System S4⁰ (S4 superscript 0) (Sobocinński)

• [Zeman, 1973]

• [Sobociński, 1962] (Original source)

Notes

Based on

The system S4⁰ (S4 superscript 0) is S1⁰ plus the axiom M9 MMp=>Mp [Zeman, 1973, p281]

Therefore the system S4⁰ is:

• 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)
  • M9: MMp=>Mp

[Zeman, 1973, p281]

Or

The system S4⁰ is S1⁰ + the axiom ~M(MMp&~p) [Feys, 1965, p114]

Basis for

S4 = S4⁰ + Axiom M6:p=>Mp [Zeman, 1973, p281]

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