Exciting headline news. Like the Gödel book, IFL2 is now available as a freely downloadable PDF. There is also an inexpensive Amazon print-on-demand book, for those who want a hard copy. UK link; US link. (Find on your local Amazon by using the ASIN identifier B08GB4BDPG in their search field.)
My apologies to anyone who bought the book at full price recently; but it was a surprise to me too that new publication arrangements became possible. Since I’m more interested in spreading the logical word than in totting up a few royalties (authors of logic books stand to make a fortune, of course …), I’ve decided to give the book away, or to provide access to a hard copy within a few pennies of the allowed Amazon minimum price. It’s a quite rotten time for students; so giving free/cheap access to decent learning materials when we can seems especially appropriate now. (Anyone thinking of using the book as a course text can be assured it will remain free.)
As I said about the Gödel book, the print-on-demand version is very decently printed, at least by Amazon’s UK printer, but the cover isn’t as good as a trade paperback’s (I don’t just mean the design, but the way it lies), and the binding is a bit tight. But on the other hand, it is a third of the price of the original, so I don’t suppose anyone will complain too loudly.
A reminder: The first edition of IFL concentrated on logic by trees. Many looking for a course text complained about this. The second edition, as well as significantly revising all the other chapters, replaces the chapters on trees with chapters on a natural deduction proof system, done Fitch-style. Which again won’t please everyone! So the chapters on trees are still available, in a revised form, here on the website. But it was considerations of space in the printed version that led to the relegation of (versions of) those chapters to the status of online supplements. This was always a second-best solution. Ideally, I would have liked to have covered both trees and natural deduction (while carefully arranging things so that the reader who only wanted to explore one of these still has a coherent path through the book). With e-publication, the question of length isn’t so vital. So over the coming months, I may be inclined to revert to the inclusive plan, and so eventually produce a third edition. We’ll see. IFL3 is for the future. For the moment, enjoy the delights of IFL2!
Soon after last comment I read some books and it seems they define ‘necessary condition’ as consequent of the material conditional, and ‘sufficient condition’ is defined as the antecedent of the material conditional. So the issue is just terminology:
The assertion ‘A is a sufficient condition for B’ or ‘B is a necessary condition for A’
is the same as asserting
A -> B
in PL by definition.
I was a bit confused because ‘necessary condition’ made me think of ‘necessary truth’ through a slippery slope:
I thought
‘B is a necessary condition for A’, ‘B is necessary for A’ etc.
is the same as
‘B necessarily follow from A’, or ‘B is necessarily true if A is true’,
but apparently these are non standard phrasings. And from then I started thinking they are the same as
‘A -> B is a necessarily truth’.
At this point it is clear that this is a claim in metalanguage: ⊨ (A -> B)
The initial assertion ‘B is a necessary condition for A’ translates to ‘A -> B’ in PL language.
‘A -> B is a necessarily truth’ translates to ‘⊨ (A -> B)’ in meta language.
In summary, A -> B being necessarily true means B is necessary condition for A in all valuations.
You are quite right to worry about the ordinary meaning of “A is a necessary condition for B”. It certainly seems to have a modal flavour (“necessary”!) which is entirely missed by rendering it by something of the form “B –> A”. That’s why the heading of Exercises 18(a) asks you to translate something of the form “A is a necessary condition for B” into PL as best you can. It’s not being said that the suggested answer is a splendidly good translation — only that it is the best we can do. (Something of the form |= (B –> A) wouldn’t be an answer as that isn’t a wff of PL, for the turnstile doesn’t belong to PL, but it shorthand for a bit of English, or whatever your metalanguage is.
I understand that’s the best translation we can do with PL. My question was intended to be outside of the context of exercise 18(a)
Generally speaking, when someone asserts “A is a necessary condition for B”:
Should I think of that assertion at the PL level (i.e they are simply asserting ‘B -> A’ )
OR
Should I think of that assertion at the metalanguage level: they are asserting ‘⊨ (B -> A)’ i.e they are making a metalinguistic claim about ‘B->A’ when they say “A is a necessary condition for B”
I understand that neither quite capture the meaning of ‘necessary’ (we need modal logic for that). e.g in the second option, B could be an absurdity. But I still think the second option is closer to ‘necessary’. That’s what I thought before my first post.
My second post above was basically an attempt to rationalize the first option, maybe “necessary condition” is just what we call the consequent of the material conditional?
I have a question. The assertion ‘A is a sufficient condition for B’ or ‘B is a necessary condition for A’
is it the same as asserting
A -> B
or is it the same as asserting
⊨ (A -> B)
?
I’m a bit confused when exercise 18 solution translates ‘R is a necessary condition for Q’ into ‘Q -> R’.
Even if ‘Q -> R’ evaluates to True in one valuation, it’s not the same as saying ‘R is necessary for Q’ right?
I just wanted to say that I’ve read this blog for really quite some years now (I think 3 or so?) and I wanted to thank you so so much for this.
I’m a pre-university student who will be attending my first lectures this September but during my teenage years have spent many an hour looking at your MathOverflow questions/posts, reading your book reviews, considering your interesting foundational thoughts etc. When you finally published IFL2 it was the first maths textbook I wanted to buy (I felt it was only fair after following your efforts on it for a *long time*) and now you’ve made it incredibly cheap!
Thank you so so much!
Thanks very much for the nice comments. Hope that the book works for you!
Yes, I think that including the material on trees in the electronic version, whether as a chapter or two or as an appendix, would be a very good move.