ON DEFINING MODALITY FERENC KIEFER 1. INTRODUCTION The term 'modality' has been used to refer to a relatively great number of apparently disparate things in linguistic and logical lit- erature. Sometimes modality has been restricted to expressions of possibility and necessity, in other cases it has been used äs a cover term for expressions of Speakers' attitudes. Modality may be iden- tified with what expressions of propositional attitudes do in sen- tences,butit may also betaken to cover the meanings of a linguisti- cally definable class of elements (modal auxiliaries, modal adver- bials, modal particles, parenthetical verbs, etc.). The logical tradition concentrates on the description of proposi- tions. Consequently, the inquiry into modality is restricted to what may be considered to be 'propositional' in modality. The linguistic tradition, on the other hand, emphasizes the 'non-propositional' aspects of modality. It is pointed out that modal expressions in natural language are mainly used to express the speaker's attitudes toward states-of-affairs. In the present paper I wish to make explicit some of the notions of modality — I hope the most important ones — which underlie current inquiries into the problems of modality. The notions which I am going to discuss are the following: (i) modality äs an expression of possibility or necessity, (ii) modality äs the meaning of propositional attitudes, (iii) modality äs expressions of speaker's attitudes. I am going to show that the logical tradition and the linguistic tradition need not be in conflict, that there is a way to reconcile the two traditions. 0165-4004/87/0021-67 $ 2,- © Moutpn Publishers, The Hague Societas Linguistica Europäer 68 2. MODALITY AS AN EXPRESSION OF POSSIBILITY OR NECESSITY The notions of 'necessarily true proposition' and 'possibly true proposition' are well-known in traditional logic.1 A proposition is necessarüy true or necessarily false if its truth or falsity does not depend on the way the world actually is, that is, if its truth or fal- sity is already guaranteed by the very meaning of the proposition. On the other hand, a proposition is possibly (or contingently) true if it is not necessarily false and it is possibly false if it is not neces- sarily true. Though they represent distinct categories, necessity and possibility are not independent notions. Either one can be elimi- nated in favor of the other one. If we denote the necessity operator by and the possibility operator by 0 we may formulate the relationship between necessity and possibility äs follows. (1) (a) D p = ~ 0 ~ p (b) 0 p = ~ D ~ p In addition, the following two implications are valid. (2) (a) D p => p (b) p =* 0 p That is, from the fact that p is necessarily true it follows that p is true, and from the fact that p is true it follows that p is possibly true. The type of modality which is exclusively concerned with the necessary or contingent truth of propositions is often referred to äs logical or alethic modality. Examples for alethic modality with must and can are given in (3) (a) —(b). (3) (a) Cats must be animals. (b) Cats can be dangerous. By making use of the equivalencies in (1) (a)—(b) the sentences in (3) (a)—(b) can be paraphrased äs (4) (a) —(b), respectively. (4) (a) It cannot be the case that cats are not animals. (b) It is not necessarily the case that cats are not dangerous. 69 Notice that the proposition Cats are animals is analytic, hence necessarily true. On the other band, the proposition Cats are dan- gerous is not analytic (it is contingent), hence possibly true. Alethic modality is a central notion in classical logic, in ordinary language, however, it is rather peripheric.2 Clear cases of alethically modal sentences do not occur frequently in everyday discourse. From a linguistic point of view the notions of epistemic and deontic modality are nrnch more important. Epistemic necessity and epistemic possibility are expressed by contingent propositions, i.e. not only 0 p but also depends on what the world is like. The proposition p is epistemically necessary iff p is entailed by what the Speaker knows about the world and the proposition p is epistemically possible iff p is compatible with what the Speaker knows about the world. The speaker's knowledge is representable äs a set of propositions.3 Epistemic necessity and epistemic possibility are interdefinable in the same way äs alethic necessity and alethic possibility are, i.e. the equivalence relations in (1) (a) —(b) hold with equal force in the case of epistemic modalities äs well. In addition, the implications (2) (a) —(b) remain valid. The following sentences may be illustrations of epistemic pos- sibility and epistemic necessity, respectively. (5) (a) It can be raining in Budapest, (b) It must be raining in Budapest. The meaning of (5) (a) can be circumscribed in the following fashion: "In view of what I know it is not excluded that it is raining in Budapest". The sentence (5) (b), on the other hand, may be given the following Interpretation: "From what I know it follows that it is raining in Budapest". The deontic modalities have to do with Obligations and permissions* Instead of the modal operators D and 0» deontic logic exploits two deontic operators usually denoted by O (for 'obligatory') and P (for 'permitted'). Something is considered to be obligatory or per- mitted with respect to some moral or legal System (with respect to a System of laws). Ä proposition^? is obligatory iff it is not permitted that ~ p and it is permitted iff it is not obligatory that ~ p. These are the deontic counterparts of the equivalence-relations (1) (a) —(b) In deontic logic, however, the implications (2) (a) —(b) are not valid. 70 Consider (6) (a) Op ^> p (b) p^Pp That is, from the fact that p is obligatory it does not follow that p is the case, and from the fact that p is the case it does not follow that p is permitted. The sentence (7) (a) may be considered äs expressing deontic possibility and the sentence (7) (b) äs expressing deontic necessity. (7) (a) Bill can go to Stockholm, (b) Bill must go to Stockholm. The close relations between deontic and modal concepts are also reflected in natural language. In most languages, the expressions of certainty, necessity and possibility are also used for obligations and permissions. The modalities discussed so far do not exhaust all possibilities. Thus, for example, possibility and necessity in the following sen- tences may have to do with the speaker's desires (boulomaic modal- tty). (8) (a) Bill can be our leader. (b) Bill must be our leader. Dlspositional modality refers to certain dispositions äs illustrated in (9) (a)-(b). (9) (a) Bill can sneeze. (b) Bill must sneeze. That is, (9) (a) may be taken to mean that in view of Bill's dispo- sitions it is not excluded that he sneezes and (9) (b) may mean that in view of BilFs dispositions it is necessary that he sneezes.5 In view of the various modalities it is important to look for a general framework which can be made use of in the definitions of all these modalities. Such a general framework is provided by possible world semantics? Possible worlds meeting a certain condition are called accessible possible worlds. The following def initions hold: 71 p is necessarily true iff p is true in all accessible possible worlds and p is possible iff there is at least one accessible possible world in which p is true. In the case of alethic modality all possible worlds are accessible from any possible world. As for epistemic modality, given a possible world w, those pos- sible worlds wl are accessible from w that are identical with respect to certain facts (what somebody knows to be the case). In the case of deontic modality the set of accessible possible worlds is the set of all worlds in which everybody acts according to some moral or legal System. Finally, in the case of boulomaic modality the accessible possible worlds are defined by the Speaker's desires: the accessible worlds are those that are compatible with the speaker's desires.7 In this way any number of modal concepts can be defined. Gen- erally, each modality will.correspond to a certain accessibility or alternativeness relation.8 It has also been argued that this relation can be replaced by the notionof background.9rfhus, what the Speaker knows defines an epistemic background, a moral or legal System defines a deontic background, the speaker's desires define a boulo- maic background, etc. The concepts of possibility and necessity can then be reconstructed in terms of logical entailment and logical compatibility. The following definitions hold: p is necessarily true iff it is logically entailed by the background and p is possibly true iff it is logically compatible with the background. To summarize, the concept of modality can be explicated in the following fashion. The modality of p means that p is necessarily true or false, or possibly true or false with respect to a certain background. Or, alternatively, the modality of p means that p is necessarily true or false, or possibly true or false in a certain set of possible worlds. This set is defined by the accessibility relation considered. For further reference we shall call this definition of modality the logical definition of modality. The logical definition of modality can easily be extended to cover graded possibilities äs well.10 We have todo with graded possibil- ities in the following sentences.