to be practicing is 0.4, and that the threshold for be likely is 0.7. Then the first sentence is false and the second true. (The likelihood for at least one of them to be practicing is about 0.8.) However, there is no condition under which the first sentence is true and the second false. (Note that someone is likely to be practicing and it is likely that someone is practicing are both ambiguous, expressing either $x (be likely (P (x))) or be likely ($x (P (x))).)

(De") it is likely that everyone is practicing entails everyone is likely to be practicing but not conversely.

Modal Operators and Quasimodal Quantifiers

(àeP) there may be many who are practicing entails there are many who may be practicing, but not conversely.

(Pe) there are many who are certain to be practicing entails it is certain that there are many who are practicing, but not conversely.

Modal and Quasimodal Quantifiers

($D) and (àP) there is some employee who pleases many managers entails there are many managers who some employee pleases, but not conversely.

(D") and (P□) there are many managers who every employee pleases entails for every employee, there are many managers who he or she pleases, but not conversely.

Modal and Quasimodal Non-Quantifier Interactions

Since (à□) does not hold for the epistemic and deontic modal systems, their extensions (àD) and (D□) also do not hold.

(àD) à (D (P)) Þ D (à (P)).

(D□) D (□ (P)) Þ □ (D (P)).

The failure of (àD) is shown by the fact that à e(De (P)) and De (à e (P)) are logically independent: it is possible to be likely to be raining neither entails nor is entailed by it is likely to be possible to be raining. Similarly, à d(Dd (P)) and Dd (à d (P)) are logically independent: it is permitted that Irene should practice neither entails nor is entailed by Irene should be permitted to practice.

The failure of (D□) is shown by the fact that be likely (be certain (P)) and be certain (be likely (P)) are logically independent: it is likely to be certain that it is raining neither entails nor is entailed by it is certain to be likely to be raining. Similarly, should (must (P (i))) and must (should (P (i))) are logically independent: Irene should be required to practice neither entails nor is entailed by it is required that Irene should practice.

Quasimodal Interactions

Quasimodal Quantifier Interactions

Let Px and Sy be quasimodal quantifiers, where P and S are not necessarily distinct. Then Px (Sy (P (x, y))) and Sy (Px (P (x, y))) are logically independent; for example, the two scopally different interpretations of two girls pushed two boys do not entail each other. For example if girl A pushed boys W and X, girl B pushed boys Y and Z, and no other pushing took place, then 2x (2y (P (x, y))) (with x ranging over girls, y ranging over boys and P = pushed) is true, but 2y (2x (P (x, y))) is false. On the other hand, if boy Y was pushed by girls A and B, boy Z was pushed by girls C and D, and no other pushing took place, then 2x (2y (P (x, y))) is false, but 2y (2x (P (x, y))) is true.

Quasimodal Quantifier and Operator Interactions

(PD) Px (D (P (x))) Þ D (Px (P (x))).

(DP) D (Px (P (x))) Þ Px (D (P (x))).

(PD) is the counterpart to the Barcan formula for the interaction of the quasimodal quantifier Px with a quasimodal operator D such as be likely, and (DP) is its converse. It is easy to verify that only (PD) is correct for this interaction: there are many who are likely to be practicing entails it is likely that there are many who are practicing, but not conversely.

Negative of Quasimodals

(ØD1a) For every P, Q in S: Ø (D (P)) Ú Ø (D (Q)) Þ Ø (D (P Ù Q)), but not conversely.

(ØD1b) For every P, Q in S: Ø (D (P Ú Q)) Þ Ø (D (P)) Ù Ø (D (Q)), but not conversely.

Be Unlikely

Be unlikely is a negative of the quasimodal be likely (and is stronger than its negation not be likely). To see this, let the threshold of be unlikely (P) be 0.3. Then if be unlikely (P Ù Q) is false (i.e., if the likelihood of (P Ù Q) is greater than 0.3), then so must be unlikely (P) Ú be unlikely (Q) (the likelihood of either P or Q must be greater than 0.3). Thus be unlikely satisfies (ØD1a). Indeed, be unlikely also satisfies the condition (ð1), which is stronger than (ØD1a). A similar argument shows that be unlikely also satisfies (ØD1b).

Be likely (Ø (P)) Þ Ø (be likely (P)), but not conversely, unless the threshold for be likely (P) is 0.5, in which case be likely (Ø (P)) Û Ø (be likely (P)) and be likely (P) Û Ø (be likely (Ø (P))); similar remarks hold for be unlikely (Ø (P)). In general, Ø (be likely (Ø (P))) Û Ø (be unlikely (P)) and Ø (be unlikely (Ø (P))) Û Ø (be likely (P)).

If a negative of a quasimodal is stronger than its negation (e.g. be unlikely is stronger than not be likely, the negation of be likely), then the negation of that negative is itself a quasimodal which is weaker than the original. For example, not be unlikely is a quasimodal which is weaker than the quasimodal be likely: Irene is likely to practice entails Irene is not unlikely to practice, but not conversely.

An Application: 'Fundamental Legal Conceptions'

The legal term 'right' has both a broad and a narrow sense. Broadly understood, it is equivalent to legal 'interest', and encompasses the narrow sense of 'right' ('constitutive right'), together with 'privilege' ('permissive right' or 'freedom'), 'power' ('facultative right'), and immunity. W. N. Hohfeld (1913) Fundamental Legal Conceptions as Applied in Judicial Reasoning. Yale Law Review 23 (Reprinted in Fundamental Legal Conceptions as Applied in Judicial Reasoning, W. W. Cook, ed., fourth printing, 1966, pp. 23-64, Yale University Press; page references are to this reprinting.) pointed out that when viewed as relations between individuals, each of these types of legal interest has a converse. The converse of 'right' stricto sensu is 'duty', the converse of 'privilege' is 'no-right' (a term coined by Hohfeld in the absence of a commonly used term for this relation), the converse of 'power' is 'liability', and the converse of 'immunity' is 'disability'. He also pointed out that these relations have opposites (negations): the opposite of 'right' is 'no-right', the opposite of 'privilege' is 'duty', the opposite of 'power' is 'disability', and the opposite of 'immunity' is 'liability'. Hohfeld called these eight legal relations 'fundamental legal conceptions'.

Right, Privilege, Duty, and No-Right

Suppose that Harry has a (legal) duty toward Irene not to push her. Harry's duty, then, amounts to an obligation, which can be expressed schematically as □d (Ø (P (h, i))), where h is the obligatee as before, and i is the 'benefiter', the individual who benefits from not undergoing P by the obligatee. Moreover, Irene's correlative right not to be pushed by Harry can be expressed schematically in exactly the same way, since Harry has a duty toward Irene not to push her if and only if Irene has a right against Harry for him not to push her.

If Harry does not have a duty toward Irene not to push her, then he has a privilege against her to push her (is 'free' to push her), and Irene has 'no-right' against Harry not to be pushed by him. Schematically, Harry's privilege and Irene's corresponding no-right can be schematized Ø (□d (Ø (P (h, i)))). By the not-not equivalence for deontic modals, this schema is equivalent to àd (P (h, i)). Similarly, Harry's privilege against Irene not to pay her for services rendered and Irene's corresponding no-right against Harry to receive such payment can be schematized either as Ø (□d (P (h, i))) or as àd (Ø (P (h, i))).

Power, Immunity, Liability, and Disability

Hohfeld defined (legal) power as one's ability to alter the legal interests (rights, privileges, powers, and immunities) among individuals in particular situations. I do not attempt an analysis of legal power, and provide only the example of the power of extinguishing one's right against trespass by others resulting from abandonment, schematically àa (cause (i, " x (Ø (□d (Ø (P (x, i))))))), equivalently à a (cause (i, " x (àd (P (x, i))))), where P is 'enter one's property'. However, this example suffices to show that legal power involves the combination of the abilitative possibility modal with a deontic modal in its scope.

The converse (correlative) of power is liability. In the preceding example, the individual with the power of abandonment has the liability of permitting others access to his former property. (Of course, if the property is transferred, the new owner has all of the legal interests of the original owner, including the right against trespass.)

The opposite (negation) of power is disability. For example, someone who is not a sheriff does not have the power to sell someone else's property under a writ of execution; that is, such a person is disabled from doing so. Finally, the converse of disability is immunity; for example, every property owner is immune from having his property sold under a writ of execution except by the sheriff so empowered.