1 Arazshura

Reflex Default Bibliography Definition

This module may require a complete rewrite in order to suit its intended audience.
You can help rewrite it. Please see the relevant discussion.

For any academic/research writing, incorporating references into a document is an important task. Fortunately, LaTeX has a variety of features that make dealing with references much simpler, including built-in support for citing references. However, a much more powerful and flexible solution is achieved thanks to an auxiliary tool called BibTeX (which comes bundled as standard with LaTeX). Recently, BibTeX has been succeeded by BibLaTeX, a tool configurable within LaTeX syntax.

BibTeX provides for the storage of all references in an external, flat-file database. (BibLaTeX uses this same syntax.) This database can be referenced in any LaTeX document, and citations made to any record that is contained within the file. This is often more convenient than embedding them at the end of every document written; a centralized bibliography source can be linked to as many documents as desired (write once, read many!). Of course, bibliographies can be split over as many files as one wishes, so there can be a file containing sources concerning topic A () and another concerning topic B (). When writing about topic AB, both of these files can be linked into the document (perhaps in addition to sources specific to topic AB).

Embedded system[edit]

If you are writing only one or two documents and aren't planning on writing more on the same subject for a long time, you might not want to waste time creating a database of references you are never going to use. In this case you should consider using the basic and simple bibliography support that is embedded within LaTeX.

LaTeX provides an environment called that you have to use where you want the bibliography; that usually means at the very end of your document, just before the command. Here is a practical example:

\begin{thebibliography}{9}\bibitem{lamport94} Leslie Lamport, \textit{\LaTeX: a document preparation system}, Addison Wesley, Massachusetts, 2nd edition, 1994. \end{thebibliography}

OK, so what is going on here? The first thing to notice is the establishment of the environment. is a keyword that tells LaTeX to recognize everything between the begin and end tags as data for the bibliography. The mandatory argument, which I supplied after the begin statement, is telling LaTeX how wide the item label will be when printed. Note however, that the number itself is not the parameter, but the number of digits is. Therefore, I am effectively telling LaTeX that I will only need reference labels of one character in length, which ultimately means no more than nine references in total. If you want more than nine, then input any two-digit number, such as '56' which allows up to 99 references.

Next is the actual reference entry itself. This is prefixed with the command. The cite_key should be a unique identifier for that particular reference, and is often some sort of mnemonic consisting of any sequence of letters, numbers and punctuation symbols (although not a comma). I often use the surname of the first author, followed by the last two digits of the year (hence lamport94). If that author has produced more than one reference for a given year, then I add letters after, 'a', 'b', etc. But, you should do whatever works for you. Everything after the key is the reference itself. You need to type it as you want it to be presented. I have put the different parts of the reference, such as author, title, etc., on different lines for readability. These linebreaks are ignored by LaTeX. The command formats the title properly in italics.


To actually cite a given document is very easy. Go to the point where you want the citation to appear, and use the following: , where the cite_key is that of the bibitem you wish to cite. When LaTeX processes the document, the citation will be cross-referenced with the bibitems and replaced with the appropriate number citation. The advantage here, once again, is that LaTeX looks after the numbering for you. If it were totally manual, then adding or removing a reference would be a real chore, as you would have to re-number all the citations by hand.

Instead of WYSIWYG editors, typesetting systems like \TeX{} or \LaTeX{}\cite{lamport94} can be used.

Referring more specifically[edit]

If you want to refer to a certain page, figure or theorem in a text book, you can use the arguments to the command:

\cite[chapter, p.~215]{citation01}

The argument, "p. 215", will show up inside the same brackets. Note the tilde in [p.~215], which replaces the end-of-sentence spacing with a non-breakable inter-word space. This non-breakable inter-word space is inserted because the end-of-sentence spacing would be too wide, and "p." should not be separated from the page number.

Multiple citations[edit]

When a sequence of multiple citations is needed, you should use a single command. The citations are then separated by commas. Here's an example:


The result will then be shown as citations inside the same brackets, depending on the citation style.

Bibliography styles[edit]

There are several different ways to format lists of bibliographic references and the citations to them in the text. These are called citation styles, and consist of two parts: the format of the abbreviated citation (i.e. the marker that is inserted into the text to identify the entry in the list of references) and the format of the corresponding entry in the list of references, which includes full bibliographic details.

Abbreviated citations can be of two main types: numbered or textual. Numbered citations (also known as the Vancouver referencing system) are numbered consecutively in order of appearance in the text, and consist in Arabic numerals in parentheses (1), square brackets [1], superscript1, or a combination thereof[1]. Textual citations (also known as the Harvard referencing system) use the author surname and (usually) the year as the abbreviated form of the citation, which is normally fully (Smith 2008) or partially enclosed in parenthesis, as in Smith (2008). The latter form allows the citation to be integrated in the sentence it supports.

Below you can see three of the styles available with LaTeX:

Here are some more often used styles:

Style NameAuthor Name FormatReference FormatSorting
plainHomer Jay Simpson#ID#by author
unsrtHomer Jay Simpson#ID#as referenced
abbrvH. J. Simpson#ID#by author
alphaHomer Jay SimpsonSim95by author
abstractHomer Jay SimpsonSimpson-1995a
acmSimpson, H. J.#ID#
authordate1Simpson, Homer JaySimpson, 1995
apaciteSimpson, H. J. (1995)Simpson1995
namedHomer Jay SimpsonSimpson 1995

However, keep in mind that you will need to use the natbib package to use most of these.

No cite[edit]

If you only want a reference to appear in the bibliography, but not where it is referenced in the main text, then the command can be used, for example:

Lamport showed in 1995 something... \nocite{lamport95}.

A special version of the command, , includes all entries from the database, whether they are referenced in the document or not.


Citation commandOutput

Goossens et al. (1993)
(Goossens et al., 1993)

Goossens, Mittlebach, and Samarin (1993)
(Goossens, Mittlebach, and Samarin, 1993)

Goossens et al.
Goossens, Mittlebach, and Samarin


Goossens et al. 1993
Goossens et al., 1993
(priv. comm.)

Using the standard LaTeX bibliography support, you will see that each reference is numbered and each citation corresponds to the numbers. The numeric style of citation is quite common in scientific writing. In other disciplines, the author-year style, e.g., (Roberts, 2003), such as Harvard is preferred. A discussion about which is best will not occur here, but a possible way to get such an output is by the package. In fact, it can supersede LaTeX's own citation commands, as Natbib allows the user to easily switch between Harvard or numeric.

The first job is to add the following to your preamble in order to get LaTeX to use the Natbib package:


Also, you need to change the bibliography style file to be used, so edit the appropriate line at the bottom of the file so that it reads: . Once done, it is basically a matter of altering the existing commands to display the type of citation you want.

plainnatProvidednatbib-compatible version of plain
abbrvnatProvidednatbib-compatible version of abbrv
unsrtnatProvidednatbib-compatible version of unsrt
apsrevREVTeX 4 home pagenatbib-compatible style for Physical Review journals
rmpapsREVTeX 4 home pagenatbib-compatible style for Review of Modern Physics journals
IEEEtranNTeX Catalogue entrynatbib-compatible style for IEEE publications
achemsoTeX Catalogue entrynatbib-compatible style for American Chemical Society journals
rscTeX Catalogue entrynatbib-compatible style for Royal Society of Chemistry journals


 :  :  : Parentheses () (default), square brackets [], curly braces {} or angle brackets <>
 : multiple citations are separated by semi-colons (default) or commas
 :  : author year style citations (default), numeric citations or superscripted numeric citations
 : multiple citations are sorted into the order in which they appear in the references section or also compressing multiple numeric citations where possible
the first citation of any reference will use the starred variant (full author list), subsequent citations will use the abbreviated et al. style
for use with the chapterbib package. redefines \thebibliography to issue \section* instead of \chapter*
keeps all the authors’ names in a citation on one line to fix some hyperref problems - causes overfull hboxes

The main commands simply add a t for 'textual' or p for 'parenthesized', to the basic command. You will also notice how Natbib by default will compress references with three or more authors to the more concise 1st surname et al version. By adding an asterisk (*), you can override this default and list all authors associated with that citation. There are some other specialized commands that Natbib supports, listed in the table here. Keep in mind that for instance does not support and will automatically choose between all authors and et al..

The final area that I wish to cover about Natbib is customizing its citation style. There is a command called that can be used to override the defaults and change certain settings. For example, I have put the following in the preamble:


The command requires six mandatory parameters.

  1. The symbol for the opening bracket.
  2. The symbol for the closing bracket.
  3. The symbol that appears between multiple citations.
  4. This argument takes a letter:
    • n - numerical style.
    • s - numerical superscript style.
    • any other letter - author-year style.
  5. The punctuation to appear between the author and the year (in parenthetical case only).
  6. The punctuation used between years, in multiple citations when there is a common author. e.g., (Chomsky 1956, 1957). If you want an extra space, then you need .

Some of the options controlled by are also accessible by passing options to the natbib package when it is loaded. These options also allow some other aspect of the bibliography to be controlled, and can be seen in the table (right).

So as you can see, this package is quite flexible, especially as you can easily switch between different citation styles by changing a single parameter. Do have a look at the Natbib manual, it's a short document and you can learn even more about how to use it.


I have previously introduced the idea of embedding references at the end of the document, and then using the command to cite them within the text. In this tutorial, I want to do a little better than this method, as it's not as flexible as it could be. I will concentrate on using BibTeX.

A BibTeX database is stored as a .bib file. It is a plain text file, and so can be viewed and edited easily. The structure of the file is also quite simple. An example of a BibTeX entry:

@article{greenwade93,author="George D. Greenwade",title="The {C}omprehensive {T}ex {A}rchive {N}etwork ({CTAN})",year="1993",journal="TUGBoat",volume="14",number="3",pages="342--351"}

Each entry begins with the declaration of the reference type, in the form of . BibTeX knows of practically all types you can think of, common ones are: book, article, and for papers presented at conferences, there is inproceedings. In this example, I have referred to an article within a journal.

After the type, you must have a left curly brace '' to signify the beginning of the reference attributes. The first one follows immediately after the brace, which is the citation key, or the BibTeX key. This key must be unique for all entries in your bibliography. It is this identifier that you will use within your document to cross-reference it to this entry. It is up to you as to how you wish to label each reference, but there is a loose standard in which you use the author's surname, followed by the year of publication. This is the scheme that I use in this tutorial.

Next, it should be clear that what follows are the relevant fields and data for that particular reference. The field names on the left are BibTeX keywords. They are followed by an equals sign (=) where the value for that field is then placed. BibTeX expects you to explicitly label the beginning and end of each value. I personally use quotation marks ("), however, you also have the option of using curly braces ('{', '}'). But as you will soon see, curly braces have other roles, within attributes, so I prefer not to use them for this job as they can get more confusing. A notable exception is when you want to use characters with umlauts (ü, ö, etc), since their notation is in the format , and the quotation mark will close the one opening the field, causing an error in the parsing of the reference. Using in the preamble to the source file can get round this, as the accented characters can just be stored in the file without any need for special markup. This allows a consistent format to be kept throughout the file, avoiding the need to use braces when there are umlauts to consider.

Remember that each attribute must be followed by a comma to delimit one from another. You do not need to add a comma to the last attribute, since the closing brace will tell BibTeX that there are no more attributes for this entry, although you won't get an error if you do.

It can take a while to learn what the reference types are, and what fields each type has available (and which ones are required or optional, etc). So, look at this entry type reference and also this field reference for descriptions of all the fields. It may be worth bookmarking or printing these pages so that they are easily at hand when you need them. Much of the information contained therein is repeated in the following table for your convenience.

articlebookbookletinbookincollectioninproceedings ≈ conferencemanualmastersthesis, phdthesismiscproceedingstech reportunpublished

+ Required fields, o Optional fields


BibTeX can be quite clever with names of authors. It can accept names in forename surname or surname, forename. I personally use the former, but remember that the order you input them (or any data within an entry for that matter) is customizable and so you can get BibTeX to manipulate the input and then output it however you like. If you use the forename surname method, then you must be careful with a few special names, where there are compound surnames, for example "John von Neumann". In this form, BibTeX assumes that the last word is the surname, and everything before is the forename, plus any middle names. You must therefore manually tell BibTeX to keep the 'von' and 'Neumann' together. This is achieved easily using curly braces. So the final result would be "John {von Neumann}". This is easily avoided with the surname, forename, since you have a comma to separate the surname from the forename.

Secondly, there is the issue of how to tell BibTeX when a reference has more than one author. This is very simply done by putting the keyword and in between every author. As we can see from another example:

@book{goossens93,author="Michel Goossens and Frank Mittelbach and Alexander Samarin",title="The LaTeX Companion",year="1993",publisher="Addison-Wesley",address="Reading, Massachusetts"}

This book has three authors, and each is separated as described. Of course, when BibTeX processes and outputs this, there will only be an 'and' between the penultimate and last authors, but within the .bib file, it needs the ands so that it can keep track of the individual authors.

Standard templates[edit]

Be careful if you copy the following templates, the % sign is not valid to comment out lines in bibtex files. If you want to comment out a line, you have to put it outside the entry.

An article from a magazine or a journal.
  • Required fields: author, title, journal, year.
  • Optional fields: volume, number, pages, month, note.
A published book
  • Required fields: author/editor, title, publisher, year.
  • Optional fields: volume/number, series, address, edition, month, note.
A bound work without a named publisher or sponsor.
  • Required fields: title.
  • Optional fields: author, howpublished, address, month, year, note.
Equal to inproceedings
  • Required fields: author, title, booktitle, year.
  • Optional fields: editor, volume/number, series, pages, address, month, organization, publisher, note.
A section of a book without its own title.
  • Required fields: author/editor, title, chapter and/or pages, publisher, year.
  • Optional fields: volume/number, series, type, address, edition, month, note.
A section of a book having its own title.
  • Required fields: author, title, booktitle, publisher, year.
  • Optional fields: editor, volume/number, series, type, chapter, pages, address, edition, month, note.
An article in a conference proceedings.
  • Required fields: author, title, booktitle, year.
  • Optional fields: editor, volume/number, series, pages, address, month, organization, publisher, note.
Technical manual
  • Required fields: title.
  • Optional fields: author, organization, address, edition, month, year, note.
Master's thesis
  • Required fields: author, title, school, year.
  • Optional fields: type (eg. "diploma thesis"), address, month, note.
Template useful for other kinds of publication
  • Required fields: none
  • Optional fields: author, title, howpublished, month, year, note.
Ph.D. thesis
  • Required fields: author, title, year, school.
  • Optional fields: address, month, keywords, note.
The proceedings of a conference.
  • Required fields: title, year.
  • Optional fields: editor, volume/number, series, address, month, organization, publisher, note.
Technical report from educational, commercial or standardization institution.
  • Required fields: author, title, institution, year.
  • Optional fields: type, number, address, month, note.
An unpublished article, book, thesis, etc.
  • Required fields: author, title, note.
  • Optional fields: month, year.

Non-standard templates[edit]

BibTeX entries can be exported from Google Patents.
(see Cite Patents with Bibtex for an alternative)
For citing arXiv.org papers in a REVTEX-style article
(see REVTEX Author's guide)

Preserving case of letters[edit]

In the event that BibTeX has been set by the chosen style not to preserve all capitalization within titles, problems can occur, especially if you are referring to proper nouns, or acronyms. To tell BibTeX to keep them, use the good old curly braces around the letter in question, (or letters, if it's an acronym) and all will be well! It is even possible that lower-case letters may need to be preserved - for example if a chemical formula is used in a style that sets a title in all caps or small caps, or if "pH" is to be used in a style that capitalises all first letters.

However, avoid putting the whole title in curly braces, as it will look odd if a different capitalization format is used:

For convenience though, many people simply put double curly braces, which may help when writing scientific articles for different magazines, conferences with different BibTex styles that do sometimes keep and sometimes not keep the capital letters:

As an alternative, try other BibTex styles or modify the existing. The approach of putting only relevant text in curly brackets is the most feasible if using a template under the control of a publisher, such as for journal submissions. Using curly braces around single letters is also to be avoided if possible, as it may mess up the kerning, especially with biblatex,[1] so the first step should generally be to enclose single words in braces.

A few additional examples[edit]

Below you will find a few additional examples of bibliography entries. The first one covers the case of multiple authors in the Surname, Firstname format, and the second one deals with the incollection case.

@article{AbedonHymanThomas2003,author="Abedon, S. T. and Hyman, P. and Thomas, C.",year="2003",title="Experimental examination of bacteriophage latent-period evolution as a response to bacterial availability",journal="Applied and Environmental Microbiology",volume="69",pages="7499--7506"}@incollection{Abedon1994,author="Abedon, S. T.",title="Lysis and the interaction between free phages and infected cells",pages="397--405",booktitle="Molecular biology of bacteriophage T4",editor="Karam, Jim D. Karam and Drake, John W. and Kreuzer, Kenneth N. and Mosig, Gisela and Hall, Dwight and Eiserling, Frederick A. and Black, Lindsay W. and Kutter, Elizabeth and Carlson, Karin and Miller, Eric S. and Spicer, Eleanor",publisher="ASM Press, Washington DC",year="1994"}

If you have to cite a website you can use @misc, for example:

@misc{website:fermentas-lambda,author="Fermentas Inc.",title="Phage Lambda: description \& restriction map",month="November",year="2008",url="http://www.fermentas.com/techinfo/nucleicacids/maplambda.htm"}

The note field comes in handy if you need to add unstructured information, for example that the corresponding issue of the journal has yet to appear:

@article{blackholes,author="Rabbert Klein",title="Black Holes and Their Relation to Hiding Eggs",journal="Theoretical Easter Physics",publisher="Eggs Ltd.",year="2010",note="(to appear)"}

Getting current LaTeX document to use your .bib file[edit]

At the end of your LaTeX file (that is, after the content, but before ), you need to place the following commands:

\bibliographystyle{plain}\bibliography{sample1,sample2,...,samplen}% Note the lack of whitespace between the commas and the next bib file.

Bibliography styles are files recognized by BibTeX that tell it how to format the information stored in the file when processed for output. And so the first command listed above is declaring which style file to use. The style file in this instance is (which comes as standard with BibTeX). You do not need to add the .bst extension when using this command, as it is assumed. Despite its name, the plain style does a pretty good job (look at the output of this tutorial to see what I mean).

The second command is the one that actually specifies the file you wish to use. The ones I created for this tutorial were called , , . . ., , but once again, you don't include the file extension. At the moment, the file is in the same directory as the LaTeX document too. However, if your .bib file was elsewhere (which makes sense if you intend to maintain a centralized database of references for all your research), you need to specify the path as well, e.g or (if the file is in the parent directory of the document that calls it).

Now that LaTeX and BibTeX know where to look for the appropriate files, actually citing the references is fairly trivial. The is the command you need, making sure that the ref_key corresponds exactly to one of the entries in the .bib file. If you wish to cite more than one reference at the same time, do the following: .

Why won't LaTeX generate any output?[edit]

The addition of BibTeX adds extra complexity for the processing of the source to the desired output. This is largely hidden from the user, but because of all the complexity of the referencing of citations from your source LaTeX file to the database entries in another file, you actually need multiple passes to accomplish the task. This means you have to run LaTeX a number of times. Each pass will perform a particular task until it has managed to resolve all the citation references. Here's what you need to type (into command line):

    (Extensions are optional, if you put them note that the bibtex command takes the AUX file as input.)

    After the first LaTeX run, you will see errors such as:

    LaTeX Warning: Citation `lamport94' on page 1 undefined on input line 21. ... LaTeX Warning: There were undefined references.

    The next step is to run bibtex on that same LaTeX source (or more precisely the corresponding AUX file, however not on the actual .bib file) to then define all the references within that document. You should see output like the following:

    This is BibTeX, Version 0.99c (Web2C 7.3.1) The top-level auxiliary file: latex_source_code.aux The style file: plain.bst Database file #1: sample.bib

    The third step, which is invoking LaTeX for the second time will see more errors like "". Don't be alarmed, it's almost complete. As you can guess, all you have to do is follow its instructions, and run LaTeX for the third time, and the document will be output as expected, without further problems.

    If you want a pdf output instead of a dvi output you can use instead of as follows:

      (Extensions are optional, if you put them note that the bibtex command takes the AUX file as input.)

      Note that if you are editing your source in vim and attempt to use command mode and the current file shortcut (%) to process the document like this:

        You will get an error similar to this:

          It appears that the file extension is included by default when the current file command (%) is executed. To process your document from within vim, you must explicitly name the file without the file extension for bibtex to work, as is shown below:

          1. (without file extension, it looks for the AUX file as mentioned above)

          However, it is much easier to install the Vim-LaTeX plugin from here. This allows you to simply type \ll when not in insert mode, and all the appropriate commands are automatically executed to compile the document. Vim-LaTeX even detects how many times it has to run pdflatex, and whether or not it has to run bibtex. This is just one of the many nice features of Vim-LaTeX, you can read the excellent Beginner's Tutorial for more about the many clever shortcuts Vim-LaTeX provides.

          Another option exists if you are running Unix/Linux or any other platform where you have make. Then you can simply create a Makefile and use vim's make command or use make in shell. The Makefile would then look like this:

          latex_source_code.pdf: latex_source_code.tex latex_source_code.bib pdflatex latex_source_code.tex bibtex latex_source_code.aux pdflatex latex_source_code.tex pdflatex latex_source_code.tex

          Including URLs in bibliography[edit]

          As you can see, there is no field for URLs. One possibility is to include Internet addresses in field of or field of , , :

          Note the usage of command to ensure proper appearance of URLs.

          Another way is to use special field and make bibliography style recognise it.

          You need to use in the first case or in the second case.

          Styles provided by Natbib (see below) handle this field, other styles can be modified using urlbst program. Modifications of three standard styles (plain, abbrv and alpha) are provided with urlbst.

          If you need more help about URLs in bibliography, visit FAQ of UK List of TeX.

          Customizing bibliography appearance[edit]

          One of the main advantages of BibTeX, especially for people who write many research papers, is the ability to customize your bibliography to suit the requirements of a given publication. You will notice how different publications tend to have their own style of formatting references, to which authors must adhere if they want their manuscripts published. In fact, established journals and conference organizers often will have created their own bibliography style (.bst file) for those users of BibTeX, to do all the hard work for you.

          It can achieve this because of the nature of the .bib database, where all the information about your references is stored in a structured format, but nothing about style. This is a common theme in LaTeX in general, where it tries as much as possible to keep content and presentation separate.

          A bibliography style file () will tell LaTeX how to format each attribute, what order to put them in, what punctuation to use in between particular attributes etc. Unfortunately, creating such a style by hand is not a trivial task. Which is why (also known as custom-bib) is the tool we need.

          can be used to automatically generate a .bst file based on your needs. It is very simple, and actually asks you a series of questions about your preferences. Once complete, it will then output the appropriate style file for you to use.

          It should be installed with the LaTeX distribution (otherwise, you can download it) and it's very simple to initiate. At the command line, type:

          latex makebst

          LaTeX will find the relevant file and the questioning process will begin. You will have to answer quite a few (although, note that the default answers are pretty sensible), which means it would be impractical to go through an example in this tutorial. However, it is fairly straight-forward. And if you require further guidance, then there is a comprehensive manual available. I'd recommend experimenting with it and seeing what the results are when applied to a LaTeX document.

          If you are using a custom built .bst file, it is important that LaTeX can find it! So, make sure it's in the same directory as the LaTeX source file, unless you are using one of the standard style files (such as plain or plainnat, that come bundled with LaTeX - these will be automatically found in the directories that they are installed. Also, make sure the name of the file you want to use is reflected in the command (but don't include the extension!).

          Localizing bibliography appearance[edit]

          When writing documents in languages other than English, you may find it desirable to adapt the appearance of your bibliography to the document language. This concerns words such as editors, and, or in as well as a proper typographic layout. The package can be used here. For example, to layout the bibliography in German, add the following to the header:


          Alternatively, you can layout each bibliography entry according to the language of the cited document:

          The language of an entry is specified as an additional field in the BibTeX entry:


          For to take effect, a bibliography style supported by it - one of , , , , , and - must be used:


          Showing unused items[edit]

          Usually LaTeX only displays the entries which are referred to with . It's possible to make uncited entries visible:

          \nocite{Name89}% Show Bibliography entry of Name89\nocite{*}% Show all Bib-entries

          Getting bibliographic data[edit]

          Many online databases provide bibliographic data in BibTeX-Format, making it easy to build your own database. For example, Google Scholar offers the option to return properly formatted output, which can also be turned on in the settings page.

          One should be alert to the fact that bibliographic databases are frequently the product of several generations of automatic processing, and so the resulting BibTex code is prone to a variety of minor errors, especially in older entries.

          Helpful tools[edit]

          See also: w:en:Comparison of reference management software
          • BibDesk BibDesk is a bibliographic reference manager for Mac OS X. It features a very usable user interface and provides a number of features like smart folders based on keywords and live tex display.
          • BibSonomy — A free social bookmark and publication management system based on BibTeX.
          • BibTeXSearch BibTeXSearch is a free searchable BibTeX database spanning millions of academic records.
          • Bibtex Editor - An online BibTeX entry generator and bibliography management system. Possible to import and export Bibtex files.
          • Bibwiki Bibwiki is a Specialpage for MediaWiki to manage BibTeX bibliographies. It offers a straightforward way to import and export bibliographic records.
          • cb2Bib The cb2Bib is a tool for rapidly extracting unformatted, or unstandardized bibliographic references from email alerts, journal Web pages, and PDF files.
          • Citavi Commercial software (with size-limited free demo version) which even searches libraries for citations and keeps all your knowledge in a database. Export of the database to all kinds of formats is possible. Works together with MS Word and Open Office Writer. Moreover plug ins for browsers and Acrobat Reader exist to automatically include references to your project.
          • CiteULike CiteULike is a free online service to organise academic papers. It can export citations in BibTeX format, and can "scrape" BibTeX data from many popular websites.

          1. Negation and opposition in natural language

          1.1 Introduction

          Negation is a sine qua non of every human language, yet is absent from otherwise complex systems of animal communication.[1] While animal “languages” are essentially analog systems, it is the digital nature of the natural language negative operator, represented in Stoic and Fregean propositional logic as a one-place sentential connective toggling the truth value of statements between T[rue] and F[alse] (or 1 and 0) and applying recursively to its own output, that allows for denial, contradiction, and other key properties of human linguistic systems.

          The simple syntactical nature of logical negation belies the profoundly complex and subtle expression of negation in natural language, as expressed in linguistically distinct categories and parts of speech (adverbs, verbs, copulas, quantifiers, affixes). As will be partly explored here (see also Horn 1989, Ladusaw 1996, Pullum 2002), the investigation of the form and meaning of negative expressions in English and other languages and of the interaction of negation with other operators (including multiple iterations of negation itself) is often far from simple, extending to scope ambiguities (Everybody didn’t leave), negative incorporation into quantifiers and adverbs (nobody, never, few), neg-raising (I don’t want to go = “I want not to go”), and the widespread occurrence of negative polarity items (any, ever, lift a finger) whose distribution is subject to principles of syntax, semantics, and pragmatics. At the core of the mental faculty of language, negation interacts in significant ways with principles of morphology, syntax, logical form, and compositional semantics, as well as with processes of language acquisition and sentence processing, whence the prominent role played by work on negation in the development of logic, semantics, linguistic theory, cognition, and psychoanalytic and literary theory.

          What sort of operation is negation? In the Categories and De Interpretatione, Aristotle partitions indicative-mood declarative sentences into affirmation and negation/denial (apophasis from apophanein “deny, say no”), which respectively affirm or deny something about something (De Int. 17a25). As a mode of predication, the “predicate denial” of Aristotelian term logic, while resulting in wide-scope negation opposed in truth value to the corresponding affirmative, is syntactically distinct from the unary “it is not the case that” connective of Stoic and Fregean logic.

          By combining subject and predicate to form a proposition, this approach can be seen as offering a more natural representation of ordinary language negation than the standard iterating operator that applies to fully formed propositions (Geach 1972; Englebretsen 1981; Horn 1989, Chap. 7; Sommers and Englebretsen 2000). Indeed, the syncategorematic negation of Montague Grammar (Montague 1973; cf. the entry on Montague semantics) is itself a means of connecting a term phrase subject with a predicate or IV (intransitive verb) phrase and thus fails to apply to its own output (see Horn 1989, §7.2 on “Aristotle as a Montague grammarian”). Cross-linguistically, the structural reflex of sentence-scope negation may be a free-standing adverb (German nicht, English not), a bound inflectional form (Japanese -na-, English -n’t), or a verb (Finnish en, ei).[2]

          Where we do not find negation is in the one place propositional logic would lead us to look, sentence- or clause-peripheral position. Furthermore, unlike speech act types (e.g., interrogation), negation never seems to be marked in natural language by a global intonation contour. Typically, sentence negation is associated directly on or near the main finite verb or predicate expression.

          1.2 Negation in natural language: markedness and asymmetry

          It has often been observed that the logical symmetry of negative and affirmative propositions in logic belies a fundamental asymmetry in natural language. It was Plato who first observed, in The Sophist, that negative sentences are less valuable than affirmative ones, less specific and less informative. The ontological, epistemological, psychological, and grammatical priority of affirmatives over negatives is supported by Aristotle:

          The affirmative proposition is prior to and better known than the negative (since affirmation explains denial just as being is prior to not-being) (Metaphysics 996b14–16)

          and St. Thomas Aquinas:

          The affirmative enunciation is prior to the negative for three reasons… With respect to vocal sound, affirmative enunciation is prior to negative because it is simpler, for the negative enunciation adds a negative particle to the affirmative. With respect to thought, the affirmative enunciation, which signifies composition by the intellect, is prior to the negative, which signifies division… With respect to the thing, the affirmative enunciation, which signifies to be, is prior to the negative, which signifies not to be, as the having of something is naturally prior to the privation of it. (St. Thomas, Book I, Lesson XIII, cited in Oesterle 1962, 64)

          Not only are negative statements (e.g., “Paris isn’t the capital of Spain”) generally less informative than affirmatives (“Paris is the capital of France”), they are morphosyntactically more marked (all languages have negative markers while few have affirmative markers)[3] and psychologically more complex and harder to process (see Just and Carpenter 1971, 248–9; and other work reviewed in Horn 1989, Chapter 3). Many philosophers, linguists, and psychologists have situated this asymmetry in logic or semantics, as in the claim that every negation presupposes a corresponding affirmative (but not vice versa).

          The strong asymmetricalist position leads to the “paradox of negative judgment”: if a positive statement refers or corresponds to a positive fact, to what state of affairs does a negative statement refer or correspond? What in fact is a negative fact? For Bergson (1911, 289), negation is necessarily “of a pedagogical and social nature”; for Wood (1933, 421) it is “infected with error and ignorance”. According to Wittgenstein (1953, §447), “the feeling is as if the negation of a proposition had to make it true in a certain sense in order to negate it”. Givón (1978: 70) points to the discourse presuppositionality of utterances like “My wife is not pregnant”. Psycholinguistic studies have shown that negation is easier to process when the denied proposition, if not already in the discourse model, is at least a plausible addition to it (e.g., “The whale is not a fish/#bird”; cf. Wason 1965; Horn 1989, Chapter 3).

          Beyond its marked status, negation has also been analyzed variously as a modality, a propositional attitude, and a speech act. The danger here is putting the pragmatic cart before the semantic horse. For example, not every negation is a speaker denial (in making this point, Frege points to the non-denial nature of embedded negation as in “If not-\(p\) then \(q\)”), nor is every speaker denial a linguistic negation. Given the repeated attempts over the centuries to liquidate or tame it—negation as positive difference, as dissimilarity or incompatibility, as falsity, as an admission of epistemic impoverishment, as the speech act of denial—and its resilience in surviving these attacks, negation qualifies as the Rasputin of the propositional calculus.

          But the prototypical use of negation is indeed as a denial of a proposition attributable to, or at least considered by, someone relevant to the discourse context. While affirmation standardly introduces a proposition into the discourse model, negation—in its “chief use” (Jespersen 1917, 4), its “most common use” (Ayer 1952, 39), its “standard and primary use” (Strawson 1952, 7)—is directed at a proposition that is already in or that can be accommodated by the discourse model.

          1.3 Matters of scope

          If we think of negation as essentially a means for opposition—the impossibility of simultaneously endorsing two incompatible options (see the entries on contradiction and the traditional square of opposition)—propositional negation is not necessarily privileged. This view is formally implemented in the Boolean algebraic model of Keenan and Faltz, on which negation is a cross-categorial operation, as are the binary connectives:

          We can directly interpret conjunctions, disjunctions, and negations in most categories by taking them to be the appropriate meet, join, and complement functions of the interpretations of the expressions conjoined, disjoined, or negated. The sense in which we have only one and, or, and not is explicated on the grounds that they are always interpreted as the meet, join, and complement functions in whatever set we are looking at. (Keenan and Faltz 1985, 6)

          Treatments of English and other languages frequently posit negative operators whose scope is narrower than the sentence or clause. This tradition dates back to Aristotle, for whom the predicate term negation in Socrates is not-wise, affirming that the predicate not-wise holds of Socrates, yields a false statement if Socrates does not exist, while the predicate denial Socrates isn’t wise denies that the predicate wise holds of Socrates and is true if Socrates does not exist. For Jespersen (1917), the subclausal “special” negation as in Nobody came, where “the negative notion…belong[s] logically to one definite idea”, is opposed to “nexal” negation, applying to “the combination of two ideas”, typically the subject-predicate nexus. Later linguists usually follow Klima (1964) and Jackendoff (1969) in allowing for constituent negation (e.g., verb phrase negation in You can [not go]) alongside sentential negation (You cannot go), utilizing various grammatical and semantic diagnostics for distinguishing the two varieties.

          A syntactic correlate of the distinction between wide- (sentential) versus narrow-scope (constituent) negation in English is that only when the negative element has clausal scope, as in the (a) examples in (1)-(3), can it trigger negative inversion. In the corresponding (b) examples, the scope of negation does not extend beyond the fronted phrase, whence the exclusion of ever, a satellite of negation (negative polarity item).[4]

          • (1)a. With no job will I be happy. [= I won’t be happy with any job]
          • b. With no job I will be happy. [= I will be happy without any job]
          • (2)a. In no clothes does Robin look good.
          • b. In no clothes Robin looks good.
          • (3)a. At no time were we (ever) alone together in the Oval Office.
          • b. In no time we were (*ever) alone together in the Oval Office.

          Negation also interacts in complicated and often surprising ways with quantification and modality. Perhaps the most analyzed interaction is with universal quantification. Despite the locus classicusAll that glitters is not gold and similar examples in French, German, and other languages, the wide scope of negation over universal subjects (or in cases like All the boys didn’t leave, the possibility of such readings, depending on the speaker, the intonation contour, and the context of utterance) is often condemned by purists, yet is not as illogical as it may appear (Horn 1989, §3.4).

          1.4 Contrariety and contradiction

          Negation as such is often semantically restricted to contradictory opposition between propositions, in which \(\neg A\) can be paraphrased (if not necessarily syntactically represented) as “it is not the case that \(A\)”. As introduced in Aristotle’s Categories (11b17), the genus of opposition (apophasis) is divided into species that include contrariety and contradiction. Contradictory opposites, whether affirmative and negative counterparts of a singular predication (Socrates is wise/Socrates isn’t wise) or quantified expressions (All pleasure is good/Some pleasure is not good), are mutually exhaustive as well as mutually exclusive, while contrary opposites (Socrates is wise/Socrates is unwise; All pleasure is good/No pleasure is good) do not mutually exhaust their domain. Contraries cannot be simultaneously true, though they may be simultaneously false. Members of a contradictory pair cannot be true or false simultaneously; contradictories “divide the true and the false between them” (see the entries on contradiction and the traditional square of opposition).

          Contrary terms (enantia) come in two varieties (Cat. 11b38ff.). In immediate or logical contraries (odd/even, sick/well), a true middle—an entity satisfying the range of the two opposed terms but falling under neither of them—is excluded, e.g., an integer neither odd nor even. But mediate contrary pairs (black/white, good/bad) allow for a middle—a shirt between black and white, a man or an act neither good nor bad. Neither mediate nor immediate contraries fall under the Law of Excluded Middle [LEM] (tertium non datur).

          For immediate contraries formed by narrow-scope predicate term negation, the rendering \(a\) is not-\(F\) in the traditional quasi-English phrasing corresponds to what Aristotle expresses through word order, utilizing the distinction between e.g., einai mê leukon “to be not-white” and mê einai leukon “not to be white” (Prior Analytics I 51b10). For Aristotle, \(a\) is neither \(F\) nor not-\(F\) can be true if \(a\) doesn’t exist (Santa is neither white nor not-white) or isn’t the kind of thing that can be F (The number 7 is neither white nor not-white), given that not-\(F\) is taken to affirm the negative property non-\(F\)-ness of the subject rather than denying a positive property.

          Other cases in which apparent contradictories can be seen as contraries, and thus immune from any application of LEM, are future contingents (There will be/will not be a sea battle tomorrow; cf. De Int. Chapter 9) and, in more recent work (Alxatib and Pelletier 2011, Ripley 2011a), vague predications. Thus a is neither F nor not-F is often judged true when F is a vague predicate (bald, rich, tall), although in the latter case speakers may also be willing to affirm that a is both F and not-F, which complicates matters (see the entries on future contingents and vagueness).

          1.5 Negation, presupposition, and singular terms

          In his classic paper on sense and reference, Frege (1892) argues that both (4a) and its contradictory (4b) presuppose that the name Kepler has a denotation. Every affirmative or negative sentence with a singular subject (name or description) presupposes the existence of a unique referent for that subject; if the presupposition fails, no assertion is made in (4a,b).

          • (4)a.Kepler died in misery.
          • b.Kepler did not die in misery.

          But this presupposition is not part of the content of the expression, and hence (4a) does not entail existence, or the negation of (4a) would not be (4b) but Kepler died in misery or the name “Kepler” has no reference, an outcome Frege seems to have taken as an absurdity but one that prefigures the later emergence of a presupposition-cancelling external or exclusion negation.

          Unwilling to countenance the truth-value gaps incurred on Frege’s analysis, Russell (1905, 485) reconsiders the status of contradictory negation with vacuous subjects:

          By the law of the excluded middle, either “A is B” or “A is not B” must be true. Hence either “the present king of France is bald” or “the present king of France is not bald” must be true. Yet if we enumerated the things that are bald and the things that are not bald, we should not find the king of France on either list. Hegelians, who love a synthesis, will probably conclude that he wears a wig.

          To resolve this (apparent) paradox while preserving a classical analysis in which every meaningful sentence is true or false, Russell banishes singular terms like the king of France from logical form, unpacking (5) and (6) as existentially quantified sentences despite their superficial subject-predicate syntax.

          • (5)The king of France is bald.
          • (6)The king of France is not bald.

          On Russell’s theory of descriptions, (5) can be represented as (5′), the (false) proposition that there is a unique entity with the property of being king of France and that this entity is bald, while (6) is ambiguous, depending on the scope of negation.

          • (5′)\(\exists x(Kx \wedge \forall y(Ky \rightarrow y=x) \wedge Bx))\)
          • (6′)\(\exists x(Kx \wedge \forall y(Ky \rightarrow y=x) \wedge \neg Bx))\)
          • (6″)\(\neg \exists x(Kx \wedge \forall y(Ky \rightarrow y=x) \wedge Bx))\)

          (6′), with narrow-scope (“internal”) negation, is the proposition that there is a unique and hirsute king of France, which is “simply false” in the absence (or oversupply) of male French monarchs. In (6″), on the other hand, the description the king of France falls within the scope of external negation and yields a true proposition. Unlike (6′), (6″) fails to entail that there is a king of France; indeed, the non-existence of a king of France guarantees the truth of (6″). This reading is more naturally expressed with the fall-rise contour and continuation characteristic of metalinguistic negation (Horn 1989) as in (7):

          • (7)The king of France isn’t \(^{{\rm v}}\)BALD—there ISN’T any king of France!

          For Strawson (1950, 1952), negation normally or invariably leaves the subject “unimpaired”. Strawson tacitly lines up with Frege and against Russell (and Aristotle) in regarding negative statements like (4b) and (6) as unambiguous and necessarily presuppositional. Someone who utters (6) does not thereby assert (nor does her statement entail) that there is a king of France. Rather, (6)—along with its affirmative counterpart (5)—presupposes it. If this presupposition fails, a statement may be made but the question of its truth value fails to arise.

          While many analysts (e.g., Wilson 1975, Atlas 1977, Gazdar 1979, Grice 1989) have since followed Russell by preserving a bivalent semantics and invoking pragmatic explanations of apparent presuppositional effects, other linguists and philosophers (e.g., Fodor 1979, Burton-Roberts 1989, von Fintel 2004) have defended and formalized theories of semantic presupposition in the Frege-Strawson spirit, allowing for the emergence of truth-value gaps or non-classical truth values when presuppositions are not satisfied.

          Non-bivalent logics of semantic presupposition, dating back to Łukasiewicz (1930) and Kleene (1952), generally posit (at least) two not-operators, the distinction arising lexically rather than (as for Russell) scopally; see the entry on many-valued logic and Section 2 below. The ordinary, presupposition-preserving internal or choice negation is the only one countenanced by Frege and Strawson; on this reading, Santa is not white, like Santa is white, is neither true nor false, given that Santa does not exist. The presupposition-cancelling or exclusion negation always determines a classical value. With exclusion negation, Santa is not white (or perhaps more plausibly It is not the case that Santa is white) is true even if there is no Santa. Thus there is no excluded middle; any affirmation and its corresponding exclusion negation are contradictories rather than contraries (see the entry on presupposition for elaboration and further details).

          1.6 From contradiction to contrariety: pragmatic strengthening of negation

          In his dictum, “The essence of formal negation is to invest the contrary with the character of the contradictory”, Bosanquet (1888) encapsulates the widespread tendency for formal contradictory (wide-scope) negation to be semantically or pragmatically strengthened to a contrary.

          We use ©\(A\) to represent any contrary of \(A\). Following the Aristotelian theory of opposition, the two contradictories \(A\) and \(\neg A\) cannot both be false, just as they cannot both be true, while a given proposition and a contrary of that proposition, \(A\) and ©\(A\), can both be false, although they cannot both be true. (Others have used \(\kappa\) or R for one-place non-truth-functional contrariety connectives; cf. McCall 1967, Humberstone 2005; see also Bogen 1991 for the distinction between linguistic and metaphysical contraries.) It should be noted that while \(\neg\) is an operator that takes one proposition into another, © is not, since a given proposition may have logically distinct contraries, while this is not the case for contradictories. Geach (1972, 71–73) makes this point with the example in (8). While (8a) has two syntactically distinct contradictories, e.g., Not every cat detests every dog and It’s not every dog that every cat detests, any such co-contradictories of a given proposition will always have the same truth conditions. But (8a) allows two contraries with distinct truth conditions, (8b) and (8c).

          • (8)a.Every cat detests every dog.
          • b.No cat detests every dog.
          • c.There is no dog that every cat detests.

          Similarly, (9a) allows three non-identical contraries:

          • (9)a.I believe that you’re telling the truth.
          • b.I believe that you’re not telling the truth.
          • c.I don’t believe that you’re telling the truth or that you’re not; I haven’t made up my mind yet.
          • d.I don’t believe that you’re telling the truth or that you’re not: I haven’t given the matter any thought.

          Thus while we can speak of the contradictory of a proposition, Geach observes, we cannot (pace McCall 1967) speak of the contrary, but only of a contrary, of a proposition. As Humberstone (1986, fn. 6) points out in response to Geach’s critique of McCall, however, the lack of uniqueness “does not prevent one from exploring the logical properties of an arbitrarily selected contrary for a given statement”. For our purposes, the crucial logical properties of contrariety are that (i) the contradictory of a proposition \(A\) is not a contrary of \(A\) and that (ii) contrariety unilaterally entails contradiction:

          • (10)a.©\(A \vdash \neg A\)
          • b.\(\neg A \not \vdash\) ©\(A\)

          For McCall (1967), contrariety is a quasi-modal notion akin to logical impossibility, \(\Box \neg\), in that \(\Box \neg A\) entails \(\neg A\) but not vice versa, but as pointed out by an anonymous reviewer, there is no intrinsic modal component of contrariety; all that is necessary is that contrariety is a non-truth-functional one-place connective. (See Humberstone 1986, 2003, 2005; Bogen 1991; and Vakarelov 1989a for additional considerations.)

          The strengthening of contradictory negation, \(\neg A\), to a contrary, ©\(A\), typically instantiates the inference schema of disjunctive syllogism or modus tollendo ponens in (11):

          • (11)\(\begin{array}{l}A \vee B \\ \neg A \\ \hline B\end{array}\)

          While the key disjunctive premise is typically suppressed, the role of disjunctive syllogism can be detected in a variety of strengthening shifts in natural language where the disjunction in question is pragmatically presupposed in relevant contexts. Among the illustrations of this pattern are the following:

          • The tendency for negation outside the scope of (certain) negated propositional attitude predicates (e.g., a does not believe that \(p\)) to be interpreted as associated with the embedded clause (e.g., a believes that not-\(p\)); this is so-called “neg-raising”, to which we return below.

          • The tendency for a semantically contradictory negation of an unmarked positive value, whether affixal (\(x\) is unfair/unhappy) or clausal (I don’t like him), to be strengthened (as either an “online” or conventionalized process) to a contrary of the positive predication. As contraries, Chris is happy and Chris is unhappy allow an unexcluded middle, since Chris can be neither happy nor strictly unhappy; similarly, I don’t like him is generally understood as stronger than a mere assertion that it’s not the case that I like him.

          • The strengthening of a negated plural definite (The kids aren’t sleeping) or bare plural (Beavers don’t eat cheese) from a contradictory to a contrary of the corresponding affirmative. In each case, the negation is understood as inside the scope of the quantified subject.

          When there are only two alternatives in a given context, as in the case of neg-raising (as stressed by Bartsch 1973; cf. Horn 1978; Horn 1989, Chapter 5), the denial of one (I don’t believe it will rain) amounts to the assertion of the other (I believe it won’t rain). The relevant reasoning is an instance of the disjunctive syllogism pattern in (11), as seen in (12), where \(F\) represents a propositional attitude and \(a\) the subject of that attitude.

          • (12)\(\begin{array}{lll} F (a, p)\vee F (a,\neg p)\, && \mbox{[the pragmatically assumed disjunction]}\\ \underline{\neg F (a, p)} && \mbox{[the sentence explicitly uttered]}\\ F (a, \neg p) && \mbox{[the stronger negative proposition conveyed]} \end{array} \)

          The key step is the pragmatically licensed disjunction of contraries: if you assume I’ve made up my mind about the truth value of a given proposition \(p\) (e.g., “it will rain”) rather than being ignorant or undecided about it, then you will infer that I believe either \(p\) or \(\neg p\), and my denial that I believe the former (“I don’t think it will rain”) will lead you to conclude that I believe the latter (“I think it won’t rain”). (See Horn 1989, Chapter 5 for more on this phenomenon; Gajewski 2007 for a neo-Bartschian analysis; and Collins and Postal 2014 for a vigorous defense of a grammatical approach to neg-raising).

          The availability of strengthened contrary readings for apparent contradictory negation has long been recognized, dating back to classical rhetoricians of the 4th century on the figure of litotes, in which an affirmative is indirectly asserted by negating its contrary (Hoffmann 1987). Litotic interpretations tend to be asymmetrical: an attribution of “not happy” or “not optimistic” will tend to convey a contrary (in this case “rather unhappy” or “fairly pessimistic”), while no analogous virtual contrariety is normally signaled by “not sad” or “not pessimistic”, which are usually understood as pure contradictories. This asymmetry is ultimately a social fact arising from the desire to respect negative face (Ducrot 1973, Brown and Levinson 1987, Horn 1989).

          For Jespersen, the tendency reflected by the neg-raised interpretation of I don’t think that \(p\) not only illustrates the general strengthening to contrariety but also participates in a more general conspiracy in natural language to signal negation as early as possible. Additional effects of this “neg-first” principle (Horn 1989, 293; after Jespersen 1917, 5) range from diachronic shifts in the expression of sentential negation (see van der Auwera 2010) and the fronting and negative inversion in (1a) or (2a) to the emergence of ambiguities arising in contexts like [neg \(S_1\) because \(S_2\)] (Jespersen 1917, 48), as in “She didn’t marry him because he’s poor”, where the “illogical” scope reading—on which his poverty was the non-cause of the wedding rather than the cause of the non-wedding—can be rendered more or less accessible by the intonation contour.

          The “neg-raised” reading of I don’t think that \(p\) as “I think that not-\(p\)” has often been deplored as an illogical placement of negation, an unfortunate ambiguity, or (in Quine’s terms) an “idiosyncratic complication” of one language:

          the familiar quirk of English whereby “\(x\) does not believe that \(p\)” is equated to “\(x\) believes that not \(p\)” rather than to “it is not the case that \(x\) believes that \(p\)”. (Quine 1960, 145–6; similar claims are made by Hintikka, Deutscher, and others)

          But this “quirk” has deep roots.

          The locus classicus is St. Anselm’s Lambeth fragments (Henry 1967, 193–94; Hopkins 1972, 231–32; Horn 1989, 308ff.). Anselm points out that “non…omnis qui facit quod non debet peccat, si proprie consideretur”—not everyone who does what he non debet (“not-should”) sins, if the matter is considered strictly (with the contradictory reading of negation as the syntax suggests). The problem is that non debere peccare is standardly used to convey the contrary meaning debere non peccare rather than the literal contradictory (“it is not a duty to sin”). It is hard to stipulate e.g., non debet ducere uxorem (= “a man is free not to marry”) without seeming to commit oneself to the stronger debet non ducere uxorem, an injunction to celibacy (Henry 1967, 193ff.; Horn 1978, 200).

          For Henry (1967, 193, §6.412), Anselm’s observations on modal/negative interaction are “complicated by the quirks of Latin usage”. But far from a Quinean quirk of English and/or Latin usage, “neg-raising”—the lower-clause understanding of negation of a believe- or ought-type predicate—is distributed widely and systematically across languages and operators.

          The raised understanding is always stronger than the contradictory (outer) negation; it applies to a proper subset of the situations to which the contradictory applies (is true in a proper subset of possible worlds). Thus neg-raising, as Anselm recognized, yields a virtual contrariety: the compositional meaning is true but too weak, and the addressee recovers a conversational implicature to “fill in” the stronger proposition.

          In some cases, the strengthened or neg-raised contrary reading may become salient enough over time to block the literal interpretation, as when French Il ne faut pas partir—literally = “one needn’t leave” (an O vertex modal)—can now be used only to express the stronger proposition that one must not-leave (E vertex). This is a modal instance of the general phenomenon of O \(>\) E drift (Horn 1989), an upward shift along the right (negative) vertical of the modal square of opposition. Such squares were constructed by Cajetan, based on Aristotle’s De Interpretatione 21b10ff. and Prior Analytics 32a18–28 (see Oesterle 1962), and by other medieval commentators.

          Figure 1

          \(\mathbf{O}>\mathbf{E}\) drift is attested cross-linguistically in the meaning shift of lexical items like Old English nealles (lit. “NEG all”) = “not at all”, Dutch nimmer (lit., “NEG always”) = “never”, or Russian nel’zja (lit. “NEG must”) = “mustn’t”. The reverse shift, in which E forms develop O meanings, appears to be unattested (cf. Horn 2012).

          In litotes and neg-raising, the interpretation of formal contradictories as contraries arises from the accessibility of the relevant disjunction, triggering the disjunctive syllogism. The homogeneity or all-or-none presupposition (Fodor 1970, 158ff.) applying to bare plurals, plural definites, and mass predications results in a comparable effect; it is natural to strengthen negative statements like Mammals don’t lay eggs, The children aren’t sleeping, or I don’t eat meat to affirmations of contraries rather than understanding them as simple wide-scope negations of the corresponding positives (Mammals lay eggs, The children are sleeping, I eat meat) as would be the case with overtly quantified universals. The relevant principle has been variously formulated:

          When a kind is denied to have a generic property P\(_k\), then any of its individuals cannot have the corresponding individual-level property P\(_i\). (von Fintel 1997, 31)

          If the predicate P is false for the NP, its negation not-P is true for the NP… Whenever a predicate is applied to one of its arguments, it is true or false of the argument as a whole. (Löbner 2000, 239)

          Once again the key step is establishing the relevant disjunction as a pragmatically inferred instance of the Law of Excluded Middle, e.g., “Either mammals lay eggs or mammals don’t lay eggs”. In fact, this practice was first identified by Aristotle (Soph. Elen. 175b40–176a17), who offered an early version of the all-or-none (or both-or-neither) in arguing that a negative answer to a “dialectical” or conjoined question like “Are Coriscus and Callias at home?” would imply that neither is at home, given the default supposition that they are either both in or both out. Once again LEM applies where it “shouldn’t”; \(A \vee © A\) behaves as though it were an instance of \(A \vee \neg A\), triggering the disjunctive syllogism:

          • (13)\(\begin{array}{l} (Fa \wedge Fb) \vee (\neg Fa \wedge \neg Fb)\\ \neg (Fa \wedge Fb)\\ \overline{(\neg Fa \wedge \neg Fb)} \end{array}\)

          1.7 Privation, affixal negation, and the markedness asymmetry

          For Aristotle, privation is an instance of opposition defined in terms of the absence or presence of a default property for a given subject:

          We say that that which is capable of some particular faculty or possession has suffered privation [sterêsis] when the faculty or possession in question is in no way present in that in which, and at the time in which, it should be naturally present. We do not call that toothless which has not teeth, or that blind which has not sight, but rather that which has not teeth or sight at the time when by nature it should. (Categories 12a28–33)

          A newborn kitten, while lacking sight, is thus no more “blind” than is a chair, nor is a baby “toothless”.

          Privation as the absence of what would be expected by nature to be present is revisited in the Metaphysics (1022b23–1023a8), where Aristotle—noting that privation can range over predictable absence, accidental removal, or deliberate “taking away by force” of the relevant property—distinguishes privation “with respect to genus”, as in the blindness of moles, from privation “with respect to self”, as in the blindness or toothlessness of an old man. In the end, Aristotle concedes, there may be as many senses of privation as there are a- prefixed terms in Greek (Met. 1022b33). Indeed, privation may be reanalyzed as the primary contrariety (1055a34).

          In a wide range of languages, affixal negation on simplex bases reflects Aristotelian privation, whence the asymmetry between possible forms (unhappy, untrue, unkind) and impossible or unlikely ones (unsad, unfalse, uncruel). We can describe a failed comedy, but not a successful tragedy, as unfunny. As Jespersen (1917, 144) observes, the tendency of semi-productive negative affixation to be restricted to unmarked or positive bases combines with that of the preference for contrariety we have reviewed:

          The modification in sense brought about by the addition of the prefix is generally that of a simple negation: unworthy = “not worthy”, etc… The two terms [X, unX] are thus contradictory terms. But very often the prefix produces a “contrary” term or at any rate what approaches one: unjust generally implies the opposite of just; unwise means more than not wise and approaches foolish, unhappy is not far from miserable, etc.

          The counter-expectation property of affixal negation extends even to contradictory, middle-excluding adjectives like alive/dead; nothing can be both and nothing capable of being either can be “in between”. But undead has been around since Bram Stoker’s Dracula (1897) as both an adjective and a zero-derived occupational noun to describe zombies, vampires, and other creatures that are “not quite dead but not fully alive, dead-and-alive” (OED). Someone or something is undead—e.g., a vampire—if it fails to conform to one’s expectation that it should be dead. But if something appears to be alive but does not quite fulfill that expectation, it is not undead but unalive, e.g., artificial flowers. Both the undead (but not quite alive) vampire and the unalive (but not dead) artificial flowers conform to Aristotle’s notion of a privative opposite, in lacking a property associated by default rules with the respective subject.

          The marked status of negative utterances has also been invoked to motivate an asymmetry in the geometry of lexicalization. Within the Square of Opposition, the Aristotelian relations of contradiction, contrariety, and subalternation are supplemented with an additional relation of subcontrariety, so called because the subcontraries are located under the contraries. As the contradictories of the two contraries, the subcontraries (e.g., Some pleasure is good, Some pleasure is not good) can both be true, but cannot both be false. For Aristotle, this was therefore not a true opposition, since subcontraries are “merely verbally opposed” (Prior Analytics 63b21–30). In pragmatic terms, the assertion of one subcontrary (Some men are bald) is not only compatible with, but actually conversationally implicates, the other (Some men are not bald), given Grice’s Maxim of Quantity (“Make your contribution as informative as is required”; see the entries on Paul Grice, pragmatics, and implicature). The fact that the two members of a subcontrary pair tend to be equipollent or mutually derivable in a given context may explain the fact that only one of the two subcontraries will lexicalize in natural language, and the markedness of negation explains why this is always the positive (I vertex, e.g., some) and not the negative (O vertex, e.g., no) value (Horn 1989, 2012). Thus the E values none, nor, and never are possible but the corresponding O values *nall (= “not all”), *nand (“or not”), and *nalways are never attested.

          1.8 Double negation

          1.8.1 “Logical” double negation

          When duplex negatio affirmat, what exactly does the double negation affirm? When a negative term is a contrary rather than a contradictory of the corresponding simple affirmative, to deny its application—Socrates isn’t a not-white log—does not result in the mutual annihilation of logical double negation, any more than does the negation of a mediate contrary (She’s not unhappy, It isn’t uncommon). While Aristotle countenanced multiple negation, to the extent of generating such unlikely sequences as Not-man is not not-just (De Int. 19b36), each proposition contains only one instance of negation as wide-scope predicate denial (juxtaposed here with both a negated subject term and a negated predicate term), since each categorical statement contains only one predicate.

          By contrast, the Stoics defined negation (apophatikon) as an iterating external operator. For Alexander of Aphrodisias, “Not: not: it is day differs from it is day only in manner of speech” (Mates 1953, 126). With their propositional connectives and one-place truth/falsity-toggling negation operator, it is the Stoics rather the Aristotelians who prefigured modern propositional logic, as well as the precepts of traditional grammar (“Duplex negatio affirmat”) and the Law of Double Negation.[5]

          Classical Fregean logic allows for but one negative operator, the contradictory-forming propositional operator applying to a proposition or open sentence, in keeping with “the thesis that all forms of negation are reducible to a suitably placed “it is not the case that”” (Prior 2006, 524). Not unexpectedly, Frege (1919, 130) proclaims the logical superfluity of double negation: “Wrapping up a thought in double negation does not alter its truth value”. Within this metaphor, \(\neg\neg A\) is simply a way of garbing the thought or proposition \(A\).

          But even a single sentence-external negation (Not: The sun is shining) is a logician’s construct rarely attested in the wild (Geach 1972; Katz 1977):

          [P]ropositional negation was as foreign to ordinary Greek as to ordinary English, and [Aristotle] never attained to a distinct conception of it. The Stoics did reach such a convention, but in doing so they violated accepted Greek usage; their use of an initial oukhi must have read just as oddly as sentences like “Not: the sun is shining” do in English. (Geach 1972, 75)

          Further, whether or not we admit the law of double negation in our logic,

          in ordinary language a doubly negated expression very seldom, if ever, has the same logical powers as the original unnegated statement. (Hintikka 1968, 47)

          It is thus worth noting that the system of dual negations described by Aristotle in Prior Analytics I, Chapter 46 is both insightful and internally consistent; its echoes are recognizable in Jespersen’s distinction between nexal negation (not happy) and special negation (unhappy), Von Wright’s (1959) distinction between weak (contradictory) versus strong (contrary) negation, and Jackendoff’s (1969) semantic reanalysis of Klima’s (1964) grammatical categories of sentential versus constituent negation. In each case, a negative marker whose scope is narrower than the proposition determines a statement logically distinct from a simple contradictory.

          If we represent the narrow-scope contrariety operator of It is not-white as ©\(A\), its contradictory, \(\neg\)©\(A\)(It isn’t not-white), does not return us to the simple positive \(A\). The result, if not the means, is similar to that in intuitionistic logic (Heyting 1956). The intuitionistic negation operator does not cancel out, given that the intuitionistic Law of Double Negation is valid in only one direction, \(A \rightarrow \neg \neg A\), while \(\neg \neg A \rightarrow A\) does not apply (see the entry on intuitionistic logic). Note too that the intuitionists posit just one negation operator that sustains double introduction but not double cancellation, while the Aristotelian system distinguishes contradictory (sentential) predicate denial from contrary (constituent) predicate term negation.

          In ordinary language, double negation (as opposed to negative concord as in I ain’t never done nothing to nobody, an agreement phenomenon in which only one semantic negation is expressed, addressed in the next section) tends not to cancel out completely. This is predictably the case when a semantic contrary is negated: not uncommon is weaker than common; one can be not unhappy without being happy. But even when an apparently contradictory negation is negated (from the unexceptionable it’s not impossible to the more unusual double-not of Homer Simpson’s concessive I’m not not licking toads [http://tinyurl.com/34jwhjz]), the duplex negatio of \(A\) doesn’t affirm \(A\), or at least provides a rhetorically welcome concealment, as Frege’s metaphor of “wrapping up a thought” in double negation might suggest. The negation in such cases (impossible, not-licking) is coerced into a virtual contrary whose negation, \(\neg\)©\(A\), is weaker than (is unilaterally entailed by) \(A\):

          Figure 2

          1.8.2 Negative concord and its relations

          In the previous section it was observed that when duplex negatio affirmat, what it affirms is often not simply the doubly negated proposition but the result of an incomplete cancellation yielded by the negation of an actual or virtual contrary (not unlikely, not impossible). But a more dramatic problem for the dictum is when duplex negatio negat, in particular in the form of negative concord, in which a single logical negation on the main verb spreads to indefinites and adverbs within the same clause (Labov 1972, Zeijlstra 2004, Penka 2011).[6]

          The grammar of negative concord is often complex and may be subject to a variety of factors. In standard Italian, for example, negative quantifiers following the main verb (whether as objects or postposed subjects) co-occur with mandatory negative marking on the verb to yield a single negative meaning, as in (14a). Such sentences express a simple negative meaning. But when a negative quantifier precedes the verb, negative concord is ruled out, as in (14b).[7]

          • (14)a.Gianni *(non) ha visto nessuno. “Gianni has seen nobody”
          • *(Non) ha telefonato nessuno. “Nobody has telephoned”
          • *(Non) ho parlato con nessuno. “I have spoken with nobody”
          • b.Nessuno (*non) ha visto Gianni. “Nobody has seen Gianni”
          • Con nessuno (*non) ho parlato. “With nobody have I spoken”

          Negative concord is a feature of many non-standard varieties of English, especially in informal speech—or music (“I can’t get no satisfaction”).[8]

          True negative concord within a given clause represents just one kind of hypernegation, the general phenomenon in which a negative marker reinforces rather than cancels the ordinary or canonical marker of sentence negation (Horn 2010). Hypernegation may extend across clause boundaries to result in the occurrence of “pleonastic” or “expletive” negative elements in the scope of inherently negative predicates. This is exemplified by the negative markers following comparatives, before clauses, or verbs of fearing in French, Russian, Yiddish, and other languages. A standard feature of earlier stages of English, pleonastic negation persists in informal English:

          • (15)a.I miss (not) seeing you around.
          • b.Don’t be surprised if it doesn’t rain. [= if it rains]
          • c.Not with my wife, you don’t.
          • d.The proposal will not be approved, I (don’t) think.

          The well-known problems encountered in processing multiple negations, verified in many psycholinguistic studies, are responsible for the appearance of other uninterpreted negations as in (16a), and the conventionalized irony or sarcasm exemplified in (16b):[9]

          • (16)a.No head injury is too trivial to ignore.
          • b.I could care less.

          Similarly, in French the expression Vous n’êtes pas sans ignorer que …, literally “You are not without being ignorant that …”, is notoriously used in the sense of “You certainly know that …”. If duplex negatio affirmat, triplex negatio confundit.

          1.9 Negative polarity

          Certain linguistic expressions in English and other languages are “polarity sensitive”, restricted in their distributions to the scope of negation or semantically analogous operators, including negative quantifiers, implicitly negative predicates or adverbs, the antecedents of conditionals, comparative clauses, and the restrictors of universals:

          • (17)a.I \(\{\)haven’t/*have\(\}\) ever eaten any kumquats at all.
          • b.\(\{\)Few/*Many\(\}\) of the assignments have been turned in yet.
          • c.The dean \(\{\)rarely/*often\(\}\) lifts a finger to help students on probation.
          • d.I \(\{\)doubt/*believe\(\}\) they’re all that pleased with the proposal.
          • e.\(\{\)All/*Many\(\}\) customers who had ever purchased any of the affected items were (*ever) contacted.

          Negative polarity items (NPIs) like those highlighted in (17) are generally restricted to downward entailing or monotone decreasing contexts, those in which inferences from sets to subsets (but not vice versa) are valid (see Fauconnier 1975; Ladusaw 1980, 1996; Peters and Westerståhl 2006; and the generalized quantifiers entry). If I’ve eaten kumquats, I’ve eaten fruit, but not necessarily vice versa; this is an upward entailing (monotone increasing) environment. On the other hand, if I haven’t eaten fruit, I haven’t eaten kumquats, but not necessarily vice versa; this is a downward entailing (monotone decreasing) environment.[10] It is just in the latter case that NPIs are licensed.

          As (17e) shows, universals like all or every license NPIs in their restrictor (the relative clause), which is a downward entailing context (if everyone who knows logic is a vegetarian, everyone who knows classical logic is a vegetarian). But universals do not license NPIs in their nuclear scope (the predicate expression), which is an upward entailing context (if everyone who knows logic is a vegan, everyone who knows logic is a vegetarian). This contrast indicates the insufficiency of an account of polarity licensing that simply marks a given lexical item as favorable to the occurrence of NPIs within its scope.

          While downward entailment may be (generally) necessary for the licensing of NPIs, it is not necessarily sufficient, depending on the nature of the context and the NPI in question. For example, some environments that permit weak NPIs like any and ever fail to license stricter ones like in weeks or until midnight.

          • (18)a.\(\{\)Nobody/Only Chris\(\}\) has ever proved any of those theorems.
          • b.\(\{\)Nobody/*Only Chris\(\}\) has been here in weeks.

          This has led to the development of more stringent algebraic conditions that some polarity items must meet, e.g., anti-additivity (Zwarts 1998). In fact, the distribution and licensing of polarity items is an important linguistic phenomenon but an extremely complex one, subject to widespread variation within and across languages; see van der Wouden 1996, Israel 2011, and Giannakidou 2011 for some complications and alternative views.

          1.10 Metalinguistic negation

          In addition to the overlapping dichotomies we have surveyed between grammatically and semantically defined varieties of negation within a given language (wide- vs. narrow-scope, sentential vs. constituent, contradictory vs. contrary, choice vs. exclusion), a “pragmatic ambiguity” has been invoked to distinguish ordinary descriptive negation from a specialized metalinguistic or echoic use (Horn 1989, chapter 6; Carston 1996; Geurts 1998; Pitts 2009).[11] In examples like (19), a speaker objects to a previous utterance on a variety of grounds, including its phonetic or grammatical form, register, or associated presuppositions or implicatures:

          • (19)a.Around here we don’t LIKE coffee-we LOVE it.
          • b.She doesn’t sell INsurance—she sells inSURance.
          • c.It’s not stewed bunny, honey, it’s civet de lapin.
          • d.I’m not HIS brother—he’s MY brother!
          • e.Mozart’s sonatas were for piano and violin, not for violin and piano.

          Seen as representing a “pragmatic ambiguity” of natural language negation, the descriptive/metalinguistic distinction is supported by converging linguistic diagnostics suggesting that metalinguistic negation operates on a different level, whence its failure to incorporate morphologically or license negative polarity items:

          • (20)a.I’m \(\{\)not happy/*unhappy\(\}\) with the plan, I’m ecstatic!
          • b.You didn’t eat \(\{\)some/*any\(\}\) of the cookies, you ate them all!

          2. The logic of negation

          The logic of negation may be presented in quite different ways, by considering various styles of proof systems (axiom systems, sequent calculi, systems of natural deduction, tableaux, etc.) or different kinds of semantics (algebraic, model-theoretic, proof-theoretic, game-theoretic, etc.). Moreover, in search of characteristics of negation as a one-place connective, several dimensions of classification are available, depending on the logical vocabulary of the language under consideration (propositional, first-order, multi-modal, etc.) and the inferential framework taken into account (single antecedents (i.e., premises) and conclusions, multiple antecedents or multiple conclusions, sets, multisets, or sequences of formulas in antecedent or succedent position).

          In a very elementary setting one may consider the interplay between just a single sentential negation \(\osim\) and the derivability relation \(\vdash\), as well as single antecedents and single conclusions. The following inferential principles are stated as proper rules with one derivability statement (sequent) or two such statements as assumption sequent(s) and a single sequent as the conclusion, or as axiomatic sequents without any assumption sequent:

          • (21) \(\begin{align*} A \vdash B &/ \osim B \vdash \osim A & \mbox{(contraposition)}\\ A &\vdash \osim \osim A & \mbox{(double negation introduction)}\\ \osim \osim A &\vdash A & \mbox{(double negation elimination)}\\ A \vdash B, \; A \vdash \osim B &/ A \vdash \osim C & (\text{negative} \textit{ ex falso})\\ A \vdash B, \; A \vdash \osim B &/ A \vdash C & (\text{unrestricted }\textit{ex falso})\\ A \vdash \osim B &/ B \vdash \osim A & \mbox{(constructive contraposition)}\\ \osim A \vdash B &/ \osim B \vdash A & \mbox{(classical contraposition) } \end{align*}\)

          The first rule, contraposition, for instance, says that if \(B\) is derivable from \(A\), then the negation of \(A\) is derivable from the negation of \(B\). All these rules and derivability statements are valid in classical logic (see the entry on classical logic); classical logic cannot distinguish between them. Some of these principles have been criticized and called into question in non-classical logic. The negated and unrestricted ex falso rules, for example, introduce an element of irrelevancy because they allow to derive a completely arbitrary formula \(C\), respectively a completely arbitrary negated formula \(\osim C\), form an assumption \(A\) if a formula \(B\) as well as its negation \(\osim B\) are derivable from \(A\), see the entries on relevance logic and paraconsistent logic. Classical contraposition has been criticized because it gives rise to non-constructive existence proofs in languages containing the existential quantifier, see the entry on intuitionistic logic. In richer vocabularies, additional negation principles can be formulated, regimenting the interaction between negation and other logical operations. Prominent examples are the De Morgan Laws. In languages without implication, one may consider the following De Morgan inference rules:

          • (22) \(\begin{align} (\osim A \vee \osim B) &\vdash \osim (A \wedge B)\\ \osim (A \vee B) &\vdash (\osim A \wedge \osim B)\\ (\osim A \wedge \osim B) &\vdash \osim (A \vee B)\\ \osim (A \wedge B) &\vdash (\osim A \vee \osim B)\\ \end{align}\)

          Whereas classical logic validates all of these rules, intuitionistic logic validates only the first three of them.

          The proof-theoretical characterization of negation is important for the use of negation connectives in derivations. To obtain a more comprehensive understanding of negation, however, the proof theory has to be supplemented by a semantics.

          2.1 Negation as a truth function

          In classical logic, the semantical principle of bivalence is assumed, saying that a formula has exactly one of two semantical values, namely either the value T[rue] or the value F[alse] (1 or 0), but not both. Negation, \(\osim \), is semantically characterized by the unary function \(f_{\osim }\) on the set \(\{1, 0\}\), defined by the following truth table:

          \[ \begin{array}{c|c} f_{\osim } & \\ \hline 1 & 0 \\ 0 & 1 \end{array} \]

          That is, if \(A\) is a formula, then \(\osim A\) is false if \(A\) is true, and \(\osim A\) is true if \(A\) is false. The function \(f_{\osim }\) is said to be a truth function because it is a function defined on the set of classical truth values \(\{1, 0\}\), see the entry on truth values.

          If negation is meant to express semantic opposition, it is clear that the remaining two-valued truth functions fail to characterize any plausible notion of semantic opposition between \(A\) and \(\osim A\):

          \[ \begin{array}{ccc} \begin{array}{c|c} f_{\textit{id}} & \\ \hline 1 & 1 \\ 0 & 0 \end{array} & \begin{array}{c|c} f_{\top} & \\ \hline 1 & 1 \\ 0 & 1 \\ \end{array} & \begin{array}{c|c} f_{\perp} & \\ \hline 1 & 0 \\ 0 & 0 \end{array} \end{array} \]

          However, if a distinction is already drawn between contradictory-forming and contrary-forming sentential negations, the ground is prepared for a pluralism with respect to negation seen as a unary connective. One might think of obtaining different concepts of negation by letting the negations interact with other logical operations in various ways, but this does not help concerning atomic formulas that do not contain any logical operation.

          There are several ways of generalizing the semantics and making room for additional sentential negations. One comes with giving up bivalence and admitting sets of truth values (truth degrees) with more than two elements, see the entry on many-valued logic. In the so-called Łukasiewicz many-valued logics, the set of values is either the whole real unit interval [0,1] or a finite set of rational numbers from [0,1], including 1 as the designated value representing True. Łukasiewicz negation \(\osim \) is defined by setting \(f_{\osim }(u) = 1- u\). Negation is thus understood in terms of subtraction from the numerical representation of True. In so-called Gödel many-valued logics, the truth function \(f_{\osim }\) for negation \(\osim \) is defined by setting \(f_{\osim }(u) = 1\) if \(u = 0\), and \(f_{\osim }(u) = 0\) if \(u \not = 0\). Here negation is understood in terms of the numerical representation of True and distinctness from the numerical representation of False.

          In Kleene’s (strong) three-valued logic K3, with \(i\) as a third value in addition to 0 and 1, the truth function \(f_{\osim }\) for negation \(\osim \) is defined by the following truth table:

          \[ \begin{array}{c|c} f\osim & \\ \hline 1 & 0 \\ i & i \\ 0 & 1 \\ \end{array} \]

          In K3 a formula \(A\) and its negation \(\osim A\) cannot both be true in the sense of both taking the designated value 1, but they both fail to be true if \(A\) receives the value \(i\). If a contrary pair of formulas is defined as a pair of formulas that cannot both be true but can both fail to be true, Kleene negation gives rise to contrary pairs.

          Falsity (understood as receiving the value 0) and non-truth (understood as taking a value different form 1) fall apart in K3. As a result, contraposition fails in K3. Another example of a logic with a non-contraposable negation is Priest’s Logic of Paradox, LP, see the entry on paraconsistent logic. If in K3 or in LP an implication \((A \supset B)\) is defined as material implication \((\osim A \vee B)\), then contraposition holds in the sense that \((A \supset B)\) entails \((\osim B \supset \osim A)\).

          The “internal”, presupposition-preserving negation \(\osim \) in K3 differs from the external, presupposition-cancelling negation \(\neg\) in Bochvar’s three-valued logic B3 by always returning a classical value. The truth function \(f_{\neg}\) is defined by the following table:

          \[ \begin{array}{c|c} f_{\neg} & \\ \hline 1 & 0 \\ i & 1 \\ 0 & 1 \\ \end{array} \]

          2.2 Negation as a modal operator

          Since modal operators are unary connectives and since there exist different notions of alethic necessity (necessary truth) and alethic possibility (possible truth), a rather natural question then is whether negations can be analyzed in a revealing way as modal operators, see the entry on modal logic.

          Very well-known modal logics are the normal modal logics that have a so-called possible worlds semantics making use of a two-place relation between possible worlds. Slightly less known are the classical (or congruential) modal logics (Segerberg 1971, Chellas 1980). The weakest requirement imposed on a necessity-like modal operator \(\Box\) in systems of classical modal logic is the congruence property: \[ \mathord{\vdash A \leftrightarrow B} \slashrel \mathord{\vdash \Box A \leftrightarrow \Box B}\] (“if \(A \leftrightarrow B\) is provable, then so is \(\Box A \leftrightarrow \Box B\)”). This property, however, is certainly not distinctive of negation.

          Classical modal logics have a semantics in terms of so-called minimal models, also known as neighbourhood models. A neighbourhood model is a structure \(\cal M\) \(=\) \((W, N, v)\), where \(W\) is a non-empty set of possible worlds, \(N\) is a function assigning to every \(w\) from \(W\) a set \(N(w)\) of subsets of \(W\), called neighbourhoods of \(w\), and \(v\) is a valuation function mapping atomic formulas to the set of worlds where they are true. Let \(\llbracket A\rrbracket\) be the set of worlds at which formula \(A\) is true. Then \( \Box A\) is defined to be true at a world \(w\) in model \(\cal M\) (in symbols: \({\cal M}, w \models \Box A\)) iff \(\llbracket A\rrbracket \in N(w)\).

          In Ripley 2009 it is suggested to use the neighbourhood semantics as a general framework for semantically capturing properties characteristic of negation connectives interpreted as a necessity operator \( \Box \), see also Yu 2010. Ripley points out, for example, that the contraposition rule \[ A \vdash B \slashrel \Box B \vdash \Box A\] is valid in a neighbourhood model \((W, N, v)\) iff for every \(w \in W\), \(N(w)\) is closed under subsets, i.e., if \(X \in N(w)\) and \(Y \subseteq X\), then \(Y \in N(w)\). It would be nice to have a convincing intuitive understanding of the neighbourhood function \(N\) in terms of a concept that explains some core aspects of negation. If \(\llbracket A\rrbracket \in N(w)\) is understood as saying that the proposition expressed by \(A\) is incompatible with world \(w\), then the above constraint emerges as reasonable because it says that if the set of worlds (the proposition) \(X\) is incompatible with \(w\) and proposition \(Y\) implies \(X\), then \(Y\) is incompatible with \(w\) as well. Whereas Ripley starts with a positive notion (\({\cal M}, w \models \Box A\) iff \(\llbracket A\rrbracket \in N(w)\)), in order to introduce a negation \(\osim \), one may also stipulate that \({\cal M}, w \models \osim A\) iff \(\llbracket A\rrbracket \not \in N(w)\), so as to obtain a connective that is more overtly a negative impossibility operator. The idea is that \(N(w)\) contains the sets of worlds compatible with \(w\), so that \(\llbracket A\rrbracket \not \in N(w)\) indicates that the proposition expressed by \(A\) is incompatible with \(w\). Negation as an “unnecessity” operator \(\neg\) in the sense of (“possibly not”) is then defined by \({\cal M}, w \models \neg A\) iff \(\overline{\llbracket A\rrbracket} \in N(w)\), where \(\overline{\llbracket A\rrbracket}\) is the complement of \(\llbracket A\rrbracket\) with respect to \(W\). As a result, \(\neg A\) is true at a state \(w\) iff the complement of the proposition expressed by \(A\) is compatible with \(w\).

          This semantics validates respective versions of congruence (\(\mathord{\vdash A \leftrightarrow B} \slashrel \mathord{\vdash \osim A \leftrightarrow \osim B}\) and \(\mathord{\vdash A \leftrightarrow B} \slashrel \mathord{\vdash \neg A \leftrightarrow \neg B}\)), but it does not yet impose any interesting constraints on negation. In order to exclude that for some world \(w\) and formula \(A\), both \(w \in \llbracket A\rrbracket\) and \(w \in \llbracket \osim A\rrbracket\), one has to stipulate that for every set of worlds \(X\), if \(w \in X\) then \(X \in N(w)\), which makes sense under the compatibility reading of the neighbourhood function \(N\) because it says that if \(X\) is true at \(w\), then \(X\) is compatible with \(w\). In order to validate contraposition, it has to be required that if \(X \subseteq Y\), then \(\{w\mid Y \not \in N(w)\}\) \(\subseteq\) \(\{w\mid X \not \in N(w)\}\). Under the compatibility reading of \(N\) this condition says that every world \(Y\) is incompatible with is also a world \(X\) is incompatible with, if proposition \(X\) implies proposition \(Y\).

          The relational semantics of normal modal logics, however, does come with a commitment to a substantial property of negation understood as impossibility or as unnecessity. The analysis of negation as a normal impossibility operator has been developed by Vakarelov (1977, 1989b) and Došen (1984, 1986, 1999) and has been further investigated in the algebraic setting of Michael Dunn’s gaggle theory (see Bimbó and Dunn 2008) by Dunn (1993, 1996, 1999) and Dunn and Zhou (2005). A relational model (or Kripke model) is a structure \(\cal M\) \(=\) \((W, R, v)\), where \(W\) is a non-empty set of information states, \(R\) is a two-place “accessibility” relation on \(W\), and \(v\) is a valuation function. Dunn denotes the accessibility relation by \(\bot\) (pronounced “perp”) and regards it as a relation of incompatibility or orthogonality between states. Negation as impossibility, denoted by \(\osim \), is then semantically defined by postulating that \(\osim A\) is true at a state \(w\) in model \(\cal M\) iff \(w\) is incompatible with all states \(u\) (from \(W\)) at which \(A\) is true: \({\cal M}, w \models \osim A\) iff (for every \(u\): \({\cal M}, u \models A \mbox{ implies } w \bot u).\) Alternatively, the relation \(R\) may be understood as a relation of compatibility between states, denoted by \(C\). \({\cal M}, w \models \osim A\) is then defined by requiring that for every \(u\): \(wCu \mbox{ implies } {\cal M}, u \not \models A.\) Negation as unnecessity, denoted by \(\neg\), is accordingly defined by the following clause: \({\cal M}, w \models \neg A\) iff (there exists \(u\) with \(wCu \mbox{ and } {\cal M}, u \not \models A)\).

          It proves useful to enrich the above relational semantics by another binary relation \(\leq\) on the set of states \(W\). The relation \(\leq\) is assumed to be a partial order (i.e., it is reflexive, transitive and anti-symmetric), which allows one to think of it as a relation of possible expansion of information states. With such a reading it is natural to assume that the truth of atomic formulas \(p\) is persistent with respect to \(\leq\): if \(w \leq u\) and \({\cal M},w \models p\), then \({\cal M},u \models p.\) The conditions on \(\leq\) and \(C\) and the truth conditions for compound formulas should then be such that persistence (also called heredity) holds for arbitrary formulas, in particular for negated formulas \(\osim A\) if negation as impossibility is considered. A compatibility model is a structure \((W,C,\leq, v)\), where \((W,C,v)\) is a Kripke model, \(\leq\) is a partial order on \(W\), and the following condition is satisfied, which guarantees the heredity of negated formulas \(\osim A\): if \(wCu\), \(w' \leq w\), and \(u' \leq u\), then \(w'Cu'\). This condition is a constraint on the compatibility frame \((W,C,\leq)\) on which a model \((W,C,\leq,v)\) is based. The condition is not only useful (as will become clear), but also plausible, because it says that two information states, expansions of which are compatible, are themselves compatible.

          We can now define that an inference (sequent) \(A \vdash B\) is valid in a compatibility model iff for every state \(w\) from that model, if \(A\) is true at \(w\), then so is \(B\); \(A \vdash B\) is called valid on a compatibility frame iff \(A \vdash B\) is valid in every model based on that frame. A rule is valid on a frame iff the validity of the premises inferences on that frame guarantees the validity of the conclusion inference on the frame. The contraposition rule from the list (21) is valid on any compatibility frame. If the order-inversion expressed by contraposition is seen as a fundamental property of negation, a hierarchy of stronger negations can be obtained syntactically by postulating further principles and semantically by characterizing these principles by means of conditions on compatibility frames \((W, C, \leq)\). This line of thought has led from a “kite” of negations in Dunn 1993 to “lopsided kites” of negations and an extended kite of negations in Shramko 2005, Dunn and Zhou 2005.

          In Dunn 1993, a negation operation satisfying the contraposition rule is called subminimal. The term “subminimal negation” had been introduced by Allen Hazen in an unpublished paper from 1992 for a richer language containing negation, conjunction, disjunction, and intuitionistic implication to denote a negation that fails to satisfy the intuitionistically valid De Morgan inference \((\osim A \wedge \osim B) \vdash \osim (A \vee B)\) and the classically but not intuitionistically valid \(\osim (A \wedge B) \vdash (\osim A \vee \osim B)\). Dunn’s use of the term “subminimal” is thus different from Hazen’s. In Dunn and Zhou 2005 only a single negation as impossibility is used, the vocabulary is enriched by conjunction and disjunction, and in both, the one with negation, conjunction, and disjunction as well as the one with negation only, subminimal negations are referred to as preminimal negations. Moreover, the minimal negations from Dunn 1993, 1996 are called quasi-minimal in Dunn and Zhou 2005, because they lack negative ex falso, a property of negation in Johansson’s so-called minimal logic, see Johansson 1936.

          If the compatibility relation is not assumed to be symmetric (although it may be argued that compatibility between states is a symmetrical relation), then one may distinguish between two negation operations \(\osim _1\) and \(\osim _2\) that are defined as follows:

          \[ \begin{align}{\cal M}, w &\models \osim _1 A \mbox{ iff } \forall u (wCu \mbox{ implies } {\cal M}, u \not \models A);\\ {\cal M}, w &\models \osim _2 A \mbox{ iff } \forall u (uCw \mbox{ implies } {\cal M}, u \not \models A). \end{align}\]

          The two negations form a so-called Galois connection, which means that \(A \vdash \osim _1 B \mbox{ iff } B \vdash \osim _2 A.\) The negations \(\osim _1\) and \(\osim _2\) are called Galois negations or split negations; they are both preminimal negations and satisfy the following interaction principles: \(A \vdash \osim _1 \osim _2 A\); \(A \vdash \osim _2 \osim _1 A.\)

          As noted in Dunn 1993, 1996, if contraposition is assumed, double negation introduction \(A \vdash \osim \osim A\) is mutually derivable with constructive contraposition \(A \vdash \osim B \slashrel B \vdash \osim A\), and if constructive contraposition is assumed, double negation elimination is mutually derivable with classical contraposition \(\osim A \vdash B \slashrel \osim B \vdash A\). The demonstrations use only reflexivity and transitivity of the derivability relation \(\vdash\). As a result, the above list of negation laws leads to the following unbalanced “kite” of negations (cf. Dunn and Zhou 2005):

          Figure 3

          The graphical arrangement in this diagram is to be understood as follows: If a sequent is assigned to a node \(n\) and node \(n'\) is placed below \(n\), then the inference assigned to \(n'\) is derivable with the aid of the sequent assigned to \(n\).

          Ortho negations satisfy all principles shown in the lopsided kite. An ortho negation in a logic with conjunction distributing over disjunction (or, equivalently, disjunction distributing over conjunction), is called a Boolean or classical negation. Boolean negation is uniquely determined in the sense that if \(\osim _1\) and \(\osim _2\) are Boolean negations, then \(\osim _1 A\) and \(\osim _2 A\) are interderivable; ortho negation is not uniquely determined, see Restall 2000.

          The negation principles of Dunn and Zhou’s lopsided kite correspond in the sense of modal correspondence theory to properties of compatibility frames. A rule \(r\) corresponds to a property \(E\) iff the rule \(r\) is valid on a compatibility frame just in case the frame satisfies \(E\). Greg Restall (2000) observed that double negation elimination corresponds to a property of both \(C\) and the relation of possible expansion of information states \(\leq\), the other negation principles have been shown to correspond to properties only of the compatibility relation \(C\), see Dunn 1996, Dunn and Zhou 2005, Berto 2014. In the following list, “&” denotes conjunction, “\(\Rightarrow\)” denotes Boolean implication, and “\(\forall\)” and “\(\exists\)” refer to universal and existential quantification, respectively, in the metalanguage:

          \[ \begin{align} A \vdash \osim \osim A & & \forall x \forall y (xCy \Rightarrow yCx) \\ A \vdash B, \; A \vdash \osim B &/ A \vdash \osim C &\forall x \forall y (xCy \Rightarrow xCx) \\ A \vdash B, \; A \vdash \osim B &/ A \vdash C &\forall x (xCx), \forall x \forall y (xCy \Rightarrow yCx)\\ \osim \osim A \vdash A & & \forall x \exists y(xCy \mathamp \forall z(yCz \Rightarrow z \leq x)) \end{align} \]

          The following first-order property of \(C\) alone also corresponds to double negation elimination: \[ \forall x \exists y (xCy \mathamp \forall z(yCz \Rightarrow (z = x))).\]

          We may observe that Dunn and Zhou’s lopsided kite of negations can be equilibrated, for example, by inserting the inference schema \(\osim \osim \osim A \vdash \osim A\). This schema corresponds to the following first-order condition on \(C\) (as calculated with the help of the SQEMA algorithm for computing first-order correspondences in modal logic due to Georgiev, Tinchev and Vakarelov (see Other Internet Resources): \[ \forall x \forall y (xCy \Rightarrow \exists z (xCz \mathamp \forall u (zCu \Rightarrow uCy))).\]

          Figure 4

          Negation as unnecessity gives rise to a dual lopsided kite of negations that can be combined with the lopsided kite into a “united” kite of negations, see Dunn and Zhou 2005. An even richer inclusion diagram of negations can be found in Ripley 2009.

          Whilst satisfying the contraposition rule \(A \vdash B \slashrel \osim B \vdash \osim A\) is a basic property of negation as a normal impossibility operator, there exist unary connectives that are referred to as negations, although they do not satisfy contraposition. Prominent examples of logics with a non-contraposable negation are Nelson’s logics N3, N4, and N4\(^{\bot}\) of constructive logic with so-called strong negation (see Nelson 1949; Gurevich 1977; Almukdad and Nelson 1984; Wansing 1993, 2001; Dunn 2000; Odintsov 2008). These logics contain intuitionistic implication as a primitive connective. Nelson (1959), however, also considers a variant of N3 with a contraposable strong negation. In this system S, the contraction axiom

          \[ (A \rightarrow (A \rightarrow B))\rightarrow (A \rightarrow B)\]

          is replaced by

          \[ (A \rightarrow (A \rightarrow (A \rightarrow B))) \rightarrow (A \rightarrow (A \rightarrow B)).\]

          This replacement avoids a collapse into classical logic. Strong negation is called “strong” because it captures a notion of negation as definite falsity and because in the system N3 the strong negation of a formula entails its intuitionistic negation. The conjunction, disjunction, and strong negation fragment of N4 coincides with the logic of first-degree entailment FDE, also known as Dunn and Belnap’s useful four-valued logic (Belnap 1977a,b; Dunn 1976). Interestingly, contraposition as stated above holds for FDE, whereas it fails in FDE for multiple-premise inferences (see Problem 7, Section 8.10, p. 162 in Priest 2008).

          The system FDE is a well-known system of relevance logic (see the entry on relevance logic) and it shares with other relevance logics the property of being a paraconsistent logic, see the entry on paraconsistent logic. Paraconsistent logics fail to satisfy the unrestricted ex falso rule, which is usually presented in a multiple-antecedent framework by the inference schema: \[ A, \osim A \vdash B.\]

          Genuine paraconsistent logics also fail to satisfy the restricted ex falso rule. Double negation elimination and classical contraposition fail to be valid in intuitionistic logic (see the entry on intuitionistic logic); if one of them is added to an axiom system of intuitionistic logic, one obtains a proof system for classical logic.

          2.3 Interactions with negation

          As already remarked, the classification of unary connectives as negations may depend on the presence or absence of other logical operations. If the propositional language to which a negation operation is added contains only conjunction and disjunction (and atomic formulas), a natural starting point is to assume that one is dealing with a so-called distributive lattice logic (cf. Dunn and Zhou 2005). A distributive lattice logic is a single-antecedent and single-conclusion proof system in the language with only conjunction \(\wedge\) and disjunction \(\vee\). In addition to reflexivity and transitivity of the derivability relation \(\vdash\), the following inferential schemata are assumed:

          • \(A \wedge B \vdash A\), \(A \wedge B \vdash B\),
          • \(A \vdash B\), \(A \vdash C\) / \(A \vdash (B \wedge C)\),
          • \(A \vdash C\), \(B \vdash C\) / \((A \vee B) \vdash C\),
          • \(A \vdash (A \vee B)\), \(B \vdash (A \vee B)\),
          • \((A \wedge (B \vee C)) \vdash ((A \wedge B) \vee (A \wedge C))\).

          Leave a Comment


          Your email address will not be published. Required fields are marked *