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22ndMarch2015, 04:24 AM  #1 
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What is the essence of mathematical logic?
?

22ndMarch2015, 04:26 AM  #2 
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Re: What is the essence of mathematical logic?
Bertrand Russell.
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22ndMarch2015, 05:45 AM  #3 
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Re: What is the essence of mathematical logic?
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.

22ndMarch2015, 10:42 AM  #4 
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Re: What is the essence of mathematical logic?
Oh I was just semijoking but cool, you got something out of it
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22ndMarch2015, 10:46 AM  #5 
straightedgy
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Re: What is the essence of mathematical logic?
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. 
22ndMarch2015, 06:46 PM  #6 
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Re: What is the essence of mathematical logic?
Modus ponens.
In short, "if A then B". Every time you reason something out, that's Modus ponens working for you.
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31stMay2015, 08:30 PM  #7 
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Re: What is the essence of mathematical logic?
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 textspecifically 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 logicexcepting computer science, where it still seems highly relevantso 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 logicFrege's Begriffsschrift and Gödel's On Formally Undecidable Propositionsbut 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? 
31stMay2015, 10:19 PM  #8  
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Re: What is the essence of mathematical logic?
Quote:
Spoiler:


1stJune2015, 12:20 AM  #9 
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Re: What is the essence of mathematical 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? 
1stJune2015, 03:14 AM  #10  
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Re: What is the essence of mathematical logic?
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:
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. Quote:


1stJune2015, 07:26 AM  #11 
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Re: What is the essence of mathematical logic?

2ndJune2015, 03:49 AM  #12 
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Re: What is the essence of mathematical logic?

2ndJune2015, 11:41 AM  #13  
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Quote:
Quote:
Quote:
x* means 1x (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? 

2ndJune2015, 09:14 PM  #14 
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Re: What is the essence of mathematical logic?
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?

2ndJune2015, 09:43 PM  #15 
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Re: What is the essence of mathematical logic?
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. 
2ndJune2015, 10:13 PM  #16 
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Re: What is the essence of mathematical logic?
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".

2ndJune2015, 10:51 PM  #17  
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Re: What is the essence of mathematical logic?
Quote:
There is a "(1  x)" in the 2nd equation, is it how it's supposed to be written or it's another not x? 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. 

3rdJune2015, 02:03 AM  #18  
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Re: What is the essence of mathematical logic?
Quote:
the "1x" 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 "1x" are the same, but I think Boole leaves it in the "1x" form to highlight that the first equation was being developed w/reference to x. and just out of curiosity, do you work in mathematics? Quote:
also, what was it about Russell that put you off? 

3rdJune2015, 03:19 AM  #19  
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Re: What is the essence of mathematical logic?
Quote:
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... 57 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. 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. 

3rdJune2015, 05:31 AM  #20  
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Re: What is the essence of mathematical logic?
Quote:


4thJune2015, 01:37 AM  #21  
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Re: What is the essence of mathematical logic?
Quote:
speaking of, think I'll give that Russell recording a listen soon. 

12thJune2015, 06:27 AM  #22 
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Re: What is the essence of mathematical logic?

12thJune2015, 11:49 AM  #23 
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Re: What is the essence of mathematical logic?
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)
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12thJune2015, 03:25 PM  #24 
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Re: What is the essence of mathematical logic?

12thJune2015, 03:33 PM  #25 
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Re: What is the essence of mathematical logic?
Is there a difference? (I'm wondering)
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12thJune2015, 03:41 PM  #26 
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Re: What is the essence of mathematical logic?
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. =) 
12thJune2015, 03:50 PM  #27 
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Re: What is the essence of mathematical logic?
All things lead to philosophy
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12thJune2015, 03:51 PM  #28 
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Re: What is the essence of mathematical logic?

7thJuly2015, 03:01 AM  #29  
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Re: What is the essence of mathematical logic?
Quote:
http://plato.stanford.edu/entries/pr...onalfunction/
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17thDecember2016, 02:25 AM  #30  
straightedgy
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Re: What is the essence of mathematical logic?
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. Quote:
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. 

17thDecember2016, 09:15 PM  #31 
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Re: What is the essence of mathematical logic?
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. 
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