2 Modal System K

Modal logic is not the name of a single logical system;there are a number of dierent logical systems thatmake use of the signs and , each with its ownset of rules.Each system reflects a dierent understanding or

conception of and , or the logical properties ofnecessity and possibility.We beginwith SystemK, which forms the basis for

many other systems. (I.e., those systems are formedby adding additional rules to K.) System K is namedafter the logician-philosopher Saul Kripke, who wasamong the first to systematize formal semantics formodal logic.The rules of K are consistent with many dier-

ent understandings of and (including, amongothers, deontic ones).It should perhaps be noted that Garsons method

for formalizing K is dierent from the usual way (orKripkes way), but it results in the same theoremsand derivability results.

2.1 Syntax

The logical language of SystemK is derived from thatof PL by adding a single primitive monadic proposi-tional operator, .It behaves syntactically precisely as does. Its

scope is only what immediately comes after it.p q means if not p then q,, whereas (p q) means its not true that if p then q. Similarly,p q means if, necessarily, p, then q,, whereas(p q) means necessarily, if p then q.The sign for possibility is not taken as primitive,

but defined in terms of ; well also define signs forso-called strict implication and strict equivalence.

(Def ) A abbreviates A(Def ) (A B) abbreviates (A B)(Def) (A B) abbreviates (A B) & (B A)

The sign was introduced by C.I. Lewis in hispioneering work onmodal logic, much of which wasmotivated by the desire to define a logical notionof if. . . then. . . that did not share the oddities ofthe material conditional. (A B) can be read, Astrictly implies B, or A entails B.

2.2 Boxed Subproofs

The inference rules of K involve the notion of a boxedsubproof, which is a subproof that is headed notwitha hypothesis, but rather simply with the sign .Informally, the-header indicates that every line in

the subproof is shown to be necessarily true.We cannot bring statements into boxed subproofs using

the (Reit) rulethat would result in our regardingevery true or assumed-true statement as necessarilytrue. Instead we have another rule, (Out).

(Out) if a line of the form A is available in thesubproof previous to a given boxed subproof,one may infer Awithin the boxed subproof.

The purpose of a boxed subproof is to be able toprove statements of the form A. Hence we have:

(In) if a statement A is derived within a boxedsubproof, the boxed subproofmay be closed andone may infer A outside of that subproof.

System K consists only of the two rules above alongwith the rules of PL.As an example proof in K, below we prove that

every instance of theDistribution schema is a theoremof K:

(Dist) K (A B) (A B)Proof:

1. (A B)2. A

3. (A B) 1 (Reit)4.

5. A 2 (Out)

6. A B 3 (Out)7. B 5,6 (MP)

8. B 47 (In)

9. A B 28 (CP)10. (A B) (A B) 19 (CP)

Using boxed subproofs and PL rules, it is possible toderive A in K for any truth-table tautology A. Forexample, here is a proof of (p (q p)):

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1.

2. p

3. q

4. p 2 (Reit)

5. q p 34 (CP)6. p (q p) 25 (CP)7. (p (q p)) 16 (In)

Indeed, any theorem of K (something that can beproven without use of premises or undischarged hy-potheses) can be proven to be necessary in K.

The Rule of Necessitation (Nec): if K A thenK A.

Indeed, the usual formulation of K is to take (Nec) asa primitive inference rule alongwith every instance of(Dist) as an axiom (i.e., something that can be writteninto any line of a proof or subproof with no other lineas justification); these two together are equivalent tohaving (In) and (Out).The rule of necessitation coheres with almost all

conceptions of necessity: if something can be provenusing the rules of logic alone, then it must be neces-sary.Do not confuse this rule with the untrue claim that

K A A.Indeed, those conceptions of necessity that are con-

sistent with the rules of System K can roughly becharacterized by two features: (i) what can be proventrue without assumptions is necessary (i.e, (Nec)),and (ii) that which necessarily follows from neces-sary truths is also necessary (i.e, (Dist)).Youll notice that fromhere onout, Ill almost never

giveproofs involving lowercase statement letters likep, q and r, but rather schematic letters like A, Band C. This makes the results proven general sothat they can be applied in other contexts. Moreover,if all instances of a certain schema are shown to betheorems, we shall allow ourselves to introduce oneinto any proof at any time, rather than duplicate thesteps of the proof already given.

2.3 Inferences Involving PossibilityBecuase is defined in terms of , no special basicrules are needed for .Here are some easy derived rules:

() From A infer A. (Really just (DN)).() From A infer A. Proof:

1. A2. A3. A 1 (Reit)4.

5. A 2 (Out)6. A 5 (DN)

7. A 46 (In)

8. 3, 7 (In)9. A 28 (IP)10. A 9 (Def )

Because such statements are actually negations ofstatements with , if occurs on both sides of aconditional, it may be easier to prove the contrapos-itive of the result youre after.

(Dist) K (A B) (A B). Proof:

1. (A B)2. B3. (A B) 1 (Reit)4.

5. B 2 (Out)6. A B 3 (Out)7. A 5,6 (MT)8. A 47 (In)9. B A 28 (CP)10. A B 9 (CN)11. A B 10 (Def )12. (A B) (A B) 111 (CP)

The above theorem schema is also important forobtaining the following derived rule, representedschematically:

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A

A...

B

B (Out)

In short, if we know that A is possible, and withina boxed subproof, can prove that B would be true ifwe assumedAwere true, we are allowed to concludethat B is possible. This is because we cold fill in theremainder of this proof schema as follows:

A

A...

B

A B (CP)(A B) (In)(A B) (A B) (Dist)A B (MP)B (MP)

Proofs of this formare quite common. So common, infact, that it isworthwhile to introduce a further abbre-viated notation for representing the two subproofsinvolved in an application of (Out) as though itwere a single subproof. Garson does this as follows:

,A

...

This kind of subproof is called a world-subproof. Ifwe understand necessity as truth in all possibleworlds, a normal boxed subproof represents whatwe can prove about all possible worlds absolutely. Aworld-subproof represents what we can prove aboutall those possible worlds in which a certain hypoth-esis holdsin this case, all those where A is true.

We shall generalize this style of abbreviation toother contexts in which we have subproofs withinsubproofs, so that, for example:

A,B,C...

D

A (B (C D)) (CP)

can be used as a convenient shorthand for three em-bedded conditional proofs of the form:

A

B

C...

D

C D (CP)B (C D) (CP)

A (B (C D)) (CP)

One must simply be careful with regard to (Reit)and (Out) for subproofs with multiple headers.(Reit) cannot be used with any header list containing. . . ,, . . ., and (Out) can only traverse past one such, unless it is applied to something of the form Aor A, etc.If it is convenient to have the multiple headers on

separate lines (so that they can be cited separately,for example), we also use the abbreviated notation:

A

B

C...

D

to the same eect.Heres (Out) in action. Lets show:(Dist&) K (A & B) (A & B)

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1. (A & B)

2. ,A & B

3. A 3 (&Out)

4. A 1,23 (Out)

5. ,A & B

6. B 5 (&Out)

7. B 1,56 (Out)

8. A & B 4,7 (&In)

9. (A & B) (A & B) 18 (CP)The converse of the above, however, does not hold,

but we do have:

K (A & B) (A & B)1. A & B

2. A 1 (&Out)

3. B 1 (&Out)

4. ,B

5. A 2 (Out), (Reit)

6. A & B 4,5 (&In)

7. (A & B) 3,46 (Out)

8. (A & B) (A & B) 17 (CP)

3 Extensions of K (T, B, S4, S5)Logical systems capturing stronger or at least morespecific conceptions of necessity can be obtained byadding additional inference rules or axioms to K.Consider the following schemata:

(M): A A(B): A A(4): A A(5): A AAccording to (M), whatever is necessary is true. This

axiom should hold for any system codifying a con-ception of alethicmodality, where necessity is under-stood in terms of necessary truth. An obvious de-rived rule, which well also abbreviate as (M), allows

us to concludeA fromAwithin the same subproof.(Notice that rule (Out) ofKdoes not allow this.) Mstands for modality, where this word is understoodin the narrow sense.According to (B), what is actually true is necessarily

possible. This is named after L. E. J. Brouwer.According to (4), if something is necessary, it is nec-

essarily necessary. This is one of the original axiomsused by C. I. Lewis in codifying modal logic, andretains the number used in his numbering of axioms.According to (5), if something is possible, it is neces-

sarily possible. This, the strongest of these principles,coheres with the understanding of necessity as truthin all possible worlds absolutely. For something tobe possible is for it to be true at some world. If it istrue at some world, it is true at all worlds that it istrue at some world, which is what it is for it to benecessarily possible. Again, the numbering comesfrom Lewis.We can define a number of systems by adding the

instances of one or more of the above to K:

System T: the system obtained from K by allowingevery instance of (M) as an an axiom.

SystemM: another name for System T.

System B: the system obtained from T by allowingevery instance of (B) as an axiom.

System S4: the system obtained from T by allowingevery instance of (4) as an axiom.

System S5: the system obtained from T by allowingevery instance of (5) as an axiom.

Note that these systems add every instance ofcertain axiom schemata, not individual axioms.Axiom (M) covers not only

p pbut everything of that form, including

p pp p(p q) (p q), etc.In the first proof scheme below, we shall writeA A, and cite (M). This is not confusinga statement with its negation but rather making useof a more specific schema all of whose instances arealso instances of a more general schema.

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