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You asked: "What I would like to know is how do you perfect your logic? Is there some sort of exercise book, or text book?"

Could you explain a little about what you mean? For example, first of all, what do you mean by "logic" here?

Second, are you asking about how to acquire knowledge of logic, as a body of rules and principles that others have identified? Or are you asking how to apply that body of ideas to thinking about big and little problems in your life?

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Hurley's "A Concise Introduction to Logic" is my favorite. It does a great job of introducing Aristotelian logic, and it seems to be the textbook most commonly used in university courses on the subject.

I, too, am undergraduate student interested in the study of logic. I'm currently taking my first course in sentential logic, which uses the book "The Logic Book" by Bergmann Moor and Nelson- which is good if you're interested in symbolic, more mathematical aspects.

Edited by Cole
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I really don't care much about philosophy, but I am absolutely fascinated with logic.

Ha, that's what they all say at first.

Anyway, here are some good books:

For just learning to work with the predicate calculus, I think that, by far, the best book is Kalish, Montague, and Mar's Logic: Techniques Of Formal Reasoning. It's had many editions, so I'd aim for the latest one. For a first book, this is the one to get. It requires virtually no mathematical background but it is rigorous enough for an instroduction. There is very little meta-theory, but that's okay, since the practice the book gives you in first order logic puts you in good stead to move on to meta-theory. This is a textbook, so it's not cheap, but it is worth the money. The book was originally written by Kalish and Montague; Mar came in for the latest edition. Sidenote: Mar is the last person to receive a doctorate sponsored by Alonzo Church.

With your grasp of predicate logic attained from the Kalish, Montague and Mar text, you'll be in a good position to get some basic set theory under your belt, and you will need set theory to study meta-logic. So, for a set theory text, I especially recommend Herbert B. Enderton's Elements Of Set Theory (online errata sheet available at his home page).

Another good set theory text, and available cheaply in paperback, is Patrick Suppes's Axiomatic Set Theory. I prefer his development of the natural numbers and ordinals to Enderton's development of these. On the other hand, I do not prefer Suppes intitial axiomatization since it allow urelements, thus requiring that 'is a set' be a predicate and that a constant for the empty set be taken as a primitive.

With enough set theory, you're ready for some meta-logic. I strongly recommend Herbert B. Enderton's A Mathematical Introduction To Logic (online errata sheet available at his home page). Much welcome is the emphasis on induction and recursion from almost the very start of the development. The prose is clear and the layout is easy to follow. The book takes you through a detailed exposition of an incompleteness proof (straight to the Rosser result) with the important stops along the way. One drawback, I think, is that the axiomatization of sentential logic is: If it's a tautology, then it's an axiom. But this is okay as long as one is already competent in sketching sentential proofs, which you will be by this time. There are some other goodies bundled, including a chapter on non-standard analysis. The logic book has a second edition.

As a reference, Alonzo Church's Introduction To Mathematical Logic has so much wisdom and information - just in the footnotes alone! The introductory chapter might as well be the definitive synopsis on those topics. It is a great book. However, I think it's a bit recondite to serve as a gateway to the subject.

As a reference, Stephen Kleene's Introduction To Metamathematics is expensive and hard to find. But this must be considered one of the great works. Kleene is one of the great logicians and among the founders of the theory of computability.

Kleene's Mathematical Logic, in its paperback edition, can be had cheaply and it is jam packed with great information. However, I find that his expositions sprawl and that the clutter - the formulas overflowing in every nook and cranny - is difficult to cope with. He just gives you so much, like he's "spilling his guts", that it's a bit much to sort through. But, considering the price, and the wealth of information, one should grab this one and refer to whenever needed. Note that his proof of incompleteness is mainly through Turing machines. I do love this book.

A beautiful book is Dirk van Dalen's Logic And Structure, and reasonably priced for a Springer book. This has some detail of rigor on topics I've not been able to find in other books. His presentation is very lucid and he deftly covers a lot of ground. He gives a natural deduction system, and covers most of the subject through incompleteness. And the book has nice chapters on intuitionist logic, including Kripke semantics.

An amazing book is Raymond Smullyan's First Order Logic. This book takes my breath away. Talk about elegant! He'll prove a major meta-theorem, and the proof is so concise, so short and sweet, that at the end of it you say, "That's it? That's all there is to it?" Then you go back through it and confirm that, sure enough, it's all there and solid as a rock. He favors trees and tableaux, by the way. However, there is quite a bit that the book does not cover, including incompleteness.

One of the most comprehensive introductory books is Joseph R. Shoenfield's Mathematical Logic as it takes you through the major meta-theorems for predicate calculus, incompleteness, recursion theory, a quick tour of set theory, and then even through forcing.

Patrick Suppes's Introduction To Logic is a bit recondite in its presentation but it's a good book, and in paperback, it's a bargain. And there is one chapter that makes the book, in my view, a must have: His chapter on the theory of definitions is the best I've come across.

Benson Mates's Elementary Logic is a real nice book for learning predicate calculus, but it's quite brief on meta-theorems.

Elliot Mendelson's Introduction To Mathematical Logic is another classic.

W.V. Quine's Mathematical Logic is a classic, and Quine really nails it on some topics, such as use-mention. This is good in conjunction with his Set Theory And Its Logic. It's nice to be apprised of Quine's NF. One drawback is the Principia style notation. Quine is, of course, an institution onto himself, and his subtle wit is always welcome.

A nice, lucid, uncluttered presentation is in L.T.F Gamut's two volume set Logic, Language, And Meaning ('L.T.F. Gamut is a collective pseudonym like 'Nicholas Bourbaki'). The first volume is basic logic and the second volume is modal logic and other topics. Not much in the way of meta-theorems, but this is a good starter kit.

For computability and recursion theory, the paperback edition of Martin Davis's Computability And Unsolvability is a steal! But what a dorky cover they put on the book!

Torkel Franzen's two books: Inexhaustibility: A Non-Exhaustive Treatment and Godel's Theorem: An Incomplete Guide To Its Use And Abuse are must reads, though they're not books to cover course material with.

Jean van Heijenoort's From Frege To Godel is the standard anthology of famous papers in the history of mathematical logic.

And Azriel Levy's Basic Set Theory is a killer book, and a bargain in paperback. It goes well past the basics (though it doesn't include forcing or even Godel's half of the independence proof). Actually, the book pretty much uses class theory as opposed to set theory, and the first chapter is a bit confusing. But the book is generally well written and has a wealth of material.

Edited by LauricAcid
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For what purpose? Learning about the foundations of mathematics? Programming computers? Formalising arguments? Understanding philosophical notions which are supposedly helped by precise symbolism? Just general interest?

Why are you asking on a specialist philosophy forum when you "dont really care about philosophy" :)

Edited by Hal
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For what purpose? Learning about the foundations of mathematics? Programming computers? Formalising arguments? Understanding philosophical notions which are supposedly helped by precise symbolism? Just general interest?
Formalising arguments, and I'm interested.

Thanks for the book list, LauricAcid, I'll look some of them up.

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If you wish to formulate arguments, you obviously have something to argue.

The act of arguing in its self presupposes some form of philosophy, particularly a specific epistemology. (a theory of the aquisition of knowledge)

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The act of arguing in its self presupposes some form of philosophy, particularly a specific epistemology.
I suppose this is true; however, logic doesn't simply generate a philosophy out of thin air. Logic is a tool to help create a philosophy.

I think it was Henri Poincare who pointed out that any consistent axiom system is just as valid as any other consistent axiom system. Every philosophy begins with axioms, whether they admit to it or not, and if the axioms are consistent then the systems are just as "valid".

How should one thereby judge a philosophy to be a good philosophy?

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I suppose this is true; however, logic doesn't simply generate a philosophy out of thin air. Logic is a tool to help create a philosophy.

I think it was Henri Poincare who pointed out that any consistent axiom system is just as valid as any other consistent axiom system. Every philosophy begins with axioms, whether they admit to it or not, and if the axioms are consistent then the systems are just as "valid".

How should one thereby judge a philosophy to be a good philosophy?

It is philosophy itself that poses and answers the question: What is the good? In the philosophy of Objectivism, (from OPAR, p. 219):

What then is the standard of moral value? A valid code of morality, Ayn Rand concludes, a code based on reason and proper to man, must hold man's life as its standard of value. "All that which is proper to the life of a rational being is the good; all that which destroys it is the evil."

For *any* reasoning to be good, by the above-named standard (which is not just a floating assertion but integrated with the rest of Objectivism), it must be both valid and true. As you are no doubt aware, one can perform a "valid" deduction from false premises, and arrive at a valid but false conclusion. (e.g. a simple syllogism: "All moons are made of green cheese. Earth has a moon. Therefore earth's moon is made of green cheese." A valid deduction that is false because based on a false major premise.) So any philosophic system that is nothing more than "valid" deductions based on false premises, is not a good philosophy. Ayn Rand frequently said "Check your premises" - check them against reality and make sure they're true.

So to answer your question: By the above standard (i.e. the standard of rational life, of Objectivism), a philosophy is good if it helps rational beings to live, and it is evil if it acts against their lives.

Objectivism does recognize 3 basic axioms: Existence, Consciousness, and Identity. From Galt's Speech in Atlas Shrugged:

Existence exists—and the act of grasping that statement implies two corollary axioms: that something exists which one perceives and that one exists possessing consciousness, consciousness being the faculty of perceiving that which exists.
And for identity:

Whatever you choose to consider, be it an object, an attribute or an action, the law of identity remains the same. A leaf cannot be a stone at the same time, it cannot be all red and all green at the same time, it cannot freeze and burn at the same time. A is A. Or, if you wish it stated in simpler language: You cannot have your cake and eat it, too.

This is covered in OPAR, 1. Reality.

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I am an undergrad student interested in logic. I really don't care much about philosophy, but I am absolutely fascinated with logic. But this in neither here nor there.

What I would like to know is how do you perfect your logic? Is there some sort of exercise book, or text book?

If you want to learn about logic, you should read "An Introduction to Logic" by H.W.B Joseph (http://www.papertig.com/Logic.htm) and listen to "Logic Thinking" by Dr. Binswanger.

If you are seriously interesting in logic, then the induction material available at the Ayn Rand bookstore is wonderful. (Remember induction preceeds deduction)

If, however, you are interested in "Predicate Logic", then you will find yourself caught between Objectivism and that "logic". "Predicate logic" divorses thought from the argument is a way that can only lead to rationalism. It pretends that the form of the argument is what makes an argument logical. Many Objectivists have rejected this claim and their material is also available at the Ayn Rand Bookstore.

Logic is a rich field in which the form of the argument is really only a small minor piece that you automate when you first learn it. Definition, categorization, conceptual common denominators -- these are all the more interesting topics in the field.

While many fewer Objectivists have written about the errors in abstract algebra, set theory, and the other inverted mathematical fields. You can find some good material on this in old Intellectual Activists (though I do not have them available to reference). The problem with the "abstract algebra" approach to mathematics is that it uses deduction in a rationalistic way by igonoring the inductive process that got them there.

Much like the classical education school in the past, abstract algebra arbitrarily looks at certain relationships in mathematics, claims, with no basis, that these are axioms, and then presumes to write completely floating systems on top of this imaginary framework.

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So to answer your question: By the above standard (i.e. the standard of rational life, of Objectivism), a philosophy is good if it helps rational beings to live, and it is evil if it acts against their lives.?
What defines a rational being to be rational?

For example, in economics there is the assumption that everyone is "rationally selif-interested" and then the homo economicus turns around and acts irrationally.

Objectivism does recognize 3 basic axioms: Existence, Consciousness, and Identity.
I would like someone's help here, for I cannot possibly understand the reasoning; with these three axioms, how does Rand deduce the standard of "rational life"?

It doesn't seem to follow her political philosophy either. If "A=A", then good philosophy helps rational beings live. If a good philosophy helps rational beings live, then capitalism is a good philosophy? I can follow the second conditional, but not the first.

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What defines a rational being to be rational?

For example, in economics there is the assumption that everyone is "rationally selif-interested" and then the homo economicus turns around and acts irrationally.

I would like someone's help here, for I cannot possibly understand the reasoning; with these three axioms, how does Rand deduce the standard of "rational life"?

It doesn't seem to follow her political philosophy either. If "A=A", then good philosophy helps rational beings live. If a good philosophy helps rational beings live, then capitalism is a good philosophy? I can follow the second conditional, but not the first.

If you are really interested, I strongly suggest that you take the time to read Atlas Shrugged (or at least Galt's Speech but then you're shortchanging yourself of a great novel), and OPAR, and go from there.

Ayn Rand's approach was not fundamentally deductive - it was inductive. She was a student of history before becoming a philosopher, and she carefully observed many instances of human action throughout history and in her lifetime, before arriving at the principles of her philosophy.

Again, I can't repeat everything in the corpus of Objectivism (and if you want to have practically everything that she wrote, plus OPAR, I sell a CDROM of her works for about $60 at www.Objectivism.net - it has the virtue of having a search engine as well.) But to touch on the point of ethics: She saw that, to even *have* an ethics, somebody has to be alive. So life is a basic precondition of an ethics, and in her philosophy, it is the *goal* of a rational man, to pursue life and the best life possible to him.

Edited by Unconquered
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She was a student of history before becoming a philosopher...

I should emphasize that being a philosopher was not her primary goal, it arose out of her desire to better write her kind of fiction - the portrayal of heroes, the ideal man at his best. Thus her unique position of being not only a great philosopher but one of the greatest writers in history.

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What I would like to know is how do you perfect your logic? Is there some sort of exercise book, or text book?

http://mathworld.wolfram.com/topics/Founda...athematics.html

You could use this page for starters, if you are interested in mathematical logic. When you get familiar with the terms, you'll be able to find books you need yourself.

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Formalising arguments

Formalizing what kinds of arguments? Predicate logic is adequate for formalization of virtually all mathematical arguments, but there are all kinds of arguments that aren't mathematical, especially inductive ones. For these there are other forms of logic, as well as, as has been mentioned, philosophy. And there are forms of logic specific to individual fields, such as legal reasoning.

I think it was Henri Poincare who pointed out that any consistent axiom system is just as valid as any other consistent axiom system. Every philosophy begins with axioms, whether they admit to it or not, and if the axioms are consistent then the systems are just as "valid".

Probably you have in mind remarks not by Poincare but rather by David Hilbert. However, your terminology is mistaken, and thus you've unintentionally misrepresented the concepts here. In the context of mathematical logic, it has become standard to reserve 'valid' not for consistency, but rather for logical truth. An axiom system then would be valid if and only if every axiom were a logical truth. In first order logic, for a complete system (completeness in the sense of all validities are provable) the only valid system is the minimal one in which there are no non-logical axioms. On the other hand, we say an argument is valid if the premises entails the conclusion. This can also be stated in terms of valid sentences, since, for a valid argument, we can bundle the premises into a conjunction to serve as the antecedent in some valid sentence.

Moreover, I do not know of Hilbert or others asserting that the principle of consistency in mathematics makes one consistent philosophy just as good as another consistent philosophy. For that matter, there are criteria other than consistency that are applied even to mathematical theories as bases for preferring some to others.

"Predicate logic" divorses thought from the argument is a way that can only lead to rationalism.

Hogwash.

It pretends that the form of the argument is what makes an argument logical.

There are logicians who describe arguments in that way, but predicate logic does not depend on accepting this premise. In fact, predicate logic includes semantics to capture the notion of validity so that, while form is an important part of the subject, the subject is not necessarily reduced to it. Usually, what is shown are proofs, not just stipulations, that certain forms are valid. And mathematical logic, as predicate logic is at the center, does not claim that there are not other valid and important methods of reasoning, such as inductive and philosophical methods.

Logic is a rich field in which the form of the argument is really only a small minor piece that you automate when you first learn it.

What does 'automate form of argument' mean?

Definition, categorization, conceptual common denominators -- these are all the more interesting topics in the field.

1. Those are more interesting or less interesting depending on the person.

2. The study of predicate logic and mathematical logic is not confined to analyses or 'automation' (whatever that means) of forms, but includes semantics and meta-theory and study of relations between mathematics and logic. Also, since definition, has been mentioned, predicate logic is very much concerned with this subject. I don't care who thinks what is more interesting than what, but the fields of study should not be mischaracterized.

3. Would you say in your own words (not just retyping quotes from ITOE) what you mean by 'conceptual common denominator'?

While many fewer Objectivists have written about the errors in abstract algebra, set theory, and the other inverted mathematical fields. You can find some good material on this in old Intellectual Activists (though I do not have them available to reference).

1. Please cite whatever writings are available on the Internet or in libraries. What writings are there that point to actual mathematical errors? What writings are there that discuss the actual mathematics? And are there any that can be read on the Internet?

2. Please defined 'inverted'.

The problem with the "abstract algebra" approach to mathematics is that it uses deduction in a rationalistic way by igonoring the inductive process that got them there.

Again, this business about rationalism is a strawman. Mathematics does not commit to rationalism. And that algebraists use deduction, as does virtually all mathematics, does not entail that algebraists or mathematicians deny the importance of induction. But proving theorems is a deductive activity. We can't go out into the world to discover whether the binomial theorem is true. But we can prove deductively that it is. To complain about this would be as silly as complaining that musicians ignore the techniques of aircraft navigation.

Much like the classical education school in the past, abstract algebra arbitrarily looks at certain relationships in mathematics, claims, with no basis, that these are axioms, and then presumes to write completely floating systems on top of this imaginary framework.

1. Usually axioms are formed with a view toward developing certain theories. The selection of axioms depends on what kind of theory the mathematician seeks. Usually, these are theories that inform mathematical understanding, especially ones that have ramifications for more than one field of mathematics.

2. What, pray tell, is 'a completely floating system'?

3. What example can you offer of Objectivist mathematics? What are the non-arbitrary Objectivist mathematical axioms and how do these account for the development of real analysis, computability theory, computer science and other mathematical disciplines that are crucial to technological progress? Would you please say where one can read, especially on the Internet, any Objectivist mathematics at all?

http://mathworld.wolfram.com/topics/Founda...athematics.html

You could use this page for starters, if you are interested in mathematical logic. When you get familiar with the terms, you'll be able to find books you need yourself.

That site might serve for reference, but it would not be a good place to start, even as a place to glean some terminology. To try to make sense or even appreciate the terms listed there requires understanding the terms and concepts as they are systematically developed in mathematics - one term defined by other terms that have been previously defined, in an orderly method. Subjecting oneself to a bunch of math terms listed alphabetically and with non-ordered definintions is of little use and is likely to breed misunderstanding due to lack of context.

Edited by LauricAcid
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"Predicate logic" divorces thought from the argument in a way that can only lead to rationalism. It pretends that the form of the argument is what makes an argument logical.

You are confused. Formal logic is not irrational. It is merely more abstract and precise than informal arguments.

Informally, one might say:

<<The temperature outside is 94 degrees Fahrenheit. Therefore, it is too hot to be comfortable outside.>>

More formally, one might say:

<<The temperature outside is 94 degrees Fahrenheit.>> implies <<It is too hot to be comfortable outside.>> major premise, inducted from experience.

<<The temperature outside is 94 degrees Fahrenheit.>> minor premise , observed.

<<It is too hot to be comfortable outside.>> conclusion, by modus ponens.

So formal logic allows you to separate out and examine each of your premises. Instead of having disputes over the validity of arguments, you have disputes over the truth of your major premises.

The validity of the rules of inference of formal logic, such as modus ponens, have been established by induction from long experience with them; and thus need not be reconfirmed in each particular instance.

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What defines a rational being to be rational?
First, this is a philosophical question and not a logical one, so you should either decide that you didn't need to ask this question, or you should decide that you really do care more about philosophy than you claim or think you do. Second, this is (probably) the wrong question. If you have a rational being, then there is no issue about figuring out that such a being is also rational. What you might have wanted to ask is, how can you tell whether some being is rational, or, what is the nature (characteristics) of the attribute "rational". The definition is less important than the identification -- the definition depends on making the correct identifications, and amounts to stating the characteristics that distinguish "rational" from "irrational". These are philosophical questions.
For example, in economics there is the assumption that everyone is "rationally selif-interested" and then the homo economicus turns around and acts irrationally.
There are two problems here. First, there is no such thing as homo economicus. There is a homo sapiens, but funny misuses of biological nomenclature does not magically lead to speciation. Second, even if economicists were to assume that all men act in their rational self interest, that would only show that they are wrong about man, which proves nothing about the nature of reason.

What you probably need to learn about, if you are interested in symbolically formalizing arguments, is something beyond FOP logic. Joseph's book is important, but you'll need something more tech, so I'd suggest looking inter alia into semantics, modal logic, and non-monotonic logic. The formal background helps a bit, and then I would suggest that you look at the question of what properties valid arguments have.

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Try to tackle modal logic and non-monotionic logic before comprehending predicate logic? And what semantics do you have in mind?

[DavidOdden:] "I would suggest that you look at the question of what properties valid arguments have."

What properties do you think valid arguments have?

Edited by LauricAcid
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Try to tackle modal logic and non-monotionic logic before comprehending predicate logic? And what semantics do you have in mind?
No, I assumed he had a decent grip on FOP logic. My suggestion for semantics is Chierchia & McConnell-Ginet Meaning and Grammar. An Introduction to Semantics, for an introduction (even though I disagree with them at various points).
What properties do you think valid arguments have?
Various -- it's an empirical question. The obvious one is that a valid argument only admits true statements (btw, don't bother whining about the sound / valid distinction: I understand it, and am totally uninterested in it, because we're talking about arguments). That means that arbitrary and false statements can't be used to derive valid conclusions. Beyond that, it gets fairly philosophical, but for example inductive generalization requires something more than just an observation of correlation, in order for a universal statement to be justified. You have to determine causal relations, not just static correlations.
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3. What example can you offer of Objectivist mathematics? What are the non-arbitrary Objectivist mathematical axioms and how do these account for the development of real analysis, computability theory, computer science and other mathematical disciplines that are crucial to technological progress? Would you please say where one can read, especially on the Internet, any Objectivist mathematics at all?

That site might serve for reference, but it would not be a good place to start, even as a place to glean some terminology. To try to make sense or even appreciate the terms listed there requires understanding the terms and concepts as they are systematically developed in mathematics - one term defined by other terms that have been previously defined, in an orderly method. Subjecting oneself to a bunch of math terms listed alphabetically and with non-ordered definintions is of little use and is likely to breed misunderstanding due to lack of context.

I am preparing a post for a seperate thread for many of your other comments. For this one, here is my original source for what I find is the most rational explination of the basis of mathematics:

"Foundation of Mathematics" by Ronald Pisaturo and Glenn D. Marcus in The Intellectual Activist Volume 8, Number 4 July 1994

"Foundation of Mathematics II" by Ronald Pisaturo and Glenn D. Marcus in The Intellectual Activist Volume 8, Number 5 September 1994

Included in these two articles is a detailed inductive approach to mathematics which has long been missing.

Edited by Jamwhite
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Predicate logic is adequate for formalization of virtually all mathematical arguments, but there are all kinds of arguments that aren't mathematical, especially inductive ones. For these there are other forms of logic, as well as, as has been mentioned, philosophy.  And there are forms of logic specific to individual fields, such as legal reasoning.

Are you saying that each specialized science has its own specialized logic? That is intriguing. What would be examples of elements of the logic of law or history, for example?

To set a context here, it would be helpful if you defined "logic" -- not any special kind of logic, but logic in general.

Edited by BurgessLau
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Are you saying that each specialized science has its own specialized logic?

Of course not, especially since I didn't say 'each'.

What would be examples of elements of the logic of law or history, for example?

I'm not interested in elaborating upon the terms and concepts of law, since my point was not to advocate that legal reasoning is profoundly different from all other reasoning. But that there is a study of legal reasoning and that it does have its own approaches to special topics in drawing inferences in legal contexts is well known. This seems pertinent to the poster's question, since he mentioned his interests in arguments, and legal argumentation is an important and rich part of modern discourse.

If, truly, you are unaware of this field of study, then a link such as below, as I casually copied it as among the first in an Internet search, might give you some idea:

http://www.uadm.uu.se/inter/education/index.php/course/503

To set a context here, it would be helpful if you defined "logic" -- not any special kind of logic, but logic in general.

I might have some quibbles with the definitions of the more primary senses given by Merriam-Webster, and wish for more detail and even greater breadth in them, but they're okay for present purposes. Edited by LauricAcid
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I"Foundation of Mathematics" by Ronald Pisaturo and Glenn D. Marcus in The Intellectual Activist Volume 8, Number 4 July 1994

"Foundation of Mathematics II" by Ronald Pisaturo and Glenn D. Marcus in The Intellectual Activist Volume 8, Number 5 September 1994

Thanks for the recommendations.

Included in these two articles is a detailed inductive approach to mathematics which has long been missing.

I'm interested in this, but does this include actual mathematics such as axioms and proofs or even statements of formal systems?

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