Metalogic: An Introduction to the Metatheory of Standard First Order Logic

by Geoffrey Hunter

Book Overview

This work makes available to readers without specialized training in mathematics complete proofs of the fundamental metatheorems of standard (i.e., basically truth-functional) first order logic. Included is a complete proof, accessible to non-mathematicians, of the undecidability of first order logic, the most important fact about logic to emerge from the work of the last half-century. Hunter explains concepts of mathematics and set theory along the...

• Format: Paperback
• Language: English
• ISBN: 0520023560
• ISBN13: 9780520023567
• Release Date: June 1996
• Publisher: University of California Press
• Length: 302 Pages
• Weight: 0.75 lbs.
• Dimensions: 0.8" x 5.5" x 8.2"

Customer Reviews

A Good Introduction to a Difficult Subject

I must respectfully disagree with many of the negative things said in the previous reviews about this book. I find it to be by far and away the easiest to understand text on metalogic that I have come across. I am, however, a professional philosopher with a good deal of training in logic. So the material is not new to me, which surely makes the book much easier for me to understand. That said, I think that some of the previous reviewers have unfairly judged this book owing to the complexity of the material it covers. One should remember that Hunter's aim is not to present rigorous arguments but to INTRODUCE metalogic. I am sure he would be satisfied if his reader is inspired to look into any of the proofs in more detail elsewhere. As far as the claim that the book contains errors, I have yet to read a logic text that didn't!

Outstanding work

This book gives examples and "answers" to the examples. It makes the clear distinction between formality or "syntax" and meaning or "semantics." This critical distinction is utilized to shed light on consistency proofs and undecidability. Part of Goedel's proof involved the liar's paradox. In other words, part of Goedel's proof required the semantic acknowledgement of a phrase that didn't go anywhere, namely, "this theorem is not provable" which, as model for the formal sentence or syntax "S", established proof that the formal sentence couldn't go anywhere. How tough is that? Goedel's proof is overrated for its complexity if one takes an inventory of the aforementioned simple fact which is the core of his argument. In other words, to establish Goedel's proof, in lieu of the tradition bequeathed to us from generations of mathematicians, such as Pappus' proof (best illustrated in D'Abro's book on "The Rise of the New Physics"), we need to take a step in the the semantic world and then take a step in the formal/syntax world. All proofs are dependent on the interchange of semantics and syntax. Some things are easier to solve in the syntax world. If not, some things are easier to solve in the semantic world. If a semantic argument is easy, such as the liar paradox, we acknowlege its mirror image in the syntax world, and by solving something in the semantic world we know we've accomplished something in the syntax world, and vice versa. That's all.

Forget "Godel Escher Bach", read "Metalogic"!

I feel compelled to write this review because I disagree utterly with the previous reviewer. This is a fantastic book! I have no specialist training in logic, but I found "Metalogic" to be a clear, interesting, well-written and definitely "painless" text. I highly recommend this book as an introduction to metalogic; in fact, I would recommend it in preference to the other popular text on metalogic, the brilliant "Godel Escher Bach". Whereas "Godel Escher Bach" introduces the field of metalogic using allegories, parables, illustrations and copious references to turtles, "Metalogic" introduces it using clear, simple arguments and gentle, step-by-step explanations. A quote from the preface: "Many elementary logic books stop just where the subject gets interesting. This book starts at that point and goes through the interesting parts...." Many of us who have struggled through the dry and boring foundations of elementary logic have become aware that not all logic is as tedious as the foundations. Beyond elementary logic lies a fascinating world inhabited by the paradoxes of Bertrand Russell, the counterintuitive proofs of Turing, Church and Godel, and the mindboggling infinite sets of Cantor. The problem is, just where logic becomes interesting, it also becomes unattainable to the average Joe. "Metalogic" undertakes to introduces this field to those without the specialist training that is normally required, and it succeeds admirably.