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What is logic?

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Re: What is logic?

#21

Post by empath75 »

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?

The classic popular treatment is: Godel Escher and Bach: An Eternal Golden Braid, by Douglas Hofstadter

Re: What is logic?

#22

Post by westoncb »

empath75 wrote: post #21
westoncb wrote: post #20

Earlier quoted context omitted.

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

The classic popular treatment is: Godel Escher and Bach: An Eternal Golden Braid, by Douglas Hofstadter

I've read about a third of it. It's too long and too focused on other topics for me personally.

I also read the book 'Gödel's Proof' a long time ago, before I had much familiarity with pure math, and remember it describing a very interesting sort of 'architecture' that Gödel had put together—but I left still wondering about any applications other than producing the incompleteness theorems. Maybe I'd get more out of it now though...

Re: What is logic?

#23

Post by kazagistar »

westoncb wrote: post #11

My take: it's a human means of systematically exploring implications of already accepted beliefs. There are implicit and explicit forms of it. The implicit form is something our brains do automatically and is a consequence of its structure. Perhaps the way it models things intrinsically does not allow for what we would call contradicting statements (though it contains many separate models which if unified would conta…

The interesting point of the given article was that pre-mathematical-formalization, logic was backwards of that purpose, and often remains so even today. The search does not start with assumptions and seek conclusions; rather, it starts with conclusions, and then seeks which sets of axioms fulfill them. A vast majority of mathematicians don't care about the axiomatic foundations of mathematics, nor mind if it gets entirely reformulated; they already know math works, and its irrelevant which particular set of axioms is used to prove that.

The assuptions => theorm structure is an artificial abstraction over a very different logic process. Symbolic logic is a framework for mechanical validation, rather then anything like how people actually do logic.

Re: What is logic?

#25

Post by pitaj »

lgas wrote: post #8
pitaj wrote: post #3

From what I know, pure logic or even math are useless without assumptions. This is what reason and science give us: a standard set of axioms from which logic and math can lead us to useful conclusions.

Where can I find this standard set of axioms?

By axioms, maybe I used the wrong word, but what I was referring to is things like the concept of "burden of proof" in reason.

Re: What is logic?

#26

Post by pitaj »

redial wrote: post #6
pitaj wrote: post #3

From what I know, pure logic or even math are useless without assumptions. This is what reason and science give us: a standard set of axioms from which logic and math can lead us to useful conclusions.

But isn't reason is the result of logic? as in something is reasonable only when it is logically sound?

A line of thought can only be reasonable if it is logically sound. Reason is built on top of logic.

All of this IIRC.

Re: What is logic?

#27

Post by aromatt »

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?

There's a chapter about it in _I Am a Strange Loop_ (Douglas Hofstadter) which I found very enjoyable and pretty enlightening.

Re: What is logic?

#28

Post by westoncb »

kazagistar wrote: post #23
westoncb wrote: post #11

My take: it's a human means of systematically exploring implications of already accepted beliefs. There are implicit and explicit forms of it. The implicit form is something our brains do automatically and is a consequence of its structure. Perhaps the way it models things intrinsically does not allow for what we would call contradicting statements (though it contains many separate models which if unified would conta…

The interesting point of the given article was that pre-mathematical-formalization, logic was backwards of that purpose, and often remains so even today. The search does not start with assumptions and seek conclusions; rather, it starts with conclusions, and then seeks which sets of axioms fulfill them. A vast majority of mathematicians don't care about the axiomatic foundations of mathematics, nor mind if it gets en…

> The search does not start with assumptions and seek conclusions; rather, it starts with conclusions, and then seeks which sets of axioms fulfill them.

Right, there was an issue with labeling justification 'logic'. An interesting point, but not as interesting as the title of this story—which I prefer to discuss :)

I was getting at the general process of inference making in the human brain (not even conscious inference making necessarily), which I see as the real root of all the different things we call logic.

Re: What is logic?

#29

Post by alasdair_ »

westoncb wrote: post #22
empath75 wrote: post #21

Earlier quoted context omitted.

The classic popular treatment is: Godel Escher and Bach: An Eternal Golden Braid, by Douglas Hofstadter

I've read about a third of it. It's too long and too focused on other topics for me personally. I also read the book 'Gödel's Proof' a long time ago, before I had much familiarity with pure math, and remember it describing a very interesting sort of 'architecture' that Gödel had put together—but I left still wondering about any applications other than producing the incompleteness theorems. Maybe I'd get more out of i…

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

Re: What is logic?

#30

Post by mafribe »

tikhonj wrote: post #16
throwaway729 wrote: post #9

Earlier quoted context omitted.

> In a sense, perhaps a certain degree of ‘artificiality’ is at the core of logic throughout history, as it operates at levels of abstraction that are at odds with ordinary language usage Of course, you could say the exact same thing for programming languages (after all, what do programmers do if not "simply working with special symbols, not with ordinary words"? And isn't this the primary criticism of programming br…

On a slightly orthogonal note, I think that not only can you think of programming languages this way, but that you should —programming languages and formal logics are ultimately instances of the same general idea. I've found this to be a consistent and powerful view for reasoning about programs, programming languages, proofs and logics in a uniform sort of way. The Curry-Howard correspondence is an important idea tha…

   programming languages and 
   formal logics are ultimately 
   instances of the same general 
   idea.
That's deeply questionable. The CH-correspondence breaks down as soon as you your computation includes non-termination, concurrency, timing, distribution etc. Classical logic doesn't really have wholly convincing CH-correspondences either.

It seems to me that constructive proofs are a special class of programs, and fall under the purview of programming language, but not the other way around.

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