# Modal Logic System N (Vredenduin)

## Notes

Vredenduin was attempting to build a modal logic without the paradoxes of strict implication that Lewis's systems had. [Vredenduin, 1939, p73-74]

## Based on

The original mixes rules and axioms in the numbering. I have left the original numbers, but have pulled the rules out separately. Axioms 1-8 and rules 9-12 were taken from Langford and Lewis, axioms 13-19 were added by Vredenduin.

• Definitions
• a<=>b =def ((a=>b)&(b=>a))
• Rules
• 9: Uniform substitution (US)
• 10: Substitution of strict equivalents
• 11: Adjunction (from |- p and |- q infer |- p&q)
• 12: Modus Ponens for => (from |- p and |- p=>q infer |- q)
• Axioms
• 1: (p&q) => (q&p)
• 2: (p&q) => p
• 3: p => (p&p)
• 4: ((p&q)&r) => (p&(q&q))
• 5: p => ~ ~ p
• 6: (p=>q) => ((q=>r) => (p=>r))
• 7: ((p&p)=>q) => q
• 8: M(p&q) => Mp
• 13: (~p=>q) => (~q=>p)
• 14: ((p=>q)=>r) => ((p&~q)=>~q)
• 15: (p=>q) => ((p&r)=>(q&r))
• 16: (p=>q) => ((r=>s) => ((p&r)=>(q&s)))
• 17: p => Mp
• 18: (p=>q) => (Mp=>Mq)
• 19: (p=>q) => ~M(p&~q)

[Vredenduin, 1939, p74]

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