# System S20 [S2 superscript 0] (Feys)

## 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, ), [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

## 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]