Page 360 - Encyclopedia of Philosophy of Language

Formal Semantics
Given the definition of '<>' in terms of 'D,' all instances of the schema
Az>OA (6)
are also theorems of the system T.
These schemas T and (6) are clearly faithful to the
intuitive notions of necessity and possibility from which this discussion began. What is necessarily the case—for example, that all grandmothers are mothers of parents—is surely true; and what is true—for exam- ple, that Jane is not in fact a grandmother—is by that very fact shown to be possible. Consequently, the system K—which does not contain these plausible schemas—may seem a counterintuitive candidate for a logic of necessity and possibility. However,there are other notions of necessity and possibility to which the resources of modal logic have been applied.
2.2 Deontic, Doxastic,and Epistemic Logics
One notion that is sometimes treated in the style of modal logic is deontic necessity: what ought to be the case, or what is obligatory. The corresponding notion of possibility is: what is allowed to be the case, or what is permissible. It is certainly not true that every- thing that ought to be the case is the case. So, in 'deontic logic,' one will require to consider modal systems that do not contain schema T (see, for exam- ple, Smiley 1963; Aqvist 1984). However, to the extent that what is obligatory is ipso facto permissible, one may want to include the weaker schema D A = O A in deontic logics.
A good deal of discussion of deontic logic centers around the schema
~(QA&D~A). (7)
For in any normal deontic logic (that is, a logic that contains K, and hence rule N), schema (7) is a conse- quence of the schema
~Q(A&~A). (8)
But while schema (8) looks innocuous, schema (7) is much more controversial, since it seems to rule out the possibility of certain kinds of moral dilemmas (see Chellas 1980; ch. 6).
The resources of modal logic have also been applied to the notions of belief and knowledge. The operator 'D ' is read, for example, as 'Jane believes that,' or as 'Jane knows that.' In the latter case—epistemic logic—the schema T is, of course, correct; but it is not correct in the former case—doxastic logic.
Normal epistemic and doxastic logics both face the problem that, according to rule N, what a person knows or believes is closed under deductive conse- quence. That is, a person knows all the deductive consequences of what he or she knows, and believes all the deductive consequences of his or her beliefs. This closure property clearly does not hold for the ordinary notions of knowledge and belief; and the
difficultythusposedisknownasthe'problemoflogi- cal omniscience.' (For a discussion of differences between the logical properties of metaphysical necess- ity and of belief or knowledge, see Forbes 1988.)
2.3 Modal Logic and Provability
There is another important example of a prepositional modal system without the schema T; this is the system G, which goes beyond K in containing the axiom schema:
D(DA=>A)=>DA.
The system G is used in the study of provability in first-order arithmetic (Peano arithmetic) (see Smiley
1963; Boolos 1979; Boolos and Jeffrey 1989).
It is possible to express, within the language of arithmetic, the claim that a given sentence of that language is provable. Gddel's Second Incompleteness Theorem states that a sentence of arithmetic that expresses the claim that arithmetic is consistent is not itself provable in arithmetic. (A sentence could express theconsistencyclaimbysayingthat0=1,forexample, is not provable in arithmetic.) Furthermore, it is poss- ible to set up a scheme of translation between the modal language and the language of arithmetic, with
the property that:
Every translation of a theorem of the modal system G is provable in arithmetic.
Finally, the translation of Q A is always the sentence of arithmetic that expresses the claim that the trans- lation of A is provable in arithmetic.
Now, suppose that this modal system, G, were to contain the schema T: DAz>A. Then, a sentence of the language of arithmetic expressing the claim:
If '0 = 1' is provable in arithmetic, then 0 = 1
would be provable in arithmetic. It can certainly be proved in arithmetic that 0^1. Consequently, by modus tollens, a sentence expressing the claim that '0 = 1' is not provable in arithmetic would be provable in arithmetic. But that would contradict Gddel's Second Incompleteness Theorem.
The interpretation of the modal operator 'D ' as expressing provability—a property of sentences— escapes some objections to modal logic that are due to Quine (1976; 1960: 195-200). Quine urges that the historical foundations of modal logic involve a con- fusion between use and mention, since 'implies' prop- erly expresses a relation between sentences. The sentences that flank the '-a' of modal logic are appar- ently used, rather than mentioned; but the correct form for a statement about (strict) implication would be, for example:
'Jane is a grandmother' implies 'Jane is the mother of a parent,'
where the two sentences about Jane are mentioned,
338