My Approach to Modal Logic - Definitions

by Dennis J. Darland
February 17, 2008
Last revised 25.02.2008 14.06 time
Copyright © 2008 Dennis J. Darland

Preliminary Definitions

1. Non-logical vocabulary from Universal point of view.
   
   • K₀ lowest type(individuals)
   • K₁ 1st order predicates
   • K₂ 2nd order predicates
   • Etc.
2. Variables from Universal point of view.
   
   • x₀,y₀,...∈ V₀ lowest type(individuals)
   • f₁,g1,... ∈ V₁1st order predicates
   • f₂,f₂,... ∈ V₂ 2nd order predicates
   • Etc.
3. Logical vocabulary
   • ¬ not [primitive]
   • ∧ and [primitive]
   • ∨ or [can be defined]
   • ⇒ materially implies [can be defined]
   • ⇔ iff [can be defined]
   • ∀ for all [primitive]
   • ∃ there exists [can be defined]
   • = equals [not sure if primitive or can be defined]
4. Modal vocabulary
   • L Necessarily [primitive]
   • M Possibly [can be defined]
5. Epistemic vocabulary [the B relations here are the belief_r relations in definition of belief and W is for knows [Wissen]][All primitive].
   • Sym₀ is symbol type 0. [E.g. Sym₀(S,now,n,o) where n ∈ K_S0 and o ∈ K₀
     [Note: there may be x_m ∈ K_m for which there is no corresponding name ∈ K_Sm.
     Likewise there may be names [or apparent names if you prefer] ∈ K_Sm for which there is no object ∈ K_m.
     
   • B₁ believes type 1
   • W₁ knows type 1
   • Sym₁ is symbol type 1
   • B₂ believes type 2
   • W₂ knows type 2
   • Sym₂ is symbol type 2
   • Etc.
6. Subject S
7. Subject S's Non-logical vocabulary
   • K_S0 names of lowest type(individuals)
   • K_S1 names of 1st order predicates
   • K_S2 names of 2nd order predicates
   • Etc.
8. Variables from Subject S's point of view.
   
   • x_S0,y_S0,... ∈ V_S0 lowest type(individuals)
   • f_S1,g_S1,... ∈ V_S1 1st order predicates
   • f_S2,f_S2,... ∈ V_S2 2nd order predicates
   • Etc.[any of these may have primes]

Formation Rules

1. If p and q are formulas then
   ¬ p
   ∨ p q
   ∧ p q
   ⇒ p q
   ⇔ p q
   are formulas.
2. If v ∈ K_m and w^N is a sequence w¹,w²,...,w^N ∈ K_n or ∈ V_n and n < m then
   vw^N is a formula.
3. If p is any formula and x_m is a variable [from Universal Point of View] then
   ∀ x_mp
   ∃ x_mp
   are formulas.
   
4. If p is a formula then
   Lp
   Mp
   are formulas.
   
5. If u^M ∈ K_Sm or ∈ V_Vm and v^N ∈ K_Sn or ∈ V_Vn and w^M is a sequence w¹,w²,...,w^M ∈ K_Si or ∈ V_Si> and i < m and x^N is a sequence x¹,x²,...,x^N ∈ K_Sj or ∈ V_Sj> and j < n then
   (S,t,u^M,w^M) is a b_formula.
   (S,t,¬,u^M,w^M) is a b_formula.
   (S,t,∧ u^M,w^M,v^N,x^N) is a b_formula.
   (S,t,∨ u^M,w^M,v^N,x^N) is a b_formula.
   (S,t,⇒ u^M,w^M,v^N,x^N) is a b_formula.
   (S,t,⇔ u^M,w^M,v^N,x^N) is a b_formula.
   
6. If p is any b_formula and x_Sm ∈ V_Sn and n < m then
   (S,t,∀,x_Sm,p) is a b_formula.
   (S,t,(∃,x_Sm,p) is a b_formula.
   (S,t,L,p) is a b_formula.
   (S,t,M ,p) is a b_formula.
   
7. If p is a b_formula with no free variables [from S's Point of View - it may contain variables from a universal point of view less than order m] then
   B_Smp is a formula.
   
8. If p is a formula with no free variables then it is a sentence.

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