Chapter 2. The System D

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 43-45.

(6) The System D

① Interpretation of L as expressing obligationess (moral necessity)

• (a) Lp ⊃ Mp is not valid in this system, since when it will then mean is that whatever ought to be the case is in fact the case.
• (b) D Lp ⊃ Mp (Denotic Interpretation)

Lp: It is obligatory that P
Mp: It is permissible that P
Lp ⊃ Mp: Whatever is obligatory is at least permissible, which souns resonable enough.

• (c) The System D

D = K + D (Lp ⊃ Mp)

 

② Translation into quantification theory

 

③ D1 M(p ⊃ p)

 

④ It is worth noting that if any wff α is a theorem of D, then so is Mα. 

1. Suppose α is a theorem of D
2. Lα                                                          1 × N
3. Lα ⊃ Mα                                                D
4.  Mα                                                         2, 3 MP

 

⑤ If any system which is an extension of K has any theorems of the form Mα, that system contains D.

1. Suppose Mα is a theorem of such a system
2. α ⊃ (p ⊃ p)                                                           PC[α/q]
3. Mα ⊃ M(p ⊃ p)                                                     2 × DR3
4. M(p ⊃ p)                                                               1, 3 MP

 

⑥ D is not a theorem of K, and T is not a theorem of D. D, however, is a theorem of T.

1. Lp ⊃ p                                                 T
2. p ⊃ Mp                                                T1
3. Lp ⊃ Mp                                              1, 2 Syll

∴ T is a proper extension of D

 

⑦ Dead Ends

• (a) There is nothing in our definition of ‘frame’ to prevent there being some worlds in a frame which cannot see any world in that frame at all.
• (b) the rule [VL] says that Lα is true in a world w iff α is true in every world that w can see, and we interpret this to mean that if there is no world at all that w can see, then Lα is (trivially) true in w, no matter what wff α may be (even if it is p ∧ ~p).
• (c) ~Lα ≡ M~α, and by [VM] any wff of form Mα can be true in w only if there is some world that w can see.

∴ ~Lα is always false in a dead end.

• (d) if a frame contains any dead end w, then D is not valid on that frame.

D: Lp ⊃ Mp 
≡ ~Lp ∨ Mp
in a dead end, Lp is 1, Mp is o
V(Lp, w)=1, V(Mp, w)=0

• (e) K has no theorems at all of the form Mα

 

⑧ Serial frames

• (a) The class of frames which contain no dead ends.
• (b) In such frames R is said to be a serial relation.
• (c) <W, R> is a serial frame iff for every w ∈ W, there is some w' ∈ W such that wRw'.
• (d) D must be valid on every serial frame.

 

⑨ D-validity

a wff is D-valid iff it is valid on every serial frame.

 

⑩ T is a proper extension of D