Chapter 2. The System T

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 41-43.

 

(5) The System T

ⓛ As we previously noted, Lp ⊃ p is not K-valid

 

② The System T

K + T (Lp ⊃ p)

 

※ T is often called Axiom of Necessity

 

③ Translation into quantification theory

 

④ T1 p  ⊃ Mp

 

⑤ T2 M(p ⊃ Lp)

 

⑥ p is not a rule of T

If P were a rule of T, then from it and T2 we could derive (p ⊃ Lp), but as we shall show in a moment, this is not a theorem of T.

 

⑦ T is valid on every frame <W, R> in which R is reflexive - i.e., in which, for every w ∈ W, wRw.

⑧ T is sound with regard to the class of all reflexive frames.

⑨ T-frame: a reflexive frame

a wff is T-valid iff it is valid on every reflexive frame.

 

⑩ T is a proper extension of K