Talk:Metalogic

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[edit] oppose merge

In my opinion Metalogic and Philosophy of Logic - as the terms are used - relate to distinct (if sometimes ralated) issues & the articles should not be merged. Godel's incompleteness theorem for instance is itself a metalogical theorem; its significance might be an issue in philosophy of logic. --Philogo 13:06, 30 August 2007 (UTC)

agreed. Pontiff Greg Bard (talk) 11:07, 14 January 2008 (UTC)
Thinking about it, the article should probably be a quasi-disambiguation dealing with formal logic, as it consists of model theory and proof theory. We also need a citation for use of the term, and some citation for its importance in the field (whatever field that might be). So a merger is inappropriate but stubification might very well be. — Arthur Rubin | (talk) 17:49, 15 January 2008 (UTC)

[edit] Dubious

Thinking about it, metalogic is difficult to distinguish from simple logic. The lead is false (logic does not deal solely, or even primarily, with truths-of-logic). — Arthur Rubin | (talk) 17:42, 15 January 2008 (UTC)

This is what I mean by hypercritical to the point of uselessness. Take a step back, take a deep breath, now say it with me Arthur: "Logic concerns itself with the truths-of-logic." You see that wasn't so bad. Your attempts to portray this straightforward statement into a muddled mess will leave the vast majority incredulous. If you have such a thesis then by all means publish it (but not here WP:OR). The source is a reliable one. It may not be good enough for you to recognize it's existence, but that doesn't matter at all. It does exist. It's a source that may or may not agree with your thesis. Live with it. Pontiff Greg Bard (talk) 21:06, 15 January 2008 (UTC)
I also don't know what "Logic concerns itself with the truths-of-logic." means; it seem tautologous. Is that a quote from the reference? The reference is fortunately available at the library, so I'm going to pick it up on the way home. — Carl (CBM · talk) 21:48, 15 January 2008 (UTC)
When juxtaposed with the other hyphenated term, the distinction being discussed is clear (even if by itself it is not). The phrase and term are from the book. This material is on the page 3 intro. It's quite a nice book. I bought my own copy on amazon recently, and that is why I have been working on some of these topics lately. I am quite fascinated as usual to see the response from the math people in general. You are one of my favorites among them Carl. I'm always interested in your perspective. Be well, Pontiff Greg Bard (talk) 22:28, 15 January 2008 (UTC)
Could you then explain why metalogic is not then model theory + proof theory? It's still the case that "Logic concerns itself with (the) truths-of-logic" is either tautological or meaningless. My calling it false was mistaken, unless truths-of-logic are further refined to mean something other than the "logical" construct. (As an aside, do all philosophers construct phrases out of apparently unrelated words, or only those from the German tradition?) — Arthur Rubin | (talk) 23:15, 15 January 2008 (UTC)
In spite of the introduction to the book, that definition clearly excludes discussions of formal languages with neither interpretation (model theory) or derivations (proof theory). — Arthur Rubin | (talk) 23:19, 15 January 2008 (UTC)
Also, there's no {{importance-inline}} tag that I can find; that is what I would have put here if I had full access to the internals of Wikipedia. — Arthur Rubin | (talk) 23:28, 15 January 2008 (UTC)
I never said that Metalogic isn't m + p theory. It seems to me that there are some formal language fundamentals that need to be understood commonly in order to move on to either Proof theory or Model theory however.
The whole sentence reads:
"While logic concerns itself with the truths-of-logic, metalogic concerns itself with the theory of sentences-used-to-express-truths-of-logic."
...and when read to the end makes complete sense. The hyphens bind together one concept which is needed to be clear here.
It is my understanding that artificial languages like Esperanto and Volapük eventually break down into dialects because their adherents are enamored with language-in-itself. Mathematicians on the other hand look past all the variations in symbols and notation rather seamlessly because they care about what lies behind the language. So why do you think the hyphenated form is so strange?
I don't know how common the hyphenated-phrase-word is in the literature, nor its distribution in Continental v. Analytic philosophy. I know they are indispensable in phenomenology with ready-at-hand, being-in-itself, etcetera. I think they are a great tool. On another aside, you may be interested to see an article in which I used words with subscripts. Pontiff Greg Bard (talk) 01:24, 16 January 2008 (UTC)
I'd like to see articles or redirects for all of the following for clarity:
  • truth-under-an-interpretation
  • proof-in-a-formal-system (formal proof)
  • proof-about-a-formal-system
  • theorem-of-a-formal-system (formal theorem)
  • theorem-about-a-formal-system (metatheorem)
  • derivation-in-a-system (formal derivation) —Preceding unsigned comment added by Gregbard (talk • contribs) 01:37, 16 January 2008 (UTC)
(←) I have no particular objections to leaving this article separate from the metamathematics article. In one reading, metamathematics is theformal study of properties of mathematical systems using mathematical logic, while metalogic is the formal study of logical systems using mathematical logic. The term metalogic seems to be well established in the literature. — Carl (CBM · talk) 01:44, 16 January 2008 (UTC)

[edit] My impression of the text by Hunter

I have the reference in front of me. Here is a summary of the table of contents
  • 1. Introduction, general notions
    • Formal languages
    • Interpretations
    • Metatheory, metatheory of logic
    • Theorem and metatheorem
    • effective methods
    • Decidable sets
    • 1-1 correspondence, cardinality
    • Finite and countable sets. Uncountable sets.
    • Proof of uncountability of the reals
    • Sequences, effective enumerations
    • Theorems about infinite sets
    • Proof of the incompleteness of the full theory of the natural numbers **
  • 2. Truth-functional propositional logic
    • Truth functions
    • semantics
    • consistency
    • deductive systems, the deduction theorem
    • semantic completeness
    • decidability
  • 3. First order predicate logic
    • A formal language
    • Semantics, satisfaction
    • model-theoretic metatheorems
    • a deductive apparatus
    • consistency
    • metatheorems
    • first-order theories
    • metatheorems:Lowenheim-Skolem, compactness
    • semantic completeness
    • nonstandard models, categoricity
    • philosphical implications
  • 4. First order logic - undecidability
    • Church's thesis
    • recursive functions
    • Representation of functions and definability of functions in formal systems
    • a formal system of arithmetic
    • proof of undecidability
    • prenex normal form, Skolem normal form
If I presented this outline to a colleague and asked them what sort of course would cover these topics, the answer would be an introductory course in mathematical logic. It appears to me that, despite the terminology employed in various parts of the book, its main purpose is to present the basic results of mathematical logic from a different perspective. I see no explanation in the book of how the results differ from metamathematics or mathematical logic, which is odd since the author makes a clear effort to avoid these words in the text. I haven't looked at other authors' use of the term, but at present I see no significant difference between the material presented in this book and the material ordinarily considered elementary mathematical logic. — Carl (CBM · talk) 01:31, 16 January 2008 (UTC)
Carl, I am a little disappointed in your recent reword of the lede. Do you really think "the theory of those formal languages that for one reason or another matter to the logician." is a more satisfying definition? It depends on the interests of certain people?! The previous formulation including the hyphenated words was certainly a more precise and rigorous account. I pieced it together from the text so as to make sense, rather than use a quote that falls way short. I am truly puzzled. Please reconsider.
I think this may be a result of a kind of a view of philosophy as some kind of namby-pamby effort. The view that logic is more about "correctness of arguments" than about the actual abstract concepts misses the more important thing going on... it's like we are trying to out-abstract each other or something. I think this characterization of logic (being a part of that namby-pamby philosophy) really relegates it to analyzing literature or rhetoric (or perhaps something even stuffier like philology?) or something fluffy like that. Logic is more about following reason. There are patterns in reason that we call principles. Logic is about identifying them. Arguments are far removed a subject matter within logic from the abstract concepts behind them. Pontiff Greg Bard (talk) 02:27, 16 January 2008 (UTC)
I found a different reference that I believe expresses the same idea as Hunter's preface, but more clearly, and added that to the lede. I also added what I think it a slightly more clear definition by Hunter himself. After looking through it, I don't see Hunter's book as particularly precise in several areas, one of which is the definition of metalogic, but I didn't want to unilaterally remove it as a reference. As I said above, I think the term is indeed established in the literature. The hyphenated phrases by Hunter, as I said above, seem tautologous rather than clear. I don't understand what you are arguing for in your second paragraph.
You can look at the other reference on google books.[1]. It's pretty common to define logic as the study of correct inference, which likely influenced my paraphrase. — Carl (CBM · talk) —Preceding comment was added at 03:04, 16 January 2008 (UTC)
I was just about to go in and remove the hyphens in favor of italicization. Thanks Carl. The whole thing looks wonderful now (although it is still a stub). What's next? Pontiff Greg Bard (talk) 04:01, 16 January 2008 (UTC)
I am planning to look up the other reference in the article, to see if it has any deeper discussion of metalogic that could be used here. — Carl (CBM · talk) 15:23, 16 January 2008 (UTC)
RE: "the author makes a clear effort to avoid these words in the text". Is this true, or is it that "mathematical logic" wasn't standard usage as a term when the book was written? I say this having studied mathematical logic in a math course, as well as having taken a philosophy class titled "Symbolic Logic" - where the professor said on the first day that the course would be more accurately described as "Mathematical Logic". Go figure. Anyway, our textbooks were, {drumroll}, Metalogic by Hunter, and Computability and Logic by Boolos. Tparameter (talk) 00:40, 19 January 2008 (UTC)
I would tend to agree with the sentiment given by your professor, that the material Hunter covers is typically called mathematical logic. Every author has a viewpoint. Hunter says in his preface,
My main aim is to make accessible to readers without any specialist training in mathematics, and with only an elementary knowledge of modern logic, complete proofs of metatheorems of standard (i.e.basically truth-functional) first-order logic, including a complete proof of the undecidability of a system of first-order predicate logic with identity.
and later,
I hope that this book will equip those who are not mathematical specialists not only to tackle more advanced works on standard logic, such as Kleene or Mendelson or Shoenfield or Smullyan, but to frame philosophically interesting systems of non-standard logic and to prove metatheorems about them.
That latter topic, about non-standard logic, does not seem to be covered in detail, based on my reading of the full table of contents. What is covered in detail is a first course in mathematical logic, occasionally with nonstandard terminology. — Carl (CBM · talk) 14:03, 19 January 2008 (UTC)