In our new "how can we help you?" thread, a reader writes:

I've always struggled with logic. I've worked my way through a few logic texts now, and every time I forget most of it shortly after and then wonder, 'what was the point of that?'. I don't enjoy logic and I'm not mathematically inclined. But some areas of my field (metaphysics) are heavy on the logic and it would be nice to be able to keep up. Does anyone know of any resources (books, software programs, etc.) that can help, or just advice in general?

Another reader submitted the following reply:

I was in your boat when I started my PhD. But I was forced to confront my distaste for formal logic because my program has a logic exam for 1st year PhD students. A few people from my cohort purchased 'Language, Proof, and Logic' by Barker-Plummer, Barwise, and Etchemendy. It comes with software that is useful for learning translations, truth tables, and derivations. The derivation software is especially useful because it tells you whenever you're taking an invalid step. I think you'd have to order a new copy to ensure that you get the software / get software that has an unused registration key. That said, it looks like it costs about $80 USD, so I'd certainly try to get a second opinion. I know it was useful for me, but perhaps more useful was the study group that I went through the book with, so the software may not have been indispensable.

Nice suggestion! Do any other readers have suggestions of their own?

You say "I've worked my way through a few logic texts now, and every time I forget most of it shortly after and then wonder, 'what was the point of that?'." Focusing specifically on the 'what was the point of that?' part, here are two things you might mean

(1) After you realize you forgot most of the material, you're not sure why you went through it in the first place, or

(2) You cannot for the life of you see the philosophical value in what you learned, even the bits that you remember.

If it's (1) that you're worried about, then don't. There are so so so many things in logic that I have to relearn every single time I encounter them. Example: in many of my publications, a form of the Lindenbaum Lemma plays a key role. Every time I prove it, there's a bit at the end that I'm convinced doesn't work. Normally it takes 24-48 hours for me to realize that it does work. At every single time, the thing I think doesn't work is really obvious and silly.

But this is a very general phenomenon in my experience. Almost all of the logic I've ever learned, I've also forgotten. And almost every time I need to use something that I vaguely remember learning, I have to go relearn it. This happens over and over and over. I don't think it means I'm bad at logic (my publication record suggests I'm at least ok at it). I think it's just part of how logic is. It's too complex, relies on too many things we casually conflate, and is features far too many long arguments for most of it to comfortably fit in a brain for any length of time. So you get what you need into your brain for a certain length of time, and then when you're done using it, you let it fall out, and then later if you need it again, you force it back in.

Of course, over time you *do* get faster at fitting things into your noggin. But generally, things don't really stay there and that's ok. At any rate, this supplies an answer to (1): the reason you go through the stuff isn't so it gets in your head and stays there. The reason you go through it is so that the next time you need to get it in your head, you can do so more quickly.

So much for (1). For (2), I have less of an answer. I gather you need the logic business for the research you care about, so perhaps (2) isn't much of an issue. But I'll report having this feeling with regard to some parts of logic and other formal fields as well. For example: I periodically teach formal epistemology. When I'm teaching it, I can for those weeks, remember what's philosophically interesting about the statistics-y stuff it foregrounds. Within a week of finishing teaching it, though, the philosophical oomph of it all just fades away. I'm left feeling like it was all just statistics, and elementary boring statistics at that---albeit with the knowledge that if I looked closely and thought hard about it, I'd know there was more to it than that.

I don't know that this latter problem is solvable. Or at least, I haven't solved it. But all it means is that you shouldn't do research in that area, and it doesn't seem like that's a problem for you in the first place.

TL;DR: I'm a logician and I also forget almost all the logic I learn as soon as I learn it. That's ok, and is just part of the game.

Posted by: Shay Logan | 02/01/2021 at 10:06 AM

I don't know if this point is covered in the OP, but: are you doing the exercises?

I never had a handle on logic until I focused less on memorising axioms/inference rules/terminology and more on applying them to particular problems. The same goes for learning mathematics, economics, and basically any subject where you have to learn a lot of new symbolisms.

Posted by: William Peden | 02/01/2021 at 12:42 PM

I'm not very good at it, either. I'll tell you what helps a lot, though: teaching it.

I think this is true more generally, as well: as Shay mentioned, when you teach something, it 'sticks' a while longer than when you just read it. (I think the same is true for using things--so articles I discuss and cite stick in my mind for longer than those I simply read.) But teaching, I think, lasts longer. Even when all you're teaching are the bare-bones basics, it helps a lot. (It doesn't have to be a class, either: you could team up with someone else who struggles, and teach each other.)

Posted by: Michel | 02/01/2021 at 01:40 PM

Are there any good resources for learning how to apply logic in the way that the early positivists (esp. Carnap) applied logic? I find it difficult to really wrap my head around how to do that level of work. How do I know which axioms to choose? How do I know that I am not making a mistake in picking out and defining the basic terms of a PC? How the heck do theorems actually work? And so on. I am able to follow along fairly well using my Symbolic Logic knowledge. But I am also not particularly mathematically inclined (I have no formal training in mathematics). What are some good texts to really grasp how to think logically in the sense of the early 20th century logicians?

Posted by: JJ | 02/01/2021 at 02:45 PM

I hear good things about this free book:

http://forallx.openlogicproject.org/

Posted by: Martin Cooke | 02/01/2021 at 03:48 PM

For how logic actually applies to mathematical proofs and mathematical practice I can't recommend Daniel J. Velleman's "How to Prove It" highly enough. And high school algebra is enough to follow most of the book. He dips into calculus every now and then in some of the optional exercises and some of the later chapters touch on geometry but really anyone with a high school degree can grasp most of the math I think (though expect to put in some work). It has lots and lots of exercises and you can learn a decent amount of set theory along the way too. The later chapters get into some pretty deep water with set theory but up through the middle ones are just really excellent in talking about general strategies in deductive logic. I'd also second the recommendation of "Language Proof and Logic" which I remember as quite good from my undergrad days. When I took logic that book and its exercises were pretty much my only teacher. My logic prof was an ambitiously bad teacher-- smart but just a mean, bullying, condescending guy-- and his TAs somehow worse so I basically skipped half the lectures and all the discussion sections and still pulled out a B+ simply on the strength of that book's well done exercises. It's a real testament to using technology as a learning aid. But honestly teaching is the best thing to really learn logic or any other subfield. I had to teach logic this spring and it really sharpened up my grasp on logic.

Posted by: Sam Duncan | 02/01/2021 at 04:25 PM

Check [this](https://oli.cmu.edu/courses/logic-proofs-copy/) out.

Then check out some books/papers by Ed Zalta, who is particularly clear and useful.

Posted by: Wes | 02/01/2021 at 05:36 PM

I second the exercise comment. If you're not doing the exercises, you're not going to learn it.

Also, I don't think you should care about being good at formal derivations. Do enough in enough different systems to get the hang of it, but focus more on actually writing (natural language, meta)proofs. Relatedly, make sure you understand the rudiments of model theory and how to interpret formal sentences in various mathematical structures. It's been like a decade, but that's the stuff I remember actually mattering in metaphysics.

I think the same approach goes for understanding why it matters. To get the point, you have to actually see it in action --- e.g., see how Kripke's models of modal sentences can be used to make precise different theories of (trans)world identity, or whatever you're into. And I don't mean "read someone else explain it to you", I mean work it out for yourself (e.g., work out which models would make an identity statement true on one interpretation, vs which models make it true on the alternative interpretation).

Because so much of this depends on just doing it yourself, I don't think the exact textbook or whatever is that important. Unless the book is terrible, it should afford the opportunities needed to do some exercises and prove some theorems. I'd recommend skimming a few books while doing the exercises, over reading any one book deeply. Perspective is good.

One thing logicians seem to love to do is write textbooks and handbooks and create cool web apps and all that. I'm sure it won't take too long (or too much Googling) to stumble over more free material than you know what to do with. (I say this as someone who tried to write a textbook and teaches logic from his own notes.)

Posted by: aphilosopher | 02/01/2021 at 07:13 PM

I'll second what Shay Logan and William Peden say above.

If OP is looking for book recommendations I would also recommend Ted Sider's "Logic for Philosophy". Compared to many other logic texts it's quite use-friendly. It is also intended for the sort of thing it sounds like OP wants--not doing research in logic, but enough understanding to follow the use of logic in service of other philosophical issues. Sider is admirably clear about important distinctions and why they matter (e.g what is the point of metalogic, what is the difference between proof theory and model theory, etc).

Posted by: ethan | 02/02/2021 at 12:56 AM

When I teach logic I use https://logiclx.humnet.ucla.edu/ for my students. It is really a remarkable piece of software. It offers helpful advice on where you went wrong and thus, allows one to sometimes figure out a problem that they otherwise couldn't, on their own.

Posted by: Christopher A. Riddle | 02/02/2021 at 08:13 AM

I haven't read it yet, but this just came out:

https://www.amazon.com/How-Logic-Works-Users-Guide/dp/0691182221

It seems to be very well received. It might be what you're looking for.

Posted by: TT | 02/05/2021 at 04:14 PM