Wikipedia talk:WikiProject Logic/Standards for notation

message: To start of with it would be useful to have an agreed set of symbols.  Not only do symbols vary from author to author, but any symbol may be written in a variety of fonts which may or may not appear on various browsers. I have started us off with a little table for the truth functional connectives & one for quantifiers.
The aim is consistency and legibility
--Philogo 13:02, 16 August 2007 (UTC)
thread_title: Starters

message: We should give the <math> codes for all the preferred symbols, even if they have a template.  When people use \forall or \exists, for example, they need to stay in math mode. MilesAgain (talk) 09:03, 1 January 2008 (UTC)
replies: text: MilesAgain (talk) 09:48, 1 January 2008 (UTC)
thread_title:  Need math tag codes 

message: We should come to agreement on the notation for distinguishing use from mention. Also whether or not we will just skirt the issue at times or attempt a rigorous adherence to a policy for clarity. I think it is a sign of good quality in the Wikipedia if we can get to this level of clarity. I have used single quotes to do this myself, however we could elect to use italics, or perhaps there is something else.
I think we could also choose to use this distinction in paragraph format text (i.e. right now it's '(PQ)') and not use it when, for instance, we have formulas that are set apart by themselves and indented, etc:
:(PQ)
--
:Proposal:
::Mention of a phrase shall be indicated by enclosing in single quotes:
::::'Alex' has four letters.
::Use of a phrase shall be indicated be the phrase itself:
::::Alex is 21 years old.
::Furthermore, iterations of being enclosed in quotes shall be allowed to indicate further levels of language.
----
Pontiff Greg Bard (talk) 17:24, 17 January 2008 (UTC)
replies: text: This is already dictated by the manual of style Wikipedia:MOS#Italics; it can't be changed here. — Carl (CBM · talk) 17:48, 17 January 2008 (UTC)
thread_title:  Use v mention 

message: I gather that the use of \supset for implication has failed for lack of a second.  Good.
I'm old enough to prefer 'cap' and 'cup' for 'and' and 'or'.  The discussion in favor of '&' is that & is widely understood.  I find that a defect rather than an advantage.  Because & for the English word "and" (and @ for the english word "at") are so common in webspeek -- I get them in term papers! -- I see a real problem with people mistaking the logical "and" for the English word "and".  Consider, for example, this sentence,
"'\wedge' & '&' are both used to mean "&" & '\vee' is used to mean "or"."
Rick Norwood (talk) 19:15, 27 May 2008 (UTC)
replies: text: On a frivolous note. I read a paper once that investigated the advantage of using symbols in computer language for operations which bore some resemblance to the maings of those  symbols (inc words) in the everyday world.  Eg Using IF and THEN and ELSE.  They taught two classes of students the same language one with the mnemonic symbols and one with completely made up symbols. Like Ek instead of IF and Ugh for AND and so on.  You would expect the students with the menmonic language to learn more quicly than the other group.  They did not.  Explanation may be that it takes  as long to shake off the surplus of the normal sense of a familiar word than to learn a new word with no previous connations. I have some really old logic books that use "." for "and". I cannot help but read "p.q" as p multiplied by q.--Philogo 22:04, 27 May 2008 (UTC)
thread_title:  2centsworth 

message: I don't agree with the recommendation of &, but I may reconsider if anybody can show me a single book on model theory, published after 1990 by a normal scientific publisher, that uses this symbol. Or an introduction to mathematical logic on the graduate level that uses it. I have never encountered the symbol in recent publications, either. I have also never seen the arrow notations for NOR and NAND outside Wikipedia. From which field do they come? (In model theory we just don't use the operations.) I haven't seen the negated left-right arrow used for exclusive disjunction in the wild either, but it's self-explanatory.
In other fields the notation is less standardised. See also the related discussion here. (I wasn't aware of this page when I started that.) For articles on Boolean logic in engineering I would agree with its use, for example, except that in that field multiplication seems to be a more standard notation, while OR is most often written as +, TRUE as 1, FALSE as 0, and NEGATION as ~ or by overlining.
I think the superset symbol used to be used for implication because there wasn't much choice before computer typesetting. I was surprised by the comment by Nahaj above claiming that it is still used in modal logic. I checked the first freely accessible hits on Google Scholar for the search term "modal logic", and I did find an apparently important book from 1996 that used this notation (Hughes and Cresswell, "A new introduction to modal logic"). It seems that this field has a subculture with diverging terminology (also including the use of &), but when people like Moshe Y. Vardi write about modal logic they use the (nowadays) more standard arrow notation. In my experience it's also the notation that finite model theorists use when giving talks related to modal logic.
The addition symbol + is used much more often for OR than for XOR. Allowing it for XOR but not for OR is absurd and likely to mislead. --Hans Adler (talk) 13:08, 1 June 2008 (UTC)
replies: text: I don't understand the symbol tables. What is the "Symbol(s)" column in the first table supposed to mean? All symbols that are in wide use for a connective, or just those that are permitted for Wikipedia? If the former, then they are horribly incomplete and misleading, see some of my remarks above. If the latter, then the choice is not good. I would like to edit the table, but for this I need to understand it. --Hans Adler (talk) 20:33, 2 June 2008 (UTC)
thread_title:  Preferred symbols 

message: Term                                      	Comments                                                                                                                                                                                                                                                 	Notable books
non-logical constants                     	Refers to functions with varying interpretation as "constants". Both authors work with a fixed set of non-logical constants.                                                                                                                             	Mendelson 1964-1997, Mates 1972
non-logical concepts                      	                                                                                                                                                                                                                                                         	Shoenfield 1967
(primitive) non-logical symbols           	                                                                                                                                                                                                                                                         	Barwise 1977
similarity type                           	Most often used in universal algebra, and often seen as a sequence of numbers (arities of function symbols) with no provision for relation symbols. (So that the primary meaning is strictly speaking something slightly different.)                     	
"stock of constants and predicate letters"	only used in passing                                                                                                                                                                                                                                     	Gamut 1991
vocabulary                                	Clashes with the standard metaphor in formal language theory: A "vocabulary" should not consist of "letters/symbols" and be a subset of an "alphabet".Gamut 1991 uses the term to refer to the entire alphabet, including the logical symbols.           	
first-order language                      	Ambiguous term: does it refer to the non-logical symbols or to the first-order formulas? Not suitable for non-first-order contexts, where exactly the same information is needed.                                                                        	
symbol set                                	As in "the symbol set of L". Abuse of language: does not include the logical symbols.                                                                                                                                                                    	Ebbinghaus et al 1996
language                                  	Ambiguous term: does it refer to the non-logical symbols or to the (first-order) formulas? Clashes with formal language. Barwise never defines the term officially, but calls the "set of non-logical symbols" L and calls it "language" in the examples.	Chang & Keisler 1973-1989, Barwise 1977
signature                                 	Term originated in computer science? Neutral: Does not suggest a logic context, is equally appropriate for purely semantic use.                                                                                                                          	Hodges 1993, Hinman 2005
An early approach in first-order logic was to have one universal language of first-order logic, containing a fixed, countably infinite, supply of non-logical symbols. For example constants c0, c1, c2, ..., and binary predicate symbols P20, P21, P22, ... . This was a major obstacle for early mathematical logicians, and many results that are now considered trivial and perfectly obvious were for them non-trivial and often very hard to prove because of this restriction.
In computer science it is a problem with the old approach that the implicit signature (i.e. the fixed set of non-logical symbols) is infinite. Many arguments in computer science only work for finite signatures, and most absolutely need a finite upper bound on the arity of relation symbols. So in computer science it is customary to restrict consideration to, e.g., only the first 3 constants, no unary predicate symbols, the first binary predicate symbol, and no predicate symbols of higher arity. This information (in the example it could e.g. be coded as 3; 0, 1, 0, 0, 0, ...) is called the signature.
In mathematics it is often necessary to extend the supply of non-logical symbols. One could of course just continue with cω, cω+1, ..., but it's much more natural to allow arbitrary sets (which are not already logical symbols) as non-logical symbols. And while we are at it, we can get rid of the "standard" symbols c0, c1, ... etc.  This covers both the original situation (just take the signature consisting of the countably many old-fashioned non-logical symbols) and what the computer scientists do (just choose finitely many of the old-fashioned non-logical symbols).
Much more importantly, the signature in the mathematical sense is exactly what we need in applications, both in mathematics and computer science.
*Let's say that an abelian group is a set A together with a group operation +, a neutral element 0, and a unary negation operation -, such that certain axioms hold. In the old-fashioned system we would code the sentence "for all x, x+(-x)=0" as \forall v_0(f^2_0(v_0,f^1_0(v_0)))=0. Now we can just code it as \forall v_0(+(v_0,-(v_0))=0, which is much more readable.
*Let's say we have an employee database with one table BASEINFO connecting personnel number, name, gender and date of birth, and another table SALARY connecting  personnel number and salary. The tables define a quaternary relation (quaternary predicate) BASEINFO and a binary relation SALARY. The fact that there are just these two tables, and no others, and how many places (columns) they have, is much more fundamental for the database than the actual content. Occasionally a database programmer will have to change this kind of information, but in normal everyday use only the strings and numbers which the database contains, and the rows in which they occur, will change. The fundamental, normally unchanging, information is the signature {BASEINFO, SALARY}.
Signatures were invented because they were needed. At first it was just a convention: People worked with them without saying so. So people started working with several first-order languages instead of just one. It took several decades for the modern standard term "signature" to be invented and to become established, and a few old-fashioned people still say "the non-logical symbols of the language" or just "the language" when they mean the signature. But in all fields that actually need signatures there is a clear trend to use the term "signature". (Sorry that this became so long.) --Hans Adler (talk) 12:25, 2 June 2008 (UTC)
thread_title:  General term for the symbols used for constants, functions and relations 

message: Last week I had some spare time when I was in a nice research library. I found two clear passages that seem to confirm some of my opinions about terminology:
A parenthetical remark from Enderton: A Mathematical Introduction to Logic (2001 edition), p. 80:
"Structures are sometimes called interpretations, but we prefer to reserve that word for another concept, to be encountered in Section 2.7." Section 2.7 is called "Interpretations Between Theories" and refers to interpretation (model theory).
From Cori and Lascar: Mathematical Logic: A course with exercises, p. 131:
"A model of the language L, or L-structure, is a structure \mathcal M consisting of: - a non-empty set M, called the domain or the base set or the underlying set [...], - for each constant symbol c of L an element [...] of M called the interpretation of the symbol c in the model \mathcal M [...] [also interpretations of function and relation symbols]."
I believe standard practice in modern model theory is exactly as in these passages. --Hans Adler (talk) 23:29, 21 January 2009 (UTC)
thread_title: Some new finds

message: Is Identity (definition) really a truth-functional connective as implied by its place on the table?  If so what's its truth table?
— Preceding unsigned comment added by Philogo (talk • contribs) 
:Of course not, but there doesn't seem to be a better place for the information. I merged it into the previous line, removed the misleading link, and added two links to content forks of logical biconditional. --Hans Adler (talk) 10:47, 23 February 2009 (UTC)
thread_title:  Identity (definition) 

message: IMHO standart should not have direct collisions like this:
 \leftrightarrow  means "equivalence". So  \nleftrightarrow  expected to be "not equivalence". But now  \nleftrightarrow  is "xor".
Sergey feo (talk) 06:58, 10 April 2010 (UTC)
replies: text: It's the same: Both non-equivalence and exclusive or are true precisely if one side is true and the other is false. Hans Adler 07:07, 10 April 2010 (UTC)
thread_title:  Current collisions