System L (Emch)

Notes

This system appeared in Volume 1, number 1 of the Journal of Symbolic Logic, a claim to odd fame.

I can't actually find that Emch explicitly named this system in that paper. But all the definitions are Lxx and the axioms Ly, so "L" seems logical. And at least one later paper [Lewis, 1936, p77] distinguished "Emch's L" from "Lewis' S", and Emch called it "L" in a number of other papers including [Emch, 1937, p78]

Based on

The system L has a lot of symbols I can't accurately represent on the web pages at the moment. In addition he uses concatenation for "and". So... I'm going to transliterate as follows: I'm using:

• & for and
• ~ for not
• + for or (as in L01)
• ~> for logicly consistent (as in L02)
• === for his "big equals" (as in L03)
• => for Strict implication (as in L04)
• = for equals (as in L05)
• > for material implication (as in L06)
• == for material equivalance (as in L07)

The system L (Emch) is:

• Special operators:
• Op = p is logicly consistent.
• Mp = p is possible
• Definitions
• L01: p + q ==def ~(~p&~q)
• L02: p ~> q ==def ~O(p&~q)
• L03: p===q ==def (p~>q) & (q~>p)
• L04: p=>q ==def ~M(p&~q)
• L05: p=q ==def (p=>q) & (q=>p)
• L06: p>q ==def ~(p&~q)
• L07: p==q ==def (p>q) & (q>p)
• Rules
• Uniform substitution (US)
• Modus Ponens for ~> (MP~>)
• Adjunction for &(given a, b, return a&b) (AD&)
• substitution of strict equivalents. (EQS)
• Axioms
• L1: (p&q) ~> (q&p)
• L2: p ~> (p&p)
• L3: ((p&q)&r) ~> (p&(q&r))
• L4: p ~> ~~p
• L5: ((p~>q)&(q~>r)) ~> (p~>r)
• L6: (p&(p~>q)) ~> q
• L7: (Mp&Mq) ~> (Op&Oq)
• L8: M(p&q) ~> Mp
• L9: exists(p, exists(q, M(p&~q) & M(p&q) ))
• L10: exists(p, exists(q, M[~M~p&~M~q) & (~(p&~q) + ~(q&~p))]))
• L11: p ~> Mp
• L12: O(p&q) ~> Op
• L13: M(p&q) ~> (M(p&~r) + M(p&r)))

[Emch, 1936, p32]

With corrections from: [Emch, 1936a, p58]

• L7 was originally Mp ~> Op in the original paper.
• L11, L12, L13 were added.

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This page was last modified on January 19th, 2007