The most important single capacity human beings have is the capacity to reason logically - to know that from given premisses, whether believed or not, follows something with necessity if these premisses are true.

There are several reasons why this is the most important single capacity human beings have:

	Some reasons why logic is of fundamental human(e) importance
	It is at the foundations of all human inferences, and therewith of human survival
	It is at the foundations of peaceful, rational and reasonable social change and all argumentation
	Together with language, logic is what makes human beings really different from other animals: only human beings can seek peaceful agreement on what may be the case and what may be done by rational discussion
	Without mathematical logic, computers are hard to think of, design, built or maintain

 

1. There are quite a few sites dedicated to logic in some sense. Most are maintained by academics who cater mostly or only to a few fellow-academics in the same specialism. And while there is nothing wrong with that, it would be pleasant if I could direct you to a site where you can learn logic at most levels and enjoy the experience. Unfortunately, sofar I have neither found that site nor made it myself. (Maybe in the future, health permitting.)

Those who want to link to some interesting sites with accessible, amusing, interesting and civilizing ideas, formulations, games, puzzles relating to (mathematical) logic and related subjects are advised to check the following site and the ones after it:

Roger Bishop Jones's Factasia

"Factasia is a philosophical fantasy about the future of society and the future of technology." and indeed it is, and it contains a lot of logic, philosophy, and many bookreferences and links. This is very well done, but to delve deeper in the logic on the site you need more than is on that site. The book-references to do so are there - and I recommend that you download Mr Jones' site in the zipped version of Factasia he provides for that purpose, if his site is even a little to your taste, for it is large and well-organized, and far easier and cheaper to access once it is on your hard disk. Also, it'll probably teach you a lot directly or indirectly if you are in any way seriously interested in philosophy, logic, computers or mechanical proofs.

At age 15 or so I probably would have committed murder to be able to read this material. Now you can do so for free on the internet.

2. If you really want to understand both the beauty and the use of mathematics and logic you have to see it applied to all manner of problems. Here are four pages that contain a great amount of links to show just this

 

Frank Potter's Science Gems - Mathematics                                               - Physical science                                               - Physical science ii                                               - Physical science iiii

3. Next - and this part of my site is VERY much "Under Construction" here are some book-references to explain what I mean by "logic". For the moment I list only authors and titles, and do not know what is in print. All titles except the last recent ones should be avaible in any decent university library.
 

Classic expositions
Bertrand Russell	Introduction to Mathematical Philosophy
Alfred Tarski	Introduction to Mathematical Logic
Hasenjaeger	Introduction to Mathematical Logic
Paul Halmos	Naïve Set Theory
Bochenski	Formale Logik
Good introductions
Evert Beth	Foundations of Mathematics
Van der Waerden	Algebra
Joseph Shoenfield	Mathematical Logic
Herbert Enderton	A mathematical introduction to logic
Geoffry Hunter	Metalogic
Marvin Minsky	Finite and infinite machines
Good recent books
F.A. Muller	Structures for everyone
Barwise & Moss	Vicious Circles
George Boolos	Logic, Logic and Logic

Here a few comments on these titles - preceded by the general comment that one fundamental criterion to list them is their clear styles of writing:

Bertrand Russell	Introduction to Mathematical Philosophy

This is an exposition of the intuitions and mathematics that went into Russell's "The Principles of Mathematics" and Whitehead & Russell's "Principia Mathematica", mostly without symbolism and accessible to anyone with a clear mind.
 

Alfred Tarski	Introduction to Mathematical Logic

In several ways the best introduction to the subject, especially because it is non-pretentious and clear about fundamentals.

However, neither Russell's or Tarski's above-mentioned texts go far mathematically (and were not meant to be). One of the best introductions to the more mathematical side of logic is
 

Hasenjaeger	Introduction to Mathematical Logic

This is expecially good because he really goes into the intuitions behind the mathematics, and also contains good expositions of stuff usually not found in other introductions, while being formally both rather clear and precise.

 

Paul Halmos	Naïve Set Theory

 

Nearly all mathematics these days at least uses the notation of set-theory and presumes an understanding of its foundations. Halmos wrote a very clear introduction
 

Bochenski	Formale Logik

There is much more to logic than modern mathematical logic. This is the best history of logic in Western thought I've seen. (There also is a fine Indian tradition, impressively summarized by - I believe at present six - volumes of Link to: Karl H. Potter (I have no idea whether he is family of Frank Potter above))
 

Evert Beth	Foundations of Mathematics

Very wide ranging survey of the subject by a great Dutch mathematical logician. Subject-wise it is a bit out of date, but stylistically and conceptually it is not.

Something similar holds for the next book, that sheds lots of light on mathematical logic from a mathematical point of view
 

Van der Waerden	Algebra

This book - in fact originally 2 volumes in German - is close in spirit (but much older) than the expositions in Muller's and Halmos's books mentioned below. It also is concerned with Algebra in the mathematical sense, which covers a lot: logic, groups, operators, matrices, fields etc.
 

Joseph Shoenfield	Mathematical Logic

There are many mathematical expositions of mathematical logic. Shoenfield I found the clearest. It also covers a lot of material in a fairly small scope.

 

Herbert Enderton	A mathematical introduction to logic

In some ways the clearest, simplest and most thorough exposition. Somewhat less fast-paced than Shoenfield.

Geoffry Hunter	Metalogic

Another fine basic exposition, especially fit for people who did not study mathematics but who want a mathematically adequate and clear exposition.

Marvin Minsky	Finite and infinite machines

This is an excellent very readable introduction to the mathematical ideas involved in computing (for which you don't need much mathematics: a clear mind is all that is necessary).

None of the books I've mentioned sofar has been recently published (or if it was, like Tarski 's text I mentioned, it is a reprint). The next few books are recent:
 

F.A. Muller	Structures for everyone

This is the recent doctoral thesis of a Dutch mathematical physicist. It covers a lot of material, including Quantum Mechanics, but has the great advantage of being very clear about what theories are supposed to be. Muller also delves quite deep into the foundations of set theory and of category theory.

In general terms, he expounds a version of Bourbaki's structuralist approach to mathematics based on a version of Ackermann's theory of sets and classes, using Sneed's, Suppes', and Stegmuller's structuralist account of what scientific theories are. As the reader may have gleaned, the general point of view is: Everything - absolutely everything - is a structure of some kind.
 

Barwise & Moss	Vicious Circles

This is a recent exposition of paradoxes and vicious circles. It contains a lot of good clear explanations of recent thinking in mathematical logic in fields related to this subject including computer programming and theories of truth.

In general terms, the authors sketch solutions (or approaches to solutions) based on the idea to give up one of the standard axioms of set theory, the Axiom of Foundation, that excludes the existence of sets that are members of themselves.

This is also interesting for psychology and philosophy of mind, since so many issues in these fields involve some kind of self-reference (such as the one that allows the reader to understand that in this sentence I am saying something about this sentence and myself using the term "I").
 

George Boolos	Logic, Logic and Logic

This is a recent collection of essays by Boolos. It consists of articles in three loose groups (whence the thrice repeated "Logic"), namely about the foundations of set theory (which is the foundation of mathematics, which is the foundation of everything else - briefly), about Frege's logical theories, and about various logical subjects, notably Gödel's theorems and things impossible or impractical in first order logic.

This is also interesting for psychology and philosophy of mind, especially because Boolos discusses higher-order logic (involved in such statements as: "There are some relations and properties thereof I can think of you cannot think of - as shown by this sentence, which you, dear reader, cannot possibly believe to be true") and gives examples of formulas computers can't compute.