What Logic Is Not | Hacker News Reader

What Logic Is Not

logicmazes.com

21 pointsColinWright23 comments

This is one of the key ideas that Gödel, Escher, Bach tries to convey accessibly. I've always thought of it like a popular science book, but for logic. Unless you're already familiar with the ideas, it's a great place to start or even just get a taste. A great read for non-mathematicians.

Viewing logic systems as objects themselves is a very important insight and yet surprisingly counter-intuitive. Logic is logic. It feels like the aether of reasoning--constant and ever-present. But in reality, you can easily design and use different systems with different constraints and philosophies, giving you different results. You can even reason logically about logic, considering a logic system as an object itself! I think this sort of realization is one of the most important steps to understanding that elusive "mathematical mindset" people like to talk about.

For me, learning all about Curry-Howard and how programming relates to logic really opened my eyes. I'm already comfortable with treating programming languages and programs as objects--I do all sorts of metaprogramming and static analysis and so on. Seeing proofs and logic systems in the same light made everything clear. A proof is just a program; a logic just a programming language and so I can treat them in the same ways. I really believe Curry-Howard is one of the most important ideas I've ever learned--it gave me a new perspective on CS, a new perspective on mathematics and a way to naturally link the two. Thinking by analogy is very powerful, and Curry-Howard is ultimately just that, a rigorous analogy.

As an aside, I think we can reasonably codify induction by talking about probabilities. See "Does Algorithmic Probability Solve the Problem of Induction"[1] which is a great paper that breaks Betteridge's law. (Sort of.)

[1]: http://world.std.com/~rjs/isis96.pdf

I just want to quickly talk about the main problem I've noticed people having when having personal conflicts and arguments (especially noticeable on MBTI forums, with the "logical thinkers" going up against the "emotional feelers").

Logic, in the practical, day-to-day sense, isn't something that replaces emotion. Without emotion, we wouldn't be able to make logical decisions (established by studies on people with brain damage that specifically took out their capacity for emotion- they can lay out all the pros and cons of a decision, but they can't actually decide)- because emotions are what allow us to atribute value to things.

So when we say "I did the logical thing and took the job that offered more pay", what goes unsaid is that we emotionally prefer more money to less money, and maybe value that over whatever the other costs much be.

TL;DR:

Logical decision-making is calculative, but it is not free of emotion. Emotions matter.

PS: I know this isn't directly related to the more mathematical, axiomatic sort of logic being discussed... but I just have a lot of feelings.

I like your point because it helps me understand why I'm a prick sometimes. It could just be that I'm looking for excuses for my overly pedantic behavior, I'm not sure. It's hard for me to read my emotions :-(

I can be a prick, too. I think it happens when I start getting carried away with communicating to others that I am right about something, rather than actually being sensitive to the context of the conversation.

and about 75% of the time, it wouldn't matter a tiny bit if I just let go.

I believe I get your point, but you seem confused about what you mean by "logic". Formally, your example is:

1. I prefer more money than less money

2. Job x is offering more money

3. I prefer job x

There is your syllogism, your "logical thing". However, the emotional part about money is just an axiom: you may call it obvious, but not logical.

... Or maybe you can, if you consider something like:

1. Money is good

2. I prefer more good than less good

3. I prefer more money than less money

However, you will always end up on an axiom: something you say is true, without any logical thinking involved, like "money is good". There is your emotional part.

I hope I understood your point correctly. In this case, my answer is: you can always separate logic from emotions by going into details of the reasoning.

Yup, you got exactly what I was trying to say. Logic is the system of reasoning, the "weighing scale". Emotions are what give the objects weight. So we could both arrive at different conlusions despite having identical logical reasoning, because we value things differently.

Happy I got it right!

This was generalized by A. Einstein in something along the lines of "a problem cannot be solved with the same state of mind it was created in". I like to understand that as:

1. You have a problem.

2. You have a situation that cannot be solved.

3. You have a situation you think cannot be solved.

4. You have a reasoning process concluding that the situation cannot be solved.

5. You have a state of mind which implies, as a logical outcome, that the situation cannot be solved.

It is not very fair to make a general criticism about the weakness of logic based on propositional logic, since that is the weakest of the various families of logic - the other two main kinds of logic being first-order logic (FOL) and term logic.

First-order logic is powerful enough that it has become the yardstick of adequacy for mathematical proofs: a mathematical proof about a mathematical domain is only adequate if (i) the mathematical domain accepts a first-order axiomatization, and (ii) we believe the proof can be rendered as a proof in FOL.

So the broader point is that we have a lot of freedom in choosing axiomatisations. But in fact we do not, at least in mathematics: there is one accepted axiomatisation (up to equivalence of inessential variations) of Peano arithmetic, one of ring theory, one of vector spaces, one of C* algebras, etc.

I always thought of axioms as "starting points" or "bases". The rules of logic require "starting points". You can't demonstrate anything from nothing. Those are the axioms.

It's worth noting that almost any statement can be used as an axiom. You can build a math on top of "2+2=5". It just won't be a very useful math. Which brings up the point of usefulness.

Some sets of axioms are more useful than others. Usually because they seem to align with how the universe appears to work. But in other cases they are just convenient.

So there are no "really, really true" axioms. But there are "really, really useful" ones.

axiom is a horrible word, as is postulate imo. i've always preferred the common man friendly 'assumption' - its honest, straightforward and doesn't obscure anything.

mathematicians have a fantastic talent for complicating language...

When you axiom, you make an ax out of you and me.

genius - will have to steal this and use :)

Nice read; R. Ellis nails it by picking out the key sentence:

My rules are the axioms, and they are axioms not because they are really, really true; they are axioms because I say they are.

Always remember: mathematics are abstract, you (the mathematician) control the system, so you pick the axioms.

I always found the study of logic to be very interesting. What I highly recommend reading first, however, is a layman's introduction to Gödel's Incompleteness Theorem [1]. The essential idea is, much like the conclusion in this article, that axioms are chosen and theorems are proven within the systems they create. The caveat is that no system can be proven to be complete.

[1] http://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/081475837...

http://lesswrong.com/lw/f4e/logical_pinpointing/

> It's not that 'axioms' are mathematicians asking for you to just assume some things about numbers that seem obvious but can't be proven. Rather, axioms pin down that we're talking about numbers as opposed to something else.

Eliezer says axioms are like scope resolution operators. I'm curious where he got this idea from, because I doubt he came up with it on his own in a vacuum.

I also think we need to distinguish logical systems and Logic. By which I mean, that anyone can create a logical system, starting with a set of axioms and deriving conclusions from it. But Logic is a very simple bases by which all such systems are judged, e.g. that two contradictory statements can't both be true; if A implies B and B implies C then A implies C, etc. I don't think anyone can dispute Logic in this first-order sense.

The above is true because it is defined to be true. The meaning of the material conditional (i.e. "->") might itself produce some surprising results for anyone attempting to intuitively equate it with the English word "implies". Interestingly, there is still a lot of debate about how the material conditional should be used to model certain aspects of reality (or at least our statement about said reality).

So for example, if we're attempting to model a causal chain, we might apply the modus ponens inference rule to deduce the truthfulness of your above proposed statement. But the inference rule is itself an axiom and there's no reason why we can't define a system of logic where it does not hold.

Would that system of logic be useful in modeling reality? It very well may be for modelling some aspects of reality.

Is it possible to definitively say that there is a "Logic" that exists apart from logical systems. I'm not sure how you could define the former without the latter. As a result, this sounds somewhat akin to the classical Greek belief that the universe fundamentally consists of numbers and is probably more of a philosophical statement than anything else.

> But the inference rule is itself an axiom

Is it? I've seen Boolean True explicitly derived from Modus Ponens. Apparently, p -> q = ~p v q which looks derivable from Aristotle's Third Law.

http://matt.might.net/articles/logical-literacy/

But Modus Ponens is the axiom here. Without it, boolean logic is just rearranging arbitrary symbols on a page. Axioms like it are necessary to link those symbols to reality in some way.

I think our Ivy league education system is not as good as advertised, if you went to Yale and only came to understand (some of) these things 40 years later.

Fascinating. Hofstadter should read Russell some time. Perhaps then he might understand just a little bit of logic. I would start with Russell's denoting phrases from 1905. Or Frege's concept script from 1879. Everyone likes to talk about Gödel instead. The end result is just confusion.

Thank you! Thank you! I finally understand what theorems are. I kinda got axioms already, but this explanation is definitely useful.

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