# System S2^{0} [S2 superscript 0] (Feys)

## Notes

Zeman apparently says "S2 naught" for the system name.

## Based on

The system S2^{0} is S1^{0} plus the
M7 axiom M(p&q)=>Mp
[Zeman,
1973, p96],
[Sobociński,
1962, p52]
(Quoting [Feys,
[1950]),
[Feys,
1965, p68]

The system S2^{0} is therefore:

- Definitions
- L a =def ~M~a
- a + b =def ~(~a & ~b)
- a > b =def ~ (a & ~b)
- a => b =def ~M(a & ~b)
- a <-> b =def (p>q) & (q>p)
[Material equivalence]
- a <=> b =def ( (a => b) & (b => a))
[Strict equivalence]

- Rules
- Uniform substitution (US)
- Strict detachment (MP=>)
- Adjunction (given a, b, return a&b)
(AD)
- substitution of strict equivalents.
(EQS)

- Axioms

## Based on (Feys)

The system S2^{0} (Feys) [S2 superscript 0] is:
S1^{0}
+ axiom p=>Mp, or in other words:

- Definitions
- L a =def ~M~a
- a + b =def ~(~a & ~b)
- a > b =def ~ (a & ~b)
- a => b =def ~M(a & ~b)
- a <-> b =def (p>q) & (q>p)
[Material equivalence]
- a <=> b =def ( (a => b) & (b => a))
[Strict equivalence]

- Rules
- Uniform substitution (US)
- Strict detachment (MP=>)
- substitution of strict equivalents.
(EQS)
- Adjunction (given |- p and |- q, infer |- a&b)
(AD)

- Axioms
- M1: (p&q)=>(q&p)
- M2: (p&q)=>p
- M3:
((p&q)&r)=>(p&(q&r))
- M4: p=>(p&p)
- M5:
((p=>q)&(q=>r))=>(p=>r)
- G1: p => Mp

[Feys,
1950, p68]

[Zeman, 1973, p281]

[Hughes and Cresswell,
1968, p217]
Remind us that many of the definitions used were actually strict
equivalences in S1 as Lewis originally presented
it. -jh]

[Sobociński,
1962, p53]
(Quoting [Feys,
1950])

## Basis for

S2 = S2^{0}
+ Axiom p=>Mp
[Zeman, 1973, p281]
[Sobociński,
1962, p53]

S6^{0} = S2^{0} + MMp [Axiom S]
[Hughes and Cresswell, 1996, p364]

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This page was last modified on January 24th, 2007