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 Þ □R) 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.

Interdefinability of Necessity and Possibility Modals

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

(Ø□Ø) 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)

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 (Ø□Ø) and (ØàØ), and satisfy (□Þà).

(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 (Ø□Ø) and (ØàØ), and satisfy (□Þ à).

(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.

(□lÞ□e) For every P in S: □l (P) Þ □e (P), but not conversely.

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

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).

(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.

□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 º 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

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

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

□n(P(i)) Irene needs to push Harry

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

□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)

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

□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

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

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 (P (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

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) Ú □p (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

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))).

("" ) "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))).

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

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

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.

Existential Quantifier and Possibility Modal

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

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

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.

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.)

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 (P (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.

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.

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.

($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.

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.

(P□e) 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.

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.

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.

(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.

(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 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)).

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))).

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.