What is the essence of mathematical logic?

[dark+matters]

?

 

[onesteptwostep]

Bertrand Russell.

 

[dark+matters]

Ooo! Thanks for the lead. People around me will talk about mathematical logic and I'm like... wut. I don't even know what that means.

 

[onesteptwostep]

Oh I was just semi-joking but cool, you got something out of it

 

[Ex-User (9086)]

Here is a page denoting the basics of formal logic.
1. Propositional Calculus
2. Formal Logic book

The essence is to arrange information and information derived from provided data in a way that is correct and/or can be understood clearly. Building coherent structures of data based on a selected subsystem of logic.

I linked a specific branch of logic using human language since it shows how logic is used more naturally.

Other interesting thing that helps visualise logic is set theory.

 

[scorpiomover]

Modus ponens.

In short, "if A then B". Every time you reason something out, that's Modus ponens working for you.

 

[letspaniclater]

hey matheletes of INTPf,

I've recently become interested in mathematical logic and have decided to begin my investigation of the subject with George Boole's The Laws of Thought. it generally takes me two readings of each chapter to grasp the concepts, but lately I have had trouble understanding the text--specifically the chapters on methods of reduction and abbreviation. a large part of the problem is that Boole omits many steps when he is developing or expanding a given proposition, leaving me at a loss as to how he arrives at certain forms.

as a bit of background, I have not taken a mathematics course in about five years, and the highest level I've taken is Calculus (roughly ten years ago!); and I'm currently relearning elementary and intermediate Algebra as a separate project. it is clear that Boole heavily relies on certain algebraic rules to manipulate and transform propositions, making it all the more frustrating when I am unable to follow his steps. and the more I fiddle with the equations, and the more I fail to arrive at the form that Boole does, the more discouraged I become.

I understand that Boole's concepts and symbolism is outdated in the context of modern logic--excepting computer science, where it still seems highly relevant--so I wonder whether Boole is even necessary. however, the historical importance of his text makes me want to understand it so I can see how the subject builds upon and expands his initial formalization. I was lucky enough to find a nifty book compiling two fundamental texts in logic--Frege's Begriffsschrift and Gödel's On Formally Undecidable Propositions--but I worry that, if I can't understand Boole, I won't understand them.


tl;dr: what is the best introductory text to mathematical logic? are there any good textbooks on the subject?

I see Blarraun recommends a book, but does it assume a certain level of mathematical understanding? that is, what concepts must be mastered before pursuing logic?

 

[Ex-User (9086)]

it is clear that Boole heavily relies on certain algebraic rules to manipulate and transform propositions, making it all the more frustrating when I am unable to follow his steps. and the more I fiddle with the equations, and the more I fail to arrive at the form that Boole does, the more discouraged I become.

Like using , De Morgan theorem/laws, transposition, etc?

Spoiler

I found this site very useful. There are also online expression minimisers to check your answers.
Boolean algebra is a part of mathematical logic. By the looks of things you are having trouble reducing boolean expressions?
[BIMG]http://puu.sh/i7CTM/8fa45040f0.png[/BIMG]
Is the above what you meant, or is it a more general form of introducing theorems and propositions in a book?

tl;dr: what is the best introductory text to mathematical logic? are there any good textbooks on the subject?
I see Blarraun recommends a book, but does it assume a certain level of mathematical understanding? that is, what concepts must be mastered before pursuing logic?

Have a look at chapter 2 and onwards and see for yourself if it's accessible, I think it shows the basics, although about sentential logic, so if you are into algebraic logic, or first order logic then you might want to look for some other resources. I found a wiki-book in english so I posted it since sometimes I rely on different languages when studying and I couldn't link what I actually used.

 

[Brontosaurie]

* there's discrete and continuous numbers

* there's positive and negative numbers

* you get to put them together but also you get to predicate one with the other, creating numbers of numbers

is the rest derivative of this or am i just clueless about advanced mathematics?

 

[letspaniclater]

thanks for the AllAboutCircuits link, Blarraun. it helped simplify the notations found in the text.

I can't place the DeMorgan theorem in the context of Boole, may not have reached that point yet. anyway, the problem isn't so much the introduction of theorems and propositions, more so in how they are applied and how he arrives at forms that allow for their reduction.

for example:

Spoiler

the chapter on reduction asserts there are two modes of reducing systems of equations. the following is an example of the first method, that of employing an arbitrary constant (expressed as c). any terms form 1-x, 1-y, etc. will be represented by x*, y*, etc.

the first step for a specific system of equations was the elimination of class symbol w. after it was eliminated, we are left with:

(xyz + cx*yz +c'z*) (xyz* + cxyz + c'x*y* - c'z*) = 0 (1)

from which x must be determined (x = ...).

in previous examples, we expanded the equation by factoring, but Boole states that doing so is not necessary. instead, he develops the first side of the equation with reference to x, arriving at

yz(yz* + cyz - c'z*)x + (cyz + c'y*) (c'y* - c'z*) (1 - x) = 0 (2)

or,

cyzx + (cyz +c'y*) (c'y* - c'z*) (1 - x) = 0 (3)

therefore,

x = (cyz + c'y*)(c'y* - c'z*)
(cyz + c'y*) (c'y* - c'z*) - cyz


what I just wrote is almost exactly how it is presented in the text. I am at a loss as to how the equation was transformed from (1) to (2).

I will read the wikibook throughout the week. took a break from Laws of Thought this past week, think I'll try again and hope some time away helped.

is the rest derivative of this or am i just clueless about advanced mathematics?

is this your way of saying that one does not require a base level knowledge of mathematics to learn mathematical logic?

 

[Brontosaurie]

is this your way of saying that one does not require a base level knowledge of mathematics to learn mathematical logic?

no, just asking if my axioms were sufficient or not.

 

[Vrecknidj]

http://plato.stanford.edu/entries/principia-mathematica/

 

[Ex-User (9086)]

http://plato.stanford.edu/entries/principia-mathematica/

I like this attempt at summarising the fundamentals, although I would argue the book is inaccessible to people without experience in first-order logic.

* there's discrete and continuous numbers

* there's positive and negative numbers

* you get to put them together but also you get to predicate one with the other, creating numbers of numbers

is the rest derivative of this or am i just clueless about advanced mathematics?

You are describing the various fields of mathematics, such as discrete mathematics, number theory, topology or algebra, it has little to do with the mathematical logic itself.

for example:

Spoiler

the chapter on reduction asserts there are two modes of reducing systems of equations. the following is an example of the first method, that of employing an arbitrary constant (expressed as c). any terms form 1-x, 1-y, etc. will be represented by x*, y*, etc.

the first step for a specific system of equations was the elimination of class symbol w. after it was eliminated, we are left with:

(xyz + cx*yz +c'z*) (xyz* + cxyz + c'x*y* - c'z*) = 0 (1)

from which x must be determined (x = ...).

in previous examples, we expanded the equation by factoring, but Boole states that doing so is not necessary. instead, he develops the first side of the equation with reference to x, arriving at

yz(yz* + cyz - c'z*)x + (cyz + c'y*) (c'y* - c'z*) (1 - x) = 0 (2)

or,

cyzx + (cyz +c'y*) (c'y* - c'z*) (1 - x) = 0 (3)

therefore,

x = (cyz + c'y*)(c'y* - c'z*)
(cyz + c'y*) (c'y* - c'z*) - cyz


what I just wrote is almost exactly how it is presented in the text. I am at a loss as to how the equation was transformed from (1) to (2).

I will read the wikibook throughout the week. took a break from Laws of Thought this past week, think I'll try again and hope some time away helped.



is this your way of saying that one does not require a base level knowledge of mathematics to learn mathematical logic?

I don't understand the symbols used in this reduction
x* means 1-x (subtraction or what is it?)
I see subtraction, so it's in normal algebra? Then x' is a different constant than x? As in x and x' prime?

 

[dark+matters]

I find it so challenging to study math on my own without that rigid structure and reward/punishment system that school imposes. Studying math isn't like studying grammar or LSAT materials for me. I started looking into Russell, but I quickly started to fall asleep during all books/videos, etc. I can enjoy most popular physics books, but again, it's so hard to get down to the nitty gritty mathematics without that disciplined school environment. Does anyone have this same problem, or does anyone totally not feel this way at all?

 

[StevenM]

I find math is a lot more fun, when applying it to something I'm curious about.

To original question, essence of logic is that either something is true, or it is false.

 

[letspaniclater]

I don't understand the symbols used in this reduction
x* means 1-x (subtraction or what is it?)
I see subtraction, so it's in normal algebra? Then x' is a different constant than x? As in x and x' prime?

x* is represented as an x with a line over it in the text, but I couldn't find the symbol; it means "not x".

 

[Ex-User (9086)]

x* is represented as an x with a line over it in the text, but I couldn't find the symbol; it means "not x".

To be fair I thought ' was negation, so it stands for prime (another constant)?
There is a "(1 - x)" in the 2nd equation, is it how it's supposed to be written or it's another not x?

Does anyone have this same problem, or does anyone totally not feel this way at all?

I sometimes don't feel like doing it at all and sometimes it's interesting on its own. To me solving things of this nature is relaxing, similar to doing a sudoku or crosswords.

 

[letspaniclater]

To be fair I thought ' was negation, so it stands for prime (another constant)?
There is a "(1 - x)" in the 2nd equation, is it how it's supposed to be written or it's another not x?

yeah, it probably is. I pulled x* out of my ass.

the "1-x" in the second equation is written in that form on purpose. it's the same class symbol, just, as you stated, a negation. both x* (or negation, or bar, or...) and "1-x" are the same, but I think Boole leaves it in the "1-x" form to highlight that the first equation was being developed w/reference to x.

and just out of curiosity, do you work in mathematics?

I find it so challenging to study math on my own without that rigid structure and reward/punishment system that school imposes[...]Does anyone have this same problem, or does anyone totally not feel this way at all?

I'll let you know how it goes, this is the first time I've attempted recreational math.

also, what was it about Russell that put you off?

 

[dark+matters]

I'll let you know how it goes, this is the first time I've attempted recreational math.

also, what was it about Russell that put you off?

Yes, do please let us know how it goes for you. I'd be very curious to hear about your experiences with it in the future.

I find drawing out physics problems relaxing if, beforehand, I have invested myself in learning the necessary concepts which build up to the problem and let me solve it (and if I have a good answer key to check my answers afterwards). If I start feeling impatient (like when I've got 3 other assignments due tomorrow) or reach beyond my abilities with nobody around to help me, I get frustrated, not relaxed. Time passes quickly when I'm doing math, but I don't think I have ever found it relaxing. I don't know if this is something that would ever change for me. It's always very hard to do things when one is just beginning, so it's hard to say. It probably takes like... 5-7 years to feel really comfortable with the basics of something.

I watched some of these types of videos on Russell and they were at least trying to be interesting, but after a while, there was just an incredible flatness to the presentation that made it hard for me to maintain attention.

https://www.youtube.com/watch?v=z5JQjcSfUO0

I saw that Russell had a strong presence in my school library, but his books on logic looked phenomenally dry. I would open the middle of the book, and it looked as though I would have had to learn a lot of jargon that was never going to go anywhere useful for me. I read some of his quotes in a philosophy book that were neat, but that was only a tiny bit of the book and the book was made for the sake of entertaining a popular audience anyway.

 

[dark+matters]

Here is a page denoting the basics of formal logic.
1. Propositional Calculus
2. Formal Logic book

By the way- these are great. Thanks for sharing them. I'll also go back to seeing if I can read Russell's work sometime this summer (hopefully- I'm sure I don't have as much vacation time as I feel like I do).

 

[letspaniclater]

I find drawing out physics problems relaxing if, beforehand, I have invested myself in learning the necessary concepts which build up to the problem and let me solve it (and if I have a good answer key to check my answers afterwards).

gotcha. is it safe to assume you felt unprepared to tackle Russell, then? it seems that even those familiar with logic find Principia impenetrable--or that it could've at least been condensed. it is partly that reputation that had me begin with Boole, but, like you, I fear I may have hit a wall. but hey, I'm on summer vacation. it's nice to have something to occupy my time besides work and youtube.

speaking of, think I'll give that Russell recording a listen soon.

 

[Lapis Lazuli]

Oh I was just semi-joking but cool, you got something out of it

I should have read further down the list, but I joined the forum when I read your response to the question. I'm glad you were joking, no disrespect to the Mad Hatter, Bertrand.

 

[onesteptwostep]

dark I think reading summaries on his Pricipia Mathamatica might be of help if you want to get anything, it's like only 20 pages all combined. I have a copy of it in my Russell's collection book (never got into tho) His views on other stuff are more interesting imo, like language, culture, religion, education.

Also I don't think we really have mathematicians on this site... or are there? (yes I'm looking at you lurker)

 

[Lapis Lazuli]

Also I don't think we really have mathematicians on this site... or are there? (yes I'm looking at you lurker)

I guess that depends on what you think a mathematician is, as distinguished from a logician?

 

[onesteptwostep]

Is there a difference? (I'm wondering)

 

[Lapis Lazuli]

Is there a difference? (I'm wondering)

Bertrand Russell didn't think there was a difference, and that led to problems for his Principia Mathematica.

What that difference is, is a philosophical question, but I am not addressing that question, at least for now. =)

 

[onesteptwostep]

All things lead to philosophy

 

[Lapis Lazuli]

All things lead to philosophy

Indeed!

 

[propianotuner1]

Here is a page denoting the basics of formal logic.
1. Propositional Calculus
2. Formal Logic book

The essence is to arrange information and information derived from provided data in a way that is correct and/or can be understood clearly. Building coherent structures of data based on a selected subsystem of logic.

I linked a specific branch of logic using human language since it shows how logic is used more naturally.

Other interesting thing that helps visualise logic is set theory.

Good references. The essence of mathematical/propositional logic is strictly defined terms, goals, and functions that can be displayed in their own concise syntax. I would also highly recommend a look at the Stanford Encyclopedia of Philosophy and popular names in propositional logic like Epictetus, Aristotle (the father of syllogisms), and Leibniz and Kripke (the two biggest names in modal logic, AKA "possible world semantics"):

http://plato.stanford.edu/entries/propositional-function/

 

[Ex-User (9086)]

yeah, it probably is. I pulled x* out of my ass.

Long overdue and probably obsolete but I had a look and read through "The Laws of Thought" by Boole and I found the examples you were talking about.

So the problem is that he was very frequently using the indeterminate quantity symbol ~ and he had a very different ruleset, which turns the evaluation of his logical expressions into incomparable to modern boolean logic, so unless you follow strictly his methodology from the book, you won't get the algebraic reduction that you should.

Based on this, while the book was an interesting read, I recommend you find a modern resource on boolean logic to study from, turns out Bool is terrible at being boolean.

If you really want to stick to his approach you'll have plenty of other puzzling and frustrating moments ahead of you as I did have.

Here's one of many basic outlines of how you reduce boolean expressions and you can find a list of all identities that could help you on the wiki and I'm sure I linked them to you back then. A modern book on boolean logic and logic circuits is recommended, going back to the sources in this case is unnecessary headache.

gotcha. is it safe to assume you felt unprepared to tackle Russell, then? it seems that even those familiar with logic find Principia impenetrable--or that it could've at least been condensed. it is partly that reputation that had me begin with Boole, but, like you, I fear I may have hit a wall. but hey, I'm on summer vacation. it's nice to have something to occupy my time besides work and youtube.

speaking of, think I'll give that Russell recording a listen soon.

Guys, please stop.

You don't begin reading about logic from Principia Mathematica.
If you want to enjoy your time and learn anything take something modern or a coursebook or something similar.

Principia is impenetrable even to its authors, Russell gave up after he realised that he can't manage to use pure logic for defining concepts. And it's not meant to be educational material either.

I can't recommend anything in your language, but if you asked in your library or a mathematician/logician you know then they would point you in the right direction.

 

[letspaniclater]

haha, better late than etc.

since I made that post, I've taken two formal logic courses at my university so it's all good now. Boole is still cool, though.