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What is logic?

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Re: What is logic?

#61

Post by mr_overalls »

empath75 wrote: post #60
fenomas wrote: post #50

Earlier quoted context omitted.

I read GEB a few years ago, and it struck me as a book full of ideas that would have transformed me had I been encountering them for the first time. I think the book may have been so influential that its big ideas had largely filtered out into places where I'd encountered them. If I'd read it at 19, say, I think it would have rocked my world. Note that I'm mostly talking about the book's first half (about incompleten…

I did read it at 19 and it did rock my world.

Same here. I read it just after Zen and the Art of Motorcycle Maintenance. Rite of passage for nerds. :-)

Re: What is logic?

#62

Post by danharaj »

mafribe wrote: post #59
danharaj wrote: post #51

Earlier quoted context omitted.

> The CH-correspondence breaks down as soon as you your computation includes non-termination, concurrency, timing, distribution etc. No it doesn't. Each of those features leads to a different logic. You can even reason about the composition of logics via category theory to see how various features interact with each other. > Classical logic doesn't really have wholly convincing CH-correspondences either. Sure it does…

can even reason about the composition of logics I'm not aware of a fully fledged approach to logical composition that tells me an interesting story about the combination of logics that arise in computation, I know that people work on this, but as far as I can see, this work is far from finished. Please point to papers if you know more than that. first class continuations As far as I'm aware, Griffin's work, which was…

It's true that the CH approach to classical logic is by no means exhausted. I see this as a feature of classical logic's rich structure. In any case, I disagree that this is a break down of the CH correspondence. Analyzing the type theories related to classical logic demonstrates to me that the CH correspondence is a gift that keeps on giving.

The approach I have in mind about composition of logics is the theory of pseudodistributive laws: http://dl.acm.org/citation.cfm?id=2612113

It's kind of cutting edge and has only yet been applied to a few situations, but it seems to be the correct notion of composing type theoretic (and hence logical) structures in compatible ways. Another approach I like that is strictly logical is Nuel Belnap's display logic, which characterizes a ridiculously broad range of logics and proves a general cut elimination theorem for them. The relationship between the categorical approach and the display logic approach is not entirely clear yet.

In any case, the CH correspondence is a rich and very active area of research. Just because we haven't nailed it down exactly yet doesn't mean it's broken! Our current understanding is a shadow of a much deeper concept.

Re: What is logic?

#63

Post by woodandsteel »

Interesting article.

I would add that Frege, Russell and Whitehead did not just want to logicize mathematics. They thought that reality itself followed this logic, and so were doing metaphysics. The later Wittgenstein is a decisive refutation of this idea, and this is a key reason logic has gone out of fashion.

Re: What is logic?

#64

Post by mafribe »

danharaj wrote: post #62
mafribe wrote: post #59

Earlier quoted context omitted.

can even reason about the composition of logics I'm not aware of a fully fledged approach to logical composition that tells me an interesting story about the combination of logics that arise in computation, I know that people work on this, but as far as I can see, this work is far from finished. Please point to papers if you know more than that. first class continuations As far as I'm aware, Griffin's work, which was…

It's true that the CH approach to classical logic is by no means exhausted. I see this as a feature of classical logic's rich structure. In any case, I disagree that this is a break down of the CH correspondence. Analyzing the type theories related to classical logic demonstrates to me that the CH correspondence is a gift that keeps on giving. The approach I have in mind about composition of logics is the theory of p…

The essence of CH is "proofs = programs" and "program execution = cut elimination". As far as I understand this does not work precisely as of Jan 2017 for most forms of computation beyond strongly normalising pure functions.

If you are broadening what you mean by CH, then fair enough, but it's shifting the argument.

Re: What is logic?

#65

Post by njohnson41 »

westoncb wrote: post #20
cousin_it wrote: post #19

Gödel's arithmetization of logic is one hell of a beautiful mathematical idea, up there with linear algebra or probability theory. One of its big selling points for me was how easily it solves the unexpected hanging paradox. (The simplest arithmetization of the judge's self-referential statement is a statement about numbers that can be shown to be self-contradictory.) It's also cool how it turns Russell's paradox int…

Do you have a recommendation for a book that covers these topics that's technical but not necessarily a text book?

I personally liked "Godel's Proof" by Ernest Nagel. It's pretty old, but it's to-the-point and inexpensive.

Re: What is logic?

#66

Post by empath75 »

westoncb wrote: post #35
alasdair_ wrote: post #29

Earlier quoted context omitted.

GEB is well worth a second attempt. It's one of the very few books that completely changed my worldview.

I lost interest when certain other ideas made me think his basic project of accounting for consciousness mathematically was deeply flawed (and it wasn't Penrose's argument about formal systems!). I did find it entertaining though, so maybe I'll revisit before too long. I'm curious what shift it caused for you, if you don't mind saying.

At least for me, it wasn't so much about computers or computer science, as it was the first time I really deeply considered what consciousness is. It's actually the book, along with The Selfish Gene that made me dismiss the necessity of any kind of 'soul', fresh out of Catholic School.

Re: What is logic?

#67

Post by westoncb »

empath75 wrote: post #66
westoncb wrote: post #35

Earlier quoted context omitted.

I lost interest when certain other ideas made me think his basic project of accounting for consciousness mathematically was deeply flawed (and it wasn't Penrose's argument about formal systems!). I did find it entertaining though, so maybe I'll revisit before too long. I'm curious what shift it caused for you, if you don't mind saying.

At least for me, it wasn't so much about computers or computer science, as it was the first time I really deeply considered what consciousness is . It's actually the book, along with The Selfish Gene that made me dismiss the necessity of any kind of 'soul', fresh out of Catholic School.

The more time I've spent reading about and considering the nature of subjectivity, the more I believe arguments from evolution or computer simulation are irrelevant. They're talking about something different.
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