# 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
- Rules
- Axioms

[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]

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© Copyright 2007, by John Halleck, All Rights Reserved.

This page is http://www.cc.utah.edu/~nahaj/logic/structures/systems/k.html

This page was last modified on July 15th, 2008