System S2⁰ [S2 superscript 0] (Feys)

[Feys, 1950] (Original Source)
[Hughes and Cresswell, 1996]
[Zeman, 1973]
[Sobociński, 1962]

Notes

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

Based on

The system S2⁰ is S1⁰ plus the M7 axiom M(p&q)=>Mp [Zeman, 1973, p96], [Sobociński, 1962, p52] (Quoting [Feys, [1950]), [Feys, 1965, p68]

The system S2⁰ 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
- 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)
- M7: M(p&q)=>p

Based on (Feys)

The system S2⁰ (Feys) [S2 superscript 0] is: S1⁰ + 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 = S2⁰ + Axiom p=>Mp [Zeman, 1973, p281] [Sobociński, 1962, p53]

S6⁰ = S2⁰ + MMp [Axiom S] [Hughes and Cresswell, 1996, p364]

System S20 [S2 superscript 0] (Feys)

Notes

Based on

Based on (Feys)

Basis for

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System S2⁰ [S2 superscript 0] (Feys)