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What is logic?

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Re: What is logic?

#41

Post by paganel »

metaphorm wrote: post #33
lngnmn wrote: post #4

A set of general rules for not losing a connection to reality. Could be used in reasoning or computer programming.

this is amusingly missing the point. logic is, by its nature, an abstraction away from reality. it's a set of specific (not general) rules for evaluating statements of language (whether that language is natural, mathematical, or artificial/computer) such that meaning can be disambiguated. "reality", whatever it is, has no such requirement that it can be expressed in unambiguous statements.

> it's a set of specific (not general) rules for evaluating statements of language

Not the OP, but you could say that no "language" exists outside of "reality", and stretching it further you could also say that all types of languages (formal, mathematical, artificial/computer), including the associated "rules", are social constructs which depend heavily on the time and place of their "existence", i.e. on "reality".

Until we meet some quantum-like entity which might "talk" to us from outside the boundaries of this physical Universe we're doomed to be solipsistic: we construct "abstractions" that are meant to distance us from "reality" only to find out in the end that "reality" was behind those "abstractions" all along.

Re: What is logic?

#42

Post by spacehacker »

I've been surprised to not find anything for ctrl-F "Bayes". It turns out that traditional deductive logic is simply a special case of applying Bayes rule in order to find out the state of some binary variables given the other variables in which the probabilities are only 0 or 1. See for example these examples from Barber's "Bayesian Reasoning and Machine Learning" (p. 39):

https://i.imgur.com/W1b0Wem.png

https://i.imgur.com/JOFvd9C.png

Re: What is logic?

#43

Post by hackits »

tel wrote: post #15

Part of the fun of logic is realizing that there are multiple forms of it which change one's language and ability to express ideas. They also don't dominate one another—some statements' truth depends upon what logic they're formalized within. Model theory is good here in that it studies the connection between logical language and more "physical" models. Some of these models can give a relatable account for what a log…

Correct me if I'm wrong, I've all ways seen Maths to be easy part. The translation of English Language (grammar, sentence structure) that can be ambiguous to a very precise meaning such as Math's to be the hardest part and where people with good translation skills excel.

Re: What is logic?

#44

Post by bluetwo »

Logic is a tool. Like all tools, they hold no value until given a purpose, then they take on the value of the task they solve.

(Just my opinion. Agree or disagree. That's cool.)

Re: What is logic?

#45

Post by GregBuchholz »

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 liked "Godels Theorem Simplified". Although it looks like it is now a lot more expensive than it used to be. Maybe get it through and interlibrary loan.

https://www.amazon.com/Godels-Theorem-Simplified-Harry-Gensl...

Re: What is logic?

#46

Post by GregBuchholz »

spacehacker wrote: post #42

I've been surprised to not find anything for ctrl-F "Bayes". It turns out that traditional deductive logic is simply a special case of applying Bayes rule in order to find out the state of some binary variables given the other variables in which the probabilities are only 0 or 1. See for example these examples from Barber's "Bayesian Reasoning and Machine Learning" (p. 39): https://i.imgur.com/W1b0Wem.png https://i.i…

I think you might enjoy E.T. Jaynes

http://bayes.wustl.edu/

https://www.amazon.com/Probability-Theory-E-T-Jaynes/dp/0521...

Re: What is logic?

#48

Post by mkehrt »

mafribe wrote: post #30
tikhonj wrote: post #16

Earlier quoted context omitted.

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 t…

Non termination is modeled as false: a function that takes an A and never terminates has type A -> F, or ~A.

There's some connection between classical logics and concurrency, though it's hard to get at. You can use callcc to implement threads, and Getzen's classical sequent calculus has multiple conclusions.

Here is a PHD thesis that uses modal logic to model distributed code in a CH fashion: http://www.cs.cmu.edu/~tom7/papers/modal-types-for-mobile-co...

Classical logic is perfectly modeled with callcc, which has the type Peirce's Law, and allows one to write terms with the types of the law of excluded middle or Double Negation Elimination.

Re: What is logic?

#49

Post by tel »

hackits wrote: post #43
tel wrote: post #15

Part of the fun of logic is realizing that there are multiple forms of it which change one's language and ability to express ideas. They also don't dominate one another—some statements' truth depends upon what logic they're formalized within. Model theory is good here in that it studies the connection between logical language and more "physical" models. Some of these models can give a relatable account for what a log…

Correct me if I'm wrong, I've all ways seen Maths to be easy part. The translation of English Language (grammar, sentence structure) that can be ambiguous to a very precise meaning such as Math's to be the hardest part and where people with good translation skills excel.

I think much of maths is that translation. You learn a lot of mathematics and you recognize powerful, subtle, deep patterns. Then you spend your time looking for those patterns in the world.

It's the same thing with even the most basic "translation problems" that kids in schools learn. They're armed with arithmetic and high-school algebra so they get arithmetic and high-school algebra "real world" scenarios to analyze.

At higher levels its the same. The problems are just harder and the patterns aren't necessarily given to you. Sometimes, oftentimes, nobody even knows what they are.

Re: What is logic?

#50

Post by fenomas »

alasdair_ wrote: post #29
westoncb wrote: post #22

Earlier quoted context omitted.

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.

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 incompleteness and canons and whatnot). The second half (ants, etc) struck me as much more dated.

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