Getting into formal logic seems essential to understand modern philosophy...

Hm I did read Language, Truth and Logic before that so thanks for reminding me.
I don't think there's anything difficult in Tractatus though

I recommend Godel Escher Bach as an introduction, especially the first half of the book.

Its bibliography alone merits your perusal. He offers some good intros to formal logic.

Identify and summarize the principle arguments of it, then.

formal logic helps the student organize arguments in syllogistic format, and having a knowledge of formal logic and operations of validity, soundness, and forumalting inductions/ or maybe it was deductions or both .

basically knowing about formal logic you can breakdown the structure of arguments pretty easily. and examine them for any formal fallacies.

*principal

well there's a whole lot of defining propositions, logical constants etc
propositional functions from a self-consistent picture that can be compared to the world to see if they are true/false
propositions are tautologies
elementary propositions form the basis of logic-space
logic is explored independently of any reality; you start from some axioms and refine the system
numbers come from successive applications of operations
objects are defined by their relations to everything else
physics and maths are the same kind of thing: general before being specific
philosophy consists of clarification/analysis of the language we use, inc. logic.

etc.. not really sure what you're after

prin·ci·ple
ˈprinsəpəl/
noun
noun: principle; plural noun: principles
1.
a fundamental truth or proposition that serves as the foundation for a system of belief or behavior or for a chain of reasoning.

There are seven in the TLP.

>propositional functions from a self-consistent picture that can be compared to the world to see if they are true/false

Wrong. Logical operations, what you mean by 'prepositional functions' here, are neither true nor false, and there is no getting outside their 'place' in language to 'see' that they hold. Their isomorphy with 'the world' must be presupposed.

>propositions are tautologies

Unbelievably wrong. Propositions form a picture of the world. A picture either is the case, or is not. A tautology is 'necessarily true' by virtue of its logical form, and does not form a picture of the world

>elementary propositions form the basis of logic-space

The fuck is 'logical space'?

>logic is explored independently of any reality; you start from some axioms and refine the system

'Philosophy [i.e. logic] does not discover anything, for nothing is hidden; everything is there, as it were, on the surface.'

Etc. Etc. Etc. Sounds like you misunderstood basically everything which is pretty embarrassing for such a simple read.

>Wrong. Logical operations, what you mean by 'prepositional functions' here, are neither true nor false, and there is no getting outside their 'place' in language to 'see' that they hold. Their isomorphy with 'the world' must be presupposed.
I do not mean "logical operations", I mean "propositional fucntions". I take it you have not encountered this phrase before.
I should have been more clear. When you compare a propositional function with the world, you substitute values for the variables; these values are parts of the world. Out comes the truth-value.

>Unbelievably wrong. Propositions form a picture of the world. A picture either is the case, or is not. A tautology is 'necessarily true' by virtue of its logical form, and does not form a picture of the world
Sorry. I meant "analytic propositions". A picture is self-consistent; it may or may not correspond to the world directly.
I think they do form pictures of the world, actually - when we substitute names for the variables. That's the whole point of creating systems.

>The fuck is 'logical space'?
Are you sure you read this book? I said "logic-space". It is the space which propositions inhabit, formed from combinations of axioms or elementary propositions.

No, I understood it fine. You seem to have had trouble, however.

I'll add something:
You can think of logic space as a vector space, and axioms are the basis vectors

>I think they do form pictures of the world, actually
I'm going to butt in and say I prefer reading pictures as something outside of logic/language. They are things whereof we cannot speak.

The whole book is something of a picture tho.

>You can think of logic space as a vector space
Can you spot the problem with that?

>I think they do form pictures of the world, actually - when we substitute names for the variables.

Let's take the tautological propositional function 'p=p', and substitute a name for the variable, 'anonymous', so we get 'anonymous is anonymous'. What picture does this form of the world?

>accepting the logic meme

It's an anology and it works. Axioms = basis vectors; propositions = linear combinations of them. The basis vectors are orthogonal, i.e. they cannot be deduced from each other. What's more, you can take a different basis and come up with a different geomety - a different system. And this is precisely what happens, until they are linked at some point (the projection is provided : r^2 = x^2 + y^2)

How about the idea that particles are probability wavefunctions?
It comes from maths entirely (a field of tautologies); the interpretation in language gives it physical meaning. Wittgenstein even talks about this towards the end of the book. You are taking an obvious tautology and trying to state its absurdity - but mathematics is all about discovering the tautologies that are not so obvious. And then physics applies them.

I'm not trying to 'state its absurdity' but show that it doesn't form a 'picture' of anything, which is exactly what Wittgenstein says.

What does it mean for physics to apply to a field of tautologies?

It is a pretty analogy, though.

>apply to

Just apply. Given that tautologies don't say anything about the world, if physics merely applied tautologies physics would likewise say nothing.

>It is a pretty analogy, though.
I'm rather happy with it. Was nice to see Witty mention it in passing. It'd be interesting to see if there is anything more than a resemblance between the ideas.

Of course the Organon, just so you have an idea about the general aims of logic.

The book I used was "Introduction to symbolic logic" by Rudolph Carnap. He was one of the major analytic thinkers of the 20th century so it seemed good to go with him. It isn't the best introduction, in fact it can be a little rigorous, but afterwards you will have a firm grasp of formal logic. I would recommend that along with other tools online. For instance you can practice solving truth tables:
ixl.com/math/geometry/truth-tables

Russel's book "Introduction to Mathematical philosophy" covers the more math intensive side of logic and focuses on his own work. I didn't read it fully so I can't say too much on it

>tfw the metamathematics of set theory is philosophy in the form of mathematical theorems

>It's an anology and it works.
It is and it isn't. Thinking of logic like a vector space is a bit of a ladder that you throw away. Because vector spaces have their basis partially in logic. You cannot then step outside the logic-space (at least not easily) and it becomes difficult to talk about the boundary of the logic-space.

>physics would likewise say nothing
Absolutely. And insofar as physics is based upon logic it does say nothing. However there are points where physics goes beyond logical propositions.

Underrated post.

P.S. I hope you're trolling.

phd in (analytic) philosophy here, ignore every post in this topic

logic is crucial for understanding analytic philosophy and helps you to understand and critically evaluate argumentative structure in philosophy texts. good call.

needless to say, whatever book you end up choosing, do the fucking exercises. try to find a book with answers (library genesis is your best friend here)

there are many entry level textbooks. i would recommend goldfarb's 'deductive logic.' it's beautifully and conversationally written, with no stupid bullshit. just the basic logic and some exercises. it's not all cluttered up like a standard textbook.

that is a solid foundation in first order logic with identity, multiple quantifiers, and decriptions. it'll get you the bedrock for everything else.

but, there are many logic textbooks online. download a bunch from lib gen and see which one fits your educational needs. which ones are written in ways you understand, which ones presume the right level of rigor, etc.

after that, i recommend ted sider's logic for philosophy. it's the next step. you learn meta logic (completeness and soundness proofs), as well as gain familiarity with set theoretic notation, and more generally, mathematical sophistication. it's harder, but worth it. again, do the exercises.

then, go nuts. van benthem's 'modal logic for open minds' is a great succinct intro to modal logic. if you want modal logic done more rigorously, fuck the standard opinions and read blackburn de rijke et al. it'll also serve as good practice with discrete structures more generally, which is useful for various things in philosophy.

michael sipser's book is an excellent introduction to automata and computability.

if you want a challenge, try peter smith's textbook on godel's theorems. but that's after the first two books i recommended, at least.

all these books are on library genesis. good luck and have fun!

>phd in (analytic) philosophy here, ignore every post in this topic
Nice paradox in the first sentence doc

lol ok fine you got me.

this reminds me that it is possible to get too carried away with logic. take breaks from doing exercises by reading some grice.

what can someone get out of these threads that they can't get out of reading a wikipedia article for an entry-point?

as far as logic, math, engineering and some sciences go, wikipedia can be a very useful reference for people who already have some training in those fields. as an introduction for people, it is at best worthless and at worst misleading and confusing.

and all wikipedia articles on philosophy should basically just be deleted. (although many articles on the stanford encyclopedia of philosophy provide accessible and serious introductions to various topics)

I don't know how it is now bit when I was an engineering undergrad a while back I didn't find wikipedia enlightening at all. It veers between oversimplified and obfuscating complexity and quite often was dead wrong on key points. I barely use it now tho.

Yeah that's true. You wouldn't learn anything new by modelling logic as a vector space, probably. It's nice to visualise though. Tbh mathematical functions are just propositional functions so yeah, nothing new

Yeah, not quite. Gödel's work in logic is way beyond what most analytics will ever use, and his actual philosophical focuses more on the continentals.

Are you familiar with Tarksi's introduction to logic or Kleene's introduction to metamathematics? I've been hoping to get into logic as a bit of a hobby. Do those seem like good books to begin with?

Gödel's work is also contested even today. A lot of people don't like it.

>It's nice to visualise though.
But then it's still logic justifying logic. I know a lot of positivist analytics hate this part of Wittgenstein, but it's important. You can look at reason from a standing of reason and everything looks neat and tidy, but it's not everything.

I'm not using it to justify anything, just to describe..

DO NOT LISTEN TO THIS user

Except for the fact that all of Russell's logic books are shite and important only as historical pieces

w-why?
The Logic Manual is used by Oxford's philosophy department
I can see why you'd disagree with the other two but not why you would do so quite as vehemently.. I guess 'Principles' might be a bit of a slog
I'd definitely add 'Language, Truth and Logic' in there before/after LM
desu though I'm in pretty much OP's situation, no expertise

you are such a pleb it hurts. it really does. and you are earnest about it, which is worse

wow what a meaningless comment

Nice try, kiddo. But your pleb is exposed. You've fucking lost it now.

I'm guessing you're an idiot that doesn't even have a formal education on logic or philosophy.

You sure are making the world a better place, o great patrician!! Keep patricing on and some day the rest of us will learn to follow your lead :^)))))

This user is the typical imbecile that throws ad hominems and dismisses ideas, without ever giving a proper argument.

Those are different anons.. and the first is responding to the imbecile

Oh yes, I (you)d the other user to support him. I thought it was rather clear from context.

So did you actually get a professorship? I'm wondering if this is a good path for me.

Earning my PhD in set theory. How much do you work in the philosophy of mathematics? I'd be interested in a technical treatise on the matter.

Would you say that user's advice is good for those who want to get started with this? As opposed to this advice
Just going for straight math?

Russell's earlier work on logic is his good work. It's his later stuff that's shit.

Paradoxes and sets and classes? Good (at first). History, Principia and p much anything after he met Wittgenstein? All shit. Principia has some interesting bits so is maybe the most polished turd.

>16 year old detected

>adds like three qualifiers to that statement
And you wonder why I just dismiss Russell wholesale. Not worth the time. Overrated because his name is familiar.

His advice is geared toward philosophical logic rather than mathematical logic. I am not an expert in the former, but it seems very sound. I would make just a couple addenda:

After predicate/first-order logic and before what he calls meta logic, one should absolutely learn model theory. Prior to learning model theory, one should briefly learn very basic group theory, just for the sake of concrete examples. Understanding what a formal structure is in the most general sense, alongside the manner in which collections of formal sentences correspond to and are interpreted in models thereof, is at the foundation of metamathematics and prerequisite to both Gödel completeness and incompleteness.

To learn basic model theory, one needs to know basic set-theoretic notation, but only after model theory should one learn axiomatic set theory. This is because the metamathematics of set theory is about, well, models of set theory. Interestingly, model theory is defined in terms of set theory, and set theory is defined in terms of model theory. The foundations of logic are ultimately circular.

From here, one should then learn Gödel completeness and incompleteness. Inextricable from the former is basic proof theory, and Gödel completeness essentially says that a sufficient notion of formal provability is equivalent to bona fide logical consequentiality in a model-theoretic sense. It is not an understatement to say that Gödel incompleteness may be the greatest theorem in all of mathematics, and indeed one of the greatest intellectual accomplishments of humankind. Very roughly, not only is no logical system complete, but also no logical system can even prove its own logical consistency.

One should then briefly return to set theory from a metamathematical vantage, and consider how there are necessarily mathematical propositions that are in some sense neither true nor false, but simply logically independent. Combined with Gödel incompleteness, we also have a hierarchy of different metamathematical foundations (comprehensive systems of logic each of which corresponds to entire universes of mathematics) stratified by both logical consequentiality and relative consistency. For instance, even if you don't believe in the Axiom of Choice, you must believe that it is logically consistent to believe it. And it is a result of Cohen that even if you do believe in the Axiom of Choice, you must accept that it is logically consistent to deny it. And by Gödel incompleteness and completeness, it is always consistent that your logical system be inconsistent, and there will always be strengthenings of fragments of your (necessarily incomplete) system whose consistency you can't realize even relative to the assumption of your own consistency. (An example is the replacement of the Axiom of Choice with the Axiom of Determinacy.) I think that understanding the dynamics of logic like this is essential to any real understanding of the philosophy of logic.

kys

Could you give some specific books then for this specific order of starting this study? Just for model theory would be fine.

>Interestingly, model theory is defined in terms of set theory, and set theory is defined in terms of model theory. The foundations of logic are ultimately circular.
Set Theory isn't Logic, it is Logic + the language of {epsilon}. Perhaps you meant "The foundations of MATHS are ultimately circular"? All popular things are for the most part flawed: among the many candidates, the Copenhagen interpretation in QM, for example, is probably the best known case. Likewise, many philosophically-minded logicians dismiss Tarski's definition of truth because they don't think that it does the job in capturing its essence; hence there are many alternative, axiomatic proposals in formalising the truth predicate. Many believe that any talk of large cardinals is bollocks, and so forth. But there are of course other, albeit less fashionable, candidates for alternate foundations of maths, too: category theory being a good example.

But aren't you showing some huge, model-theoretic signs of bias (typical since it is something of an orthodoxy)? Not all logicians prefer to work in a model-theoretic framework; some think model-theoretic thinking is deeply flawed and use proof-theoretic ideas instead.

Before model theory, teach first learn mathematical notation and proofs. I have heard very good things about "How to Prove It" by Vellerman. This will teach you mathematical notation, propositional logic, set theory, and group theory, and assumes essentially no prior mathematical knowledge. This one book takes care of all of the prerequisites.

My recommendation for both model theory and completeness/incompleteness is Leary's "A Friendly Introduction to Mathematical Logic." It is nice because it covers all of the topics and should be accessible after "How to Prove It." Other books tend to be both more specialized and aimed at graduate students.

So really, that's only two books, and should be sufficient for one interested in philosophical rather than mathematical logic. These two books are my recommended addenda to , after whose foundations you can read his Sider and Van Benthem for the philosophical direction and proceed from there.

Ah, that was my bias as a mathematician. I identify logic and mathematics. Any objectively logical thing is mathematics, and mathematics is the study of the logically objective. If one makes a distinction between objective logic and "logic", then there is logic that is not mathematical.

From there, I identify mathematics with set theory. This is because the entirety of mathematics is expressible in the language of set theory (yes, including category theory), and under this syntactic translation, every mathematical theorem is a consequence of ZF/ZFC — call having this property "globality". This is what it means to be foundational. While set theory may not be the only possible foundation of mathematics, it and any other foundation must be both inter-translatable and global.

As for your second paragraph, model-theoretic reasoning and proof-theoretic reasoning are equivalent by Gödel completeness. Even if you're one of those strange constructivists whose proof-theoretic reasoning rejects the law of the excluded middle, models with truth values in a free Heyting algebra satisfy a completeness theorem with respect to intuitionist deduction, though this is unwieldy.

Fuck yes I just bought How to Prove It, looking forward to it with my other Set Theory book

What set theory book, and what is your aim?

op of that post here. kleene, tarski and church all have "classic" introductions to logic and metalogic/metamathematics, but they're really hard going.

first, the notation is old, non-standardized, and often times needlessly complex. one of the benefits of reading a textbook written by contemporary authors is that exposition, notation and proofs have been beautifully simplified. i'm not familiar with the tarski book but i do know the kleene and church.

another thing is that they require a fair amount of mathematical sophistication.

if you want to take mathematical logic dead fucking seriously, like a mathematician would, the standard text is schoenfield's mathematical logic. it's fucking hard as fuck. but it will make you into a logician. if you want a serious challenge, use that, and read the kleene/church/tarski for further developments/curios/historical interest.

Set theory is fucking rad. My friend is really into Koellner Woodin shit, but I don't really have the time for it, unfortunately.

I don't really know philosophy of mathematics very well, sorry. I work in other fields. But you should scope the work of Koellner, he takes the philosophy part just as seriously as he takes the mathematical part.

Introduction to Set Theory third edition by Hrbacek and Jech

I graduated with a BA in philosophy a year ago and had to put off grad school while helping out my family. In university I wanted to focus on phenomenology but later realized what I was interested in are the tools of philosophy, which I see as logic and language.

From that, goals are to study logic, phil. of maths and language, figure out specifically what I want to study, then apply to some grad schools. Why do you ask?

I took a class in set theory that used Jech's Set Theory. Hard as fuck but you fucking learn set theory.

Was it at UCI?

It's funny. I haven't even read either of the books I recommended. But they seemed appropriate and well-regarded for texts aimed at those who don't intend to study mathematics.

Jech's Set Theory (not to be confused with his Introduction to Set Theory) is an extraordinary textbook. It's excellent your class used that. Yes, it assumes mathematical maturity, but it is the single comprehensive reference for truly learning the subject.

Thanks, m80, I'll check that out. No point in half-assing it, really.

Did my undergrad in math here. Don't get into formal logic it's a giant inescapable hellhole. You won't survive the descent and little is gained from pursuing the topic.

Before u know it youll be asking questions about inaccessible cardinals, pathological models of zfc, whether or not the real numbers should exist, alternative axiomizations of set theory and even worse questions than these

And you will have wasted countless precious years of your life without anything to show for it

It is both extremely technical and completely ugly and totally useless, even by the standards of theoretical math

On the outside the questions you deal with in formal logic are very interesting but the answers are almost never as illuminating as you would hope them to be.

Formal logic is about as rewarding as hammering a nail through your thumb

I can't even understand this. It's like calling beautiful ugly.

>Reverse Mathematics was too hard for the baby
Aww at least you tried!

What does everyone think of Mendelsens Introduction to Mathematical Logic? I picked it up from my uni library and worked through the first couple of chapters but if it's regarded as not good I'm fine with changing textbook.

I haven't read it, but I can see the table of contents on amazon. It seems to require mathematical maturity, so I judge it on that level and assume you can read graduate-level math.

The book seems extremely lacking. In its attempt to cover the basics of the various branches of logic, it does an extravagantly slipshod job at all of them. The only thing that may be worthwhile about the book is its chapter 3 on Gödel Incompleteness, since that's not a field but a result.

For model theory, read Hodge's "A Shorter Model Theory."

For set theory, read Jech's "Set Theory."

For recursion theory, read Soare's "Recursively Enumerable Sets and Degrees."

Why Soare and not the one by Hartley Rogers?

Waits far too long to get into foundational notions. For instance, the recursion theorem should be on page 36, not 180, and the arithmetic hierarchy on page 60, not 301.

How are logic and philosophy of language tied together?

Language Proof and Logic was my textbook for my first year first order logic course, it also has some good set theory and comes with some programs for the exercises to do with possible worlds, truth tables, formal proofs etc.

It may be haughty but whenever I see someone mention truth tables as an aspect of logic I grimace because they're completely trivial.

True, we only used them to display Tautological consequence/equivalency, although outside of early Wittgenstein and Frege I've never seen them come up in any contemporary analytic stuff.

thanks for recs

Aren't you?

Thank you! I'm in physics so yes - the maths is fine for me. I'll check out those recommendations next time I'm in the uni library.

For what it's worth, as that poster, I find set theory to be the most philosophically scintillating mathematics. Furthermore, Jech's Set Theory is the best math book I've ever read.

I would like to draw this thread's attention to the maverick philosophy of the autodidact Christopher Langan. Known for having the highest verified IQ of about 200 (IQ is untestable at this level), his crippling hubris held him back from academia. However, his philosophy of reality is scintillating. It is relevant to this thread because it draws heavily on philosophical extensions of mathematical ideas, employing mathematical vocabulary together with defined neologisms. Intuitive understanding of graduate mathematics is a prerequisite.

megafoundation.org/CTMU/Articles/Langan_CTMU_092902.pdf

It has a learning curve like a cliff. I strongly encourage you to try to decipher the primary material first, but at the request of Veeky Forums, which has taken a curiosity in it for years, I typed an informal explanation of a way to approach some of the basic concepts. pastebin.com/WknXipZ0

>megafoundation
I'll read it because you piqued my interest but I suspect I will not enjoy it.

The first 10 pages make it look like gibberish, unfortunately. He tries to give a hand-wavy explanation but fails. He also explains the core primitive concept of the paper terribly. These are likely circumstances of his not truly knowing how to write academically. I retract what I said about reading the primary material first and recommend you read my explanation of the basic concepts.

I'm reading the introduction and following fine. I've had very similar thoughts myself recently. However I suspect this will end up being more of an index of other shit to look at more than anything that profound.

Having a very quick skim through his syndiffeonesis bit looks to be, uh, a little pedestrian? Or something like that anyway. It looks interesting enough, thanks for bringing it to my attention.

I suggest you read the first couple paragraphs of my explanation to learn what else means by a unit of information, because he does an abysmal job of explaining it.

It feels a lot like he's trying to get at an idea of language as described in the architectural work "A Pattern Language". I reserve judgement for the quality until I get my head around what he says later on. I'd also need to reread A Pattern Language tbqph

Well the ideas are so original I highly doubt it. Read my intro.

I've pretty much gotten through it. Knee jerk reaction is it's boring in a philosphical sense, that it's a less decent index than I expected (but still not bad I guess), and it's even a little blinkered regarding science and the whole processsing like paradigm (I know I know, it isn't purely information theory but he does fall back on the state machine paradigm which I'll ask about). I withhold true judgement tho. I think he knows there's something to his explanation because it works on at least two levels at the same time, but really he promises a lot in that intro and doesn't deliver.

>While linguistic processing is dynamically paralleled by changes in the internal and external states of processors, the processors are still considered separate from the language and grammar being processed.
I really feel we have a fundamental misunderstanding here. Like what is a turing machine all about fundamental.

I reserve judgement on conspansion but I suspect there's an issue there.

So eh.

I forget to point out:
And what are the implications for death?
The implications for socialisation?
There was something else but I forget.

I can't understand your indifference to a philosophy that offers a basis for a unit of information to engage in global transduction without input, an a priori basis for the positivity of the cosmological constant, and a mechanism of distributed ex nihilo creation and self-determinacy.