Talk:Logic/archive-1

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Archive This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive-1 created Charles Stewart 07:44, 10 Nov 2004 (UTC)

Contents

Cut starts here

I took the liberty of removing some of the older entries from before 2004, and the ones that were irrelevent to the article, in part to try and make this page less than 32 kb. This page could benefit greatly from an explanation of how logic relates to artificial languages, predicate, sentential and modal logic as prime examples. This would also naturally involve a small digression on how logic relates to natural languages and linguistics in general, including say informal versus formal methods, and how natural deduction can be used to infer rules of inference.

Stuff

Why does compound sentence redirect to logic?

Seems to be a big mess of changing double redirects, I'll try to sort it out. Thanks for pointing that out —siroχo 04:52, Sep 8, 2004 (UTC)

Who invented truth tables, Charles Pierce or Wittgenstein? Who should get the credit?

Shouldn't this page include the basic logic relationships? Why don't you add them to truth tables, connective, and quantifier?

Also the logic of relations, or polyadic quantification, which Quine said is what brought serious stature to logic as a field. It might also be nice if there were a striking & concise way to characterize the difference between formal or mathematical logic, & algebra (as the theory of calculation), i.e., the heart of the difference between formal reasoning & calculation, as opposed to lists of respective subfields.

A Calculus Of Ignorance?

Thinking about statistics and the frequentist formulation of confidence intervals ( The population parameter is "fixed but unknown") leads me to wonder if anyone has ever put the idea that something in a mathematical problem can be "given to be unknown" on a rigorous logical footing. Suppose we have an a symbol p representing a a "quantity" in a mathematical problem, be it a statistical problem or otherwise. Some examples of thethings that we might be "given" about p are:

1. p is an unknown number.

2. p is a normally distributed random variable with mean 0 and std. dev.= 1

3. p is a normally distributed random variable with unknown mean and std. dev.= 1

4. p is a normally distributed random variable with unknown mean and unknown std. dev.

5. p is selected by a person who has flipped a fair coin. If the coin landed heads then he chose p to be an unknown but fixed number. Otherwise p was chosen from a normal distribution with mean 0 and std. dev. 1. How the coin landed is unknown. (i.e. There is a 0.5 probability that p is "fixed but unknown" )

Lacking such a calculus, I , nevertheless attempt the following deductions:

If we are given 1) alone then an example of a valid conclusion is "The probability that p = 23.2 is either 1 or 0" (this follows famous textbook examples from statistics). If we are given 2) alone then the previous conclusion is invalid. If we are given 3) alone then we may say "The probability that {the probability that p = 23.2 is either 0 or 1} is 0". If we are given 5) alone then I don't think the previous conclusion is valid.

I'm sure these examples suggest even more mind boggling ones that make the frequentist assumption look like simplicity itself. Has some obscure branch of logic worked all this out for us? (I'm not talking about Bayesian statistics, which involves adding additional assumptions.)

Stephen Tashiro

Domain Theory

Dana Scott's theory of domains should handle the above kinds of uncertainty to your satisfaction. The essential idea is that it is a kind of theory of types that is naturally related to a logic without the principle of excluded middle. Chalst 23:43, 11 Aug 2004 (UTC)

Law of identity

Should not the logic page include the Law of Identity as one of the three laws of classical logic? That is: A is A or A = A. It seems to me that this law of Identity is crucial to the application of the Law of Non-Contracdictories and the Law of Excluded Middle in determining the truth or error of a proposition. Afterall, Logic really is the mechanism by which we pursue truth and distinguish it from error.

Perhaps the three laws of logic could be more understandable to those who are not aquainted with them as follows:

Law of Identity: A is A, or A = A. Something (a proposition or a thing) is what it is. Example: An apple is an apple. However, all apples are not identical, and any particular apple is not any other particular apple. Stating the law is not without difficulty. For example, a paricular apple is only a single example within the broad spectrum of the class concept of "apple-ness." In this case, the law of identity is valid in making the detemination that, "this particular apple is an example of what is subsummed under the class concept of "apple-ness.""

Law of Non Contradiction: A is not non-A. Something is what it is and not what it is not. Example: Applying the Law of Identity, an apple is an apple. Then applying the Law of Non-Contradiction, an apple is not anything an apple is not. The law does not preclude the possibility that an apple could become something else, applesauce, for instance, but that an apple cannot be an apple and applesauce at the same time.

Law of Excluded Middle: A is B or non-B. Usually, when considering propositions, the Law of Excluded Middle is applied by stating that the proposition P is either true or false, with no third possibility (tertium quid) allowed. However, in keeping with our "apple" example in explaining the first two laws, we could state that, "an apple is either a fruit or a non-fruit (a vegetable, or an animal)"

It seems that the application of the laws as given in the examples above have a direct bearing on what the could be considered "significant speech" and in resolving the ever present problem of communicating propositions "meaningfully" by addressing the "one within the many."

The communication of a proposition "meaningfully" from one person to another is absolutely dependent upon the definitions of the terms in the minds of both. Since we've been using "apples" in our other examples, I'll stick with apples here. Unless both have the same class concept of what an apple is, then applying the Law of Non-Contradictories is meaningless. For the term "apple" to have meaning and relevance, both the author of the proposition and the one to whom the proposition is communicated must have agreement about what consitutes the class concept of "apple-ness." The more agreement within the class concept, the more "meaningfullness" being communicated.

If both the author of the proposition and the one to whom it is communicated do not have a significant alignment on Identify, then applying the Law of Non-Contraditories is of no value in determining the "truth" or "error" of the proposition and neither is the application of the Law of Excluded Middle.

I have removed this, which I don't understand, from the formal logic section.

In the calculus of classes, disjunction elimination and conjunction elimination are also invalid, for, from the statement that the class a \cap b is null, we can't conclude that one of the classes a and b is null; and from the statement that the sum a \cup b is equal to the whole we cannot conclude that either is equal to the whole. But these implications are true in propositional logic.

I don't see that it is consistent with the linked definitions, for one thing.

Charles Matthews 11:15, 28 Jan 2004 (UTC)

The proof of non-contradiction in the Aristotelian Logic section contains the premise "it is only the law of non-contradiction that prevents " can be" from necessarily becoming "is"). ". This needs to be elaborated upon. "can be" has no translation in propositional logic as far as I know.

As someone who studied mathematics at the university level, I think the statement copied below, and the "proofs" that follow do not prove that the laws of non-contradiction and the excluded middle aren't axioms in a formal mathematical system:

Some have considered classical logic to be just like a mathematical theory, and in particular the laws of non-contradiction and the excluded middle to be simply axioms of the theory, which have to be assumed without proof. In fact this is not so...

The reasoning in the "proofs" which follow that statement belong to something that I would characterize as meta-mathematics. That is the "proofs" implicity assume that you know what is meant by the atoms "true" and "false". But if you consider Aristotelian logic as a mathematical system then the atoms are only defined by their use in the two axioms. In a mathematical system there is no way to define such basic terms as "true" and "false" without the use of axioms.

Since the "proofs" given are interesting on philisophical level, I don't think erasing them would be warrented, but I was wondering if anyone had any ideas on how to make it clear that the "proofs" given don't satisfy actual mathematical rigor. millerc 21:14, 19 Mar 2004 (UTC)

Compuetr logic

User:152.163.252.131 added:

  • The basic definition for computer LOGIC is that it is the sequence of operations performed by hardware or software.

There is perhaps something useful to be said about computer logic (e.g. TTL), but this is simply confusing (and non-wikified) as a first paragraph. Bovlb 12:35, 2004 Apr 28 (UTC)

I'm just a curious beginner who found the logic article interesting, but would someone be kind enough to say whether Bill is a 'knight' or a 'knave' in example 3:

>Logician: Are you both knights? John: Yes or No. Logician: Are you both >knaves? John: Yes or No.

>Who is who?

I tried using the notation given for example 1, and went from

(J \wedge ((J \wedge B) \vee (J \wedge B)') \wedge ((J' \wedge B') \vee (J' \wedge B')')) \vee (J' \wedge ((J \wedge B) \vee (J \wedge B)') \wedge ((J' \wedge B') \vee (J' \wedge B')')') = \mbox{tautology}

to

J = tautology

which seemed sensible as John is obviously a knight and there doesn't seem to be any information about Bill? ..Help :)

I suppose a point I'm trying to raise, is how beginner friendly should the wiki be. Should any rational thinking newbie be able to understand any given topic in the wiki by following enough links? ..or would this make the wiki too verbose?

Page Goals and design

This article needs more careful editing.


  • For instance, predicate logic is a kind of formal logic (as well as propositional logic, temporal logic, Hoare logic, the calculus of constructions etc.) this isn't clear from the article and some at least cursory mention of this fact should be made.
  • The distinction between formal logic and mathematical logic needs clarification. Is it because it mathematics applied to logic (which I think is the answer) and in particular can include model theory and proof theory.
  • Aristotelian logic is important historically, and I am reasonably happy with essentially a link to the article of the same name which is informative. That probably could stay as it currently stands.

CSTAR 00:25, 5 Jun 2004 (UTC)

Yeah, this article has some good info but definately needs more editing. I think that the page itself should not be overly technical, which is the path it has taken recently. Besides serving as a definition and glorified disambiguation page, it should definately give a good background of logic, probably even its history.
  • Devote the page "Logic" itself to a few things.
    1. What logic is, definition (this should obviously be on the page)
    2. The general, various goals of logic (to explain to people why it exists)
    3. The recently made informal vs. formal clarification.(Could be part of the definitions section)
    4. The current links to the various types of logic with short descriptions (Helps to disambiguate clearly)
    5. The background of logic, and history
  • I think the History of logic section should be the real heart of the article, actually. It'll keep the article interesting to those browsing, and people just looking for a general idea of logic, and it'll also be a nice central place for information about logic in general. It also won't get in the way of people that were looking for somethign more technical, because they'll find one of the disambiguation links first. The history section will also be excellent to help allow this page to link to the important people and results of logic, Aristotle, Descartes, Frege, Godel, Turing, etc...
  • In general, This page should be the defining Wikipedia page on logic, and basically be able to take you wherever you want to go, while providing lots of interesting information on the subject
siroxo 23:56, Jun 10, 2004 (UTC)
I recently removed the History part out becuase the information did not fit well with what was there. I'm not sure I understand what you mean by the background of logic is. I think the reason to keep the history section out, other than a list of bulleted items and links, is that Historical questions and debates can be very difficult. This way we can safely relegate that debate to a different page. One thing I disliked with the previous versions of this article was the presence of too many topics, some redundant and some far too wordy.
We also need to decide on some philosophical issues: An important distinction is between deductive and inductive logic. This article I believe should be mainly deductive logic. Inductive logic is more complex. This is different in my view from the distinction between empirical knowledge and a-priori knowledge. I'm perfectly happy to entertain the idea that logic is emprirical but discussions of this kind should not be a part of this article.
CSTAR 01:22, 11 Jun 2004 (UTC)
Yeah, i saw that you took the history out, and that is certianly fine with me, since it didn't fit well. I'd advocate developing the history page more, in parallel with this page, and then perhaps reuniting them if it seems like a good idea.
I guess my main priority is to make sure this page remains useful beyond just a disambiguation and clarification page. It should be useful as those, but should provide some more interesting content about what logic is, how it relates to other subjects (esp. philosophy), and how it developed over time. Technical pieces should not be included, those should be relegated to whatever sub-topic they belong to, as they are for the most part. Logic is a broad topic, so this should not try to hard to explain any one piece of it, but should try hard to put these pieces together, and then link to them.
siroxo 03:58, Jun 11, 2004 (UTC)

I've created a possible outline for the logic page, I believe it can provide an excellent general idea of what logic is, with enough specifics to distinguish topics for the reader, and lead him/her to appropriate topics. I also belive it provides a broad enough description of logic that it has the nature of an encyclopedia article. Please critique and make suggestions!

  1. Intro (Above TOC)
    • Defintions
      • "Laymans" general definition (marked as a non-technical def)
      • A more techinical definition
    • General Study of logic (background stuff about relation to Philosophy, Math, Computer Science, etc) more detailed explanations follow, and exist in other articles.
  2. Clarifications
    • Formal vs. Informal logic
    • Deductive versus Inductive reasoning
  3. A short background of the history of logic with links into the history of logic as necessary, hopefully relating the following "logics"
  4. Types of logic (each has link to "Main article, as currently, with a short description)
    • Formalisms
      1. Propositional/Sentential
      2. Predicate/First Order
      3. Higher Order
      4. Mathematical
      5. Modal
    • Less formal (and informal) systems and topics (Many of these contain formal elements, but are not formalisms themselves)
      1. Aristotelian logic
      2. Philosophical logic
      3. Logic and Computation
      4. Multi Valued logic
      5. Logic and Computation
  5. See also (incorporate as many of these into above text as possible)

siroxo 15:03, Jun 12, 2004 (UTC)

I am in general agreement with the outline. However, my most cherished principle of writing is this:

Lo bueno si breve, dos veces bueno,

which I learned in my Spanish Jesuit high school, meaning The good if brief twice good. This principle is particularly important in an article on logic!

CSTAR 16:35, 12 Jun 2004 (UTC)

I've seen the page in a number of versions - none of them really good. I really think it suffers every time it gets closer to the kind of computer science thinking (roughly speaking, you can have any kind of formal system you want and call it the XYZ logic, if you think you have an application).

I'm not an expert, but the history tells one something. There was an old meaning of formal or symbolic logic (already in Kant?). In the nineteenth century logic was formalised first for pedagogic reasons (Boole, Lewis Carroll) - because it was still considered a pedagogic subject. Only with Frege did it really become once more a research area (after 2000 years). I don't believe there really was a subject called mathematical logic, before Gödel. The need to write what he had done in a way that could be understood made for recursion theory first, and then model theory, as disciplines within mathematics. So basically one can say that the incompleteness theorem directly caused those developments. For example, for the Polish school at its foundation, logic was not 'inside' mathematics at all; but a separate subject. And before that the philosophers were sorting out philosophical logic, i.e. parts of the logic of natural language that are difficult to formalise.

After 1950 it really is all less clear ...

Charles Matthews 18:50, 12 Jun 2004 (UTC)

Well I agree that specialized formal logics, particularly for reasoning about a specialized subjects such as concurrency or computer programs really should have a subordinate position in the article. I was trying to achieve that goal--- for this reason I introduced the section header highlighting the distinction between formal and informal logic. Apparently I didn't quite achieve that yet.

However, I would stress the following points:

  • Though history of logic is exceedingly important, it is also very specialized so that that this subject should be clearly delineated and preferably discussed somewhere else.
  • The logic page should have a practical purpose, for someone trying to distinguish between good and bad argument. In this regard the article on logical fallacy is extremely important. See the paragraph in the intro to David H. Fischer's book Historians Fallacies dealing with the argument that emphasizing fallacies in logic is explaining how to get from Boston to New York by describing the roads that don't go there. His response: It is useful to know places where one might get lost in this trajetory. (Having done this trip many times, this is very good advice to have, particular near NY)
  • The concept of a valid inference is important. This article currently stresses deductive validity as opposed to inductive validity. This is OK, in my view since induction is much more complicated and uses deductive validity
  • Mathematical logic as we currently understand it (as is correctly formulated in the article mathematical logic: application of mathematics to logic and inference) probably does originate with Godel. However, Peano, Poincare, Hilbert, Russell and Whitehead did refer to something called mathematical logic which today we might think of as sterile formalizations of mathematics. Symbolic logic is a term which is probably even older.

I noticed that the Routledge Encyclopedia of Philosophy does not have an entry on Logic pure and simple. It has lots of entries logic of XYZ.

CSTAR 20:26, 12 Jun 2004 (UTC)

Plan of action for the logic page

Is there one? CSTAR 16:14, 20 Jun 2004 (UTC)

  • I was busy the last couple weeks, but I have time to work on this page again. I suggest we start working on setting up a good introductory definition and make the clarifications very clear, then attack the rest of the page —siroχo 03:31, Jun 24, 2004 (UTC)
One of the things which is not clear from the introduction currently is the difference between inference and argument. Argument I guess is the linguistic or symbolic expression of inference. One can infer something (in one's head) without an argument.CSTAR 04:12, 24 Jun 2004 (UTC)

I suggest: traditional logic discussed in general terms, up to the time of Boole and Mill (on induction) when things started to shift. Then mathematical logic Frege up to about 1940 (when the structure of the subject had become reasonably clear). Philosophical logic had a definite meaning for Russell, which then moved somewhat; but this is still a key area for analytic philosophers.

Charles Matthews 07:21, 24 Jun 2004 (UTC)

I just reworked the intro a bit. Tried to make it flow better, as well as adding an idea or two. One note, I think the first sentance, as it is in the new version, is required. In common conversation, logic basically just refers to the reasoning used to reach a conclusion. I think we should allude to that, and then introduce the formal concept. This will even help to reinforce the formal view of logic to people who have not studied it, as well as easing people into the article.

I also resectionized the first couple sections, and dropped the deductive/inductive thing into there, I hope it fits better like that.

Regarding inference/argument, I'm not sure how to convey that in this page, but it might be an interesting addition.

—siroχo 09:08, Jun 24, 2004 (UTC)

I made some small changes to the first paragraph.

  • Replaced at its most basic with ordinary language. I think that's what we mean; although go ahead and revert if you disagree. I'm not trying to be ideological. I have a nasty habit of pondering every word, which means I never get very far.
  • I used inference instead of argument-- argument implies communication. I've spent some time working on the article logical argument where that distinction is made.
  • Corrected two typos (assumption was mispelled)

CSTAR 14:27, 24 Jun 2004 (UTC)

  • Good points all around, i like the ordinary language especially —siroχo 19:54, Jun 24, 2004 (UTC)

The most recent edit (Lorenzo Martelli) tries to clarify the situation in which the assumptions are inconsistent --- is this really necessary? If they are inconsistent, the premises will never hold and by the meaning of material impication, validity is vacuously true. Mainly I don't think this should be put into the introduction, maybe some other place. Otherwise the introduction is very rapidly going to get unwiedly. CSTAR 15:36, 24 Jun 2004 (UTC)

Reply from Lorenzo: I agree in general with your comments CSTAR. Maybe then we should just leave in the qualification 'for most intents and purposes'? What I think would be wrong would be to have a statement beginning 'Validity means .....' in the introduction to an article on Logic, if that statement is not 100% accurate. After all, half of the reason behind formal logic was to get rid of the ambiguities...!I also think we should put somewhere in the article a strict definition of validity, of the form 'an argument is valid if and only if...' to ensure it is clear and referenceable, and that all that follows (the paradox of entailment etc, why truth tables are so important)can be explained through/referred back to the definition. Hope you agree and if so feel free to move my comments to a more appropriate place.

Validity is important enough to define on the page without going into the detail of the validity article. Where it is right now seems quite appropriate, as a reader interested mainly in validiity will click the link in the intro, and a reader interested in logic will still get a chance to see basically what it means. Now we shoudl figure out how to approach all these types of logic. I like having small introductions with "Main article" links. It helps to give a deeper background of what logic is, and to disambiguate, but we need to organize them appropriately and give them a good introduction explaingin why they are divided such —siroχo 19:54, Jun 24, 2004 (UTC)
I think one way one can argue that logic is an empirical science is that various notions of validity are models of an ideal notion of validity. This isn't too hard to argue for inductive validity and definiong deductive validity in some semantic way is also an approximation. I agree these should be placed in the article, but not in paragraph 1.CSTAR 20:09, 24 Jun 2004 (UTC)

Specialized logics

Could we collapse the sections on Multi-valued logic and Logic in Computer science into a single section, say specialized logics. Also I prefer to use inference instead of reasoning in the section currently titled deductive and inductive reasoning. For example. inference could be procedural, i.e., apply Matlab to infer something. I think also a sentence or two on the distinction (and also relations) between inference and logical argument is necessary. CSTAR 16:07, 1 Jul 2004 (UTC)

  • I made minor expansions to multi-valued logic, but I believe that since several major concepts in logic, and related concepts are founded in multi-valued logic, it should have its own section. I divided otu the CS/AI section from what I now call "Types of logic" (needs a better name), and I think it sits well as a separate idea. Also, IMHO, you should feel free to make any changes regarding inference/reasoning/logical argument. If they are confusing, just make an explanation. The page will evolve to incorporate them even better, as well. —siroχo 11:40, Jul 5, 2004 (UTC)

Possible revision of first paragraph

I know we have beat this subject to death, but I am still unhappy with some of our characterizations of logic given in the 1st paragraph, particularly, since in the broad sense that logic traditionally has had in philosophy, it is concerned with the structure of inference. Thus I propose a change starting at the second sentence:

More formally, logic is concerned with inference, that is, the process whereby new assertions are produced from already established ones. Of particular concern in logic is first, the structure of inference, that is the formal relations between the the newly produced assertions and the previously established ones (formal, in the sense of being independent of the assertions themselves) and second the investigation of validity of inference, including various possible definitions of validity and practical conditions for its determination. It is thus seen that logic plays an important role in epistemology, that is it provides a mechanism for extension of knowledge.

CSTAR 04:33, 31 Jul 2004 (UTC)

Excellent, worth beating to death if it can be clarified better (; Yours is much clearer about the general idea of formal logic. I've repasted it here, with a couple grammatical changes I suggest for clarity.
More formally, logic is concerned with inference—the process whereby new assertions are produced from already established ones. As such, of particular concern in logic is the structure of inference—the formal relations between the the newly produced assertions and the previously established ones. Where formal relations are independent of the assertions themselves. Just as important is the investigation of validity of inference, including various possible definitions of validity and practical conditions for its determination. It is thus seen that logic plays an important role in epistemology in that it provides a mechanism for extension of knowledge.
—siroχo 10:28, Jul 31, 2004 (UTC)

Dialectic and rhetoric

I'm not sure I agree with the opposition of rhetoric (and dialectic) to logic. The relation between the is complex, that's clear, but to say it is one opposition is also misleading.CSTAR 14:11, 12 Aug 2004 (UTC)

Quite so, and a brief discussion of the Organon should make this clear ---- Charles Stewart 15:44, 14 Aug 2004 (UTC)

Missing core topics

  • Central tools of logicians: proof theory, model theory, and possibly more sophistcatedly mathematical tools such as set theory, recursion theory and category theory.
  • Controversies in logic: eg. intuitionism, relevantism.
  • The scope of logic: non-deductive inference forms, eg. induction and abduction (shouldn't say too much here, hive off on an AI or knowledge representation page).
  • Modern research innovations: substructural logics, hybrid logics, non-commutative logics.
  • Quantification: relate to problem of multiple generality in Aristotelian logic.

Any problem with me charging ahead and adding these? --- Chalst 23:55, 11 Aug 2004 (UTC)

Keep in mind there are (or at least should be) specialized articles on all of these. There are certainly articles on quantification, model theory, proof theory, set theory, recursion theory the subjects you mention. I suggest you poke around first. ALso this logic article took a long time to stabilized to its current form. CSTAR 00:12, 12 Aug 2004 (UTC)
Most have topics, some do not. I think all of these are floating near enough th e top that a good overview should at least point to the topics in an informative way. Also, I think a weakness of the article as it stands is that the place of logic is not as clearly defined as it should be (the Organon took a central place in Aristotelianism), and also some mention should be made of in virtue of what do logicians think logic to be true. I think I can avoid vandalising the page. nCharles Stewart 01:20, 12 Aug 2004 (UTC)
Well good luck! CSTAR 04:20, 12 Aug 2004 (UTC)


Intuitionism and relevantism probably deserve a mention. I'm not too familiar with the modern research you allude to. Everything else i'm not sure yet, might as well add some stuff and we'll see where it goes. Just remember that this is not the main article on any of those topics. —siroχo 01:13, Aug 12, 2004 (UTC)

I've created two short articles semantics of logic and proof-theoretic semantics (among others) prior to editing the Logic page proper. Close readers may notice that there are some disagreements of fact between those articles and this one. Comments welcome. ---- Charles Stewart 23:35, 18 Aug 2004 (UTC)

Although my own preference is not to regard proof-theoretic semantics as semantics, you are right to include them both. However, I think it would be far too narrow to exclude informal logic within the broadest scope of logic. I for one am not sure where to draw the line between logic and non-logic; For this reason I prefer to think of logic as the structure of inference and as a particular case the structure of argument. For instance Structure of argumentative dialogue can be (and recently has been) formalized as a communication protocol where messages sent or received by a protocol participant depend on current commitments and pending challenges. Also structure of argument is important in just about any field of endeavor -- economics and political science come to mind. The play between formal models and informal reasoning is intriguing.
None of this is intended to dissuade you from including those articles in the logic page. They are well-written, but I don't think it would be accurate to entirely supplant logic with this formalized idealization.CSTAR 02:30, 19 Aug 2004 (UTC)
Thanks for the comments. These other pages are meant to stand on their own, they are really stubs and need much more development, and I plan on putting shorter summaries of the topics on the main logic page. Regarding informal logic: I am a partisan on this issue -- the opening sentence of my doctoral thesis runs "Logic is the study of arguments correct in virtue of their form", and the first footnote says, in effect, that the term informal logic is perhaps a misnomer -- it is the study of arguments whose form is concealed in the deep structure of language. ---- Charles Stewart 09:43, 19 Aug 2004 (UTC)
That view of informal logic is fine by me; I just try to be careful not to fall into naive reductionism --- that natural language reasoning is either meaningless or reducible to reasoning in a formal sequent calculus (I tend to fall into this, being a mathematician ). Maybe you should have at the logic page, and while your at it, also the logical argument page. I also started a political argument page which is in need of serious attention. CSTAR 21:30, 19 Aug 2004 (UTC)

Types of Logic

I don't like the current organisation of this section. It's possible to organise logic by its formal type, when one gets a list along the lines of:

  • Syllogistic
  • Stoic (propositional)
  • Boolean
  • Predicate
  • Other

Or one can organise it by motivation/application, and get:

  • Philosophical
  • Dialectical (Aristotle's motivation for the Organon was as a tool for reasoning and argument)
  • Mathematical
  • Computational

Mixing the two axes, as is done now, though, seems like a recipe for confusion. The first list seems more important to me.

Furthermore, "Multi-valued logic" as a category of logic, is just wrong. It's an issue in the semantics of logic: one can give multi-valued semantics to FOL, and there are non-bivalent logics, such as intuitionistic logic, that are not commonly treated as being truth-valued at all. The existence of multi-valued logics is maybe appropriate in a semantics section, or in the section I proposed earlier on controversies in logic.

If there are no objections, I'm planning on applying a series of changes in the next few days, starting from tomorrow evening ---- Charles Stewart 14:31, 17 Aug 2004 (UTC)

I agree that Multi-valued logic as a category is wrong, although the preferability of one "axis" to another is not clear to me - yet; I also think that the distinction currently made between "informal logic" and "formal logics" at the beginning of the article is useful.CSTAR 14:44, 17 Aug 2004 (UTC)
Since some of the changes I am thinking of applying will be controversial, would it be a good idea to make a mock-up in a page called, say, Sandbox/logic? ---- Charles Stewart 16:14, 17 Aug 2004 (UTC)
I suppose it doesn't really matter, since one can always revert. CSTAR 21:09, 17 Aug 2004 (UTC)

Big changes applied

Now applied the restructring I've been threatening... Two issues:

Aristotelian logic is sometimes referred to as formal logic because it specifically deals with forms of reasoning, but is not formal in the sense we use it here or as is common in current usage. It can be considered as a precursor to formal logic.

  • I've deleted this, because I think it is wrong: it is perfectly easy to see how syllogistic is formal, c.f. the work of John Corcoran expressing syllogistic in modern notation. Do any authorities seriously dispute the formal nature of syllogistic? I would be surprised. I've completely rewritten these paragraphs.

Mathematical logic refers to two distinct areas of research: The first, primarily of historical interest, is the use of formal logic to study mathematical reasoning

Not at all! The application of logic to the study of mathematics and mathematical proofs is still one of the main driving forces of mathemcatical logic. Just look at applied model theory, reverse mathematics, Borel spaces. I've completely rewritten this paragraph.
I obect! I wrote that sentence! By that I meant the following: once it became clear (in historical time possibly in the late 1920s but I'm guessing) that informal arguments in mathematics can be reduced to purely formal proofs, hardly anyone regarded purely formal proofs as useful (caveat of course: formal verification does stress formal proof and it is useful, I suppose for many critical applications, but it is highly debatable whether formal proof-checking is mathematics at all! Despite MIZAR and Isabelle, in the current state of the mathematical sciences it would be misleading, I think, to regard this area is of any interest outside of a relatively small group of specialists. This of course is a sociological fact more than anything else, but in fact the assertion in question is also sociological). Notice that I stated
use of formal logic to study mathematical reasoning
The area of research you describe in Borel spaces or model theory (which I think would include non-standard analysis) could be called formal logic only by a stretch, I think. In other words, though logic may occur in the subject of these investigations, logic is itself regarded as an object of study (e.g., somewhat fancifully, say one could regard formal structures as a kind of space on which a group acts such as a Tits building or some such thing -- and actually as you point out above somewhere domain theory is an area where the mathematization of formal systems becomes more visible ). CSTAR 02:55, 26 Aug 2004 (UTC)
I guess it's possible to argue that both what I changed this section from and what I changed it to are not NPOVs. I hope not, because I like what I wrote. OK, quibbles with what you wrote first: all the stuff about Borel sets, analytic sets, projective sets did change the systems that set theorists studied, since they gave rise to new large cardinal axioms via the work on descriptive set theory. By applied model theory, I mean the stuff that MacIntyre was doing -- applying model theoretic techniques to algebra to get new mathematical results algebraists think are valuable; it really couldn't be a clearer application of logic to mathematics; whilst in reverse mathematics, you need to really go back and think about how you prove things one is really recreating mathematics according to logically imposed constraints. With Macintyre's work, I think it couldn't be clearer that the objectives are mathematical ones (indeed, see [1] why this might not be logic anymore...), and withthe other two, while we may have foundational motivations for studying them, it should be clear that the objects of study themselves are mathematical. I think from this point of view, what I wrote is OK (do I need to say more so that this is clear?)
There is another issue you raise, namely that few logicians (the applied model theorists being exceptions) are genuinely close to mathematical practice. In fact I'm reading a nice book by David Corfield at the moment "Towards a Philosophy of Real Mathematics", and it is abundantly clear that he would agree with you. When a logician studies mathematics, he learns something, but not something that is of much use to the mathematician, and this is something that the pioneers of mathematical logic did not expect.
Resolution: maybe we need to divide up mathematical logic into more kinds of activity: both foundations of mathematics and applied logic would fall under the use of formal logic to study mathematical reasoning, and maybe we need to make clear that these are two different kinds of activity. ---- Charles Stewart 10:01, 26 Aug 2004 (UTC)
Do I have the right indentation here?... :)
Yes I generally I am very happy with what you wrote -- though I think the point I raise is more than a minor quibble. In applying logic (i.e. model theory) to mathematics in the way you mention is hardly different than noting that the transfer principle is useful in translating certain nonstandard statements to standard ones. I use this equivalence just because I am very familiar with non-standard analysis. Now where does logic begin or end in non-standard analysis? This is the perennial question, what are the limits of pure logic? I certainly don't know.
But my main point is that the study of logic and logical systems is more like the emprirical study of physics: There are these systems of inference, we propose models for them and study them using mathematics. I believe a very weak version of this in that I am not a Putnamist on this, i.e., I am not prepared to argue as does Putnam that logic is empirical. See Quantum logic.
BTW the logic page as it now stands is of very high quality.CSTAR 13:49, 26 Aug 2004 (UTC)

Comments welcome on all changes ---- Charles Stewart 00:36, 26 Aug 2004 (UTC)

I merged the "types of logic" and "motivations for logic" sections, because it was confusing to have types split between the two. The specific motivations are inherent to what type of logic being considered, though so I think they go well together this way. Perhaps a little general idea of motivation could be placed in the introductory secont above the TOC if you wish. [[User:Siroxo|—siroχo

—siroχo]] 05:03, Aug 26, 2004 (UTC)

Hmmm. I separated them out because it was clear to me that they were different kinds of subsection: in particular, I don't think that, say, mathematical logic is a type of logic, but rather a set of interests and concerns that certain kinds of logician bring to whatever branch of logic they study. Philosophical logicians and mathematical logicians study one and the same first-order logic, and they study one and the same S4 modal logical system. Instead, they just ask different questions about them, and do different kinds of investigations into them. I agree, though, that "motivations for logic" was a clumsy section heading. Maybe "Types of logician" would be better...
On rereading what I wrote, the preamble to the types of logic section is not consistent with the contents, and so needs to be rewritten anyway.
And thanks for the cleanup work with the all the red I left on the pages, and sorry there was so much: I actually 'corrected' many of these links, but corrected them to new broken links. ---- Charles Stewart 06:32, 26 Aug 2004 (UTC)
You make a good point regarding "types" of logic and all. I changed the name to "paradigms of logic" in hopes that it would imply a more general idea that these were various ideas that different logicians followed. [[User:Siroxo|—siroχo

—siroχo]] 07:48, Aug 26, 2004 (UTC)

Mathematical logic

The article currently states:

Both the statement of Hilbert's Program and it's refutation by Gödel depended upon their work establishing the second area of mathematical logic, the application of mathematics to logic in the form of proof theory.


Is model theory usually considered part of proof theory?

No, but in context, Hilbert's program centred on the role of consistency statements, which are considered a part of proof theory. Goedel's completeness theorem states a key result in model theory, but the machinery he used to prove it is proof-theoretic (the original theorem used esentially a proof search algorithm).

If so that's news to me. Model theory certainly is mathematics applied to logic.

If proof theory and model theory have been the foundation of mathematical logic, they have been but two of the four pillars of the subject. Set theory originated in the study of the infinite by Cantor, and it has been the source of many of the most challenging and important issues in mathematical logic, from Cantor's theorem, through the status of the Axiom of Choice and the question of the independence of the continuum hypothesis, to the modern debate on large cardinal axioms.

These seem like fairly subjective evaluations. Are we sure it's four pillars and say not...sixty five pillars? (I'm being a little facetious here purposely to spice up the discussion) . I concede that these boundaries are really murky, but is set theory really logic?

Indeed, it is subjective, so maybe a remark to that end is in order. But it's a pretty standard way of dividing up the field since Barwise's 1977 Handbook of Math. logic.
Ok that is an important fact and probably should be stated here (or in some referenced article).CSTAR 15:33, 29 Aug 2004 (UTC)

And I think we should be less harsh on the logicist programme. Saying it was a failure really gives an erroneous impression. The logicist program did succeed in one way: As a result of the logicist programme it became abundantly clear that mathematical reasoning is in principle reducible to a formal calculus (if anybody has enough time or funding on their hands to do it)

It failed according to its own criteria, but as I said, there was a silver lining ---- Charles Stewart 15:02, 29 Aug 2004 (UTC)
P.S. Actually as I said earlier I like very much what you added, it's just that I'm not sure the limits of mathematics and logic are adequately discussed (and I don't claim to know the answer either)CSTAR 04:52, 28 Aug 2004 (UTC)

Dialectics and rhetoric

I think the article still gives short-shrift to Dialectic and rhetoric. It ignores the work of Chaim Perelman and Tyteca-Olbrechts on argument. This is briefly mentioned in logical argument.

I've put a comment on the talk page of that article. There is a section, so far with no body, that I created, entitled dialectical logic, which seems the natural home for the work of Perelman & Tyteca-Olbrechts as you have described them. ---- Charles Stewart 13:07, 30 Aug 2004 (UTC)