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Formal Logic: Its Scope and Limits
The first beginning logic text to employ the tree method--a complete formal system of first-order logic that is remarkably easy to understand and use--this text allows students to take control of the nuts and bolts of formal logic quickly, and to move on to more complex and abstract problems. The tree method is elaborated in manageable steps over five chapters, in each of which its adequacy is reviewed; soundness and completeness proofs are extended at each step, and the decidability proof is extended at the step from truth functions to the logic of nonoverlapping quantifiers with a single variable, after which undecidability is demonstrated by example. The first three chapters are bilingual, with arguments presented twice, in logical notation and in English. The last three chapters consider the discoveries defining the scope and limits of formal methods that marked logic’s coming of age in the 20th Godel’s completeness and incompleteness theorems for first and second-order logic, and the Church-Turing theorem on the undecidability of first-order logic. This new edition provides additional problems, solutions to selected problems, and two new Truth-Functional Equivalence reinstates material on that topic from the second edition that was omitted in the third, and Variant Methods, in which John Burgess provides a proof regarding the possibility of modifying the tree method so that it will always find a finite model when there is one, and another, which shows that a different modification―once contemplated by Jeffrey--can result in a dramatic speed--up of certain proofs.
192 pages, Hardcover
First published October 1, 1967
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Displaying 1 - 12 of 12 reviews
June 11, 2023
This is a great first text on formal logic.
It contains a thorough and practical description of what he calls the 'tree method' for determining validity of sentences in propositional and first order predicate logic. ( And therefore also methods for satisfiability of sentences, validity of arguments etc ).
It then shows clearly that the methods are sound and adequate. ( the terms 'consistent' are 'complete' are somewhat overloaded ).
In the case of predicate logic adequate is taken to mean an adequate 'yes machine'. He shows that it is not an adequate 'no machine'. The program might never terminate. This is a form of Church's theorem on the undecidability of predicate logic ( relying of course on a definition/intuition of what constitutes computation - or effective procedure - long since firmly established.
This is all easy to read and understand.
As an added bonus he moves on from Church's theorem to establish that there are no formal axiomatic theories for Arithmetic - or that any attempted axiomatic theory is necessarily incomplete. AKA Godel's theorem. This is a bit rushed and less thorough but it is a nice taste of things to come in a second textbook on logic.
I only have one complaint. He casually asserts that Godel's theorem established that truth in mathematics could no longer be taken to be the same as provability. Often repeated. And frequently overly enthused on. But simply not true. And he should have known better. The correct way of looking at Godel's theorem is that it establishes the incompleteness of the logical apparatus - and all apparatuses of a similar very broad kind. The challenge then is to extend the definition of 'derive' or 'prove' - taken to be the same thing - to get around these limitations. Godel and his colleagues were already hard at work on just this. Basically looking at various forms of higher infinity axioms. The very notion that finite/recursive theorizing could possibly map out the infinities beyond infinities of mathematical truth was absurd to begin with. And in the century since there have been some intriguing results along these lines. Though the story is far from 'complete'.....
It contains a thorough and practical description of what he calls the 'tree method' for determining validity of sentences in propositional and first order predicate logic. ( And therefore also methods for satisfiability of sentences, validity of arguments etc ).
It then shows clearly that the methods are sound and adequate. ( the terms 'consistent' are 'complete' are somewhat overloaded ).
In the case of predicate logic adequate is taken to mean an adequate 'yes machine'. He shows that it is not an adequate 'no machine'. The program might never terminate. This is a form of Church's theorem on the undecidability of predicate logic ( relying of course on a definition/intuition of what constitutes computation - or effective procedure - long since firmly established.
This is all easy to read and understand.
As an added bonus he moves on from Church's theorem to establish that there are no formal axiomatic theories for Arithmetic - or that any attempted axiomatic theory is necessarily incomplete. AKA Godel's theorem. This is a bit rushed and less thorough but it is a nice taste of things to come in a second textbook on logic.
I only have one complaint. He casually asserts that Godel's theorem established that truth in mathematics could no longer be taken to be the same as provability. Often repeated. And frequently overly enthused on. But simply not true. And he should have known better. The correct way of looking at Godel's theorem is that it establishes the incompleteness of the logical apparatus - and all apparatuses of a similar very broad kind. The challenge then is to extend the definition of 'derive' or 'prove' - taken to be the same thing - to get around these limitations. Godel and his colleagues were already hard at work on just this. Basically looking at various forms of higher infinity axioms. The very notion that finite/recursive theorizing could possibly map out the infinities beyond infinities of mathematical truth was absurd to begin with. And in the century since there have been some intriguing results along these lines. Though the story is far from 'complete'.....
May 12, 2019
If there was a result that defined an entire field of study you think more people would know about it, we've heard of e=mc^2 and evolution, yet few know (and less understand) Godels incompleteness theorem. This book is essentially one long project to reach that theorem. Good read.
September 14, 2022
Text too short. The book was very popular before, but the year our lecturer decided to change to Jeffrey's book, the author had decided to revise the book, reducing not the content but the amount of explanatory text. That made it harder to read.
June 16, 2021
Takes you from TFL to Godel's Incompleteness... And all in 200 pages! At many times too compact, but otherwise a good introductory text.
September 1, 2023
A book that can ruin your friendships and social life. Approach it with great caution!
January 2, 2024
One of the best introductory logic textbook out there.
January 24, 2023
It’s ok - to be honest I think the truth trees really hamstrung this book it made the presentation of the material more confusing than it needed to be and I really don’t feel like I got a ton out of this book because of it it’s relatively short and the discussion of functions feels very full circle but overall it’s lacking
November 25, 2013
I have to admit that I wasn't excited about the methodology of Jeffrey's system. I've always used natural deduction systems (mostly Forbes) to check validity of arguments. The biggest problem was that you could show validity but harder to show invalidity. With Jeffrey's use of trees you can quickly show an argument to be valid or invalid and also quickly create a counterexample. Using this method takes some getting used to for one who is used to the scores of rules and sequents needed needed for a robust natural deduction system, but it greatly simplifies things by stripping all of these away except for a handful of very easy to learn rules.
The hardest things wasn't mastering it, but learning to trust the results. For a while I would do a proof and then check it with either a truth table or a natural deduction argument. After a while this goes away and even very complex arguments can quickly be checked.
The hardest things wasn't mastering it, but learning to trust the results. For a while I would do a proof and then check it with either a truth table or a natural deduction argument. After a while this goes away and even very complex arguments can quickly be checked.
July 26, 2009
I took logic before this book became the textbook for symbolic logic at vcu. I have to say, I did not like the LPL text, and from what I read of this, it was overly technical for an introductory logic course. There are other textbooks in Logic that effectively explain the same inference rules, laws, definitions, and concepts just as well, whilst avoiding over technicality *when it is unnecessary*. Sometime I get the hunch that Logician-text-book-writers are either sincerely writing the logic texts with a view that it is overly easy, and hence they fail to see the difficulty it can be for incoming logic students, or else they are insecure about their field. Actually, I don't know what the fuck I think about this book.
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August 17, 2007This book is interesting because it introduces and makes extensive use of "truth trees." These devices are really easy to understand and use. They allow one to check for validity and invalidity at once. So they are like truth tables. However, unlike truth tables, truth trees are adequate for checking arguments involving quantifiers. Another virtue of the trees is that they allow easy demonstrations of completeness and soundness. Thus, the book is able to get very, very far in a short space. It's like 150 pages and it gets to Godel's incompleteness results. I don't think this would be a good text for a standard symbolic logic course.
January 27, 2014
Compared to other logic texts I have read, this one is horrifying. It's miserable at best if used as intended as an introductory text.
It would be okay for people already very familiar with logic concepts, and only as a refresher. In the first chapter, I struggled trying to understand his points despite this being my third class on the subject. I had never heard of truth trees before reading this for a logic class, and I had to use outside sources just to get a semblance of what he was talking about.
I highly recommend Discrete Mathematics with Applications by Susanna Epp over this for the beginner, although I don't think she covers truth trees.
It would be okay for people already very familiar with logic concepts, and only as a refresher. In the first chapter, I struggled trying to understand his points despite this being my third class on the subject. I had never heard of truth trees before reading this for a logic class, and I had to use outside sources just to get a semblance of what he was talking about.
I highly recommend Discrete Mathematics with Applications by Susanna Epp over this for the beginner, although I don't think she covers truth trees.
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November 30, 2016dalmar
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