Page 361 - Encyclopedia of Philosophy of Language

rather than used. Similarly, Quine urges that the notion of necessity itself is innocent of confusion only if it is construed as a property of sentences.
Montague (1963) proves that, even in a very weak theory of arithmetic, no predicate of sentences can be awarded the basic properties of the modal operator 'D ' without resulting in inconsistency. Montague's result certainly shows that no predicate of sentences of arithmetic can consistently have the properties enjoyed by 'D ' in the system T; and he concludes that the interpretation of the modal operator as a predicate of sentences would involve sacrificing the greater part of modal logic. Work on the system G demonstrates, however, that a substantial and interesting system of modal logic—though not including T—can be based upon an interpretation of 'D' as a predicate of sentences. Whether Quine's objections should be regarded as casting suspicion on other systems of modal logic is, of course, a further question.
2.4 Modal Systems that Extend T
The language of prepositional modal logic permits a modal operator to occur embedded within the scope of another modal operator. Thus, for example, instances of the schema D(Av<>B) are WFFS of modal logic. In particular, modal operators can occur alongside one another to form 'iterated modalities': DQA, DOA,ODA,OOA,andsoon.
All instances of the schemas o n A D O A and DOA=> OA are already instances of the schema T. The two best-known prepositional modal systems extending T are obtained by adding to T the converses of these schemas. Thus, S4 is the system that results by adding to T the schema
S4. nA=>nnA,
and S5 is the system that results by adding to T the schema
S5. OA=DOA.
The S4 schema is in fact a theorem schema in S5, so that the system S5 contains the system S4. In fact, S5 can be characterized as the system that results by adding to S4 the schema
B. Az>QOA.
The schema B is called the 'Brouwersche' axiom because of connections with Brouwer's work on the foundations of intuitionist mathematics. If the schema B is added directly to T, then the resulting system is B. Thus, S4 and B are systems that each extend T, and S5 is the smallest system that includes them both.
In S3, iterated modalities collapse into the final member of the string. Thus, for example, ODDOODOA=OA. This isbecause S5contains the equivalences:
nnA=nA (9)
DOA = OA (10) ODA = DA (11) OOA = OA. (12)
Schemas (9) and (10) are immediate from T and S4, and T and S5, respectively. To establish schema (11), consider that by T one obtains DA=>OQA; so one only needs to prove O QAr> £]A. It suffices to prove the contrapositive, ~ QA=> ~ ODA. Because of the way that 'O ' and 'Q ' are related, ~ QA isequivalent to O ~ A; likewise, ~ O D A is equivalent to D O ~ A. Consequently, it suffices to prove
o~A^no~A.
But all instances of this schema are already instances of the schema S5; so we are done. Schema (12) is established similarly, but using S4 instead of S5.
In fact, S5 has an even stronger property. Any WFF of S5 is equivalent to a WFF in which no modal oper- ator occurs within the scope of another modal oper- ator. For example, D(A v OB), in which 'O ' occurs within the scope of'D/ isequivalentto QA v OB, in which neither modal operator is within the scope of the other (for all cases of this reduction, see Hughes and Cresswell 1968: 50-54).
The systems K, G, T, B, S4, and S5 are just a few from the host of prepositional modal systems that have been studied (Hughes and Cresswell 1968; Bull and Segerberg 1984). There have been, for example, detailed investigations of many systems that are inter- mediate between S4 and S5. As it happens, the schema S4 is also a theorem schema of the system G (Boolos 1979: 30), although G—as already noted—does not contain T. The system intermediate between K and G is K4, resulting by the addition of the schema S4 directly to K. Further systems could be obtained by adding the schema B or S5 directly to K.
Despite this great variety of modal systems weaker than S5, it is very plausible that the axioms of S5 are all intuitively valid for the notion of necessity that has loomed large in philosophical work: metaphysical necessity (Kripke 1980), or 'broadly logical' necessity (Plantinga 1974). Consequently, S5 is widely used as the formal basis for philosophical discussions of necessity.
3. Model-theoretic Semantics for Prepositional Modal Logic
The semantics for nonmodal prepositional logic is very simple. A model or valuation is determined by an assignment of truth-values to sentence letters. By way of the familiar truth-tables for the connectives, the assignment is extended to evaluate each WFF as true or false. A WFF is then said to be 'valid'—or a tautology—if it is evaluated as true on every assign- ment: if it is true in every model. An argument from A,,..., An to Bisthen said to bevalidifthe conclusion
Modal Logic
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