(Part 2) Best mathematical logic books according to redditors
We found 329 Reddit comments discussing the best mathematical logic books. We ranked the 107 resulting products by number of redditors who mentioned them. Here are the products ranked 21-40. You can also go back to the previous section.
Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.
Amazon search for Dover Books on mathematics
There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
Pre-Calculus / Problem-Solving
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
I started from scratch on the formal CS side, with an emphasis on program analysis, and taught myself the following starting from 2007. If you're in the United States, I recommend BookFinder to save money buying these things used.
On the CS side:
On the math side, I was advantaged in that I did my undergraduate degree in the subject. Here's what I can recommend, given five years' worth of hindsight studying program analysis:
Final bit of advice: you'll notice that I heavily stuck to textbooks and Ph.D. theses in the above list. I find that jumping straight into the research literature without a foundational grounding is perhaps the most ill-advised mistake one can make intellectually. To whatever extent that what you're interested in is systematized -- that is, covered in a textbook or thesis already, you should read it before digging into the research literature. Otherwise, you'll be the proverbial blind man with the elephant, groping around in the dark, getting bits and pieces of the picture without understanding how it all forms a cohesive whole. I made that mistake and it cost me a lot of time; don't do the same.
You need to develop an "intuition" for proofs, in a crude sense.
I would suggest these books to do that:
Proof, Logic, and Conjecture: The Mathematician's Toolbox by Robert Wolf. This was the book I used for my own proof class at Stony Brook - (edit: when I was a student.) This book goes down to the logic level. It is superbly well written and was of an immense use to me. It's one of those books I've actually re-read entirely, in a very Wax-on Wax-off Mr. Miyagi type way.
How to Read and Do Proofs by Daniel Slow. I bought this little book for my own self study. Slow wrote a really excellent, really concise, "this is how you do a proof" book. Teaching you when to look to try a certain technique of proof before another. This little book is a quick way to answer your TL:DR.
How to Solve it by G. Polya is a classic text in mathematical thinking. Another one I bought for personal collection.
Mathematics and Plausible Reasoning, Vol 1 and Mathematics and Plausible Reasoning, Vol 2 also by G. Polya, and equally classic, are two other books on my shelf of "proof and mathematical thinking."
I'm in literally the same boat with you. Full time software engineer and I graduated almost a year ago.
I think two things motivate me to self-study:
To a degree, 2 is kind of a subset of 1. Anyway, when I come home I just find an hour to set aside and work through a book, ebook, or video to develop the skill in question. Recently it's been this book. And someday's I just wanna play video games and be lazy. So I think a large part of it is understanding that it's a slow process for the most part.
There are very few true textbooks - i.e. books designed to teach the material to those who don't already know the classical versions - written in this style.
While we're at it, a quick skim through the algebra chapter of Troelstra: Constructivism in Mathematics, vol. 2 should explain why there are no textbooks on abstract algebra written in the purely constructive tradition.
Read this book: Proof, Logic and Conjecture. I read and studied it for a class that was a bridge course going from basic calculus types of mathematics into formal proof and analysis. If you understand this material, there pretty much is nothing stopping you from understanding anything you want afterwards.
Congrats on your foray deeper into logic! This turned out rather long, so if you want to continue your education you may simply want to skip to the end for my book recommendation. However, I'm keeping my rundown of your particular example here in case you just want this one question answered now.
You can start dealing with this sort of thing with what's called predicate logic or first-order logic. In predicate logic, you can have predicates or functions that output truth values when given different variables. For example, I could define the predicate T_, which returns true for all x that exist, and F_, which returns true for all x which do not exist. This lets you write expressions like Tx -> ~Fx, which means "x exists implies x doesn't not exist." Furthermore, you get quantifiers over whole expressions, like ∀ (for all) and ∃ (there exists). This lets you be more specific with those sorts of expressions. The following are both true:
The former says that there is some object x such that if that particular x exists, it doesn't not exist. The latter says that for any possible object x, if that object exists, that same object doesn't not exist.
Let's apply this to your argument. We can formulate it like this:
(1) ∀x: Gx -> Ax
(2) ~Aa
(C) ~Ga
What's happening here? In premise 1, I'm saying that for all possible x, if that x is good (G_), then that x is amazing (A_). I haven't quantified a, so it's a free variable. In predicate logic, free variables are interpreted as particulars. We would read premise 2 as "a isn't amazing," and we'd have to simply refer back to semantic interpretation to know what a is (in this case, it's "I"). Free variables always refer to the same thing, so the a in premise 2 is the a in the conclusion. It reads, "a is not good."
Now for determining if it's valid. Essentially the same rules that apply in propositional logic also apply to the predicates in predicate logic. Just as we have the rule (x -> y) <-> (~y -> ~x), we have the analogous rule (∀x: Fx -> Gx) <-> (∀x: ~Gx -> ~Fx). (I use F and G here because they are common letters to use for predicates, which comes from F being a popular letter for functions in math). We can apply universally quantified expressions to particulars. We can get the following from premise 1:
(3) ∀x: ~Ax -> ~Gx
This immediately gives us
(4) ~Aa -> ~Ga
And then we can derive the conclusion directly from this.
If you continue to be curious, there are very many texts that cover this topic in much further detail with much better pedagogical organization. In particular, I recommend Ray Smullyan's great little book first-order logic for further reading.
It's a fair point; applied category theory is really in its infancy. For a long time, it was considered pretty inaccessible and obscure. I think that's starting to change, e.g. with some new pedagogically oriented books (Cheng, Fong-Spivak), new international conferences, new journal, etc. But it might take time.
The most successful application so far is certainly Haskell, OCAML, and other similar functional programming languages. These were built entirely on category-theoretic principles, and have become quite popular (Haskell is used at AT&T, Amgen, Apple, Bank of America, Facebook, Google, Verizon, etc.).
There are control theory researchers such as Paulo Tabuada, robotics researchers such as Aaron Ames and Andrea Censi, and others who have explicitly used category theory in their work. For-profit companies such as Kestrel, Statebox, R-Chain, Conexus, etc. all use category theory more or less explicitly.
Whether or not electrical engineers—or others of that sort—will use CT depends on whether there are enough interested parties who can drive it more deeply into that domain. So far, the work has been at a very surface level because category theorists have to "go to them" instead of them "coming to us". As category theorists, we don't know enough about the depths of these fields to make a direct and immediate impact without preparing the ground. It takes time and effort, and we need more people on the case.
But if we continue—and I think we will—my guess is that in the future, people will use category theory to learn lots of different fields and connect their knowledge from one to another. A major value proposition of category theory is its ability to transfer information and problem specification from one field to another. I think that will eventually be broadly useful.
If we look at the "set" of all Cardinal Numbers, then that "set" can be indexed by the "set" of all Ordinal Numbers, call it V. Ordinal numbers are a standard way to talk about Well-Ordered sets. But it turns out that V has all the properties of a well ordered set, so it is a well ordered set and, additionally, it is strictly bigger than every other ordinal. But we can make the ordinal V+1, since V+1 != V we have V+1<V, but by definition of the successor ordinal V<V+1 which is a contradiction, V cannot be a set. Therefore, because the set of all ordinals cannot exist and the set of all cardinals is bijective with all the ordinals, it follows that the set of all cardinals cannot exist. Also, this does not depend on Choice or the GCH, if you don't assume them, then there are even more Cardinals so the result still holds.
This essentially says that the number of all infinites is too big for any infinity to count! Crazy, huh? Read about it here. Also if you want to start to learn more Set Theory and things I recommend looking for Suppes' book. It's a nice introduction to these ideas and will get you acquainted with the language and how things work. Once you have a bigger vocabulary and a bit more mathematical maturity I recommend Jech's book, it is gigantic, dense and tough but has everything that someone who isn't actively doing Set Theory research could ever need about Set Theory in it, if you can get through it...
Despite being offered by the philosophy department, logic is much like a math class. It's best to treat it that way when it comes to studying. Do as many practice problems as you need -- there are plenty exercises around online and in textbooks. Schaum's Outlines is dirt cheap.
The pace of the class shouldn't be too bad -- UCD dedicates a full quarter to just propositional logic. It's common to see 12 and 112 combined in a single class.
The course will follow this rough outline:
I would advise you not to start with category theory, but abstract algebra. Mac Lane and Birkhoff's book Algebra is excellent and well worth the money in hardback. It covers things like monoids, groups, rings, modules and vector spaces, all of which are -- not coincidentally -- typical examples of structures that form categories. Saunders Mac Lane invented category theory along with Samuel Eilenberg, and Birkhoff basically founded universal algebra, so you cannot find a more authoritative text.
Edit: The other thing that will really help you is a basic understanding of preorders and posets. I don't have a book that deals exclusively with this topic, but any introduction to lattice theory, logical semantics or denotational semantics of programming languages will treat it. I would recommend Paul Taylor's Practical Foundations of Mathematics, though the price on Amazon is very steep. You can look through it here: http://paultaylor.eu/~pt/prafm/
What is Mathematical Logic? by Crossley will give you the foundations to understand the theorem in full.
What this expressions says
First of all let's specify that the domain over which these statements operate is the set of all people say.
Let us give the two place predicate P(x,y) a concrete meaning. Let us say that P(x,y) signifies the relation x loves y.
This allows us to translate the statement:
∀x∀yP(x,y) -> ∀xP(x,x)
What does ∀x∀yP(x,y) mean?
This is saying that For all x, it is the case that For all y, x loves y.
So you can interpret it as saying something like everyone loves everyone.
What does ∀xP(x,x) mean?
This is saying that For all x it is the case that x loves x. So you can interpret this as saying something like everyone loves themselves.
So the statement is basically saying:
Given that it is the case that Everyone loves Everyone, this implies that everyone loves themselves.
This translation gives us the impression that the statement is true. But how to prove it?
Proof by contradiction
We can prove this statement with a technique called proof by contradiction. That is, let us assume that the conclusion is false, and show that this leads to a contradiction, which implies that the conclusion must be true.
So let's assume:
∀x∀yP(x,y) -> not ∀xP(x,x)
not ∀xP(x,x) is equivalent to ∃x not P(x,x).
In words this means It is not the case that For all x P(x,x) is true, is equivalent to saying there exists x such P(x,x) is false.
So let's instantiate this expression with something from the domain, let's call it a. Basically let's pick a person for whom we are saying a loves a is false.
not P(a,a)
Using the fact that ∀x∀yP(x,y) we can show a contradiction exists.
Let's instantiate the expression with the object a we have used previously (as a For all statement applies to all objects by definition) ∀x∀yP(x,y)
This happens in two stages:
First we instantiate y
∀xP(x,a)
Then we instantiate x
P(a,a)
The statements P(a,a) and not P(a,a) are contradictory, therefore we have shown that the statement:
∀x∀yP(x,y) -> not ∀xP(x,x) leads to a contradiction, which implies that
∀x∀yP(x,y) -> ∀xP(x,x) is true.
Hopefully that makes sense.
Recommended Resources
Wilfred Hodges - Logic
Peter Smith - An Introduction to Formal Logic
Chiswell and Hodges - Mathematical Logic
Velleman - How to Prove It
Solow - How to Read and Do Proofs
Chartand, Polimeni and Zhang - Mathematical Proofs: A Transition to Advanced Mathematics
It depends what you're trying to get out of it.
There are literally hundreds of introductory texts for first-order logic. Other posters can cover them. There's so much variety here that I would feel a bit silly recommending one.
For formal tools for philosophy, I would say David Papineau's Philosophical Devices. There's also Ted Sider's Logic for Philosophy but something about his style when it comes to formalism rubs me the wrong way, personally.
For a more mathematical approach to first-order logic, Peter Hinman's Fundamentals of Mathematical Logic springs to mind.
For a semi-mathematical text that is intermediate rather than introductory, Boolos, Burgess, and Jeffrey's Computability and Logic is the gold standard.
Finally, if you want to see some different ways of doing things, check out Graham Priest's An Introduction to Non-Classical Logic.
Peter Smith is a philosopher and his Godel's Theorems book seems to me like what you are looking for. It considers the implications of the theorems and comes with a careful bibliography. Most university libraries would have it.
A shorter book is Torkel Franzén's.
Halmos is good.
Others at the undergraduate level:
[Enderton's Elements of Set Theory] (http://www.amazon.com/Elements-Set-Theory-Herbert-Enderton/dp/0122384407)
Moschavakis' Notes on Set Theory
If you are just looking at books on sigma algebras, you should look at measure theory books. Royden, Halmos are the standard texts.
https://www.amazon.com/Schaums-Outline-Logic-Second-Outlines/dp/0071755462 if you really need a crash course.
If you grew up in the US under "no child left behind" and are not mathematically / scientifically inclined it might be a challenge.
At least at my university the logic courses replace math requirements. Many students end up taking them thinking they must be easy because "hey, it's not math!" However, they kind of miss the implications of being able to take logic as a math equivalent.
Moreover, for various reasons philosophy is not usually taught at the K-12 level in public schools. I find that students are generally able to summarize readings and regurgitate lectures, but struggle with actually putting it all together in analysis and application.
If you want a head start pick up something like Schaums.
Edit: when I can choose the textbook I usually use a version of this. This seems to be a nice balance between logic and informal logic.
Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.
Example,
Linear Algebra for freshmen: some books that talk about manipulating matrices at length.
Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler
Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman
Basically, math is all interconnected and it doesn't matter where exactly you enter it.
Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.
Books you might like:
Discrete Mathematics with Applications by Susanna Epp
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Building Proofs: A Practical Guide by Oliveira/Stewart
Book Of Proof by Hammack
Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al
How to Prove It: A Structured Approach by Velleman
The Nuts and Bolts of Proofs by Antonella Cupillary
How To Think About Analysis by Alcock
Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash
Problems and Proofs in Numbers and Algebra by Millman et al
Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi
Mathematical Concepts by Jost - can't wait to start reading this
Proof Patterns by Joshi
...and about a billion other books like that I can't remember right now.
Good Luck.
You're welcome! But upon reflection, "Category Theory in Context," I think, presupposes too much background in abstract algebra to really qualify as "introductory." And I criminally overlooked An Invitation to Applied Category Theory: Seven Sketches in Compositionality! BTW, it's even available online.
> Never read it, will google them after this reply.
It's so fucking cool it's unreal. Not up to date with recent developments but wanna check it out again properly soon.
>Mendelson can be useful but, heck, you need some strong background. There's a lot of books mistitled as "introductions", mendelson is one of them.
That'd explain why it was so dense lol - I dived from no mathematical logic (apart from like basic predicate calculus) and using first order symbols sparingly.
>There's actually no perfect book to serve as introduction to mathematical logic, but I highly recommendthat you check out https://www.amazon.com/Mathematical-Logic-Oxford-Texts/dp/0199215626
>
>Also get this little fella here: https://www.amazon.com/Mathematical-Logic-Dover-Books-Mathematics/dp/0486264041 for a nice, short survey.
Thanks :D I'll check it out. Given your breadth of knowledge on it I imagine your background is pure mathematics?
https://www.amazon.com/Proof-Theory-Second-Dover-Mathematics/dp/0486490734
Since mathematical logic has split after the Second World War in four or five branches, it is uncommon for more advanced texts to have a broad focus on logic in the general sense like beginner's books.
That being said, the most 'advanced' (and quite recent) logic book that attempts at doing justice to the main (though pretty introductory) results of all the major branches of logic is Hinman's Fundamentals of Mathematical Logic (it claims to be an update on Shoenfield's Mathematical Logic, which has some more info on proof theory).
In the spirit of the answers below, where 'advanced' is taken to mean most unreadable or with the most prerequisites, I would nominate another of Shelah's books, Classification Theory.
I found this book immensely helpful.
>I worry about the kinds of logical fallacies that I'm missing. I want to pick up a text that will give me a good introduction to formal logic.
Generally, fallacies occur in informal logic, or critical thinking. Technically yes, a mistake in formal logic does constitute a fallacy, but those don't come up nearly as often. Moreover, learning formal logic comes with a potential risk. In most systems taught in universities and introductory texts, you end up learning bivalent logic in which every proposition gets treated as true or false. Habitual immersion in such a system can lead to black and white thinking when students fail to see the system as a tool and identifiy it with the whole of "correct thinking."
I think you might get more mileage out of a good critical thinking book. I wanted to recommend Attacking Faulty Reasoning by T. Edward Damer, which reads like an encyclopedia of fallacies and how to combat them, but apparently it costs a fortune. (WTF! I paid like $20 for it back in '94!)
> Second and third semester calculus
Is this vector calc? If so I enjoyed this book as it's very geometric, not at all rigorous and has lots of worked examples and exercises. Sorry it seems to be so expensive -- it wasn't when I bought it, and hopefully you can find it a lot cheaper if it's what you're looking for.
In general Stewart's big fat calculus book is a nice thing to have for autodidacts.
Obviously what you describe might include analysis, which these books won't help with.
>Formal logic theory (Think Kurt Godel)
I've heard Peter Smith's book on Godel is good, but haven't read it. Logic is a huge field and it depends a lot on what your background is and what you want to get out of it. You may need a primer on basic logic first; I like this one but again it's quite personal.
The best introductory logic text you will ever find: Logic: Techniques of Formal Reasoning, 2nd Edition Donald Kalish, Richard Montague.
This book is especially good if you have done any programming. The structure of main and sub-proofs corresponds to main program and subroutine calls. You can pick up a used copy for around $23 here: https://www.abebooks.com/book-search/author/kalish-montague-mar/ and you can see the table of contents here: https://www.powells.com/book/logic-techniques-of-formal-reasoning-9780195155044 (but, obviously, don't buy it for $133!)
For meta-theory, take a look at: Metalogic: An Introduction to the Metatheory of Standard First Order Logic by Geoffrey Hunter, https://www.amazon.com/Metalogic-Introduction-Metatheory-Standard-First/dp/0520023560. This book explains things in a clear way using ordinary English, before setting out the proofs.
And, if you are interested in model theory, take a look at Model Theory by C.C. Chang and H. Jerome Keisler, https://www.amazon.com/Model-Theory-Third-Dover-Mathematics/dp/0486488217 and you should get a good idea of what additional mathematics you might want to pursue.
Enderton and Schoenfield are two typical books on mathematical logic. Both cover the completeness of propositional logic, the incompleteness theorems and the basics of both model theory and recursion theory. I don't like the style of Enderton's book--I don't think it does a good job of helping develop fluency with the material, but it's certainly a more accessible introduction than Shoenfield's.
I actually found Hughes & Cresswell's A New Introduction to Modal Logic to be a great text.
Hey cool, thanks a ton!
I have been looking at logic and set theory a lot lately so this is all pretty helpful.
If you were interested, I could add to the list a textbook I'd recommend - http://www.amazon.com/gp/product/0716730502/ref=oh_aui_detailpage_o03_s00?ie=UTF8&amp;psc=1
EDIT: the book really goes through logic in a way i can understand, it may be too elementary for you though.
Either way, I hope that is a useful weapon to add into the ol' arsenal!
If you want to learn elementary set theory from the "inside", I recommend Axiomatic Set Theory. It's old, but everything in it is still valid and relevant to modern mathematics, and it's easy for self-study. You can learn "the" set-theoretical construction of the integers (i.e., not the Zermelo version), the reals, and so on, in a very rigorous context, yet it's not difficult for anyone who's had a basic course in logic.
This book does not treat set theory from the "outside", e.g., with modern techniques like forcing. That's a good thing, tho, 'cause the fundamental theorem of forcing is about 100x more difficult to fully understand than the fundamental theorem of algebra.
> Why do people push Velleman's "How to Prove It" so much?
Well the alternative book to this rather narrow subject (textbook speaking) is the classic (i.e. has been around for ever - the early eighties) is this book, "How to Read and Do Proofs". The advantage Velleman's book has are the included exercises which the former does not even have. Reading Amazon's reviews indicates that Velleman's book has a more favorable reception with reviewers liking the prose.
>I thought the book was very lacking, especially in the exercises.
I really haven't met many exercises/"math problems" that I really "liked"! Maybe that's why some people call them "problems".
Bitcoin mining programs are pretty advanced pieces of software, especially due to the rise of ASICs and other esoteric hardware that must be supported. You might want to consider an easier project for the time being.
Fundamentally, you need to have a good grasp on data structures before you can really get going with Merkle trees and cryptocurrency mining in general.
Here are some free resources to get started
If you want to consider grabbing a book, Algorithms (4th ed) is generally regarded as a decent book to get started on thinking with and using algorithms and data structures.
edit: depending on the time you have, you may also want to consider reading this on set theory. It tends to make later algorithms a bit more clear as a lot of it draws from set theory.
I don't know if there is any useful distinction between the two. Metamathematics may be thought of as strictly the reduction of math to a single, consistent formal system, whereas philosophy of mathematics can also be concerned with epistemology and existence (as well as metamathematics).
Well, anyways, good reading for this stuff that's also "pragmatic" (usefulness of the concept of pragmatic questionable in this context) starts with theoretical computer science.
No, wait, it starts with internet encyclopedias. I'm not going to link to Wiki, you can do that yourself. You however might not know about Stanford's encyclopedia of philosophy:
http://plato.stanford.edu/entries/philosophy-mathematics/
Okay, now for books. You should know logic first. I suggest Hinman, "Fundamentals of Mathematical Logic". It's a big part of fundamental mathematics, and touches a lot of its philosophy.
For computer science, you want to look at complexity theory I think. That should cover things like decidability and the Halting problem. You should be able to find a good book for free from the Prof. Arora:
http://www.cs.princeton.edu/~arora/
For more philosophy-oriented books, I don't know. Good luck!
Sorry. The sentence is not circular. It only appears to be circular. The ideas are clearly explained in text books on computability theory. Or if you are smart and patient you can just read Turing's original paper, or if you are really smart and really patient Goedel's work as well. Take your time and use your own mind to form your own opinion.
https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
My introduction was in this book:
https://www.amazon.com/Computability-Unsolvability-Prof-Martin-Davis/dp/0486614719
For more original sources:
https://www.amazon.com/Undecidable-Propositions-Unsolvable-Computable-Mathematics/dp/0486432289#reader_0486432289
I would just dive into it to see if it makes more sense! Here is a guide about delta epsilon proofs, which is one of the most common basic proofs you learn about in pure mathematics. Real Mathematical Analysis is a great textbook about real analysis. Also, if you're worried about the math, I would look into philosophical logic—Logic by Hodges is a good text for that and it won't involve any necessary background in math.
Thank you very much for your reply and explanation. It's starting to make more sense now - I think I just need to study Model Theory on its own. (I saw that you praised this book in another thread. Would you recommend it for self study?)
> The distinction is that in math, all foundational meta-theories are require to get the right answers on simple object-level questions like "What's 1 + 1?". If your mathematical metatheory answers, "-3.7" rather than "2", then it is not "different", it is simply wrong. We can thus say that Foundations of Mathematics is always done with a realist view.
The natural numbers are an interesting example, which goes back to ADefiniteDescription's point about a privileged model. The basic axioms of arithmetic are categorical, i.e. have only one model, up to isomorphism. Not all theories have this property, though.
If it could be shown that some moral theory similarly has only one correct interpretation - that all alternative interpretations end up being isomorphic - then that could support a kind of realism, at least in the context of that theory. A lot would depend on the nature and scope of the theory in question, and its interpretation.
So perhaps Parfit's position would be better captured by saying that he believes there are unique true answers to moral questions, as there are for questions in categorical mathematical theories.
> What's a good textbook for that field, anyway?
The books I studied are quite outdated now, but a classic modern text is Model Theory by Chang & Keisler. That might be more comprehensive than you're looking for. You could try Model Theory: An Introduction - its first chapter is quite a concise basic intro. There's also A Shorter Model Theory.
I haven't used the set theory books myself so I can't comment on their quality, but anytime I hear someone looking for reasonably priced math books I immediately think of the Dover Books on Mathematics series.
https://www.amazon.com/Theory-Logic-Dover-Books-Mathematics/dp/0486638294
https://www.amazon.com/Axiomatic-Theory-Dover-Books-Mathematics/dp/0486616304/ref=pd_bxgy_2/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486616304&pd_rd_r=c84f7c07-350b-4ec3-8fc5-adf30ae9b20c&pd_rd_w=74PSR&pd_rd_wg=RG95z&pf_rd_p=a2006322-0bc0-4db9-a08e-d168c18ce6f0&pf_rd_r=RKXGP4020J1B5GVED0PH&psc=1&refRID=RKXGP4020J1B5GVED0PH
https://www.amazon.com/Book-Theory-Dover-Books-Mathematics/dp/0486497089/ref=pd_sbs_14_2/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486497089&pd_rd_r=82e4d26d-281c-4eb6-982f-5811be6be764&pd_rd_w=gx29l&pd_rd_wg=O6GtQ&pf_rd_p=43281256-7633-49c8-b909-7ffd7d8cb21e&pf_rd_r=8TQ89WSVK726CHBY6N96&psc=1&refRID=8TQ89WSVK726CHBY6N96
https://www.amazon.com/Naive-Theory-Dover-Books-Mathematics/dp/0486814874/ref=pd_sbs_14_1/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486814874&pd_rd_r=82e4d26d-281c-4eb6-982f-5811be6be764&pd_rd_w=gx29l&pd_rd_wg=O6GtQ&pf_rd_p=43281256-7633-49c8-b909-7ffd7d8cb21e&pf_rd_r=8TQ89WSVK726CHBY6N96&psc=1&refRID=8TQ89WSVK726CHBY6N96
edit: added more books
My department uses Enderton's Elements of Set Theory in our introductory Set Theory course. It starts at the basics and has some large cardinal and forcing stuff towards the end, I believe.
http://www.amazon.com/Elements-Set-Theory-Herbert-Enderton/dp/0122384407
For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.
Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:
Discrete Mathematics with Applications by Susanna Epp
Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Numbers and Proofs by Allenby
Mathematics: A Discrete Introduction by Edward Scheinerman
How to Prove It: A Structured Approach by Daniel Velleman
Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi
Some special topics(elementary treatment):
Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby
A Friendly Introduction to Number Theory Joseph Silverman
Elements of Number Theory by John Stillwell
A Primer in Combinatorics by Kheyfits
Counting by Khee Meng Koh
Combinatorics: A Guided Tour by David Mazur
Just a nice bunch of related books great to have read:
generatingfunctionology by Herbert Wilf
The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule
A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger
If you wanna do graphics stuff, you wanna do some applied Linear Algebra:
Linear Algebra by Allenby
Linear Algebra Through Geometry by Thomas Banchoff, John Wermer
Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman
Best of Luck.
Never read it, will google them after this reply.
Mendelson can be useful but, heck, you need some strong background. There's a lot of books mistitled as "introductions", mendelson is one of them.
There's actually no perfect book to serve as introduction to mathematical logic, but I highly recommend that you check out https://www.amazon.com/Mathematical-Logic-Oxford-Texts/dp/0199215626
The price is outrageous, so get a pdf here if available.
Also get this little fella here: https://www.amazon.com/Mathematical-Logic-Dover-Books-Mathematics/dp/0486264041 for a nice, short survey.
Have you tried the print shop at the university? they are used to binding dissertations for people and have some nice options (at least at my local one) .....in that general vein its a printing service you want, not a publisher , so try local print and copy shops ....ones with digital printing are better for short run things and one offs. Most of these places wont care what you are printing , they might ask you to sign a box saying you have the right to print, but that's as far as it goes.
Also seems like there are a few books available of his work:
http://www.turing.org.uk/sources/biblio.html
https://www.amazon.com/exec/obidos/ASIN/0198250800/alanturingwebsit
https://www.amazon.com/exec/obidos/ASIN/0486432289/alanturingwebsit
https://www.amazon.com/Alan-Turings-Systems-Logic-Princeton/dp/0691155747/alanturingwebsit
Consider First-Order Logic by Raymond M. Smullyan, or forall x: An Introduction to Formal Logic by P. D. Magnus of SUNY Albany.
Of course, reporting to the police is good too. There is also a line for what to report. If you witness a drug deal or are offered drugs, it might be something to report. If you just see some guy who looks sketchy, that's not the best use of a police's time.
But, the problems you state is the reason it's good to get politicians involved.
When police forces aren't under the watch of political leaders, and consequently, the public, you can get degradation of their services. Complacency, thoughtlessness, filing away tips into black holes; these are all symptoms of an out-of-touch police centre with no outside stimulus. When they understand that political leaders are getting involved, it makes them more self-concious and will also weed out bad officers.
You also say that police reporting for smaller things is a resource hog, but it's not really the case. It doesn't really take much more work time to get officers to record tips. In fact, that work can be passed off to cheaper office secretaries.
The idea that small details and reports don't become useful until after the crime doesn't hold much water either, because of the steady rate in which criminals raise the stakes. Almost all the big crimes happen after a string of lesser crimes. It's also a big win to catch a criminal before he commits too many crimes, because they can be dealt with before they become pathological.
And even after the crime, more information gives the courts more to go on, so they can more accurately asses what should happen after a charge.
Also, 9/11 is a bad example. They're such different scenarios that you can't really apply one to the other confidently. When you make comparisons, look for the most similar examples you can find, like another city's statistics. Using vastly different events to shed light on municipal problems is inaccurate and can lead to wrong conclusions.
Also, I'm going to go out on a limb and assume you haven't had a lot of research into critical thinking. I'd highly recommend you read a few books on it. You can rent both of these at the public library.
There's even a kickstarter on a college level book on it. I can send you a copy when it's done.
Really? Nolt's Logics? Besides the numerous errors, it's telling that the book has not come out in a second edition.
I think Quine's Methods of Logic remains a fantastic text, if it is a bit dated and filled with Quinean quirks. A more recent text, Ted Siders' Logic for Philosophy is also very good, although the exercises are sometimes quite difficult. I would combine Sider's text with a book on metalogic, since he skips over some of that. Kleene's Mathematical Logic is a classic text by a real giant in the history of 20th century logic. Those should keep someone busy for a good year of study. If you want to branch out, Graham Priest's Introduction to Non-classical Logics will get you started in modal, tense, epistemic, paraconsistent and dialethic logics, also by a contemporary giant in the field.
After that, I would go on to set theory, and stop when I had a grasp of forcing.
Physicist here so don't pretend I don't know what science is. (Though like the ancient Pythagoreans I'm sure as soon as I discuss something that has been proven that goes against a purely scientific worldview out comes the pitchforks.) And though I love science, unlike some people here I am willing to admit to the limits of science. Science can lead to all truth in the same way that rational numbers define all numbers: it can't! and Godel proved it.
The real problem with science is that it has been mathematically proven by Godel that there are more things that are true then are provable and thus you can't ever have a scientific theory that can determine the truth or falsity of all things. As soon as you write down that theory, assuming it allows for arithmetic, Godel's incompleteness theorem immediately shows if the theory is true there will be true statements about reality that are beyond provability. Read Godel Esher Bach or Incompleteness or work through it yourself in this textbook as I have.
So like I said above, science is great in it's sphere (and in that sphere let me emphasize it is awesome!) but leads to all truth in the same way that rational numbers leads to all numbers. (And the analogy is precise since Godel used the famous diagonizational argument in his proof.) Russell and Whitehead set out to show in the early 1900s that if we could determine the axioms of reality then through logic work out everything that was true and Godel spoiled the party.
It it would be one thing if these truths were trivial things, but they are not. Some examples of true or false statements that may fall into this category of being unprovable are:
Now, at this point critics almost always tell me: but Joe, Godel's incompleteness theorem is only relative to your set of logic. (Ie... we can prove Goldbach by just adding axioms needed to do so.) Fine. But two things: (first) adding axioms to prove what you want willy nilly is not good science. (Two) You now have a new set of axioms and by Godel's theorem there is now a new uncountable set of things that are true (and non-trivial things like I listed) that are beyond proof.
Now usually comes the second critique: But Joe, this doesn't prove God exists. And this is true. But at least it has been proven God gives you a chance. It has been proven that an oracle machine is free from the problems that hold science and logic back from proving the truth of all things. At least something like God gives you a chance (whereas science falls short).
Or, like Elder Maxwell says so well: it may only be by the "lens of faith" that we can ever know the truth of all things. He maybe be right, and hence the importance to learn by study, and also by faith...