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I've asked before about good recommendations to study algebra for the sake of algebraic logic and I've got very good recommendations.

I wonder if you have some recommendations to start studying algeraic logic itself.

I've come across some books but not sure which one to start with.

Now, What do you recommend? and why?

Some of the books that I've seen are:

  • An Algebraic introduction to mathematical logic, Barnes
  • The Mathematics of Metamathematics, Helena Rasiowa & Roman Sikorski
  • An Algebraic approach to non-classical logics, Rasiowa
  • Algebraic logic, Andreka and Monk
  • Algebraic logic, Halmos
  • Algebraic methods in philosophical logic, Dunn.
  • Algebraic Methods of mathematical logic, Ladisalv Rieger

Which one do you recommend? feel free to recommend books that I've not mentioned above.

By the way, I know elements of algebraic logic as I've read most of Paul Halmos' Logic via Algebra

FNH
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  • I suggest to add (before Rasiowa's book) : Helena Rasiowa & Roman Sikorski, The Mathematics of Metamathematics (1963) – Mauro ALLEGRANZA Jan 04 '15 at 17:29
  • @MauroALLEGRANZA, Done, I've added another text too "Check the last one" – FNH Jan 04 '15 at 22:50
  • Since 2015, the book "Abstract Algebraic Logic, An Introductory Textbook" by Josep Maria Font appeared. I did not yet read lots of it or compare it to the other books mentioned here. But I think it's worth mentioning. – 8bc3 457f Nov 11 '22 at 07:03

2 Answers2

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Some comments...

I suggest you to consider also :

it is a "standard" textbook with : Part I: Recursive Function Theory; Part II: Elements of Logic; Part III: Decidable and Undecidable Theories; Part IV: Model Theory, but the treatment of mathematical logic has a strong "algebraic" flavour, with a section dedicated to Cylindric algebras.

Personally, I do not reccomend Paul Halmos, Algebraic logic (1962), because is a collection of separate papers.

About :

it is a "modern classic", very rigorous and complete for classical and intuitionistic logics; unfortunately, it is quite old (there is no Kripke semantics) and also the symbols used are quite "old-fashioned".

About :

it is a good textbook, but "limited" to classical logic.

Mauro ALLEGRANZA
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  • Do you have any comment on "Algebraic methods" by Dunn? or "Algebraic methods in mathematical logic" by Rieger?. For Donald, I think it's not a drawback to consider only classical logic as a "first step into the subject". For "The mathematics of metamathematics" I've checked it and found that it uses topological methods too, not only the algebraic methods. Do you think it's a good text for self-study even with its old-fashioned notation? At least, to study Algebraic classical logic from it? – FNH Jan 07 '15 at 14:22
  • @MathsLover - I think that Donald Barnes & John Mack is simpler to begin with ... But R&S Math of Metamath is abcolutely self-contained and rigorous; it ia huge but not "difficult". – Mauro ALLEGRANZA Jan 07 '15 at 14:27
  • Ah, cool, Do you think I need some training with topology before reading it? I've already studied some basics of topology from Munkres, I don't know if what I've is already sufficient. – FNH Jan 07 '15 at 14:32
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    @MathsLover - I think that the "quantity of topology" needed is quite little; you can browse your topology textbook if needed. – Mauro ALLEGRANZA Jan 07 '15 at 14:41
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Well, as always the choice of appropriate book depends on your motivations for studying algebraic logic. Although my knowledge of it is very superficial it seems to me you have not received many recommendations so let me comment briefly.

I only skimmed through Algebraic methods in philosophical logic by Dunn and Hardegree but it seemed to me that it is easy to read. However, it does not cover the algebraization of FOL. For that see cylindric algebras (e.g. two volumes by Henkin, Monk, and Tarski), complete lattices (e.g. book by Rasiowa and Sikorski), Halmos (polyadic) algebras or some other approaches.

The mathematics of metamathematics by Rasiowa and Sikorski and An algebraic approach to non-classical logics by Rasiowa are classics. If I remember it correctly the later provides a more general approach and the former deals mainly with classical and intuitionistic logic.

It is worth of mentioning that Rieger's book Algebraic methods of mathematical logic is based on complete but unlikely final manuscript found after author's premature death.

If you are interested in so-called abstract algebraic logic in general setting then Protoalgebraic logic by Czelakowski is a useful source. There are also some survey papers and influential monograph Algebraizable logics by Blok and Pigozzi.

Karel Chvalovský
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