Record of Recent Changes to these pages.

If you were directed here for information, you probably want to go
directly to the **List of Logic Systems**.
(The "guts" of these pages.)

Table of contents

- Overview
- Notation notes
- Other information
- Some Propositional Logics
- Some Deontic Logics
- Some Temporal Logics
- Some Alethic Logics
**Full List of Logic Systems covered**- List of standard axioms
- List of standard inference rules
- Bibliography

There are lots of logic systems (especially modal logic systems) that are related to each other by the addition of an axiom or two.

I have only been able to find a few summaries of such interrelationships, and none of those on-line.

What few cross references that I have seen give "facts" but no clue as to where they came from. If a reference is given, it only points to a book, and you have no idea where in the book the fact is given. I suspect that the main contribution that these pages can make is to provide a cross reference table that one can actually use to go look up the interesting facts that one may find here.

There are important facts about systems that I have not yet had time to enter, and there are important facts about systems that I don't happen to know.

If there is an important fact of a system that you think I should have listed, and I didn't, please send me that fact and a reference where I can look up the information.

**If you have comments (positive or negative) about these
pages, I would like to hear about them. If you spot errors, I would
like to hear about it. I can be contacted at: John.Halleck@utah.edu If there
are facts you think that need to be mentioned on a system, sending me
mail may get that system moved up the todo list so that it gets done
earlier. This is largely a feedback driven effort.**

I have made references as I find them, because of this, I may not always have a pointer to the "best" references to a system. If you know better ones than I have here, please contact me.

I think it is time to make the cross references on these pages automatically generated. It would nice to have machine readable/processable files for the various axiom sets also.

I can't do these pages in all possible notations, so the abbreviated notation I try to use here is listed below:

- And: &
- Or: +
- Not: -
- Implication: >
- Strict implication: =>
- Equivalence: ==
- Strict equivalence: <=>
- Quantification
- Universal quantification: forall(x,...)
- Existential quantification: exists(x,...)

- Modalities
- Alethic
- Possibility: M
- Necessity: L

- Deontic
- Obligatory: O
- Permission: P

- Temporal
- It will always be: G
- It has always been: H
- Past: P
- Future: F

- Doxastic Logic
- Bx x believes that ...

- Epistemic
- Kx it is known that

- Provability
- Px: x is provable

- Alethic

I will sometimes give other notations also, so you'll sometimes see something like p>(q>p) [CpCqp]

Different books use different names for axioms and deduction rules. Therefore there is no one set of names that is going to agree with all the sources. I have made an effort here to have a consistent naming, so my naming will not always have the same name as the original source, although I've made some effort to at least note the original names.

A list of the inference rules used is on my Rules Page.

A skimpy list of the axioms used is on my Axioms Page.

A limited bibliography is available.

I reserve the right to change the name of any page below this one (and often do so as I find two systems are the same). Please don't directly link to pages below this one.

- Hilbert's Positive Propositional Logic (PC without negation)
- Various "Nonstandard" variations of negation.
- "Standard" PC (Has the "usual" notion of negation)

- Ernst Mally's 1929 Deontik
- G.H. Von Wright's 1951 Deontic System
- G.H. Von Wright's 1956 Dyadic Deontic System
- G.G. Von Wright's 1965 Dyadic Deontic System.
- The Standard Deontic System
- System D

- Built for the task
- Some that can be taken that way.

There are *many* Alethic modal logic systems. Below you'll find
some collections with interconnection diagrams.

James W. Garson has allowed me to use his very nice diagram of the interrelationships of most of the more popular modal logic systems: He has also written a very good quick Introduction to Modal Logics (Where I found the diagram) for the Stanford Encyclopedia of Philosophy.

K. E. Pledger's Enumeration of systems from his 1972 Paper "Modalities of Systems Containing S3". The article proves the systems distinct. It also provides a consistent naming, to avoid confusions (such as Anderson and Åqvist both having a system called S7.5, and those systems not being equivalent) R. I. Goldblatt has published a paper [Goldblatt, 1973] "A Model-Theoretic Study of some Systems Containing S3" that gives completeness proof's for all the systems in Pledger's enumeration. The relationships between the main systems are: (This diagram from [Pledger, 1972, 273], and is used here by permission of Dr. Pledger.)

- In K2, but not in S5
- In S5, but not in K2
- In BOTH S5 and K2
- In NEITHER S5 nor K2
- 0p is called the Trivial System
- 6p
- 8p
- 10p
- 12p

The distinct irregular systems are related as follows: (Diagram from [Pledger, 1972, 278], and is used here by permission of Dr. Pledger.)

- Irregular systems formed by adding MMp to one of Pledger's systems above. (The names are the names of the base system with a trailing "a")
- Irregular systems from adding MLMMp to one of Pledger's systems above. (The names are the names of the base system with a trailing "b")
- Irregular systems from adding LMMp to one of Pledger's regular systems above. (The names are the names of the base system with a trailing "c")

- The Trivial system (Lp == p == Mp)
- KF (Lp == Mp)
- The VER Verum system (L means true)
- Inconsistent systems

- S1
^{0} - S2
^{0} - S3
^{0} - S4
^{0} - S5
^{0}doesn't really exist, it collapses to S5 [Zeman, 1973, p181] - S6
^{0} - S7
^{0} - T
^{0}

Quick diagrams are good for a quick feel, but for serious work you need more detail... so here it is:

**Full List of Logic Systems covered**- List of standard axioms
- List of standard inference rules
- Mechanicaly generated/checked proofs

One has to start somewhere, I'd recommend

History: [Goldblatt, 2005]

- Full List of Logic Systems covered
- List of standard axioms
- List of standard inference rules
- Logic Page
- John Halleck's Home Page

© Copyright 2005,2006,2007,2008 by John Halleck, All Rights Reserved. This page is http://www.cc.utah.edu/~nahaj/logic/structures/ This page was last modified on July 17th, 2008.