System S3⁰ [S3 - superscript 0] (Sobociński)

• [Hughes and Cresswell, 1996]
• [Zeman, 1973]
• [Sobociński, 1962]

Notes

Zeman apparently says "S3 naught" for the system name.

Based on

The system S3⁰ is S1⁰ plus the M8 axiom (p=>q)=>(Mp=>Mq) [Zeman, 1973, p281] [Sobociński, 1962, p53]

The system S3⁰ is therefore:

• Definitions
  • ?
• 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)
  • M8: (p=>q)=>(Mp=>Mq)

[Zeman, 1973, p281]

Basis for

S3 = S3⁰ + Axiom M6 p=>Mp [Zeman, 1973, p281]

The system S8 is: S3 + SP [LMMp] [Hughes and Cresswell, 1996]

S7⁰ = S3⁰ + MMp [Axiom S] [Hughes and Cresswell, 1996, p354]

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