Talk:First-order logic/Archive 1

message: In adding a title at the beginning of Talk:First-order predicate calculus (section), I ended up with an extra edit link, one to this part (which was blank).  I am making it not be blank, and perhaps this is a good place to capture some ideas about overall organization of the article (and it's Talk).--Orcmid 16:51, 20 Mar 2004 (UTC)
thread_title: FPC

message: I've trawled Wikipedia, and I still don't know what a predicate is. Well, actually, I'm almost sure that I do, but no thanks to this article. I'm only beginning to really grok this stuff, and not to have a clear, concise definition of 'predicate' in any of the pages dealing with predicate logic (no matter what you call it) is soooooooooooooooooo FRUSTRATING. Go on, I know you don't believe me, it has to be defined somewhere, right? Nope.
:This is a bona-fide hole in the logic pages.  I'll put starting this article on my tasks list. --- Charles Stewart 15:37, 27 October 2005 (UTC)
thread_title:  What is a Predicate 

message: The use of the letters of the Greek alphabet is not defined. I must assume they are propositions from the propositional calculus (and stand for complete propositions), whereas the remainder of the grammar defines the notation for the new constants, variables and functions. Could someone verify this?--[]
:An emerging convention is that Greek letters are metalogical devices whose first job is to reduce (not eliminate--nothing can achieve that!) use-mention confusions. Once this is established, you don't need Quine corners or to enclose all sorts of expressions in fastidous single quotes. If the entry intends this usage, it should say so explicitly. A recent text that makes heavy metalogical use of Greek letters is Bostock (1997).202.36.179.65 20:20, 31 March 2006 (UTC)
thread_title: Grammar

message: I just changed some of the notation used in this entry and I feel I should justify myself.  I changed two things.
Previous version:
:∀ x ∀ y : (x < y) ⇒ (y > x)
:∀ x ∃ y : y > x
Current version:
:∀ x : ∀ y : (x < y) ⇒ (y > x)
:∀ x : ∃ y : y > x
A mathematician would say I added extra colons.  A grammatician would say the previous version elided some colons.
Omitting the extra colons is very common practice, but considering the audience I think the more explicit notation is better.  The punctuation is a nice visual cue to the structure of the language. It suggests pluggability.  Also, in some cases, not having the colon can lead to serious semantic mistakes when trying to translate the symbols into English (see below).
Hofstadter never elides colons in GEB, which is written for raw beginners.
:In any standard textbook on (mathematical) logic, colons are not part of the language and so they should not be used in formulas. The only time colons and dots are used is as a replacement for parentheses. The colons look ridiculous---they do not add anything to the (unique) readability of formulas. Nortexoid 19:16, 24 December 2005 (UTC)
replies: text: I certainly have no problem with that; I just used the notation I found.  My favorite approach is (\forall x.\phi) and (\exists x.\phi); I think that scope is so important that explicit parentheses are justified.  Randall Holmes 22:27, 3 January 2006 (UTC)
replies: text: I would suggest (\forall x)\phi and (\forall x)(A(x_1,\dots ,x_n)\dots) to exclude any symbols that are not part of the language. If we use parentheses, dots and colons aren't needed. Nortexoid 01:02, 5 January 2006 (UTC)

text: I'm not about to engage in an editing war about this, but the definition of equality which I gave as an example would benefit from a dot after the quantifier; the repeated P is distracting.  As I said, in my own writing I never omit parentheses or an intermediate dot where quantifiers are concerned, though this is a matter of taste... Randall Holmes 03:29, 5 January 2006 (UTC)
replies: text: For improved readability simply place each quantifier and associated variable in parentheses, and enclose the matrix in parentheses as well--e.g. (\forall x)(\forall y)(x=y). This is probably the most common practice (in conjunction with using brackets [,], and braces {, }). It's not a big deal but I'm opposed to using notation within the object language that is not officially part of the object language. I also prefer to keep the primitive notation at a minimum. Nortexoid 05:36, 5 January 2006 (UTC)

text: That works (the result is readable) but it should be noted that there is no strictly correct notation here:  there are variants which appear in the published literature.  For example, referees read my (\forall x.\phi) without difficulty (and that is how I specify a well-formed universal quantifier sentence when I present a grammar for first-order logic, which I do from time to time).  All that is necessary is that the grammar in use in any given text be clearly specified (particularly if it is not standard).  That said, I have no objection to Nortexoid's favored grammar -- particularly given permission to add parentheses for clarity.  Randall Holmes 06:19, 5 January 2006 (UTC)
thread_title: Notation

message: Previous version of article:
:∃ y ∀ x : y > x
:(ie: thereExists y forAll x suchThat y greaterThan x )
Current version:
:∃ y : ∀ x : y > x
:(ie: thereExists y suchThat forAll x, y isGreaterThan x )
Previous version:
# ∀ x : x * 0=0
#* ie: forAll x suchThat x * 0=0
Current version:
# ∀ x : x * 0=0
#* ie: forAll x, x * 0=0
In both cases, I think the previous version is clearly broken.  In English, "such that" only makes sense after "there exists".  It makes no sense after "for all".
So the English-speaking mind parses "thereExists y forAll x suchThat y greaterThan x" as something like "There exists some y, for any given x, such that y is greater than x."  In symbols, ∀ x : ∃ y : y>x.
(I guess it could be argued that the problem is trying to transliterate math to English symbol-by-symbol in the first place, but I'll pass on that argument.)
-- Jorend
====History====
When did predicate calculus first appear?
:It's "in" Principia but buried under an avalanche of symbolism and other notions, such as the theory of types. A full awareness of the centrality of FOL had to await Principles of Theoretical Logic  by Hilbert and Wilhelm Ackermann (1928). The predicate calculus is very clear in Willard Quine's 1940 Mathematical Logic and Alonzo Church's 1944 Introduction to Mathematical Logic.
That FOL was very important was evident as soon as it became clear that ZFC, Peano arithmetic, and nearly all algebraic structures could be formulated as first order theories (if FOL admits axiom schemata), although I am unsure how this realization came about. Tarski formulated high school algebra and Euclidean geometry as first order theories around 1930, although did not publish this result until 1948.
Who first devised/used it?
:Frege in his 1879 Begriffsschrift, which did not clearly demarkate first from higher order logic.
:In complete ignorance of Frege, Charles Peirce's student Oscar Mitchell and Peirce himself, discovered quantification 1882-85. This work culminated in a classic 1885 article by Peirce alone, which does demarcate first from higher order logic. Frege's notation is utterly weird and was never taken up by any one else; the Peirce-Mitchell notation closely resembles that used today, and was that of Leopold Loewenheim, Skolem, many Polish logicians, and even Kurt Goedel (1931).
Why?
:To formalize the foundations of mathematics. That predicate letters and quantified variables might be a powerful way of reasoning about natural languages emerged in the work of Carnap, Henry Hiz, Noam Chomsky, and Chomsky's teachers, Zellig Harris and Richard Milton Martin.
What were its precursors?
:Frege's precursors are poorly understood; see Hans Sluga's 1980 monograph. There is also no biography of Frege. Peirce's road to quantification, on the other hand, has been scrutinized; see Geraldine Brady's 2000 monograph, who highlights, among other things, Peirce's fascination with Augustus DeMorgan's pioneering work on relations.202.36.179.65 09:28, 1 April 2006 (UTC)
thread_title: Transliteration

message: When did predicate calculus first appear?
:It's "in" Principia but buried under an avalanche of symbolism and other notions, such as the theory of types. A full awareness of the centrality of FOL had to await Principles of Theoretical Logic  by Hilbert and Wilhelm Ackermann (1928). The predicate calculus is very clear in Willard Quine's 1940 Mathematical Logic and Alonzo Church's 1944 Introduction to Mathematical Logic.
That FOL was very important was evident as soon as it became clear that ZFC, Peano arithmetic, and nearly all algebraic structures could be formulated as first order theories (if FOL admits axiom schemata), although I am unsure how this realization came about. Tarski formulated high school algebra and Euclidean geometry as first order theories around 1930, although did not publish this result until 1948.
Who first devised/used it?
:Frege in his 1879 Begriffsschrift, which did not clearly demarkate first from higher order logic.
:In complete ignorance of Frege, Charles Peirce's student Oscar Mitchell and Peirce himself, discovered quantification 1882-85. This work culminated in a classic 1885 article by Peirce alone, which does demarcate first from higher order logic. Frege's notation is utterly weird and was never taken up by any one else; the Peirce-Mitchell notation closely resembles that used today, and was that of Leopold Loewenheim, Skolem, many Polish logicians, and even Kurt Goedel (1931).
Why?
:To formalize the foundations of mathematics. That predicate letters and quantified variables might be a powerful way of reasoning about natural languages emerged in the work of Carnap, Henry Hiz, Noam Chomsky, and Chomsky's teachers, Zellig Harris and Richard Milton Martin.
What were its precursors?
:Frege's precursors are poorly understood; see Hans Sluga's 1980 monograph. There is also no biography of Frege. Peirce's road to quantification, on the other hand, has been scrutinized; see Geraldine Brady's 2000 monograph, who highlights, among other things, Peirce's fascination with Augustus DeMorgan's pioneering work on relations.202.36.179.65 09:28, 1 April 2006 (UTC)
thread_title: History

message: This is symantec, but important to help garner respect for Wikipedia: modern logic uses →, rather than ⇒ as its conditional within WFFs.
:Not true, both forms are widely used.  The most established symbol for implication is still the most used by philosophers:
::\supset
:It would be good to have a default standard specifying symbols for logical connectives. ---- Charles Stewart 17:45, 15 Sep 2004 (UTC)
replies: text: This is a bit of an oversimplification.  For example, in categorical logic, "=>" is used for functional notation, because "->" is normally used to express morphisms.  Using arrows to express function spaces is not universal; often superscripts are used (because of the parallelism with exponentiation). --- Charles Stewart(talk) 19:03, 9 February 2006 (UTC)
thread_title: Conditional

message: A recent post by Sandy Hodges on the FOM list  points out that the way first-order calculus is specified on this page is wrong for almost all commonly used formalisations of FOL, because the rule of  generalisation, simply read, does not take theorems to theorems. The rule needs correction.  I plan to have a go in the next few days. ---- Charles Stewart 17:53, 15 Sep 2004 (UTC)
thread_title: Definition of first-order calculus is wrong

message: :(first-order logic) is a stronger theory than sentential logic
This statement is true only if we assume that sentential logic is zeroeth order; this ignores the existence of second-order propositional logic, which is a much stronger theory than first-order predicate logic.
I think that we need a more careful and consistent terminology across the mathematical logic pages when referring to logical systems and calculi, and also we need a common place to talk about the many conventions we have in the logic pages.  I mentioned the idea of a Wikiproject for logic in the Talk:Logic pages, but received a less than heartfelt agreement.  I'd like to propose the idea again. ---- Charles Stewart 06:28, 13 Oct 2004 (UTC)
replies: text: I've the beginnings of a User:Chalst/WikiProject Logic proposal; comments and suggestions welcome. ---- Charles Stewart 07:29, 13 Oct 2004 (UTC)
thread_title: Refining terminology

message: I think you guys are doing this page a serious disservice in using only mathematical examples.  Maybe you should have a separate page for predicate logic and a seperate page for first-order predicate calculas, the first can be more the philosophy of logic and the latter can be used for math.  Or maybe you can just used the current page and expand the scope quite a bit with more standard, non-mathematical, examples.
Thanks for allowing my comment.
:I think quantification is the right place for more philosophical discussion of predicate logic.  I think, though, that predicate logic should be a more superficial page mostly concerned with linking to the more technically formal and/or philosophical pages, especially since FOL is not the only predicate logic.
:Since the name has come up, shouldn't this page be called First-order logic, on the grounds that (i) this and its abbreviation FOL are the terms I most commonly encounter for the logic, and (ii) FOPC is a misnomer in any case, since there is no consensus what"calculus" should be associated with the consequence relation, the Hilbert-Ackermann formulation being only the most commonly cited. ---- Charles Stewart 11:54, 13 Feb 2005 (UTC)
replies: text: Yes, it should be moved to First-order logic. Nortexoid 00:54, 22 Mar 2005 (UTC)
thread_title:  Why is this page so math oriented? 

message: The axioms are non-logical -- i.e., they are not valid in the predicate calculus (hence the name 'non-logical'). PA is just one first-order theory and should not be included in this article at all. Link it. I have a very strong urge to remove it, but I will wait for some comments. Nortexoid 08:44, 23 Mar 2005 (UTC)
replies: text: Very much agreed.202.36.179.65 18:55, 30 March 2006 (UTC)
thread_title: Peano Axioms?

message: I'll just dive in:
#The axiomatisation of arithmetic, and hence the Peano rules is of particular interest to FOL, because the intended theory is usually understood to be beyond the ability of FOL to axiomatise (depends on how your read the incompleteness theorems), its the simplest such theorym and hence the "limits of FOL" section this article ought to have would best treat arithmetic.
:Yes, first-order arithmetic is very interesting indeed, but does it merit an entire section in an encyclopedic article on FOL (and when it has its own article)? I still find it best to place a link, perhaps in the Metalogical theorems section that states that FOL cannot provide an axiomatization of arithmetic, linking to the appropriate corresponding articles. Nortexoid 10:46, 26 Mar 2005 (UTC)
replies: text: I wasn't aware of that. However, in a previous edit, it was stated that just plain old propositional logic (i.e. classical propositional logic) was of higher order than predicate logic. Hence my editing comment. Nortexoid 10:46, 26 Mar 2005 (UTC)

text: My hand is still raised. Nortexoid 10:46, 26 Mar 2005 (UTC)

text: I agree with the name change to first-order logic. Deco 02:39, 1 Apr 2005 (UTC)
thread_title: Some comments, mostly in response to comments & edits of Nortexoid

message: I just noticed an edit in the Metalogical theorems section that refers readers to Gödel's incompleteness theorem in regards to the undecidability of FOL. That is misleading. Results conerning the undecidability of FOL come from Church, while Godel's theorem conerns specifically arithmetical theories of FOL, like Russell and Whitehead's (or generally N, or PA), and not FOL generally. That is, Godel's theorem is not a result about FOL generally. Nortexoid 03:11, 27 Mar 2005 (UTC)
replies: text: My mistake — you're right. Deco 04:45, 31 Mar 2005 (UTC)
thread_title: Undecidability of FOL

message: At present, this article introduces the calculus by layering it upon the definition of propositional calculus:
:The predicate calculus is an extension of the propositional calculus.  If the propositional calculus is defined with eleven axioms and one inference rule (modus ponens), not counting some auxiliary laws for the logical equivalence operator, then the predicate calculus can be defined by appending four additional axioms and one additional inference rule.
This definition appears to assume that the propositional calculus is given in a Hilbert-style, whereas the calculus is given in natural deduction style (esp. no axioms, only rules).  Natural deduction makes the rule of universal generalisation problematic: the rule is valid only if it is interpreted as making restrictions on open assumptions, which inexperienced users of the logic are unlikely to grasp.
I suggest we change this article to formulate the calculus in natural deduction, for example as is done in the Stanford Enclyclopedia of Philosophy article (I added the link).  An alternative would be to create a page documenting propositional logic in a Hilbert style and build on that.  Perhaps we should do both. --- Charles Stewart 16:25, 20 May 2005 (UTC)
thread_title: Interference between treatments of propositional and predicate logic

message: Most of the content of Mathematical logic is about First order logic, which is not surprising given the universality and historical development of logic.  But we should decide where to put what.  Perhaps mathamtical logic should be about standard results in first order logic, which is now mostly the case.  First-order logic should point to Mathematical logic, and discuss the differences with other logics.  vkuncak, Sun Jun 19 11:04:38 EDT 2005
:I'd say that any logic that is not, in principle at least, first-orderisable has a dubious claim to being mathematical logic.  But there are some tricky caveats about that: is modal logic first-orderisable?  What about important non-first-orderisable logics that have played an important role (and indeed continue to play) in the development of ML?  I'm not quite sure what I would want to write in fulfilment of your request. --- Charles Stewart 15:34, 27 October 2005 (UTC)
thread_title: [[First-order logic]] versus [[Mathematical logic]]

message: the intro says that it is difficult to use first order logic to formulate topology, but never really expands on that in the article.  is it because the domain of discource in topology usually includes (quantification over) sets (of points) and not just points? -Lethe | Talk 20:47, August 7, 2005 (UTC)
: This statement confuses me, too, and I was a topologist at one time.  Topology needs set theory, but set theory in a first-order framework is adequate. Randall Holmes 15:58, 27 December 2005 (UTC)
replies: text: It was 128.2.70.48 t c.  A CMU address.  Made a few dozen edits in May 2004.  Mostly minor, a few contributions of mathematical content which seem to be correct.  I doubt if he's watching his talk page though.  I'm inclined to remove it, as it doesn't seem true to me, I'd want a source. -lethe talk 16:50, 27 December 2005 (UTC)
thread_title:  difficult for topology? 

message: I'm confused... starting with the "peano axioms are not valid".  Does that mean that in a "valid" predicate calculus, all axioms must be in the form of the 4 axiom schemas?  And set theory can be put in this form, but peano axioms cannot?  There are no predicates with 2 parameters in the schemea, which would disallow things like a "equals" predicate. 
Perhaps a example of a valid (and trivial) extention of predicate calculus, (and maybe a example of a invalid (and trivial) one too)
Also, if resolution is refutation complete, then how do we not have a decision procedure for first order predicate calculus?
*Where does it say any of these things? The only notion of "valid" I am familiar with is the validity of a formula; for example, (Ax) (Ay)[x=y => y=x] is valid, but (Ax)(Ay)[x=y] is not, since it fails in any model with more than one element. The refutation completeness of resolution allows the theorems of FOPC to enumerated, but it may take an arbitrarily long time for the refutation to be generated, so there is no point at which one can conclude that a statement is not a theorem. User:Ben Standeven at 70.249.214.16 02:04, 14 October 2005 (UTC)
thread_title:  confused 

message: I rewrote the introductory section, which I found really muddy; I think I preserved all the links that are needed.
I corrected mistakes in several places in the balance of the article.  I removed the assertion
that the Peano axioms are not valid for first-order logic (correct but confusing).  I removed the incorrect definition of atomic sentence (all sentences P(t_1,...,t_n) are atomic; it has nothing to do with arity of P).  I dealt with the inconsistency between the statement that modus ponens is the only inference rule and the additional (necessary) provision of universal generalization as an inference rule.  The axioms are (I think) correct, but it is not adequate to construe W and Z as predicate letters:  the correct statement of how W and Z are to be understood is supplied.  Have a happy holiday season.  Randall Holmes 17:10, 27 December 2005 (UTC)
thread_title:  rewrote blurb, corrected errors 

message: In my experience, when people say "(the) predicate calculus" they almost always mean (classical) first order logic.  I followed a predicate calculus link expecting to find a nice, detailed article on FOL and instead found this stumpy, pathetic little article with unorthodox notation that I doubt many people would find helpful at all. Dbtfz 06:22, 19 January 2006 (UTC)
replies: text: I've left Charles a note on his talk page. Paul August ☎ 02:58, 31 January 2006 (UTC)
thread_title: Proposal to merge [[Predicate Calculus]] article into this one

message: I added references to the technical possibility of eliminating predicates and functions of arity > 2 if an ordered pair or its projection relations are provided (with suitable axioms).  I also added a reference to the equivalence between FOL with the same extension of vocabulary and a natural system of relation algebra with projections of an ordered pair as special relations; I will eventually add an article on this relation algebra.  This is from a book by Tarski and Givant, and apparently has excited some interest in CS.  Randall Holmes 16:50, 31 January 2006 (UTC)
replies: text: In complete ignorance of what you wrote above, I created the entry relation algebra! Kalmar (1936) and Quine (1954) proved that all predicates of arity > 2 can be reexpressed using predicates of arity 2. One would like to think this an elegant fact; nevertheless, it has drawn little attention in the literature; good to see you thinking like me. (Peircians argue that Peirce's reduction thesis, proved by Hertzberg in 1981, is a closely related result.) Warning: nearly all of the material in Wikipedia on relations is by one Jon Awbrey.202.36.179.65 19:18, 30 March 2006 (UTC)
thread_title:  references to pairing and relation algebra 

message: Intrigued in the topic of FOL, I was curious as to what made this logic different from other types of logic and why it is "first order".  While this Article does a great job of answering my first question, it made a meager attempt to describe what a first order theory is by providing a link to an article "first-order theory" and briefly talking about ZFC and axioms.  Unfortunately, the link to first-order theory redirected right back to first order logic, leaving me with a few examples of first order theories, but not a very clear definition of what it is.  Surely First-Order Theory ahould not redirect to first-order logic if it is not the only theory of this kind.  Apparently, ZFC and Peaso Arithmetic are also theories of this kind, so I believe that first-Order theory should have it's own article with a basic overview of what defines a theory of this sort followed by some links to the example theories briefly mentioned in the FOL aricle. Of course, I would write the article myself, but I came here asking the question, not knowing the answer.   lokoluis15      69.254.149.179 04:34, 2 March 2006 (UTC)
thread_title:  First-Order Theory redirect 

message: * JA: We'll eventually have to sort out some of the confusion that is currently well-distributed throughout WP logic-related articles over the distinction (that some writers, e.g. Quine, are sticklers about) between connectives, which operate on syntactic thingies, and mathematical operations, which are mathematical objects.  The word "operator" is widely used in a way that vacillates between these two poles.  Not everybody makes the distinction, but it is necessary to be aware of it in any case.  Probably not ready for that just yet, but I do see the day coming.  In the meantime, let's all try not to compound the confusion.  Jon Awbrey 19:46, 18 March 2006 (UTC)
thread_title: On the distinction (made by &exist;) between connectives &and; operations

message: The first bullet point in the Vocabulary section contains the sentence: "The minimum number of primitive symbols needed is one, but if we restrict ourselves to the operators listed above, we need three; for example, ¬, ∧, and ∀ suffice." Should that ∀ be a ∨ for "logical and"?
--jivy 03:21, 22 April 2006 (UTC)
replies: text: Firstly, ∨ stands for logical or.  Secondly, no.  You can get ∨ from ∧ using ¬ (see de Moivre).  You need a quantifier, or else you're not doing first order logic, you're doing sentential logic.  So the ∀ definitely cannot be replaced by a ∨, though it can probably be replaced by a ∃. -lethe talk [ +] 03:33, 22 April 2006 (UTC)
replies: text: Damn.  I think I got de Moivre and de Morgan mixed up. -lethe talk [ +] 03:35, 22 April 2006 (UTC)
thread_title:  Typo? 

message: The intro states that "it is generally accepted that all of classical mathematics can be formalized in ZFC". Well, yes, since the article Classical mathematics defines this as "mathematics constructed and proved on the basis of classical logic and ZFC set theory". So, given that definition, the claim from the intro reduces to:
:It is generally accepted that all mathematics constructed and proved on the basis of classical logic and ZFC set theory can be formalized in ZFC.
The subtext meaning of phrases like "generally accepted" is usually something like: "although contended, widely supported". But here there is nothing to contend, and acceptance is not support; the statement is vacuously true. 
Suggestions for an NPOV formulation having substance, and yet standing a chance of being consensual, will be graciously accepted. --LambiamTalk 10:01, 3 June 2006 (UTC)
replies: text: The Peano arithmetic article suggests that the term generally refers to the first-order system, even though Peano originally formulated his induction axiom as second order. 192.75.48.150 17:48, 5 September 2006 (UTC)
thread_title:  Vacuous statement 

message: Is there any reason that the names of these rules are not given? Adam Acosta 22:19, 12 July 2006 (UTC)
thread_title: Inference Rules

message: The article uses the term valence. Valence has no lemma in this meaning in wikipedia whereas arity has. Therefore I propose to use arity and add valence as another synonym for arity in the article arity. Otto 12:28, 5 August 2006 (UTC)
thread_title: Valence or arity

message: I found that the lead of this article is very confusing and unhelpful for a reader unaware with the topic. Can anyone rewrite it to make it clearer. 202.63.233.8 16:02, 8 August 2006 (UTC)
thread_title:  Confusing Lead 

message: I moved a abbreviated set of definitions for first order logic from the page mathematical logic to Talk:First-order logic/Technical reference.  I don't know if it is helpful here but it wasn't helpful in the other place.  The guideline WP:SUBPAGE discourages linking directly from articles to subpages of talk pages, so please move the reference somewhere else before linking to it from an article. CMummert 12:21, 29 October 2006 (UTC)
thread_title:  Technical reference 

message: Is there any way that, well, a layman can understand this article (or at least the introduction and first paragraph) without going link-hunting through all the notation used, or is the subject just that dense?  I am, of course, unqualified to do anything about it (if it can be done, or should be, or whatever,) but I was wondering if there could be an introduction for the uneducated.  TwilightDragon 01:40, 30 December 2006 (UTC)
thread_title: Density

message: Well, I found it informative and a little more transparent at first glance than this article! I wonder if someone with a grounding in logic might be able to use this? (You might need to scroll down a little to get into the meat of the article. Mark 23:39, 28 January 2007 (UTC)
thread_title:  Good intro to FOPL 

message: Just a heads-up:
The problem with Internet Explorer (and other browsers) not displaying certain math symbols on this page correctly (also mentioned in another post above) is described extensively at Help:Special_characters. Specifically, see the Viewing and Displaying Special Characters sections. The unicode template can be used to force IE into using a font that can display Unicode / UTF-8 characters properly. An alternative is to use  wikimarkup (see WP:MATH), although this is often avoided, mostly due to the font appearing overly large.
I thought it might be a good idea to "tag" affected articles with the same "Technical note" box displayed at the top of the Help:Special_characters page. At least this will alert affected users about the problem and its cause. If you decide to make use of this, the code is:

Technical note: Due to technical limitations, some web browsers may not display some special characters in this article. Such characters may be rendered as boxes, question marks, or other replacement symbols, depending on your browser, operating system, and installed fonts. Even if you have ensured that your browser is interpreting the article as UTF-8 encoded and you have installed Unicode font(s) that supports a wide range of Unicode blocks, fonts such as  Arial Unicode MS, Code2000, Code2001, TITUS Cyberbit Basic, Lucida Sans Unicode or one of the Free software Unicode fonts, you may still need to use a different browser, as browser capabilities in this regard tend to vary.


This will display:
Technical note: Due to technical limitations, some web browsers may not display some special characters in this article. Such characters may be rendered as boxes, question marks, or other replacement symbols, depending on your browser, operating system, and installed fonts. Even if you have ensured that your browser is interpreting the article as UTF-8 encoded and you have installed Unicode font(s) that supports a wide range of Unicode blocks, fonts such as  Arial Unicode MS, Code2000, Code2001, TITUS Cyberbit Basic, Lucida Sans Unicode or one of the Free software Unicode fonts, you may still need to use a different browser, as browser capabilities in this regard tend to vary.
—XhantarTalk 11:19, 9 February 2007 (UTC)
thread_title:  IE and math symbols: more details 

message: Coming onto this page with, formally, only a grade-school strength knowledge of the subject, I was able to depart it with a greater understanding of the highly ingroup-intensive subject without any confusion.  The page is well organized, verifiable, (subjectively) interesting, and compositionally sound.  I believe recognition is in order. .Absolution. 09:17, 23 April 2007 (UTC)
thread_title:  Article Recognition