Logic System Interrelationships

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

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:

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.

Other information

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.

Some Propositional Logics

Some Deontic Logics

Some Temporal Logics

Some Alethic Logics

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

Garson's list of popular systems

James W. Garson has allowed me to use his very nice diagram of the interrelationships of most of the more popular modal logic systems: Complex diagram of common 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.

Pledger's Enumeration of systems containing S3

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.) Relationships of main systems containing S3

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

Some degenerate systems

Naught systems

More detailed information on this site

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

Where to get started

One has to start somewhere, I'd recommend

  • [Hughes and Cresswell, 1996]
  • [Chellas, 1980]
  • History: [Goldblatt, 2005]

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    © 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.