# 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]

## Basis for

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