Chapter 3. Iterated Modalities

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 51-53.

 

(1) Iterated Modalities

① Constructing a system stronger than T

when we ask, informally, whether Lp ⊃ LLp is valid, the issue we are raising is this: is whatever is necessary necessarily necessary? when something is necessarily so, is the fact that it is necessarily so always itself something that is necessarily so? Now this is both a disputed question and one of some obscurity, for it is not at all clear under what conditions we should say that something is necessarily necessary. It is, however, at least a reputable and plausible view that in certain well-established senses of ‘necessary’ it should be answered in the affirmative; it is, for example, plausible to maintain that whenever a proposition is logically necessary, this is never a matter of accident but is always something which is logically bound to be the case. We do not, however, need to try to settle the issue definitely here; for what we have just said about Lp ⊃ LLp is enough to give us a motive for constructing a system stronger than T, in which that formula would be a theorem, and for seeing what such a system would be like. (pp. 51-52)

 

② Lp ⊃LLp contains the sequence LL. Such sequences are known as iterated modalities.

③ By using LMI, we can obtain (LLP ⊃ Lp) or (LMp ⊃ Mp) from (Lp ⊃ p).

④ LLp ⊃ Lp is a substitution-instance of T, and is therefore a theorem of T and all its extension; so the new system would have Lp ≡ LLp as a theorem

⑤ A reduction law

• (a) An equivalential theorem such as this, which entitles us to replace some sequence of modal operators by a shorter sequence, we shall call a reduction law of any system of which it is a theorem.
• (b) Taking the reduction law Lp ≡ LLp as valid would be one way of resolving the perplexity about ‘necessarily necessary’, for we should then say that p is necessarily necessary whenever p is necessary, and not otherwise.
• (c) 

R1 Mp ≡ LMp
R2 Lp ≡ MLp
R3 Mp ≡ MMp
R4 Lp ≡ LLp

• (d) one important feature of T is that it contains no reduction laws whatsoever.

 

⑥ Theorems of T

1. Lp ⊃ p                              T
2-1. LLP ⊃ Lp                      T[Lp/p]
2-2. LMp ⊃ Mp                    T[Mp/p]

1. Lp ⊃ p                             T
2. ~M~p ⊃ p                        1, Def M
3. ~Mp ⊃ ~p                        2[~p/p]
4. p ⊃ Mp                            3, DN
5-1. Lp ⊃ MLp                     4[Lp/p]
5-2. Mp ⊃ MMp                   4[Mp/p]

Theorems of T
1. LLP ⊃ Lp   
2. LMp ⊃ Mp
3. Lp ⊃ MLp           
4. Mp ⊃ MMp    

 

⑦ So one half of each equivalence is in T already. And it would be sufficient to add the converse:

R1a Mp ⊃ LMp ( R1a is not derivable from R4a)
R2a MLp ⊃ Lp (R2a can be derived from R1a)
R3a MMp ⊃ Mp (R3a can be derived from R4a)
R4a  Lp ⊃ LLp (R4a can be derived from R1a)

※ Namely, if we have R1a as an axiom, then we can have all the reduction laws.
※※ If we have R4a as an axiom, then we can have R3 and R4, respectively. 

All this suggests the construction of two axiomatic systems, each stronger than T and one of them stronger than the other. The first of these, obtained by adding Lp ⊃ LLp (R4a) as a new axiom to T, is known as the system S4. The second, obtained by adding Mp ⊃ LMp (R1a) to T, is known as the system S5.

 

⑧ S4 and S5

S5 = T + R1a (Mp ⊃ LMp)* 

 

*with R2a, R3a, and R4a (Stronger)

 

S4= T + R4a (Lp ⊃ LLp)*

 

*with R3a (weaker)