# System K

[Hughes and Cresswell, 1996]

[Chellas, 1980]

[Sobociński, 1964]

[Sobociński, 1964a]

## Notes

K is characterized by irreflexive frames. [Hughes and Cresswell, 1996, p176]

K is the weakest Normal system. [Chellas, 1980, p236] (And a system that contains K is normal -JH)

With an appropriate mapping, any Non-degenerate Normal system is inherently a paraconsistent logic (In the same sense that intuitionist logic can be mapped onto S4) [João Marcos, "Modality and Paraconsistency", 2004]

K is also called EMCN, MCN, RN [Chellas, 1980, p236]

## Based on

The system K (named after Kripke) is the result of PC + K axiom[L(p>q)>(Lp>Lq)] + the primitive rule of Necessitation.

In other words, it is:

• PC
• Definitions
• Ma is ~L~a
• Rules
• Axioms
• K: L(p>q)>(Lp>Lq)

[Hughes and Cresswell, 1996, p25]

K is also System E + Axiom M [L(p&q) > (Lp & Lq)] + Axiom C[(Lp&Lq) > L(p&q)] [Chellas, 1980, p237]

K is also System E + Axiom N: L(true) [Chellas, 1980, p237]

K is System R + Axiom N: L(true) [Chellas, 1980, p237]

K is System EMN + Axiom C[(Lp&Lq) > L(p&q)] [Chellas, 1980, p237]

K is System ECN + Axiom M [L(p&q) > (Lp&Lq)] [Chellas, 1980, p237]

## Basis for

K + the D axiom (Lp>Mp) gives the system D (Lemmon and Scott). [Hughes and Cresswell, 1996, p43]

K + the T axiom (Lp>p) gives the system T. [Hughes and Cresswell, 1996, p42]

K + the axiom W [L(Lp>p)>Lp] = KW [Hughes and Cresswell, 1996]