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I want to study Mathematical Logic. One concept that confuses me, is that implication is equivalent to 'P or -Q'. So, I want to start from the book where this idea first started; but I'm not looking only for this idea, but also other basic ideas of Mathematical Logic.

I guess Boole's Boolean Algebra helped build Mathematical Logic. Can you give a brief explanation of how it and other ideas did (Like the previously mentioned implication definition), where they first started (in which books), and what other classic books talk about them?

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What might you have found out so far? –  Joseph Weissman Feb 25 '13 at 19:12
    
(If you really want to study mathematical logic, I suggest going to Math.SE) About your confusion: "A implicates B" = "not (A and not B)" = "not A or not (not B)" = "not P or Q". –  zaarcis Feb 26 '13 at 0:01
    
I can't help with book, but one way to look at your question is to simply consider the truth tables of both propositions and you can see that they are the same. Does that help in any way? –  Timotej Feb 26 '13 at 1:43
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Are you looking for a historical treatment about the development of logic, or a systematic introduction to the subject? The systematic introduction that I know of is Barwise and Etchemendy's "Language, Proof and Logic" published with Stanford. (A new edition has just come out.) The best book on the history of logic is probably, Kneale and Kneale, "The Development of Logic" (Oxford University Press, 1985). Be warned though, that the history of logic is very, very long. If you are interested specifically in mathematical logic, then work through Barwise, then Kleene's "Mathematical Logic". –  shane May 5 at 12:30
    
Also: mathoverflow.net/questions/33096/… –  shane May 5 at 12:31
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5 Answers 5

implication is equivalent to 'P or -Q'

That is not quite precise: Material implication Q -> P is equivalent to -Q or P.

Material implication is only a very specific conception of implication, though not one that is often used in day to day language.

There are other, richer, forms of implications, such as modal implication, indicative conditional, consequential implication which tries to better capture the implication that are more commonly used in day to day language, but they cannot be represented fully using just a truth table.

Another thing you need to be aware about in Classical/Mathematical logic is the OR. In real life, people more often uses OR when they actually mean XOR. For example, the question "tea or coffee?", most people would not expect the answer "both", but in Classical/Mathematical logic, both would be a perfectly fine correct response.

I believe those two are the biggest gotcha on classical/mathematical logic.

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If you want to read works on Logic where the ideas of modern Mathematical Logic were first stated, I would recommend reading works of Gottlob Frege. The first paragraphs of the Begriffschrift by Frege are easy to read and understand, and were very important for the development of Mathematical Logic. I do recommend however reading a brief introduction about Frege before you start reading his original writings. I think the following page will be a very good introduction about his views and influences on Mathematical Logic, especially Chapter 2 (Contributions to Logic): http://www.iep.utm.edu/frege/. At this web page you will also find comments about Boole's contribution to Mathematical Logic.

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Judging by your comment on implications and Boolean algebra, you may want to consider a textbook devoted to elementary propositional and predicate logic. From my personal experience, a book on mathematical logic may not be the best place to first acquire those skills. It would be a bit like learning to drive from a textbook on automotive engineering.

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I've just had a look at the Wikipedia Entry:

http://en.wikipedia.org/wiki/Mathematical_logic

There you will find a useful overview of key ideas, protagonists and books.

Hope that helps as a starting point.

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Using mathematical logic, we often find that statements that seem quite different on the surface are equivalent, which is to say, they have the same truth value under all distinguishable conditions. Or, to put it another way, if statement A is true in all and only the cases that statement B is true, they are equivalent.

One simple and basic case of two statements that are equivalent, but that may not look equivalent is

P IMPLIES Q

and

NOT P OR Q.

Both are true in all and only the cases where Q is true or P is false. This is because the way IMPLIES is defined (in logics such as the one you are studying), is as meaning that if P is true, Q must also be true (if P is false, Q can be either true or false), and the way OR is defined is as meaning that at least one of the two sides of the OR is true (but both can be).

In some ways, we might say that the whole purpose of mathematical logic is to go beyond the dictates of intuition in cases such as these.

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