The Principles of Mathematics Revisited

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Summary: Exposition of "independence-friendly" logic
Review: Hintikka argues in this book that a novel extension of the predicate calculus, called "independence-friendly" (IF) logic, has a better claim to being the "core" of mathematical logic than the predicate calculus (PC) itself. Besides presenting his formalism, he rethinks the entire history and philosophical foundation of logic based on insights from independence-friendly logic. IF logic is similar to the PC except that variables bound by a new kind of independence-friendly existential quantifier refer to values that need not be functionally dependent on outer universal quantifiers. This is contrary to the PC, in which an inner existentially-quantified variable always depends functionally on all outer universal quantifiers. The definition of truth of a (closed) interpreted formula in IF is based not on Tarski's classic definition, but on the existence of a winning strategy in a formal two-person game associated with the formula. The game associated with a PC formula is always a perfect-information game, and the game-theoretic truth definition is equivalent to Tarski's. But for those IF formulae containing the independence-friendly existential quantifiers the associated game is not generally perfect-information, and truth is not definable by a Tarski-like definition.
IF is a pure extension of the PC in the sense that the PC formulae (those with only standard--not independence-friendly--existential quantifiers) are a subset of IF formulae, and all valid PC formulae are valid as IF formulae. But the resemblance between the two logics ends there. For example, IF, unlike PC, is not recursively axiomatizable, and the law of excluded middle fails. Hintikka argues convincingly that these facts are not defects in IF, but actually liberating.
I am a computer scientist with a wide background in logics. I found the book exhilerating, and convincing in its claims. I would prefer an improved notation for IF, but that is a quibble; the logic itself is fascinating and deeply thought-provoking. Despite its unexpected properties IF is not a fringe or exotic logic; I find it reasonable to agree with Hintikka when he says that IF is indeed the true core of formal logic. Required reading for anyone interested in logic.

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Summary: BORING !
Review: I am a working mathematician with a PhD in Differential Geometry who has recently become interested in the foundations of Mathematics. I came to this book circuitously via a pdf by Jaakko called "A revolution in Logic". Jaakko is on the offensive in this book, and he certainly can be polemic, to say the least. Having said that, the book is a tour de force: as a book on first order logic and IF, on the philosophy of mathematics, and the nature of mathematics and its objects and structures---it has gotten me thinking on many different levels. I would guess English is not his first language (due to easily corrected spelling, grammar, Etc), but despite this the writing style is quite smooth, with some interesting turns of phrases. I get the feeling this is one of the most brilliant books that I will ever have the pleasure of reading AGAIN and AGAIN! My 2cents David Thompson Assistant Professor of Mathematics and Computer Science Olivet College, Michigan

Rating:
Summary: BORING !
Review: I read this book for preparation to enter City College of San Francisco's Math 840-Elementary Algebra Class. It took me like 3 months to get through this and it didn't help me out at all. On top of that it was so boring I was afraid I might die of boredom at times. This book is of no help at all for understanding the principles of mathematics. If this guy was my teacher, I'd give him a lousy evaluation.

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Summary: A philosophical and logical milestone
Review: I read this book last year and I found it both entertaining and interesting, but only recently I was struck by how profound it is. I find Mr. Hintikka's insight into the dual use of logic (description vs. inference) and the conflict it caused in the development of logics (expressivity vs. axiomatic completeness) fascinating. He decisively resolves this conflict in favour of expressivity by introducing a brand of logic which is amazingly expressive yet non-axiomatizable. So, instead of proving propositions, we validate them through calculations in the semantic model.

I don't think this book will have the impact it should, only because the philosophical-logical establishment is already entrenched in certain ways of thinking that it can not abandon. And I think Mr. Hintikka is painfully aware of this. His tone is polemical, almost vitriolic at times, and it has a certain voice-of-reason-crying-in-the-wilderness streak to it. While entertaining, the style detracts from the importance of the book.

I consider "The Principles of Mathematics Revisited" one of the most important books on logic ever. Its impact will not be immediate but it should eventually be momentous.

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Summary: The importance is evident even for an amateur
Review: I'm no matematician, no philosopher, just found the book in a bookstore and got curious about the presentation of a 'new logic' (the little I know was from philosophy class in high school). As an amateur you are faced with formulae and notation that you have to figure out on your own, which makes it harder to read it, but I did manage to figure it out with the help of the Equation Editor in the Office Suit. Just a single page explaining the notation would have been enough to make the book self-contained, even for an amateur. The language is simple enough even for a non-native English speaker (it is typically easier reading books written by non-native speakers, since their language is less complicated).
Regarding the subject, it is very, very interesting, because of the profound implications this will have--I'm sure of that--on the development of sicence, once we get a new generation of scientists who are not entrenched. It was like reading something from the future. It's been years since I read it, but I still have it in fresh memory, and the information-independent logic is the only logic I would consider using, if it came to that. When working with incomplete information, I realize that it is essential to be aware of the lack of information. And what better way to introduce the tools for handling incomplete information, than on the basic, logical level?
I submit, that in a few decades, the ideas of this book will pervade software in our computers.

Rating:
Summary: A pleasure to read
Review: This book re-examines first-order logic as it has been applied to the foundations of mathematics. But it is much more than that. If you are interested in the human decisions behind why logics were built as they were, if you want to understand the impact of Godel's Incompleteness Theorem, if you are interested in understanding philosophy of inferential systems in general, then you will find this book quite profound and useful.

Requirements for reading this book are roughly: a general knowledge of syllogistic and first-order predicate logic, an idea of what Godel's theorem is about and the concept of godel-numbering, some philosophy (e.g., ontology vs. epistemology), but mostly a keen interest at learning about logic and it's foibles and potentials.

Chapter 1 begins with the Hilbert program, and the attempt at axiomatization in general. Chapter 5 clears up alot of confusion about the Godel Incompleteness theorem and what it really means. He delineates between descriptive, semantic, deductive and Hilbertian completeness notions, and describes their inter-relatedness and Godel's theorem's role. These chapters alone are useful for gaining deeper understanding of the problems that arise in syntactic axiomatic deductive systems.

Chapter 7 is on the Liar Paradox, and he offers a unique solution to that based not upon Austinian notions, but rather based upon Hintikka's IF ("independence-friendly") first-order logic which avoids resorting to infinities or relying on any semantic re-interpretation (Hintikka uses a simple formal statement "~T[d]" where d is the godel-number of that statement, as the basis of the discussion).

He then goes on to discuss the presumed role of axiomatic set theory and chips away at it's pretense as a secure foundational approach.

But this merely scratches the surface. The book is primarily about the human decisions that were made, the reasoning behind them and why/where they failed. This is part of what makes it so readable and engaging. For Hintikka, logic and math seem to be very human activities, and there is no attempt to sanitize logic as being something pure or absolute.

As an explication of human decision-making in logic, I think this book has important insights buried within and consequences for the inferential world of logic and mathematics, as well as reasoning in general. It will take several readings to grasp it's profound implications.

'IF logic' itself (chapter 3) is a ridiculously simple and brilliant enhancement to first-order predicate logic, produced merely by lifting the mandatory left-to-right scoping restrictions Frege had placed on quantifiers in the syntax. And he extends (no pun intended) that notion by similarly lifting restrictions on mandatory scoping across operators as well. What arises looks very much like ordinary predicate logic, but the scoping independence opens up new vitality to the logic that makes it's applicability broader, as well as philosophically more interesting.

IF logic, in particular, is more amenable to being about imperfect information, and information independence (hence "independence-friendly logic"). Hintikka's version of truth-definition is about a verification game (as in game theory), not a Tarskian retreat to a metalevel of formalism. Throughout, there are these kinds of comments and concepts on relating logic back to the world.

IF logic is an intriguing example of how a subtle change in rules of syntax can have large consequences, and Hintikka is definitely pushing for it as -the- preferable first-order logic (actually, family of logics) over standard predicate logic. (And for game theoretical semantics and model theory as his preferred meta framework.) However, Hintikka's salesmanship aside, the insights in the book are not dependant on IF as being -the- alternative, but as a demonstration of those insights.

As a non-mathematician/non-logician, I had braced myself for a slog through a dry, tough read (particularly since there are nearly two decades of rust accumulated on my predicate logic skills) despite the positive reviews I had read on Amazon, but was pleasantly surprised at the lively writing style and also the modicum of formulae, with no tedious proofs to sweat over. Even the final chapter on "Epistemology of Mathematical Objects" is quite readable. And with some chapter headings like "Who's Afraid of Alfred Tarski?" and "Axiomatic Set Theory: Fraenkelstein's Monster?" you know the author enjoys his subject matter. :)

Rating:
Summary: One of the most importnat books you will ever read
Review: This book was profoundly satisfying. I found the arguments convincing, and the subject important and original. The first review does a good job of describing the book, but doesn't seem to instill the sheer importance of the book. I believe that people who do not read this book are missing out.

With IF logic one doesn't need to worry about set existence and one can understand the role and nature of the "axiom of choice".