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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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The equivalence of a conditional statetement and contrapositive is easy to understand if you think in terms of sets.For example , "all rabbits are white" means "anything that is not white can't be a rabbit". –  Mohan Feb 26 '13 at 1:36
Pages 12–16 of Graham Priest's An Introduction to Non-Classical Logic: From If to Is has a very nice discussion of implication and $\lnot P\lor Q$ and whether this makes sense and why. –  MJD Feb 26 '13 at 2:59
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3 Answers 3

What you are referring to with your example is the propositional calculus, also knowns as the sentential calculus, which can be traced back to the ancient Greeks and the Aristotelian logic that emerged from that era.

Boole's study of Boolean algebras is an algebraization of Aristotelian logic, providing an algebraic model and rules of algebra that faithfully mimic Aristotelian deductions.

Mathematical logic is a very broad subject, of which propositional calculus is a very elementary part of, and is very well understood, and is in fact quite trivial. A large portion of modern mathematical logic is concerned with Model Theory, and to some extent is a study of the expression power of formal languages.

Related to Boole's boolean algebras are Heyting algebras. These provide an algebraization of a different type of logical system, known as Intuitionism.

I hope this gives you a very rough idea about what mathematical logic might be, and what it certainly is not.

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It is NOT a fact that two-valued propositional calculus is trivial. And even an opinion that such qualifies as trivial seems rather uninformed. Just because people often teach a subject as if it were trivial, that does not imply it qualifies as such. Let $\delta$ stand for a variable functor of one argument in two-valued logic. I challenge anyone who claims such as trivial to deduce say Frege's axioms for classical propositional calculus from C. A. Meredith's six-symbol axiom C $\delta$ $\delta$ 0 $\delta$ p. And n-valued, n>2, propositional calculi are not trivial. –  Doug Spoonwood Jul 15 '13 at 2:00
@DougSpoonwood what I refer to as being trivial is such results as OP mentions, like $P\implies Q$ being equivalent to $\neg P\vee Q$. Determining such an equivalence is trivial. I tried to give the OP a sense for that triviality. I tried to formulate my answer in relation to the OP's question. I doubt OP has any knowledge of the issues you raise. In any case, if my formulation aggravated you, I'm sorry. Thanks for the clarification you provided. –  Ittay Weiss Jul 15 '13 at 2:16
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This addresses mostly your title question.

Before moving on to Mathematical Logic, I'd suggest you get a firm grasp of propositional logic, as Ittay Weiss suggests. You'll want to master predicate logic, and to develop a thorough understanding of quantifiers.

One very helpful, credible (and freely available!) resource is Paul Teller's Logic Primer. (The link will take you to Paul Teller's website for the Primer, which you can download in pdf format.) There are two volumes, and together they should provide a firm foundation in first order logic and the basis from which to pursue mathematical logic.

Only then, when you've mastered the fundamentals of formal logic listed above, does it make much sense to dig into Mathematical Logic.

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"Mathematical Logic", these days, connotes a quite advanced study (it is often the title of third-level logic courses in universities, for example). What the OP means actually wants, I guess, is a lower-level introduction to formal logic (a.k.a. symbolic logic).

There are a lot of good books out there (and quite a lot of not-quite-so-good ones too!). @amWhy mentions a good one that is now freely available. You'll find a few more suggestions at various levels in the introduction to the Teach Yourself Logic study guide which you can download from http://logicmatters.net/students/tyl/

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I'm glad you posted: I was trying to recall the link to your site, to reference your "Teach Yourself Logic" guide in my post, but "blanked out" on the URL. All for good, though, as it provided the opportunity for this post from the master himself! :-) –  amWhy Feb 26 '13 at 14:42
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