Page 359 - Encyclopedia of Philosophy of Language

For the material implication ' = ' of classical logic, the schemas in (1-3) are all valid.
Ar>(B=>A) (1) ~Ao(A^B) (2) * (A=B)v(B =A) (3)
However, for the intuitive notion of entailment, or strict implication, one may first consistently accept that Jane is a grandmother (A), while denying that the proposition that penguins waddle (B) entails that Jane is a grandmother (A). Second, one may consistently accept that Jane is not a politician ( ~ A ) while denying that the proposition that Jane is a politician (A) entails that penguins fly (B). And third, one may consistently deny that the proposition that Jane is a politician (A) entails that penguins waddle (B) while also denying that the proposition that penguins waddle (B) entails that Jane is a politician (A). In short, the three schemas that are valid for material implication are intuitively not valid for entailment, or strict implication. (Schemas (1) and (2) are sometimes called the 'para- doxes of material implication.') Modal logic was designed to permit the formalization of statements about strict implication, such as the statement that the proposition that Jane is a grandmother strictly implies that Jane is the mother of a parent.
The notation of modal logic results from adding to the notation of ordinary propositional or predicate logic the one-place sentence operators 'D ' (box) and 'O ' (diamond), and the two-place sentence operator '-3' (fish-hook), expressing necessity, possibility, and strict implication, respectively. Recall that in the case of propositional logic one can define ' v ' in terms of '&'and'~,'forexample.Similarly,inthecaseofthe modal operators, one does not have to take all three as newprimitive operators. If'D ' istaken as primitive, thenonecandefine'OA'as'~D~A'and'A-sB'as 'D(A = B).' Similarly, if 'O ' is taken as primitive, then o n e c a n define ' D A ' a s ' ~ O ~ A ' a n d ' A - s B ' a s '~O(A&~B).' Finally, if '-3' is taken as primitive, then one can define 'DA' as '((Av ~A)-aA)' and 'O A 'as'~(A -a(A & ~ A )).'
2. PropositionalModalLogic
The language of propositional modal logic is the result of adding some of the modal operators to the language of propositional logic. In fact, 'D ' will here be added as the only new primitive operator; and the definition of 'well-formed formula' (WFF) will be extended by allowing 'D ' the same privileges of occurrence as' ~ .'
2.1 The Systems K and T
There are many different systems of propositional modal logic, differing in their proof-theoretic resources. But all so-called 'normal' modal prop- ositional systems are based upon a common core. This core is made up of three components. First, there is
some complete proof system for classical prep- ositional logic, including the rule of modus ponens:
From: AandA:=>B Infer: B.
Second, there is an axiom schema: K. D(A=>B)=j(nA3DB).
Third, there is a rule of proof, called 'Necessitation': If h A, then h QA.
Here, the turnstile 'X hY' is to be read as 'there is a proof of Y from assumptions X.' The rule of Neces- sitation thus says that if there is a proof of A from no assumptions, then there is also a proof of DA from no assumptions.
It has already been seen that one can prove all instances of the schemas (1) and (2)—the paradoxes of material implication—from no assumptions, given propositional calculus resources alone. Consequently, one can prove all instances of D(Ai3(B=>A)) and D(~A=>(A^B)) by the rule of Necessitation. Thence, by the schema K and the definition of '-a,' one obtains (4) and (5):
DA = (B-3A) (4)
D~A:=>(A-3B). (5)
These schemas are known as the 'paradoxes of strict implication.'
The rule of Necessitation certainly does not say that from A as an assumption one can prove DA. If it did, then it would trivialize modal logic. However, given the rule of Necessitation and the axiom schema K, one can derive a rule about proofs from assumptions, namely:
N. IfA,,...,AnhB,thenDA,,...,QAnhDB.
For suppose that A , , . . . , An h B. Then, just as in non- modal propositional logic, one obtains
h(A1=.(...(An = B)...)).
Thence, by the rule of Necessitation,
hn(A, = (...(AnaB)...)).
By n —1 applications of K, one obtains
i-DA,=(...(0^=08)...);
and so, by n—1 applications of modus ponens, DA,,...,DAn I-DB, as required. Rule N says that the class of necessary truths is closed under deductive consequence.
The propositional modal system which contains just the core comprising propositional logic, Neces- sitation, and the schema K, is itself called 'K .'
The first extension of K to consider is the system T, which results by adding to the system K the further axiom schema:
T. QA=>A.
Modal Logic
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