Logic Systems Axiom List

There are a limited number of axioms used to build various modal and non-standard logic systems.


There are any number of numbering schemes for the many of the axioms below. In fact, a different scheme in just about every book on the topic. There is no way to combine them and resolve the conflicts without having to rename some.

I've tried to have a consistent naming (numbering) scheme...

There were also a number of axioms I kept having to refer to that don't even have a name. I've given them at least something one can refer to.

Further reading

[Hackstaff, 1966]

[Hughes and Cresswell, 1996]

Logic Structures Pages

Logic Systems Pages


Non-modal axioms

Positive Propositional Logic

Russell and Whitehead basis for PC

Standard PC

Intuitionist PC (Scholz and Schröter)

Heyting's actual Intuitionist PC (Heyting)

[Hackstaff, 1966, p223]

Fitch Calculus

Johansson Minimal Calculus

Misc. axioms.

Modal Logic

Lewis systems:

"Standard" Modal logics

Misc. Lewis style axioms. (Following the numbering of [Zeman, 1973]


[Sobociński, 1962, p53] (With Sobociński's typo of F2 fixed - JH)

Hacking's systems

[Sobociński, 1962, p53]

von Wright's systems (Following Hilpinen's numbering)

Misc. (Some of these really need names.)

Cross Reference

(Grossly in progress)

Meta Proof Notes

If a system having the rule Modus Ponens for an operator ">", and can prove:

Then it has the Deduction theorem, and therefore allows hypothetical syllogism. [Hackstaff, 1966, p121]

Note that the first axiom p>p [Cpp] is actually provable from PL1 and PL2, given the rules modus ponens (for >) and universal substitution.

Any system having Modus Ponens for an implicational operator > and the theorem (A > (B > (A & B))) has the derived rule of Adjunction for &. [Simons, 1953, p313]

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This page was last modified on October 13th, 2009.