What is logic? • Hacker News Board

Re: What is logic?

#31

Post by SomeStupidPoint » Tue, Jan 17, 2017, 9:04 PM UTC

kensai wrote: post #2

"The idea that ordinary language is expressively inadequate to account for mathematical (or even logical) reasoning became a recurring theme in the ensuing tradition of mathematical logic, so much so that the term ‘symbolic logic’ became synonymous with this tradition. Doing logic came to mean simply working with special symbols, not with ordinary words. In this respect, it is worth noting that the humanist authors h…

The whole drive of mathematics (and logic) is ultimately that syntax is the semantics.

This, of course, is annoying for almost all communication, particularly when we want to use fuzzy, hard to define terms. Languages are useful because you can pretty heavily butcher the syntax, but still be understood. (And arguably, that fuzziness is what drives new thoughts.)

Re: What is logic?

#32

Post by abtinf » Tue, Jan 17, 2017, 9:10 PM UTC

> “It’s logical, but logic has nothing to do with reality.” Logic is the art or skill of non-contradictory identification. Logic has a single law, the Law of Identity, and its various corollaries. If logic has nothing to do with reality, it means that the Law of Identity is inapplicable to reality. If so, then: a. things are not what they are; b. things can be and not be at the same time, in the same respect, i.e., reality is made up of contradictions. If so, by what means did anyone discover it? By illogical means. (This last is for sure.) The purpose of that notion is crudely obvious. Its actual meaning is not: “Logic has nothing to do with reality,” but: “I, the speaker, have nothing to do with logic (or with reality).” When people use that catch phrase, they mean either: “It’s logical, but I don’t choose to be logical” or: “It’s logical, but people are not logical, they don’t think—and I intend to pander to their irrationality.”

http://aynrandlexicon.com/lexicon/logic.html

Re: What is logic?

#33

Post by metaphorm » Tue, Jan 17, 2017, 9:13 PM UTC

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.

Re: What is logic?

#34

Post by westoncb » Tue, Jan 17, 2017, 9:36 PM UTC

abtinf wrote: post #32

> “It’s logical, but logic has nothing to do with reality.” Logic is the art or skill of non-contradictory identification. Logic has a single law, the Law of Identity, and its various corollaries. If logic has nothing to do with reality, it means that the Law of Identity is inapplicable to reality. If so, then: a. things are not what they are; b. things can be and not be at the same time, in the same respect, i.e., r…

> If logic has nothing to do with reality, it means that the Law of Identity is inapplicable to reality.

That may be the case. If our brain is only capable of forming self-consistent schemas (consider how it reacts when encountering new facts that contradict existing schemas: https://en.wikipedia.org/wiki/Cognitive_dissonance), then the world's apparent consistency may only be a consequence of our contemplating it through the human brain. (To be clear though, it's consistency within single schemas—not between, where contradictions abound.)

> If so, then: a. things are not what they are; b. things can be and not be at the same time

Can someone be a hero and not be a hero at the same time? I think there are some subtleties to this issue arising from the nature of language. Certainly at our scale (i.e. not quantum scales—although even there I don't really think 'real' inconsistency is happening, but it's less clear cut) a certain kind of consistency has very strong empirical support (it's probably our best supported hypothesis!), so it would be adaptive for us to build models of the world that would reflect that property—but this doesn't mean we've hit on something deep about the nature of reality.

Re: What is logic?

#35

Post by westoncb » Tue, Jan 17, 2017, 9:40 PM UTC

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 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.

Re: What is logic?

#36

Post by tikhonj » Tue, Jan 17, 2017, 9:57 PM UTC

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…

The point is that they're both formal systems of reasoning, not that they have exactly the same properties. The techniques and abstractions from both PL and formal logic often generalize well to both.

Re: What is logic?

#37

Post by danielam » Tue, Jan 17, 2017, 10:31 PM UTC

Speaking of classical logic, I came across a book on Fred Sommers' work entitled "The Old New Logic" (David Oderberg is editor). I haven't read it yet, so I can't comment, but based on some excerpts and reviews, it sounds like a good read. I thought I'd mention it here for those interested. Link: https://mitpress.mit.edu/books/old-new-logic

Also, those interested in the fallout of Russell's paradox are encouraged to look into Lesniewski's mereology (this is a topic that occurs in my master's thesis where I attempted to formalize the structure of Aristotelian living things mereologically using many-sorted logic). In short, Lesniewski held that the axiomatic set theory of the day betrayed the intuitions of naive set theory and resulted in paradoxes. He attempted to dispense with the these paradoxes by proposing a formal system known as mereology that replaced the primitive element relation in set theory with the parthood relation (other relations can be defined in terms of parthood, e.g., overlap, disjoint, proper/improper parthood depending on which one you assume, and vice versa). Link: https://plato.stanford.edu/entries/lesniewski/

Re: What is logic?

#38

Post by marcosdumay » Tue, Jan 17, 2017, 10:39 PM UTC

613style wrote: post #13

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.

I don't think I would agree with the general statement that science gives us axioms. In normal usage, "axioms" are not the same as scientific "givens" or "data" even though it makes sense to interpret the word that way. In math and logic, the axioms we use are typically very foundational. We might say, "the empty set exists," or "for all x, it holds that x = x." There's no need to invoke science for these sorts of ax…

You can create any set of axioms you want, but unless they carry a reference to something that happen "outside of Math", at the "Real World", they are only useful for nerd-sniping.

Re: What is logic?

#39

Post by spynxic » Tue, Jan 17, 2017, 11:34 PM UTC

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…

> (though it contains many separate models which if unified would contain contradictions)

Contradictions suggest that you don't have a complete picture

Re: What is logic?

#40

Post by abtinf » Tue, Jan 17, 2017, 11:39 PM UTC

westoncb wrote: post #34

abtinf wrote: post #32

> “It’s logical, but logic has nothing to do with reality.” Logic is the art or skill of non-contradictory identification. Logic has a single law, the Law of Identity, and its various corollaries. If logic has nothing to do with reality, it means that the Law of Identity is inapplicable to reality. If so, then: a. things are not what they are; b. things can be and not be at the same time, in the same respect, i.e., r…

> If logic has nothing to do with reality, it means that the Law of Identity is inapplicable to reality. That may be the case. If our brain is only capable of forming self-consistent schemas (consider how it reacts when encountering new facts that contradict existing schemas: https://en.wikipedia.org/wiki/Cognitive_dissonance ), then the world's apparent consistency may only be a consequence of our contemplating it t…

To be honest with you, I just about fell out of my chair reading your comment. Seriously, this is the first time I've posted a relevant quote from Ayn Rand on the web and received a substantive and thoughtful response.

I will have to think on the points you raise.