
What Logic Is Not - ColinWright
http://logicmazes.com/g4g7.html?HN_repost
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tikhonj
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](http://world.std.com/~rjs/isis96.pdf)

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visakanv
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.

~~~
nmc
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.

~~~
visakanv
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.

~~~
nmc
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.

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chalst
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.

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otikik
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.

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jheriko
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...

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

~~~
jheriko
genius - will have to steal this and use :)

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nmc
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.

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agentultra
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://www.amazon.com/Gödels-Proof-Ernest-
Nagel/dp/0814758371/ref=sr_1_1?ie=UTF8&qid=1387379844&sr=8-1&keywords=gödels+proof)

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Double_Cast
[http://lesswrong.com/lw/f4e/logical_pinpointing/](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.

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transfire
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.

~~~
hetman
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.

~~~
Double_Cast
> 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/](http://matt.might.net/articles/logical-literacy/)

~~~
hetman
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.

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transfire
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.

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bachback
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.

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

