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Wednesday, August 2, 2023

GNU General Public License

GNU General Public License
AuthorRichard Stallman
Latest version3
PublisherFree Software Foundation
Published25 February 1989; 34 years ago
SPDX identifier
  • GPL-3.0-or-later
  • GPL-3.0-only
  • GPL-2.0-or-later
  • GPL-2.0-only
  • GPL-1.0-or-later
  • GPL-1.0-only
Debian FSG compatibleYes
FSF approvedYes
OSI approvedYes
CopyleftYes Binary or compilation-based
Linking from code with a different licenceSoftware licensed under GPL compatible licenses only, depending on the version used.
Websitewww.gnu.org/licenses/gpl.html 

The GNU General Public License (GNU GPL or simply GPL) is a series of widely used free software licenses that guarantee end users the four freedoms to run, study, share, and modify the software. The license was the first copyleft for general use and was originally written by the founder of the Free Software Foundation (FSF), Richard Stallman, for the GNU Project. The license grants the recipients of a computer program the rights of the Free Software Definition. These GPL series are all copyleft licenses, which means that any derivative work must be distributed under the same or equivalent license terms. It is more restrictive than the Lesser General Public License and even further distinct from the more widely used permissive software licenses BSD, MIT, and Apache.

Historically, the GPL license family has been one of the most popular software licenses in the free and open-source software domain. Prominent free software programs licensed under the GPL include the Linux kernel and the GNU Compiler Collection (GCC). David A. Wheeler argues that the copyleft provided by the GPL was crucial to the success of Linux-based systems, giving the programmers who contributed to the kernel the assurance that their work would benefit the whole world and remain free, rather than being exploited by software companies that would not have to give anything back to the community.

In 2007, the third version of the license (GPLv3) was released to address some perceived problems with the second version (GPLv2) which were discovered during the latter's long-time usage.

To keep the license up to date, the GPL license includes an optional "any later version" clause, allowing users to choose between the original terms or the terms in new versions as updated by the FSF. Software projects licensed with the optional "or later" clause include the GNU Project, while the Linux kernel, for instance, is licensed under GPLv2 only.

The "or any later version" clause is sometimes known as a lifeboat clause since it allows combinations between different versions of GPL licensed software to maintain compatibility.

History

The GPL was written by Richard Stallman in 1989, for use with programs released as part of the GNU project. The original GPL was based on a unification of similar licenses used for early versions of GNU Emacs (1985), the GNU Debugger, and the GNU C Compiler. These licenses contained similar provisions to the modern GPL, but were specific to each program, rendering them incompatible, despite being the same license. Stallman's goal was to produce one license that could be used for any project, thus making it possible for many projects to share code.

The second version of the license, version 2, was released in 1991. Over the following 15 years, members of the free software community became concerned over problems in the GPLv2 license that could let someone exploit GPL-licensed software in ways contrary to the license's intent. These problems included tivoization (the inclusion of GPL-licensed software in hardware that refuses to run modified versions of its software), compatibility issues similar to those of the Affero General Public License, and patent deals between Microsoft and distributors of free and open-source software, which some viewed as an attempt to use patents as a weapon against the free software community.

Version 3 was developed to attempt to address these concerns and was officially released on 29 June 2007.

Version 1

GNU General Public License, version 1
Published25 February 1989
Websitewww.gnu.org/licenses/old-licenses/gpl-1.0.html

Version 1 of the GNU GPL, released on 25 February 1989, prevented what were then the two main ways that software distributors restricted the freedoms that define free software. The first problem was that distributors may publish only binary files that are executable, but not readable or modifiable by humans. To prevent this, GPLv1 stated that copying and distributing copies of any portion of the program must also make the human-readable source code available under the same licensing terms.

The second problem was that distributors might add restrictions, either to the license or by combining the software with other software that had other restrictions on distribution. The union of two sets of restrictions would apply to the combined work, thus adding unacceptable restrictions. To prevent this, GPLv1 stated that modified versions, as a whole, had to be distributed under the terms of GPLv1. Therefore, software distributed under the terms of GPLv1 could be combined with software under more permissive terms, as this would not change the terms under which the whole could be distributed. However, software distributed under GPLv1 could not be combined with software distributed under a more restrictive license, as this would conflict with the requirement that the whole be distributable under the terms of GPLv1.

Version 2

GNU General Public License, version 2
PublishedJune 1991
Websitewww.gnu.org/licenses/old-licenses/gpl-2.0.html

According to Richard Stallman, the major change in GPLv2 was the "Liberty or Death" clause, as he calls it – Section 7. The section says that licensees may distribute a GPL-covered work only if they can satisfy all of the license's obligations, despite any other legal obligations they might have. In other words, the obligations of the license may not be severed due to conflicting obligations. This provision is intended to discourage any party from using a patent infringement claim or other litigation to impair users' freedom under the license.

By 1990, it was becoming apparent that a less restrictive license would be strategically useful for the C library and for software libraries that essentially did the job of existing proprietary ones; when version 2 of the GPL (GPLv2) was released in June 1991, therefore, a second license – the GNU Library General Public License – was introduced at the same time and numbered with version 2 to show that both were complementary. The version numbers diverged in 1999 when version 2.1 of the LGPL was released, which renamed it the GNU Lesser General Public License to reflect its place in the philosophy. The GPLv2 was also modified to refer to the new name of the LGPL, but its version number remained the same, resulting in the original GPLv2 not being recognised by the Software Package Data Exchange (SPDX).

The license includes instructions to specify "version 2 of the License, or (at your option) any later version" to allow the flexible optional use of either version 2 or 3, but some developers change this to specify "version 2" only.

Version 3

GNU General Public License, version 3
Published29 June 2007
Websitewww.gnu.org/licenses/gpl-3.0.html

In late 2005, the Free Software Foundation (FSF) announced work on version 3 of the GPL (GPLv3). On 16 January 2006, the first "discussion draft" of GPLv3 was published, and the public consultation began. The public consultation was originally planned for nine to fifteen months, but finally stretched to eighteen months with four drafts being published. The official GPLv3 was released by the FSF on 29 June 2007. GPLv3 was written by Richard Stallman, with legal counsel from Eben Moglen and Richard Fontana from the Software Freedom Law Center.

According to Stallman, the most important changes were in relation to software patents, free software license compatibility, the definition of "source code", and hardware restrictions on software modifications, such as tivoization. Other changes related to internationalization, how license violations are handled, and how additional permissions could be granted by the copyright holder. The concept of "software propagation", as a term for the copying and duplication of software, was explicitly defined.

The public consultation process was coordinated by the Free Software Foundation with assistance from Software Freedom Law Center, Free Software Foundation Europe, and other free software groups. Comments were collected from the public via the gplv3.fsf.org web portal, using purpose-written software called stet.

During the public consultation process, 962 comments were submitted for the first draft. By the end of the comment period, a total of 2,636 comments had been submitted.

The third draft was released on 28 March 2007. This draft included language intended to prevent patent-related agreements such as the controversial Microsoft-Novell patent agreement, and restricted the anti-tivoization clauses to a legal definition of a "user" and a "consumer product". It also explicitly removed the section on "Geographical Limitations", whose probable removal had been announced at the launch of the public consultation.

Richard Stallman at the launch of the first draft of the GNU GPLv3 at MIT, Cambridge, Massachusetts, United States. To his right is Columbia Law Professor Eben Moglen, chairman of the Software Freedom Law Center.

The fourth discussion draft, which was the last, was released on 31 May 2007. It introduced Apache License version 2.0 compatibility (prior versions are incompatible), clarified the role of outside contractors and made an exception to avoid the perceived problems of a Microsoft–Novell style agreement, saying in Section 11 paragraph 6 that:

You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license ...

This aimed to make future such deals ineffective. The license was also meant to cause Microsoft to extend the patent licenses it granted to Novell customers for the use of GPLv3 software to all users of that GPLv3 software; this was possible only if Microsoft was legally a "conveyor" of the GPLv3 software.

Early drafts of GPLv3 also let licensors add an Affero-like requirement that would have plugged the ASP loophole in the GPL. As there were concerns expressed about the administrative costs of checking code for this additional requirement, it was decided to keep the GPL and the Affero license separated.

Others, notably some high-profile Linux kernel developers such as Linus Torvalds, Greg Kroah-Hartman, and Andrew Morton, commented to the mass media and made public statements about their objections to parts of discussion drafts 1 and 2. The kernel developers referred to GPLv3 draft clauses regarding DRM/Tivoization, patents, and "additional restrictions", and warned of a Balkanisation of the "Open Source Universe". Linus Torvalds, who decided not to adopt the GPLv3 for the Linux kernel, reiterated his criticism several years later.

GPLv3 improved compatibility with several free software licenses such as the Apache License, version 2.0, and the GNU Affero General Public License, which GPLv2 could not be combined with. However, GPLv3 software could only be combined and share code with GPLv2 software if the GPLv2 license used had the optional "or later" clause and the software was upgraded to GPLv3. While the "GPLv2 or any later version" clause is considered by FSF as the most common form of licensing GPLv2 software, Toybox developer Rob Landley described it as a lifeboat clause. Software projects licensed with the optional "or later" clause include the GNU Project, while a prominent example without the clause is the Linux kernel.

The final version of the license text was published on 29 June 2007.

Terms and conditions

The terms and conditions of the GPL must be made available to anybody receiving a copy of the work that has a GPL applied to it ("the licensee"). Any licensee who adheres to the terms and conditions is given permission to modify the work, as well as to copy and redistribute the work or any derivative version. The licensee is allowed to charge a fee for this service or do this free of charge. This latter point distinguishes the GPL from software licenses that prohibit commercial redistribution. The FSF argues that free software should not place restrictions on commercial use, and the GPL explicitly states that GPL works may be sold at any price.

The GPL additionally states that a distributor may not impose "further restrictions on the rights granted by the GPL". This forbids activities such as distributing the software under a non-disclosure agreement or contract.

The fourth section for version 2 of the license and the seventh section of version 3 require that programs distributed as pre-compiled binaries be accompanied by a copy of the source code, a written offer to distribute the source code via the same mechanism as the pre-compiled binary, or the written offer to obtain the source code that the user got when they received the pre-compiled binary under the GPL. The second section of version 2 and the fifth section of version 3 also require giving "all recipients a copy of this License along with the Program". Version 3 of the license allows making the source code available in additional ways in fulfillment of the seventh section. These include downloading source code from an adjacent network server or by peer-to-peer transmission, provided that is how the compiled code was available and there are "clear directions" on where to find the source code.

The FSF does not hold the copyright for a work released under the GPL unless an author explicitly assigns copyrights to the FSF (which seldom happens except for programs that are part of the GNU project). Only the individual copyright holders have the authority to sue when a license violation is suspected.

Printed GPL statements for consumer entertainment devices which incorporate GPL components

Use of licensed software

Software under the GPL may be run for all purposes, including commercial purposes and even as a tool for creating proprietary software, such as when using GPL-licensed compilers. Users or companies who distribute GPL-licensed works (e.g. software), may charge a fee for copies or give them free of charge. This distinguishes the GPL from shareware software licenses that allow copying for personal use but prohibit the commercial distribution or proprietary licenses where copying is prohibited by copyright law. The FSF argues that freedom-respecting free software should also not restrict commercial use and distribution (including redistribution).

In purely private (or internal) use—with no sales and no distribution—the software code may be modified and parts reused without requiring the source code to be released. For sales or distribution, the entire source code needs to be made available to end users, including any code changes and additions—in that case, copyleft is applied to ensure that end users retain the freedoms defined above.

However, software running as an application program under a GPL-licensed operating system such as Linux is not required to be licensed under GPL or to be distributed with source-code availability—the licensing depends only on the used libraries and software components and not on the underlying platform. For example, if a program consists only of original source code, or is combined with source code from other software components, then the custom software components need not be licensed under GPL and need not make their source code available; even if the underlying operating system used is licensed under the GPL, applications running on it are not considered derivative works. Only if GPLed parts are used in a program (and the program is distributed), then all other source code of the program needs to be made available under the same license terms. The GNU Lesser General Public License (LGPL) was created to have a weaker copyleft than the GPL, in that it does not require custom-developed source code (distinct from the LGPL'ed parts) to be made available under the same license terms.

The fifth section of version 3 states that no GPL-licensed code shall be considered an effective "technical protection measure" as defined by Article 11 of the WIPO Copyright Treaty, and that those who convey the work waive all legal power to prohibit circumvention of the technical protection measure "to the extent such circumvention is effected by exercising rights under this License with respect to the covered work". This means that users cannot be held liable for circumventing DRM implemented using GPLv3-licensed code under laws such as the U.S. Digital Millennium Copyright Act (DMCA).

Copyleft

The distribution rights granted by the GPL for modified versions of the work are not unconditional. When someone distributes a GPL'ed work plus their own modifications, the requirements for distributing the whole work cannot be any greater than the requirements that are in the GPL.

This requirement is known as copyleft. It earns its legal power from the use of copyright on software programs. Because a GPL work is copyrighted, a licensee has no right to redistribute it, not even in modified form (barring fair use), except under the terms of the license. One is only required to adhere to the terms of the GPL if one wishes to exercise rights normally restricted by copyright law, such as redistribution. Conversely, if one distributes copies of the work without abiding by the terms of the GPL (for instance, by keeping the source code secret), they can be sued by the original author under copyright law.

Copyright law has historically been used to prevent distribution of work by parties not authorized by the creator. Copyleft uses the same copyright laws to accomplish a very different goal. It grants rights to distribution to all parties insofar as they provide the same rights to subsequent ones, and they to the next, etc. In this way the GPL and other copyleft licenses attempt to enforce libre access to the work and all derivatives.

Many distributors of GPL'ed programs bundle the source code with the executables. An alternative method of satisfying the copyleft is to provide a written offer to provide the source code on a physical medium (such as a CD) upon request. In practice, many GPL'ed programs are distributed over the Internet, and the source code is made available over FTP or HTTP. For Internet distribution, this complies with the license.

Copyleft applies only when a person seeks to redistribute the program. Developers may make private modified versions with no obligation to divulge the modifications, as long as they do not distribute the modified software to anyone else. Copyleft applies only to the software, and not to its output (unless that output is itself a derivative work of the program). For example, a public web portal running a modified derivative of a GPL'ed content management system is not required to distribute its changes to the underlying software, because the modified web portal is not being redistributed but rather hosted, and also because the web portal output is also not a derivative work of the GPL'ed content management system.

There has been debate on whether it is a violation of the GPL to release the source code in obfuscated form, such as in cases in which the author is less willing to make the source code available. The consensus was that while unethical, it was not considered a violation. The issue was clarified when the license was altered with v2 to require that the "preferred" version of the source code be made available.

License versus contract

The GPL was designed as a license, rather than a contract. In some Common Law jurisdictions, the legal distinction between a license and a contract is an important one: contracts are enforceable by contract law, whereas licenses are enforced under copyright law. However, this distinction is not useful in the many jurisdictions where there are no differences between contracts and licenses, such as Civil Law systems.

Those who do not accept the GPL's terms and conditions do not have permission, under copyright law, to copy or distribute GPL-licensed software or derivative works. However, if they do not redistribute the GPL'ed program, they may still use the software within their organization however they like, and works (including programs) constructed by the use of the program are not required to be covered by this license.

Software developer Allison Randal argued that the GPLv3 as a license is unnecessarily confusing for lay readers, and could be simplified while retaining the same conditions and legal force.

In April 2017, a US federal court ruled that an open-source license is an enforceable contract.

In October 2021 SFC sued Vizio over breach of contract as an end user to request source code to Vizio's TVs, a federal judge has ruled in the interim that the GPL is an enforceable contract by end users as well as a license for copyright holders.

Derivations

The text of the GPL is itself copyrighted, and the copyright is held by the Free Software Foundation.

The FSF permits people to create new licenses based on the GPL, as long as the derived licenses do not use the GPL preamble without permission. This is discouraged, however, since such a license might be incompatible with the GPL and causes a perceived license proliferation.

Other licenses created by the GNU project include the GNU Lesser General Public License, GNU Free Documentation License, and Affero General Public License.

The text of the GPL is not itself under the GPL. The license's copyright disallows modification of the license. Copying and distributing the license is allowed since the GPL requires recipients to get "a copy of this License along with the Program". According to the GPL FAQ, anyone can make a new license using a modified version of the GPL as long as they use a different name for the license, do not mention "GNU", and remove the preamble, though the preamble can be used in a modified license if permission to use it is obtained from the Free Software Foundation (FSF).

Linking and derived works

Libraries

According to the FSF, "The GPL does not require you to release your modified version or any part of it. You are free to make modifications and use them privately, without ever releasing them." However, if one releases a GPL-licensed entity to the public, there is an issue regarding linking: namely, whether a proprietary program that uses a GPL library is in violation of the GPL.

This key dispute is whether non-GPL software can legally statically link or dynamically link to GPL libraries. Different opinions exist on this issue. The GPL is clear in requiring that all derivative works of code under the GPL must themselves be under the GPL. Ambiguity arises with regards to using GPL libraries, and bundling GPL software into a larger package (perhaps mixed into a binary via static linking). This is ultimately a question not of the GPL per se, but of how copyright law defines derivative works. The following points of view exist:

Point of view: dynamic and static linking violate GPL

The Free Software Foundation (which holds the copyright of several notable GPL-licensed software products and of the license text itself) asserts that an executable that uses a dynamically linked library is indeed a derivative work. This does not, however, apply to separate programs communicating with one another.

The Free Software Foundation also created the LGPL, which is nearly identical to the GPL, but with additional permissions to allow linking for the purposes of "using the library".

Richard Stallman and the FSF specifically encourage library writers to license under the GPL so that proprietary programs cannot use the libraries, in an effort to protect the free-software world by giving it more tools than the proprietary world.

Point of view: static linking violates GPL but unclear as of dynamic linking

Some people believe that while static linking produces derivative works, it is not clear whether an executable that dynamically links to a GPL code should be considered a derivative work (see weak copyleft). Linux author Linus Torvalds agrees that dynamic linking can create derived works but disagrees over the circumstances.

A Novell lawyer has written that dynamic linking not being derivative "makes sense" but is not "clear-cut", and that evidence for good-intentioned dynamic linking can be seen by the existence of proprietary Linux kernel drivers.

In Galoob v. Nintendo, the United States Ninth Circuit Court of Appeals defined a derivative work as having "'form' or permanence" and noted that "the infringing work must incorporate a portion of the copyrighted work in some form", but there have been no clear court decisions to resolve this particular conflict.

Point of view: linking is irrelevant

According to an article in the Linux Journal, Lawrence Rosen (a one-time Open Source Initiative general counsel) argues that the method of linking is mostly irrelevant to the question about whether a piece of software is a derivative work; more important is the question about whether the software was intended to interface with client software and/or libraries. He states, "The primary indication of whether a new program is a derivative work is whether the source code of the original program was used [in a copy-paste sense], modified, translated or otherwise changed in any way to create the new program. If not, then I would argue that it is not a derivative work," and lists numerous other points regarding intent, bundling, and linkage mechanism. He further argues on his firm's website that such "market-based" factors are more important than the linking technique.

There is also the specific issue of whether a plugin or module (such as the NVidia or ATI graphics card kernel modules) must also be GPL, if it could reasonably be considered its own work. This point of view suggests that reasonably separate plugins, or plugins for software designed to use plugins, could be licensed under an arbitrary license if the work is GPLv2. Of particular interest is the GPLv2 paragraph:

You may modify your copy or copies of the Program or any portion of it, thus forming a work based on the Program, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions: ...

b) You must cause any work that you distribute or publish, that in whole or in part contains or is derived from the Program or any part thereof, to be licensed as a whole at no charge to all third parties under the terms of this License. ... These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the Program and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the Program, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it.

The GPLv3 has a different clause:

You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of Section 4, provided that you also meet all of these conditions: ...

c) You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable Section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it. ... A compilation of a covered work with other separate and independent works, which are not by their nature extensions of the covered work, and which are not combined with it such as to form a larger program, in or on a volume of a storage or distribution medium, is called an "aggregate" if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation's users beyond what the individual works permit. Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate.

As a case study, some supposedly proprietary plugins and themes/skins for GPLv2 CMS software such as Drupal and WordPress have come under fire, with both sides of the argument taken.

The FSF differentiates on how the plugin is being invoked. If the plugin is invoked through dynamic linkage and it performs function calls to the GPL program then it is most likely a derivative work.

Communicating and bundling with non-GPL programs

The mere act of communicating with other programs does not, by itself, require all software to be GPL; nor does distributing GPL software with non-GPL software. However, minor conditions must be followed that ensures the rights of GPL software are not restricted. The following is a quote from the gnu.org GPL FAQ, which describes to what extent software is allowed to communicate with and be bundled with GPL programs:

What is the difference between an "aggregate" and other kinds of "modified versions"?

An "aggregate" consists of a number of separate programs, distributed together on the same CD-ROM or other media. The GPL permits you to create and distribute an aggregate, even when the licenses of the other software are non-free or GPL-incompatible. The only condition is that you cannot release the aggregate under a license that prohibits users from exercising rights that each program's individual license would grant them.

Where's the line between two separate programs, and one program with two parts? This is a legal question, which ultimately judges will decide. We believe that a proper criterion depends both on the mechanism of communication (exec, pipes, rpc, function calls within a shared address space, etc.) and the semantics of the communication (what kinds of information are interchanged).

If the modules are included in the same executable file, they are definitely combined in one program. If modules are designed to run linked together in a shared address space, that almost surely means combining them into one program.

By contrast, pipes, sockets, and command-line arguments are communication mechanisms normally used between two separate programs. So when they are used for communication, the modules normally are separate programs. But if the semantics of the communication are intimate enough, exchanging complex internal data structures, that too could be a basis to consider the two parts as combined into a larger program.

The FSF thus draws the line between "library" and "other program" via 1) "complexity" and "intimacy" of information exchange and 2) mechanism (rather than semantics), but resigns that the question is not clear-cut and that in complex situations, case law will decide.

Legal status

The first known violation of the GPL was in 1989, when NeXT extended the GCC compiler to support Objective-C, but did not publicly release the changes. After an inquiry they created a public patch. There was no lawsuit filed for this violation.

In 2002, MySQL AB sued Progress NuSphere for copyright and trademark infringement in United States district court. NuSphere had allegedly violated MySQL's copyright by linking MySQL's GPL'ed code with NuSphere Gemini table without complying with the license. After a preliminary hearing before Judge Patti Saris on 27 February 2002, the parties entered settlement talks and eventually settled. After the hearing, FSF commented that "Judge Saris made clear that she sees the GNU GPL to be an enforceable and binding license."

In August 2003, the SCO Group stated that they believed the GPL to have no legal validity and that they intended to pursue lawsuits over sections of code supposedly copied from SCO Unix into the Linux kernel. This was a problematic stand for them, as they had distributed Linux and other GPL'ed code in their Caldera OpenLinux distribution, and there is little evidence that they had any legal right to do so except under the terms of the GPL. In February 2018, after federal circuit court judgement, appeal, and the case being (partially) remanded to the circuit court, the parties restated their remaining claims and provided a plan to move toward final judgement. The remaining claims revolved around Project Monterey, and were finally settled in November 2021 by IBM paying $14.25 million to the TSG (previously SCO) bankruptcy trustee.

In April 2004, the netfilter/iptables project was granted a preliminary injunction against Sitecom Germany by Munich District Court after Sitecom refused to desist from distributing Netfilter's GPL'ed software in violation of the terms of the GPL. Harald Welte, of Netfilter, was represented by ifrOSS co-founder Till Jaeger. In July 2004, the German court confirmed this injunction as a final ruling against Sitecom. The court's justification was that:

Defendant has infringed on the copyright of plaintiff by offering the software 'netfilter/iptables' for download and by advertising its distribution, without adhering to the license conditions of the GPL. Said actions would only be permissible if the defendant had a license grant. ... This is independent of the questions whether the licensing conditions of the GPL have been effectively agreed upon between plaintiff and defendant or not. If the GPL were not agreed upon by the parties, defendant would notwithstanding lack the necessary rights to copy, distribute, and make the software 'netfilter/iptables' publicly available.

This exactly mirrored the predictions given previously by the FSF's Eben Moglen. This ruling was important because it was the first time that a court had confirmed that violating terms of the GPL could be a copyright violation and established jurisprudence as to the enforceability of the GPLv2 under German law.

In May 2005, Daniel Wallace filed suit against the Free Software Foundation in the Southern District of Indiana, contending that the GPL is an illegal attempt to fix prices (at zero). The suit was dismissed in March 2006, on the grounds that Wallace had failed to state a valid antitrust claim; the court noted that "the GPL encourages, rather than discourages, free competition and the distribution of computer operating systems, the benefits of which directly pass to consumers". Wallace was denied the possibility of further amending his complaint, and was ordered to pay the FSF's legal expenses.

On 8 September 2005, the Seoul Central District Court ruled that the GPL was not material to a case dealing with trade secrets derived from GPL-licensed work. Defendants argued that since it is impossible to maintain trade secrets while being compliant with GPL and distributing the work, they are not in breach of trade secrets. This argument was considered without ground.

On 6 September 2006, the gpl-violations.org project prevailed in court litigation against D-Link Germany GmbH regarding D-Link's copyright-infringing use of parts of the Linux kernel in storage devices they distributed. The judgment stated that the GPL is valid, legally binding, and stands in German court.

In late 2007, the BusyBox developers and the Software Freedom Law Center embarked upon a program to gain GPL compliance from distributors of BusyBox in embedded systems, suing those who would not comply. These were claimed to be the first US uses of courts for enforcement of GPL obligations. (See BusyBox GPL lawsuits.)

On 11 December 2008, the Free Software Foundation sued Cisco Systems, Inc. for copyright violations by its Linksys division, of the FSF's GPL-licensed coreutils, readline, Parted, Wget, GNU Compiler Collection, binutils, and GNU Debugger software packages, which Linksys distributes in the Linux firmware of its WRT54G wireless routers, as well as numerous other devices including DSL and Cable modems, Network Attached Storage devices, Voice-Over-IP gateways, virtual private network devices, and a home theater/media player device.

After six years of repeated complaints to Cisco by the FSF, claims by Cisco that they would correct, or were correcting, their compliance problems (not providing complete copies of all source code and their modifications), of repeated new violations being discovered and reported with more products, and lack of action by Linksys (a process described on the FSF blog as a "five-years-running game of Whack-a-Mole") the FSF took them to court.

Cisco settled the case six months later by agreeing "to appoint a Free Software Director for Linksys" to ensure compliance, "to notify previous recipients of Linksys products containing FSF programs of their rights under the GPL," to make source code of FSF programs freely available on its website, and to make a monetary contribution to the FSF.

In 2011, it was noticed that GNU Emacs had been accidentally releasing some binaries without corresponding source code for two years, in opposition to the intended spirit of the GPL, resulting in a copyright violation. Richard Stallman described this incident as a "very bad mistake", which was promptly fixed. The FSF did not sue any downstream redistributors who also unknowingly violated the GPL by distributing these binaries.

In 2017 Artifex, the maker of Ghostscript, sued Hancom, the maker of an office suite which included Ghostscript. Artifex offers two licenses for Ghostscript; one is the Affero GPL License and the other is a commercial license. Hancom did not acquire a commercial license from Artifex nor did it release its office suite as free software. Artifex sued Hancom in US District Court and made two claims. First, Hancom's use of Ghostscript was a violation of copyright; and second, Hancom's use of Ghostscript was a license violation. Judge Jacqueline Scott Corley found the GPL license was an enforceable contract and Hancom was in breach of contract.

On 20 July 2021, the developers of the open-source Stockfish chess engine sued ChessBase, the creator of chess software, for violating the GPLv3 license. It was claimed that Chessbase had made only slight modifications to the Stockfish code and sold the new engines (Fat Fritz 2 and Houdini 6) to their customers. Additionally, Fat Fritz 2 was marketed as if it was an innovative engine. ChessBase had infringed on the license by not distributing these products as Free Software in accordance with the GPL.

A year later on 7 November 2022, both parties reached an agreement and ended the dispute. In the near future ChessBase will no longer sell products containing Stockfish code, while informing their customers of this fact with an appropriate notice on their web pages. However, one year later, Chessbase's license would be reinstated. Stockfish did not seek damages or financial compensation.

Compatibility and multi-licensing

Quick guide of license compatibility with GPLv3 according to the FSF. Dashed line indicates that the GPLv2 is only compatible with the GPLv3 with the clause "or any later version".

Code licensed under several other licenses can be combined with a program under the GPL without conflict, as long as the combination of restrictions on the work as a whole does not put any additional restrictions beyond what GPL allows. In addition to the regular terms of the GPL, there are additional restrictions and permissions one can apply:

  1. If a user wants to combine code licensed under different versions of GPL, then this is only allowed if the code with the earlier GPL version includes an "or any later version" statement. For instance, the GPLv3-licensed GNU LibreDWG library cannot be used anymore by LibreCAD and FreeCAD who have GPLv2-only dependencies.
  2. Code licensed under LGPL is permitted to be linked with any other code no matter what license that code has, though the LGPL does add additional requirements for the combined work. LGPLv3 and GPLv2-only can thus commonly not be linked, as the combined Code work would add additional LGPLv3 requirements on top of the GPLv2-only licensed software. Code licensed under LGPLv2.x without the "any later version" statement can be relicensed if the whole combined work is licensed to GPLv2 or GPLv3.

FSF maintains a list of GPL-compatible free software licenses containing many of the most common free software licenses, such as the original MIT/X license, the BSD license (in its current 3-clause form), and the Artistic License 2.0.

Starting from GPLv3, it is unilaterally compatible for materials (like text and other media) under Creative Commons Attribution-ShareAlike 4.0 International License to be remixed into the GPL-licensed materials (prominently software), not vice versa, for niche use cases like game engine (GPL) with game scripts (CC BY-SA).

David A. Wheeler has advocated that free/open source software developers use only GPL-compatible licenses, because doing otherwise makes it difficult for others to participate and contribute code. As a specific example of license incompatibility, Sun Microsystems' ZFS cannot be included in the GPL-licensed Linux kernel, because it is licensed under the GPL-incompatible Common Development and Distribution License. Furthermore, ZFS is protected by patents, so distributing an independently developed GPL-ed implementation would still require Oracle's permission.

A number of businesses use multi-licensing to distribute a GPL version and sell a proprietary license to companies wishing to combine the package with proprietary code, using dynamic linking or not. Examples of such companies include MySQL AB, Digia PLC (Qt framework, before 2011 from Nokia), Red Hat (Cygwin), and Riverbank Computing (PyQt). Other companies, like the Mozilla Foundation (products include Mozilla Application Suite, Mozilla Thunderbird, and Mozilla Firefox), used multi-licensing to distribute versions under the GPL and some other open-source licenses.

Text and other media

It is possible to use the GPL for text documents instead of computer programs, or more generally for all kinds of media, if it is clear what constitutes the source code (defined as "the preferred form of the work for making changes in it"). For manuals and textbooks, though, the FSF recommends the GNU Free Documentation License (GFDL) instead, which it created for this purpose. Nevertheless, the Debian developers recommended (in a resolution adopted in 2006) to license documentation for their project under the GPL, because of the incompatibility of the GFDL with the GPL (text licensed under the GFDL cannot be incorporated into GPL software). Also, the FLOSS Manuals foundation, an organization devoted to creating manuals for free software, decided to eschew the GFDL in favor of the GPL for its texts in 2007.

If the GPL is used for computer fonts, any documents or images made with such fonts might also have to be distributed under the terms of the GPL. This is not the case in countries that recognize typefaces (the appearance of fonts) as being a useful article and thus not eligible for copyright, but font files as copyrighted computer software (which can complicate font embedding, since the document could be considered 'linked' to the font; in other words, embedding a vector font in a document could force it to be released under the GPL, but a rasterized rendering of the font would not be subject to the GPL). The FSF provides an exception for cases where this is not desired.

Adoption

Historically, the GPL license family has been one of the most popular software licenses in the FOSS domain.

A 1997 survey of MetaLab, then the largest free software archive, showed that the GPL accounted for about half of the software licensed therein. Similarly, a 2000 survey of Red Hat Linux 7.1 found that 53% of the source code was licensed under the GPL. As of 2003, about 68% of all projects and 82.1% of the open source industry certified licensed projects listed on SourceForge.net were from the GPL license family. As of August 2008, the GPL family accounted for 70.9% of the 44,927 free software projects listed on Freecode.

After the release of the GPLv3 in June 2007, adoption of this new GPL version was much discussed and some projects decided against upgrading. For instance the Linux kernel, MySQL, BusyBox, AdvFS, Blender, VLC media player, and MediaWiki decided against adopting GPLv3. On the other hand, in 2009, two years after the release of GPLv3, Google open-source programs office manager Chris DiBona reported that the number of open-source project licensed software that had moved from GPLv2 to GPLv3 was 50%, counting the projects hosted at Google Code.

In 2011, four years after the release of the GPLv3, 6.5% of all open-source license projects are GPLv3 while 42.5% are GPLv2 according to Black Duck Software data. Following in 2011 451 Group analyst Matthew Aslett argued in a blog post that copyleft licenses went into decline and permissive licenses increased, based on statistics from Black Duck Software. Similarly, in February 2012 Jon Buys reported that among the top 50 projects on GitHub five projects were under a GPL license, including dual licensed and AGPL projects.

GPL usage statistics from 2009 to 2013 was extracted from Freecode data by Walter van Holst while analyzing license proliferation.

Usage of GPL family licenses in % on Freecode
2009 2010 2011 2012 2013 2014-06-18
72% 63% 61% 59% 58% approx. 54%

In August 2013, according to Black Duck Software, the website's data shows that the GPL license family is used by 54% of open-source projects, with a breakdown of the individual licenses shown in the following table. However, a later study in 2013 showed that software licensed under the GPL license family has increased, and that even the data from Black Duck Software has shown a total increase of software projects licensed under GPL. The study used public information gathered from repositories of the Debian Project, and the study criticized Black Duck Software for not publishing their methodology used in collecting statistics. Daniel German, Professor in the Department of Computer Science at the University of Victoria in Canada, presented a talk in 2013 about the methodological challenges in determining which are the most widely used free software licenses, and showed how he could not replicate the result from Black Duck Software.

In 2015, according to Black Duck, GPLv2 lost its first position to the MIT license and is now second, the GPLv3 dropped to fourth place while the Apache license kept its third position.

Usage of GPL family licenses in the FOSS domain in % according to Black Duck Software
License 2008-05-08 2009-03-11 2011-11-22 2013-08-12 2015-11-19 2016-06-06 2017-01-02 2018-06-04
GPLv2 58.69% 52.2% 42.5% 33% 23% 21% 19% 14%
GPLv3 1.64% 4.15% 6.5% 12% 9% 9% 8% 6%
LGPLv2.1 11.39% 9.84% ? 6% 5% 4% 4% 3%
LGPLv3 ? (<0.64%) 0.37% ? 3% 2% 2% 2% 1%
GPL family together 71.72% (+ <0.64%) 66.56% ? 54% 39% 36% 33% 24%

A March 2015 analysis of the GitHub repositories revealed, for the GPL license family, a usage percentage of approximately 25% among licensed projects. In June 2016, an analysis of Fedora Project's packages revealed the GNU GPLv2 or later as the most popular license, and the GNU GPL family as the most popular license family (followed by the MIT, BSD, and GNU LGPL families).

An analysis of whitesourcesoftware.com in April 2018 of the FOSS ecosystem saw the GPLv3 on third place (18%) and the GPLv2 on fourth place (11%), after MIT license (26%) and Apache 2.0 license (21%).

Reception

Legal barrier to application stores

The GPL is incompatible with many application digital distribution systems, like the Mac App Store, and certain other software distribution platforms (on smartphones as well as PCs). The problem lies in the right "to make a copy for your neighbour", as this right is violated by digital rights management systems embedded within the platform to prevent copying of paid software. Even if the application is free in the application store in question, it might result in a violation of that application store's terms.

There is a distinction between an app store, which sells DRM-restricted software under proprietary licenses, and the more general concept of digital distribution via some form of online software repository. Virtually all modern Unix systems and Linux distributions have application repositories, including NetBSD, FreeBSD, Ubuntu, Fedora, and Debian. These specific application repositories all contain GPL-licensed software apps, in some cases even when the core project does not permit GPL-licensed code in the base system (for instance OpenBSD). In other cases, such as the Ubuntu App Store, proprietary commercial software applications and GPL-licensed applications are both available via the same system; the reason that the Mac App Store (and similar projects) is incompatible with GPL-licensed apps is not inherent in the concept of an app store, but is rather specifically due to Apple's terms-of-use requirement that all apps in the store utilize Apple DRM restrictions. Ubuntu's app store does not demand any such requirement: "These terms do not limit or restrict your rights under any applicable open source software licenses."

Microsoft

In 2001, Microsoft CEO Steve Ballmer referred to Linux as "a cancer that attaches itself in an intellectual property sense to everything it touches". In response to Microsoft's attacks on the GPL, several prominent Free Software developers and advocates released a joint statement supporting the license. Microsoft has released Microsoft Windows Services for UNIX, which contains GPL-licensed code. In July 2009, Microsoft itself released a body of around 20,000 lines of Linux driver code under the GPL. The Hyper-V code that is part of the submitted code used open-source components licensed under the GPL and was originally statically linked to proprietary binary parts, the latter being inadmissible in GPL-licensed software.

"Viral" nature

The description of the GPL as "viral", when called 'General Public Virus' or 'GNU Public Virus' (GPV), dates back to a year after the GPLv1 was released.

In 2001, the term received broader public attention when Craig Mundie, Microsoft Senior Vice President, described the GPL as being "viral". Mundie argues that the GPL has a "viral" effect in that it only allows the conveyance of whole programs, which means programs that link to GPL libraries must themselves be under a GPL-compatible license, else they cannot be combined and distributed.

In 2006, Richard Stallman responded in an interview that Mundie's metaphor of a "virus" is wrong as software under the GPL does not "attack" or "infect" other software. Accordingly, Stallman believes that comparing the GPL to a virus is inappropriate, and that a better metaphor for software under the GPL would be a spider plant: if one takes a piece of it and puts it somewhere else, it grows there too.

On the other hand, the concept of a viral nature of the GPL was taken up by others later too. For instance, a 2008 article stated: "The GPL license is 'viral,' meaning any derivative work you create containing even the smallest portion of the previously GPL licensed software must also be licensed under the GPL license."

Barrier to commercialization

The FreeBSD project has stated that "a less publicized and unintended use of the GPL is that it is very favorable to large companies that want to undercut software companies. In other words, the GPL is well suited for use as a marketing weapon, potentially reducing overall economic benefit and contributing to monopolistic behavior" and that the GPL can "present a real problem for those wishing to commercialize and profit from software."

Richard Stallman wrote about the practice of selling license exceptions to free software licenses as an example of ethically acceptable commercialization practice. Selling exceptions here means that the copyright holder of a given software releases it (along with the corresponding source code) to the public under a free software license, "then lets customers pay for permission to use the same code under different terms, for instance allowing its inclusion in proprietary applications". Stallman considered selling exceptions "acceptable since the 1990s, and on occasion I've suggested it to companies. Sometimes this approach has made it possible for important programs to become free software". Although the FSF does not practice selling exceptions, a comparison with the X11 license (which is a non-copyleft free software license) is proposed for suggesting that this commercialization technique should be regarded as ethically acceptable. Releasing a given program under a non-copyleft free software license would permit embedding the code in proprietary software. Stallman comments that "either we have to conclude that it's wrong to release anything under the X11 license—a conclusion I find unacceptably extreme—or reject this implication. Using a non-copyleft license is weak, and usually an inferior choice, but it's not wrong. In other words, selling exceptions permits some embedding in proprietary software, and the X11 license permits even more embedding. If this doesn't make the X11 license unacceptable, it doesn't make selling exceptions unacceptable".

Open-source criticism

In 2000, developer and author Nikolai Bezroukov published an analysis and comprehensive critique of GPL's foundations and Stallman's software development model, called "Labyrinth of Software Freedom".

Version 2 of the WTFPL (Do What The Fuck You Want To Public License) was created by Debian project leader Sam Hocevar in 2004 as a parody of the GPL.

In 2005, open source software advocate Eric S. Raymond questioned the relevance of GPL then for the FOSS ecosystem, stating: "We don't need the GPL anymore. It's based on the belief that open source software is weak and needs to be protected. Open source would be succeeding faster if the GPL didn't make lots of people nervous about adopting it." Richard Stallman replied: "GPL is designed to ... ensure that every user of a program gets the essential freedoms—to run it, to study and change the source code, to redistribute copies, and to publish modified versions ... [Raymond] addresses the issue in terms of different goals and values—those of 'open source,' which do not include defending software users' freedom to share and change software."

In 2007, Allison Randal, who took part in the GPL draft committee, criticized the GPLv3 for being incompatible with the GPLv2 and for missing clarity in the formulation. Similarly, Whurley prophesied in 2007 the downfall of the GPL due to the lack of focus for the developers with GPLv3 which would drive them towards permissive licenses.

In 2009, David Chisnall described in an InformIT article, "The Failure of the GPL", the problems with the GPL, among them incompatibility and complexity of the license text.

In 2014, dtrace developer and Joyent CTO Bryan Cantrill called the copyleft GPL a "Corporate Open Source Anti-pattern" by being "anti-collaborative" and recommended instead permissive software licenses.

GPLv3 criticism

Already in September 2006, in the draft process of the GPLv3, several high-profile developers of the Linux kernel, for instance Linus Torvalds, Greg Kroah-Hartman, and Andrew Morton, warned on a splitting of the FOSS community: "the release of GPLv3 portends the Balkanisation of the entire Open Source Universe upon which we rely." Similarly Benjamin Mako Hill argued in 2006 on the GPLv3 draft, noting that a united, collaborating community is more important than a single license.

Following the GPLv3 release in 2007, some journalists and Toybox developer Rob Landley criticized that with the introduction of the GPLv3 the split between the open source and free software community became wider than ever. As the significantly extended GPLv3 is essentially incompatible with the GPLv2, compatibility between both is only given under the optional "or later" clause of the GPL, which was not taken for instance by the Linux kernel. Bruce Byfield noted that before the release of the GPLv3, the GPLv2 was a unifying element between the open-source and the free software community.

For the LGPLv3, GNU TLS maintainer Nikos Mavrogiannopoulos similarly argued, "If we assume that its [the LGPLv3] primary goal is to be used by free software, then it blatantly fails that", after he re-licensed GNU TLS from LGPLv3 back to LGPLv2.1 due to license compatibility issues.

Lawrence Rosen, attorney and computer specialist, praised in 2007 how the community using the Apache license was now able to work together with the GPL community in a compatible manner, as the problems of GPLv2 compatibility with Apache licensed software were resolved with the GPLv3. He said, "I predict that one of the biggest success stories of GPLv3 will be the realization that the entire universe of free and open-source software can thus be combined into comprehensive open source solutions for customers worldwide."

In July 2013, Flask developer Armin Ronacher draws a less optimistic conclusion on the GPL compatibility in the FOSS ecosystem: "When the GPL is involved the complexities of licensing becomes a non fun version of a riddle", also noting that the conflict between Apache License 2.0 and GPLv2 still has impact on the ecosystem.

Deductive reasoning

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Deductive_reasoning

Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. it is impossible for the premises to be true and the conclusion to be false.

For example, the inference from the premises "all men are mortal" and "Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is sound if it is valid and all its premises are true. Some theorists define deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning.

Psychology is interested in deductive reasoning as a psychological process, i.e. how people actually draw inferences. Logic, on the other hand, focuses on the deductive relation of logical consequence between the premises and the conclusion or how people should draw inferences. There are different ways of conceptualizing this relation. According to the semantic approach, an argument is deductively valid if and only if there is no possible interpretation of this argument where its premises are true and its conclusion is false. The syntactic approach, on the other hand, holds that an argument is deductively valid if and only if its conclusion can be deduced from its premises using a valid rule of inference. A rule of inference is a schema of drawing a conclusion from a set of premises based only on their logical form.

There are various rules of inference, like the modus ponens and the modus tollens. Invalid deductive arguments, which do not follow a rule of inference, are called formal fallacies. Rules of inference are definitory rules and contrast to strategic rules, which specify what inferences one needs to draw in order to arrive at an intended conclusion. Deductive reasoning contrasts with non-deductive or ampliative reasoning. For ampliative arguments, like inductive or abductive arguments, the premises offer weaker support to their conclusion: they make it more likely but they do not guarantee its truth. They make up for this drawback by being able to provide genuinely new information not already found in the premises, unlike deductive arguments.

Cognitive psychology investigates the mental processes responsible for deductive reasoning. One of its topics concerns the factors determining whether people draw valid or invalid deductive inferences. One factor is the form of the argument: for example, people are more successful for arguments of the form modus ponens than for modus tollens. Another is the content of the arguments: people are more likely to believe that an argument is valid if the claim made in its conclusion is plausible. A general finding is that people tend to perform better for realistic and concrete cases than for abstract cases. Psychological theories of deductive reasoning aim to explain these findings by providing an account of the underlying psychological processes. Mental logic theories hold that deductive reasoning is a language-like process that happens through the manipulation of representations using rules of inference. Mental model theories, on the other hand, claim that deductive reasoning involves models of possible states of the world without the medium of language or rules of inference. According to dual-process theories of reasoning, there are two qualitatively different cognitive systems responsible for reasoning.

The problem of deductive reasoning is relevant to various fields and issues. Epistemology tries to understand how justification is transferred from the belief in the premises to the belief in the conclusion in the process of deductive reasoning. Probability logic studies how the probability of the premises of an inference affects the probability of its conclusion. The controversial thesis of deductivism denies that there are other correct forms of inference besides deduction. Natural deduction is a type of proof system based on simple and self-evident rules of inference. In philosophy, the geometrical method is a way of philosophizing that starts from a small set of self-evident axioms and tries to build a comprehensive logical system using deductive reasoning.

Definition

Deductive reasoning is the psychological process of drawing deductive inferences. An inference is a set of premises together with a conclusion. This psychological process starts from the premises and reasons to a conclusion based on and supported by these premises. If the reasoning was done correctly, it results in a valid deduction: the truth of the premises ensures the truth of the conclusion. For example, in the syllogistic argument "all frogs are amphibians; no cats are amphibians; therefore, no cats are frogs" the conclusion is true because its two premises are true. But even arguments with wrong premises can be deductively valid if they obey this principle, as in "all frogs are mammals; no cats are mammals; therefore, no cats are frogs". If the premises of a valid argument are true, then it is called a sound argument.

The relation between the premises and the conclusion of a deductive argument is usually referred to as "logical consequence". According to Alfred Tarski, logical consequence has 3 essential features: it is necessary, formal, and knowable a priori. It is necessary in the sense that the premises of valid deductive arguments necessitate the conclusion: it is impossible for the premises to be true and the conclusion to be false, independent of any other circumstances. Logical consequence is formal in the sense that it depends only on the form or the syntax of the premises and the conclusion. This means that the validity of a particular argument does not depend on the specific contents of this argument. If it is valid, then any argument with the same logical form is also valid, no matter how different it is on the level of its contents. Logical consequence is knowable a priori in the sense that no empirical knowledge of the world is necessary to determine whether a deduction is valid. So it is not necessary to engage in any form of empirical investigation. Some logicians define deduction in terms of possible worlds: A deductive inference is valid if and only if, there is no possible world in which its conclusion is false while its premises are true. This means that there are no counterexamples: the conclusion is true in all such cases, not just in most cases.

It has been argued against this and similar definitions that they fail to distinguish between valid and invalid deductive reasoning, i.e. they leave it open whether there are invalid deductive inferences and how to define them. Some authors define deductive reasoning in psychological terms in order to avoid this problem. According to Mark Vorobey, whether an argument is deductive depends on the psychological state of the person making the argument: "An argument is deductive if, and only if, the author of the argument believes that the truth of the premises necessitates (guarantees) the truth of the conclusion". A similar formulation holds that the speaker claims or intends that the premises offer deductive support for their conclusion. This is sometimes categorized as a speaker-determined definition of deduction since it depends also on the speaker whether the argument in question is deductive or not. For speakerless definitions, on the other hand, only the argument itself matters independent of the speaker. One advantage of this type of formulation is that it makes it possible to distinguish between good or valid and bad or invalid deductive arguments: the argument is good if the author's belief concerning the relation between the premises and the conclusion is true, otherwise it is bad. One consequence of this approach is that deductive arguments cannot be identified by the law of inference they use. For example, an argument of the form modus ponens may be non-deductive if the author's beliefs are sufficiently confused. That brings with it an important drawback of this definition: it is difficult to apply to concrete cases since the intentions of the author are usually not explicitly stated.

Deductive reasoning is studied in logic, psychology, and the cognitive sciences. Some theorists emphasize in their definition the difference between these fields. On this view, psychology studies deductive reasoning as an empirical mental process, i.e. what happens when humans engage in reasoning. But the descriptive question of how actual reasoning happens is different from the normative question of how it should happen or what constitutes correct deductive reasoning, which is studied by logic. This is sometimes expressed by stating that, strictly speaking, logic does not study deductive reasoning but the deductive relation between premises and a conclusion known as logical consequence. But this distinction is not always precisely observed in the academic literature. One important aspect of this difference is that logic is not interested in whether the conclusion of an argument is sensible. So from the premise "the printer has ink" one may draw the unhelpful conclusion "the printer has ink and the printer has ink and the printer has ink", which has little relevance from a psychological point of view. Instead, actual reasoners usually try to remove redundant or irrelevant information and make the relevant information more explicit. The psychological study of deductive reasoning is also concerned with how good people are at drawing deductive inferences and with the factors determining their performance. Deductive inferences are found both in natural language and in formal logical systems, such as propositional logic.

Conceptions of deduction

Deductive arguments differ from non-deductive arguments in that the truth of their premises ensures the truth of their conclusion. There are two important conceptions of what this exactly means. They are referred to as the syntactic and the semantic approach. According to the syntactic approach, whether an argument is deductively valid depends only on its form, syntax, or structure. Two arguments have the same form if they use the same logical vocabulary in the same arrangement, even if their contents differ. For example, the arguments "if it rains then the street will be wet; it rains; therefore, the street will be wet" and "if the meat is not cooled then it will spoil; the meat is not cooled; therefore, it will spoil" have the same logical form: they follow the modus ponens. Their form can be expressed more abstractly as "if A then B; A; therefore B" in order to make the common syntax explicit. There are various other valid logical forms or rules of inference, like modus tollens or the disjunction elimination. The syntactic approach then holds that an argument is deductively valid if and only if its conclusion can be deduced from its premises using a valid rule of inference. One difficulty for the syntactic approach is that it is usually necessary to express the argument in a formal language in order to assess whether it is valid. But since the problem of deduction is also relevant for natural languages, this often brings with it the difficulty of translating the natural language argument into a formal language, a process that comes with various problems of its own. Another difficulty is due to the fact that the syntactic approach depends on the distinction between formal and non-formal features. While there is a wide agreement concerning the paradigmatic cases, there are also various controversial cases where it is not clear how this distinction is to be drawn.

The semantic approach suggests an alternative definition of deductive validity. It is based on the idea that the sentences constituting the premises and conclusions have to be interpreted in order to determine whether the argument is valid. This means that one ascribes semantic values to the expressions used in the sentences, such as the reference to an object for singular terms or to a truth-value for atomic sentences. The semantic approach is also referred to as the model-theoretic approach since the branch of mathematics known as model theory is often used to interpret these sentences. Usually, many different interpretations are possible, such as whether a singular term refers to one object or to another. According to the semantic approach, an argument is deductively valid if and only if there is no possible interpretation where its premises are true and its conclusion is false. Some objections to the semantic approach are based on the claim that the semantics of a language cannot be expressed in the same language, i.e. that a richer metalanguage is necessary. This would imply that the semantic approach cannot provide a universal account of deduction for language as an all-encompassing medium.

Rules of inference

Deductive reasoning usually happens by applying rules of inference. A rule of inference is a way or schema of drawing a conclusion from a set of premises. This happens usually based only on the logical form of the premises. A rule of inference is valid if, when applied to true premises, the conclusion cannot be false. A particular argument is valid if it follows a valid rule of inference. Deductive arguments that do not follow a valid rule of inference are called formal fallacies: the truth of their premises does not ensure the truth of their conclusion.

In some cases, whether a rule of inference is valid depends on the logical system one is using. The dominant logical system is classical logic and the rules of inference listed here are all valid in classical logic. But so-called deviant logics provide a different account of which inferences are valid. For example, the rule of inference known as double negation elimination, i.e. that if a proposition is not not true then it is also true, is accepted in classical logic but rejected in intuitionistic logic.

Prominent rules of inference

Modus ponens

Modus ponens (also known as "affirming the antecedent" or "the law of detachment") is the primary deductive rule of inference. It applies to arguments that have as first premise a conditional statement () and as second premise the antecedent () of the conditional statement. It obtains the consequent () of the conditional statement as its conclusion. The argument form is listed below:

  1.   (First premise is a conditional statement)
  2.   (Second premise is the antecedent)
  3.   (Conclusion deduced is the consequent)

In this form of deductive reasoning, the consequent () obtains as the conclusion from the premises of a conditional statement () and its antecedent (). However, the antecedent () cannot be similarly obtained as the conclusion from the premises of the conditional statement () and the consequent (). Such an argument commits the logical fallacy of affirming the consequent.

The following is an example of an argument using modus ponens:

  1. If it is raining, then there are clouds in the sky.
  2. It is raining.
  3. Thus, there are clouds in the sky.

Modus tollens

Modus tollens (also known as "the law of contrapositive") is a deductive rule of inference. It validates an argument that has as premises a conditional statement (formula) and the negation of the consequent () and as conclusion the negation of the antecedent (). In contrast to modus ponens, reasoning with modus tollens goes in the opposite direction to that of the conditional. The general expression for modus tollens is the following:

  1. . (First premise is a conditional statement)
  2. . (Second premise is the negation of the consequent)
  3. . (Conclusion deduced is the negation of the antecedent)

The following is an example of an argument using modus tollens:

  1. If it is raining, then there are clouds in the sky.
  2. There are no clouds in the sky.
  3. Thus, it is not raining.

Hypothetical syllogism

A hypothetical syllogism is an inference that takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form:

  1. Therefore, .

In there being a subformula in common between the two premises that does not occur in the consequence, this resembles syllogisms in term logic, although it differs in that this subformula is a proposition whereas in Aristotelian logic, this common element is a term and not a proposition.

The following is an example of an argument using a hypothetical syllogism:

  1. If there had been a thunderstorm, it would have rained.
  2. If it had rained, things would have gotten wet.
  3. Thus, if there had been a thunderstorm, things would have gotten wet.

Fallacies

Various formal fallacies have been described. They are invalid forms of deductive reasoning. An additional aspect of them is that they appear to be valid on some occasions or on the first impression. They may thereby seduce people into accepting and committing them. One type of formal fallacy is affirming the consequent, as in "if John is a bachelor, then he is male; John is male; therefore, John is a bachelor". This is similar to the valid rule of inference named modus ponens, but the second premise and the conclusion are switched around, which is why it is invalid. A similar formal fallacy is denying the antecedent, as in "if Othello is a bachelor, then he is male; Othello is not a bachelor; therefore, Othello is not male". This is similar to the valid rule of inference called modus tollens, the difference being that the second premise and the conclusion are switched around. Other formal fallacies include affirming a disjunct, denying a conjunct, and the fallacy of the undistributed middle. All of them have in common that the truth of their premises does not ensure the truth of their conclusion. But it may still happen by coincidence that both the premises and the conclusion of formal fallacies are true.

Definitory and strategic rules

Rules of inferences are definitory rules: they determine whether an argument is deductively valid or not. But reasoners are usually not just interested in making any kind of valid argument. Instead, they often have a specific point or conclusion that they wish to prove or refute. So given a set of premises, they are faced with the problem of choosing the relevant rules of inference for their deduction to arrive at their intended conclusion. This issue belongs to the field of strategic rules: the question of which inferences need to be drawn to support one's conclusion. The distinction between definitory and strategic rules is not exclusive to logic: it is also found in various games. In chess, for example, the definitory rules state that bishops may only move diagonally while the strategic rules recommend that one should control the center and protect one's king if one intends to win. In this sense, definitory rules determine whether one plays chess or something else whereas strategic rules determine whether one is a good or a bad chess player. The same applies to deductive reasoning: to be an effective reasoner involves mastering both definitory and strategic rules.

Validity and soundness

Argument terminology

Deductive arguments are evaluated in terms of their validity and soundness.

An argument is “valid” if it is impossible for its premises to be true while its conclusion is false. In other words, the conclusion must be true if the premises are true. An argument can be “valid” even if one or more of its premises are false.

An argument is “sound” if it is valid and the premises are true.

It is possible to have a deductive argument that is logically valid but is not sound. Fallacious arguments often take that form.

The following is an example of an argument that is “valid”, but not “sound”:

  1. Everyone who eats carrots is a quarterback.
  2. John eats carrots.
  3. Therefore, John is a quarterback.

The example's first premise is false – there are people who eat carrots who are not quarterbacks – but the conclusion would necessarily be true, if the premises were true. In other words, it is impossible for the premises to be true and the conclusion false. Therefore, the argument is “valid”, but not “sound”. False generalizations – such as "Everyone who eats carrots is a quarterback" – are often used to make unsound arguments. The fact that there are some people who eat carrots but are not quarterbacks proves the flaw of the argument.

In this example, the first statement uses categorical reasoning, saying that all carrot-eaters are definitely quarterbacks. This theory of deductive reasoning – also known as term logic – was developed by Aristotle, but was superseded by propositional (sentential) logic and predicate logic.

Deductive reasoning can be contrasted with inductive reasoning, in regards to validity and soundness. In cases of inductive reasoning, even though the premises are true and the argument is “valid”, it is possible for the conclusion to be false (determined to be false with a counterexample or other means).

Difference from ampliative reasoning

Deductive reasoning is usually contrasted with non-deductive or ampliative reasoning. The hallmark of valid deductive inferences is that it is impossible for their premises to be true and their conclusion to be false. In this way, the premises provide the strongest possible support to their conclusion. The premises of ampliative inferences also support their conclusion. But this support is weaker: they are not necessarily truth-preserving. So even for correct ampliative arguments, it is possible that their premises are true and their conclusion is false. Two important forms of ampliative reasoning are inductive and abductive reasoning. Sometimes the term "inductive reasoning" is used in a very wide sense to cover all forms of ampliative reasoning. However, in a more strict usage, inductive reasoning is just one form of ampliative reasoning. In the narrow sense, inductive inferences are forms of statistical generalization. They are usually based on many individual observations that all show a certain pattern. These observations are then used to form a conclusion either about a yet unobserved entity or about a general law. For abductive inferences, the premises support the conclusion because the conclusion is the best explanation of why the premises are true.

The support ampliative arguments provide for their conclusion comes in degrees: some ampliative arguments are stronger than others. This is often explained in terms of probability: the premises make it more likely that the conclusion is true. Strong ampliative arguments make their conclusion very likely, but not absolutely certain. An example of ampliative reasoning is the inference from the premise "every raven in a random sample of 3200 ravens is black" to the conclusion "all ravens are black": the extensive random sample makes the conclusion very likely, but it does not exclude that there are rare exceptions. In this sense, ampliative reasoning is defeasible: it may become necessary to retract an earlier conclusion upon receiving new related information. Ampliative reasoning is very common in everyday discourse and the sciences.

An important drawback of deductive reasoning is that it does not lead to genuinely new information. This means that the conclusion only repeats information already found in the premises. Ampliative reasoning, on the other hand, goes beyond the premises by arriving at genuinely new information. One difficulty for this characterization is that it makes deductive reasoning appear useless: if deduction is uninformative, it is not clear why people would engage in it and study it. It has been suggested that this problem can be solved by distinguishing between surface and depth information. On this view, deductive reasoning is uninformative on the depth level, in contrast to ampliative reasoning. But it may still be valuable on the surface level by presenting the information in the premises in a new and sometimes surprising way.

A popular misconception of the relation between deduction and induction identifies their difference on the level of particular and general claims. On this view, deductive inferences start from general premises and draw particular conclusions, while inductive inferences start from particular premises and draw general conclusions. This idea is often motivated by seeing deduction and induction as two inverse processes that complement each other: deduction is top-down while induction is bottom-up. But this is a misconception that does not reflect how valid deduction is defined in the field of logic: a deduction is valid if it is impossible for its premises to be true while its conclusion is false, independent of whether the premises or the conclusion are particular or general. Because of this, some deductive inferences have a general conclusion and some also have particular premises.

In various fields

Cognitive psychology

Cognitive psychology studies the psychological processes responsible for deductive reasoning. It is concerned, among other things, with how good people are at drawing valid deductive inferences. This includes the study of the factors affecting their performance, their tendency to commit fallacies, and the underlying biases involved. A notable finding in this field is that the type of deductive inference has a significant impact on whether the correct conclusion is drawn. In a meta-analysis of 65 studies, for example, 97% of the subjects evaluated modus ponens inferences correctly, while the success rate for modus tollens was only 72%. On the other hand, even some fallacies like affirming the consequent or denying the antecedent were regarded as valid arguments by the majority of the subjects. An important factor for these mistakes is whether the conclusion seems initially plausible: the more believable the conclusion is, the higher the chance that a subject will mistake a fallacy for a valid argument.

An important bias is the matching bias, which is often illustrated using the Wason selection task. In an often-cited experiment by Peter Wason, 4 cards are presented to the participant. In one case, the visible sides show the symbols D, K, 3, and 7 on the different cards. The participant is told that every card has a letter on one side and a number on the other side, and that "[e]very card which has a D on one side has a 3 on the other side". Their task is to identify which cards need to be turned around in order to confirm or refute this conditional claim. The correct answer, only given by about 10%, is the cards D and 7. Many select card 3 instead, even though the conditional claim does not involve any requirements on what symbols can be found on the opposite side of card 3. But this result can be drastically changed if different symbols are used: the visible sides show "drinking a beer", "drinking a coke", "16 years of age", and "22 years of age" and the participants are asked to evaluate the claim "[i]f a person is drinking beer, then the person must be over 19 years of age". In this case, 74% of the participants identified correctly that the cards "drinking a beer" and "16 years of age" have to be turned around. These findings suggest that the deductive reasoning ability is heavily influenced by the content of the involved claims and not just by the abstract logical form of the task: the more realistic and concrete the cases are, the better the subjects tend to perform.

Another bias is called the "negative conclusion bias", which happens when one of the premises has the form of a negative material conditional, as in "If the card does not have an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card has an A on the left". The increased tendency to misjudge the validity of this type of argument is not present for positive material conditionals, as in "If the card has an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card does not have an A on the left".

Psychological theories of deductive reasoning

Various psychological theories of deductive reasoning have been proposed. These theories aim to explain how deductive reasoning works in relation to the underlying psychological processes responsible. They are often used to explain the empirical findings, such as why human reasoners are more susceptible to some types of fallacies than to others.

An important distinction is between mental logic theories, sometimes also referred to as rule theories, and mental model theories. Mental logic theories see deductive reasoning as a language-like process that happens through the manipulation of representations. This is done by applying syntactic rules of inference in a way very similar to how systems of natural deduction transform their premises to arrive at a conclusion. On this view, some deductions are simpler than others since they involve fewer inferential steps. This idea can be used, for example, to explain why humans have more difficulties with some deductions, like the modus tollens, than with others, like the modus ponens: because the more error-prone forms do not have a native rule of inference but need to be calculated by combining several inferential steps with other rules of inference. In such cases, the additional cognitive labor makes the inferences more open to error.

Mental model theories, on the other hand, hold that deductive reasoning involves models or mental representations of possible states of the world without the medium of language or rules of inference. In order to assess whether a deductive inference is valid, the reasoner mentally constructs models that are compatible with the premises of the inference. The conclusion is then tested by looking at these models and trying to find a counterexample in which the conclusion is false. The inference is valid if no such counterexample can be found. In order to reduce cognitive labor, only such models are represented in which the premises are true. Because of this, the evaluation of some forms of inference only requires the construction of very few models while for others, many different models are necessary. In the latter case, the additional cognitive labor required makes deductive reasoning more error-prone, thereby explaining the increased rate of error observed. This theory can also explain why some errors depend on the content rather than the form of the argument. For example, when the conclusion of an argument is very plausible, the subjects may lack the motivation to search for counterexamples among the constructed models.

Both mental logic theories and mental model theories assume that there is one general-purpose reasoning mechanism that applies to all forms of deductive reasoning. But there are also alternative accounts that posit various different special-purpose reasoning mechanisms for different contents and contexts. In this sense, it has been claimed that humans possess a special mechanism for permissions and obligations, specifically for detecting cheating in social exchanges. This can be used to explain why humans are often more successful in drawing valid inferences if the contents involve human behavior in relation to social norms. Another example is the so-called dual-process theory. This theory posits that there are two distinct cognitive systems responsible for reasoning. Their interrelation can be used to explain commonly observed biases in deductive reasoning. System 1 is the older system in terms of evolution. It is based on associative learning and happens fast and automatically without demanding many cognitive resources. System 2, on the other hand, is of more recent evolutionary origin. It is slow and cognitively demanding, but also more flexible and under deliberate control. The dual-process theory posits that system 1 is the default system guiding most of our everyday reasoning in a pragmatic way. But for particularly difficult problems on the logical level, system 2 is employed. System 2 is mostly responsible for deductive reasoning.

Intelligence

The ability of deductive reasoning is an important aspect of intelligence and many tests of intelligence include problems that call for deductive inferences. Because of this relation to intelligence, deduction is highly relevant to psychology and the cognitive sciences. But the subject of deductive reasoning is also pertinent to the computer sciences, for example, in the creation of artificial intelligence.

Epistemology

Deductive reasoning plays an important role in epistemology. Epistemology is concerned with the question of justification, i.e. to point out which beliefs are justified and why. Deductive inferences are able to transfer the justification of the premises onto the conclusion. So while logic is interested in the truth-preserving nature of deduction, epistemology is interested in the justification-preserving nature of deduction. There are different theories trying to explain why deductive reasoning is justification-preserving. According to reliabilism, this is the case because deductions are truth-preserving: they are reliable processes that ensure a true conclusion given the premises are true. Some theorists hold that the thinker has to have explicit awareness of the truth-preserving nature of the inference for the justification to be transferred from the premises to the conclusion. One consequence of such a view is that, for young children, this deductive transference does not take place since they lack this specific awareness.

Probability logic

Probability logic is interested in how the probability of the premises of an argument affects the probability of its conclusion. It differs from classical logic, which assumes that propositions are either true or false but does not take into consideration the probability or certainty that a proposition is true or false. The probability of the conclusion of a deductive argument cannot be calculated by figuring out the cumulative probability of the argument’s premises. Dr. Timothy McGrew, a specialist in the applications of probability theory, and Dr. Ernest W. Adams, a Professor Emeritus at UC Berkeley, pointed out that the theorem on the accumulation of uncertainty designates only a lower limit on the probability of the conclusion. So the probability of the conjunction of the argument’s premises sets only a minimum probability of the conclusion. The probability of the argument’s conclusion cannot be any lower than the probability of the conjunction of the argument’s premises. For example, if the probability of a deductive argument’s four premises is ~0.43, then it is assured that the probability of the argument’s conclusion is no less than ~0.43. It could be much higher, but it cannot drop under that lower limit.

There can be examples in which each single premise is more likely true than not and yet it would be unreasonable to accept the conjunction of the premises. Professor Henry Kyburg, who was known for his work in probability and logic, clarified that the issue here is one of closure – specifically, closure under conjunction. There are examples where it is reasonable to accept P and reasonable to accept Q without its being reasonable to accept the conjunction (P&Q). Lotteries serve as very intuitive examples of this, because in a basic non-discriminatory finite lottery with only a single winner to be drawn, it is sound to think that ticket 1 is a loser, sound to think that ticket 2 is a loser,...all the way up to the final number. However, clearly, it is irrational to accept the conjunction of these statements; the conjunction would deny the very terms of the lottery because (taken with the background knowledge) it would entail that there is no winner.

Dr. McGrew further adds that the sole method to ensure that a conclusion deductively drawn from a group of premises is more probable than not is to use premises the conjunction of which is more probable than not. This point is slightly tricky, because it can lead to a possible misunderstanding. What is being searched for is a general principle that specifies factors under which, for any logical consequence C of the group of premises, C is more probable than not. Particular consequences will differ in their probability. However, the goal is to state a condition under which this attribute is ensured, regardless of which consequence one draws, and fulfilment of that condition is required to complete the task.

This principle can be demonstrated in a moderately clear way. Suppose, for instance, the following group of premises:

{P, Q, R}

Suppose that the conjunction ((P & Q) & R) fails to be more probable than not. Then there is at least one logical consequence of the group that fails to be more probable than not – namely, that very conjunction. So it is an essential factor for the argument to “preserve plausibility” (Dr. McGrew coins this phrase to mean “guarantee, from information about the plausibility of the premises alone, that any conclusion drawn from those premises by deductive inference is itself more plausible than not”) that the conjunction of the premises be more probable than not.

History

Aristotle, a Greek philosopher, started documenting deductive reasoning in the 4th century BC. René Descartes, in his book Discourse on Method, refined the idea for the Scientific Revolution. Developing four rules to follow for proving an idea deductively, Descartes laid the foundation for the deductive portion of the scientific method. Descartes' background in geometry and mathematics influenced his ideas on the truth and reasoning, causing him to develop a system of general reasoning now used for most mathematical reasoning. Similar to postulates, Descartes believed that ideas could be self-evident and that reasoning alone must prove that observations are reliable. These ideas also lay the foundations for the ideas of rationalism.

Related concepts and theories

Deductivism

Deductivism is a philosophical position that gives primacy to deductive reasoning or arguments over their non-deductive counterparts. It is often understood as the evaluative claim that only deductive inferences are good or correct inferences. This theory would have wide-reaching consequences for various fields since it implies that the rules of deduction are "the only acceptable standard of evidence". This way, the rationality or correctness of the different forms of inductive reasoning is denied. Some forms of deductivism express this in terms of degrees of reasonableness or probability. Inductive inferences are usually seen as providing a certain degree of support for their conclusion: they make it more likely that their conclusion is true. Deductivism states that such inferences are not rational: the premises either ensure their conclusion, as in deductive reasoning, or they do not provide any support at all.

One motivation for deductivism is the problem of induction introduced by David Hume. It consists in the challenge of explaining how or whether inductive inferences based on past experiences support conclusions about future events. For example, a chicken comes to expect, based on all its past experiences, that the person entering its coop is going to feed it, until one day the person "at last wrings its neck instead". According to Karl Popper's falsificationism, deductive reasoning alone is sufficient. This is due to its truth-preserving nature: a theory can be falsified if one of its deductive consequences is false. So while inductive reasoning does not offer positive evidence for a theory, the theory still remains a viable competitor until falsified by empirical observation. In this sense, deduction alone is sufficient for discriminating between competing hypotheses about what is the case. Hypothetico-deductivism is a closely related scientific method, according to which science progresses by formulating hypotheses and then aims to falsify them by trying to make observations that run counter to their deductive consequences.

Natural deduction

The term "natural deduction" refers to a class of proof systems based on self-evident rules of inference. The first systems of natural deduction were developed by Gerhard Gentzen and Stanislaw Jaskowski in the 1930s. The core motivation was to give a simple presentation of deductive reasoning that closely mirrors how reasoning actually takes place. In this sense, natural deduction stands in contrast to other less intuitive proof systems, such as Hilbert-style deductive systems, which employ axiom schemes to express logical truths. Natural deduction, on the other hand, avoids axioms schemes by including many different rules of inference that can be used to formulate proofs. These rules of inference express how logical constants behave. They are often divided into introduction rules and elimination rules. Introduction rules specify under which conditions a logical constant may be introduced into a new sentence of the proof. For example, the introduction rule for the logical constant "" (and) is "". It expresses that, given the premises "" and "" individually, one may draw the conclusion "" and thereby include it in one's proof. This way, the symbol "" is introduced into the proof. The removal of this symbol is governed by other rules of inference, such as the elimination rule "", which states that one may deduce the sentence "" from the premise "". Similar introduction and elimination rules are given for other logical constants, such as the propositional operator "", the propositional connectives "" and "", and the quantifiers "" and "".

The focus on rules of inferences instead of axiom schemes is an important feature of natural deduction. But there is no general agreement on how natural deduction is to be defined. Some theorists hold that all proof systems with this feature are forms of natural deduction. This would include various forms of sequent calculi or tableau calculi. But other theorists use the term in a more narrow sense, for example, to refer to the proof systems developed by Gentzen and Jaskowski. Because of its simplicity, natural deduction is often used for teaching logic to students.

Geometrical method

The geometrical method is a method of philosophy based on deductive reasoning. It starts from a small set of self-evident axioms and tries to build a comprehensive logical system based only on deductive inferences from these first axioms. It was initially formulated by Baruch Spinoza and came to prominence in various rationalist philosophical systems in the modern era. It gets its name from the forms of mathematical demonstration found in traditional geometry, which are usually based on axioms, definitions, and inferred theorems. An important motivation of the geometrical method is to repudiate philosophical skepticism by grounding one's philosophical system on absolutely certain axioms. Deductive reasoning is central to this endeavor because of its necessarily truth-preserving nature. This way, the certainty initially invested only in the axioms is transferred to all parts of the philosophical system.

One recurrent criticism of philosophical systems build using the geometrical method is that their initial axioms are not as self-evident or certain as their defenders proclaim. This problem lies beyond the deductive reasoning itself, which only ensures that the conclusion is true if the premises are true, but not that the premises themselves are true. For example, Spinoza's philosophical system has been criticized this way based on objections raised against the causal axiom, i.e. that "the knowledge of an effect depends on and involves knowledge of its cause". A different criticism targets not the premises but the reasoning itself, which may at times implicitly assume premises that are themselves not self-evident.

Hydrogen-like atom

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