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Argumentum a fortiori (literally "argument from the stronger [reason]") (UK: /ˈɑː fɔːrtiˈoʊri/,^[1] US: /ˈeɪ fɔːrʃiˈɔːraɪ/) is a form of argumentation that draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be implicit in, and even more certain than, the first.^[2]

Usage

American usage

In Garner's Modern American Usage, Garner says writers sometimes use a fortiori as an adjective as in "a usage to be resisted". He provides this example: "Clearly, if laws depend so heavily on public acquiescence, the case of conventions is an a fortiori [read even more compelling] one."^[3]

Jewish usage

A fortiori arguments are regularly used in Jewish law under the name kal va-chomer,^[4] literally "mild and severe", the mild case being the one we know about, while trying to infer about the more severe case.^{[further explanation needed]}

Relation to ancient Indian logic

In ancient Indian logic (nyaya), the instrument of argumentation known as kaimutika or kaimutya nyaya is found to have a resemblance with a fortiori argument. Kaimutika has been derived from the words kim uta meaning "what is to be said of".^[5]

Islamic usage

In Islamic jurisprudence, a fortiori arguments are proved utilising the methods used in qiyas (reasoning by analogy).^[6]

Examples

If a person is dead (the stronger reason), then one can, with equal or greater certainty, argue a fortiori that the person is not breathing. "Being dead" trumps other arguments that might be made to show that the person is dead, such as "he is no longer breathing"; therefore, "he is no longer breathing" is an extrapolation from his being dead and is a derivation of this strong argument.^[7]
If it is known that a person is dead on a certain date, it may be inferred a fortiori that he is exempted from the suspect list for a murder that took place on a later date, viz "Allen died on April 22nd, therefore, a fortiori, Allen did not murder Joe on April 23rd."
If driving 10 mph over the speed limit is punishable by a fine of $50, it can be inferred a fortiori that driving 20 mph over the speed limit is also punishable by a fine of at least $50.
If a teacher refuses to add 5 points to a student's grade because the student does not deserve an additional 5 points, it can be inferred a fortiori that the teacher will also refuse to raise the student's grade by 10 points.

In mathematics

Consider the case where there is a single necessary and sufficient condition required to satisfy some axiom. Given some theorem with an additional restriction imposed upon this axiom, an "a fortiori" proof will always hold. To demonstrate this, consider the following case:^[8]

For any set A, there does not exist a function mapping A onto its powerset P(A). (Even if A were empty, the powerset would still contain the empty set.)
There cannot exist a one-to-one correspondence between A and P(A).

Because bijections are a special case of functions, it automatically follows that if (1) holds, then (2) will also hold. Therefore, any proof of (1) also suffices as a proof of (2). Thus, (2) is an "a fortiori" argument.

Types

A maiore ad minus

In logic, a maiore ad minus describes a simple and obvious inference from a claim about a stronger entity, greater quantity, or general class to one about a weaker entity, smaller quantity, or specific member of that class:^[9]

From general to particular ("What holds for all X also holds for one particular X")
From greater to smaller ("If a door is big enough for a person two metres high, then a shorter person may also come through"; "If a canister may store ten litres of petrol, then it may also store three litres of petrol.")
From the whole to the part ("If the law permits a testator to revoke the entirety of a bequest by destroying or altering the document expressing it, then the law also permits a testator to revoke the portion of a bequest contained in a given portion of a document by destroying or altering that portion of the document.")
From stronger to weaker ("If one may safely use a rope to tow a truck, one may also use it to tow a car.")

A minore ad maius

The reverse, less known and less frequently applicable argument is a minore ad maius, which denotes an inference from smaller to bigger.^[10]

In law

"Argumentum a maiori ad minus" (from the greater to the smaller) – works in two ways:

"who may more, all the more so may less" (qui potest plus, potest minus) and relates to the statutory provisions that permit to do something
"who is ordered more, all the more so, is ordered less" and relates to the statutory provisions that order to do something

An a fortiori argument is sometimes considered in terms of analogical reasoning – especially in its legal applications. Reasoning a fortiori posits not merely that a case regulated by precedential or statutory law and an unregulated case should be treated alike since these cases sufficiently resemble each other, but that the unregulated case deserves to be treated in the same way as the regulated case in a higher degree. The unregulated case is here more similar (analogues) to the regulated case than this case is similar (analogues) to itself.^{[citation needed]}

References

^ Morwood, James (1998). A Dictionary of Latin Words and Phrases. Oxford: Oxford University Press. pp. x–xii. ISBN 978-0-19-860109-8.
^ Purtill, Richard (2015). "a fortoriori argument". In Audi, Robert (ed.). The Cambridge Dictionary of Philosophy (Third ed.). New York City: Cambridge University Press. p. 14. ISBN 978-1-139-05750-9. OCLC 927145544.
^ Garner, Bryan A. (2009). Garner's Modern American Usage (3rd ed.). Oxford: Oxford University Press. p. 28. ISBN 978-0-19-538275-4.
^ Abramowitz, Jack. "Torah Methodology #1 – Kal v'Chomer". Orthodox Union. Retrieved 20 July 2016.
^ Sion, Avi (2013-11-24). A Fortiori Logic: Innovations, History and Assessments. Avi Sion.
^ Hallaq, Wael (2009). Sharī'a: Theory, Practice, Transformations (1st ed.). Cambridge: Cambridge University Press. p. 105. ISBN 978-0521678742.
^ Grabenhorst, Thomas Kyrill (1990). Das argumentum a fortiori: eine Pilot-Studie anhand der Praxis von Entscheidungsbegründungen (in German). Lang. ISBN 978-3-631-43261-7.
^ Kaplansky, Irving (1977). Set Theory and Metric Spaces (2nd ed.). Chelsea, NYC: AMS Publishing. p. 29. ISBN 978-0-8284-0298-9.
^ Fellmeth, Aaron Xavier; Horwitz, Maurice (2009). Guide to Latin in international law (1 ed.). Oxford: Oxford University Press. pp. 2–3. ISBN 978-0-19-536938-0. Retrieved 21 October 2023.
^ Fellmeth, Aaron Xavier; Horwitz, Maurice (2009). Guide to Latin in international law (1 ed.). Oxford: Oxford University Press. pp. 3–4. ISBN 978-0-19-536938-0. Retrieved 21 October 2023.

[1] Morwood, James (1998). A Dictionary of Latin Words and Phrases. Oxford: Oxford University Press. pp. x–xii. ISBN 978-0-19-860109-8.

[2] Purtill, Richard (2015). "a fortoriori argument". In Audi, Robert (ed.). The Cambridge Dictionary of Philosophy (Third ed.). New York City: Cambridge University Press. p. 14. ISBN 978-1-139-05750-9. OCLC 927145544.

[3] Garner, Bryan A. (2009). Garner's Modern American Usage (3rd ed.). Oxford: Oxford University Press. p. 28. ISBN 978-0-19-538275-4.

[4] Abramowitz, Jack. "Torah Methodology #1 – Kal v'Chomer". Orthodox Union. Retrieved 20 July 2016.

[5] Sion, Avi (2013-11-24). A Fortiori Logic: Innovations, History and Assessments. Avi Sion.

[6] Hallaq, Wael (2009). Sharī'a: Theory, Practice, Transformations (1st ed.). Cambridge: Cambridge University Press. p. 105. ISBN 978-0521678742.

[7] Grabenhorst, Thomas Kyrill (1990). Das argumentum a fortiori: eine Pilot-Studie anhand der Praxis von Entscheidungsbegründungen (in German). Lang. ISBN 978-3-631-43261-7.

[8] Kaplansky, Irving (1977). Set Theory and Metric Spaces (2nd ed.). Chelsea, NYC: AMS Publishing. p. 29. ISBN 978-0-8284-0298-9.

[9] Fellmeth, Aaron Xavier; Horwitz, Maurice (2009). Guide to Latin in international law (1 ed.). Oxford: Oxford University Press. pp. 2–3. ISBN 978-0-19-536938-0. Retrieved 21 October 2023.

[10] Fellmeth, Aaron Xavier; Horwitz, Maurice (2009). Guide to Latin in international law (1 ed.). Oxford: Oxford University Press. pp. 3–4. ISBN 978-0-19-536938-0. Retrieved 21 October 2023.

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+{{Short description|Argument from a yet stronger reason}}
-{{Refimprove|date=May 2016}}{{Italic title|noerror}}
+{{Italic title|noerror}}
-'''''Argumentum a fortiori''''' ({{IPAc-en|UK|ˈ|ɑː|_|f|ɔːr|t|i|ˈ|oʊ|r|i}},<ref>{{cite book|title=A Dictionary of Latin Words and Phrases|url=https://archive.org/details/dictionaryoflati00morw|url-access=registration|last=Morwood|first=James|publisher=Oxford University Press|year=1998|isbn=978-0-19-860109-8|location=Oxford|pages=x–xii}}</ref> {{IPAc-en|US|ˈ|eɪ|_|f|ɔːr|ʃ|i|ˈ|ɔːr|aɪ}}; [[Latin]]: "from a/the stronger [thing]") is a form of [[Argumentation theory|argumentation]] that draws upon existing [[confidence]] in a [[proposition]] to argue in favor of a second proposition that is held to be [[Logical consequence|implicit]] in the first. The second proposition may be considered weaker, and therefore the arguer utilizes the former as the stronger proposition from which the second proposition is deduced.
+'''''Argumentum a fortiori''''' (literally "argument from the stronger [reason]") ({{IPAc-en|UK|ˈ|ɑː|_|f|ɔːr|t|i|ˈ|oʊ|r|i}},<ref>{{cite book|title=A Dictionary of Latin Words and Phrases|url=https://archive.org/details/dictionaryoflati00morw|url-access=registration|last=Morwood|first=James|publisher=Oxford University Press|year=1998|isbn=978-0-19-860109-8|location=Oxford|pages=x–xii}}</ref> {{IPAc-en|US|ˈ|eɪ|_|f|ɔːr|ʃ|i|ˈ|ɔːr|aɪ}}) is a form of [[Argumentation theory|argumentation]] that draws upon existing confidence in a [[proposition]] to argue in favor of a second proposition that is held to be [[Logical consequence|implicit]] in, and even more certain than, the first.<ref>{{Cite book|last=Purtill|first=Richard|title=[[The Cambridge Dictionary of Philosophy]]|publisher=[[Cambridge University Press]]|year=2015|isbn=978-1-139-05750-9|editor-last=Audi|editor-first=Robert|edition=Third|location=New York City|pages=14|chapter=''a fortoriori'' argument|oclc=927145544}}</ref>
-== Examples ==
+==Usage==
-* If a person is dead (the stronger reason), then one can with equal or greater certainty argue ''a fortiori'' that the person is not [[Breathing#Control of breathing|breathing]]. "Being dead" trumps other arguments that might be made to show that the person is dead, such as "he is no longer breathing"; therefore, "he is no longer breathing" is an extrapolation from his being dead and is a derivation of this strong argument.
-* If driving 10&nbsp;mph over the speed limit is punishable by a fine of $50, it can be inferred ''a fortiori'' that driving 20&nbsp;mph over the speed limit is also punishable by a fine of at least $50.
-*If a teacher refuses to add 5 points to a student's grade, on the grounds that the student does not deserve an additional 5 points, it can be inferred ''a fortiori'' that the teacher will also refuse to raise the student's grade by 10 points.
+=== American usage ===
+In ''[[Garner's Modern American Usage]]'', [[Bryan A. Garner|Garner]] says writers sometimes use ''a fortiori'' as an [[adjective]] as in "a usage to be resisted". He provides this example: "Clearly, if laws depend so heavily on public acquiescence, the case of conventions is an ''a fortiori'' [read ''even more compelling''] one."<ref>{{cite book |title=Garner's Modern American Usage |last=Garner |first=Bryan A.|edition=3rd |year=2009 |publisher=Oxford University Press |location=Oxford |isbn=978-0-19-538275-4 |page=28 }}</ref>
+=== Jewish usage ===
-==Usage==
+''A fortiori'' arguments are regularly used in [[Halakha|Jewish law]] under the name [[Talmudical hermeneutics#Kal va-chomer (קל וחומר)|kal va-chomer]],<ref>{{cite web|last1=Abramowitz|first1=Jack|title=Torah Methodology #1 – Kal v'Chomer|url=https://www.ou.org/torah/gemara/the-13-rules-of-rabbi-yishmael/methodology1/|website=Orthodox Union|access-date=20 July 2016}}</ref> literally "mild and severe", the mild case being the one we know about, while trying to infer about the more severe case.{{Explain|date=March 2024}}
+=== Relation to ancient Indian logic ===
-In the [[English language]], the phrase ''a fortiori'' is most often used as an [[adverbial phrase]] meaning "by even greater force of logic" or "all the more so".{{Citation needed|date=May 2016}}
+In [[Indian logic|ancient Indian logic]] ([[nyaya]]), the instrument of argumentation known as ''kaimutika'' or ''kaimutya nyaya'' is found to have a resemblance with ''a fortiori'' argument. K''aimutika'' has been derived from the words ''kim uta'' meaning "what is to be said of".<ref>{{Cite book|last=Sion|first=Avi|url=https://books.google.com/books?id=2E9CAgAAQBAJ&dq=kaimutika+or+kaimutya+nyaya&pg=PA323|title=A Fortiori Logic: Innovations, History and Assessments|date=2013-11-24|publisher=Avi Sion|language=en}}</ref>
+=== Islamic usage ===
-In ''[[Garner's Modern American Usage]]'', [[Bryan A. Garner|Garner]] says writers sometimes use ''a fortiori'' as an [[adjective]] as in "a usage to be resisted". He provides this example: "Clearly, if laws depend so heavily on public acquiescence, the case of conventions is an ''a fortiori'' [read ''even more compelling''] one."<ref>{{cite book |title=Garner's Modern American Usage |last=Garner |first=Bryan A.|edition=3rd |year=2009 |publisher=Oxford University Press |location=Oxford |isbn=978-0-19-538275-4 |page=28 }}</ref>
+In [[Fiqh|Islamic jurisprudence]], ''a fortiori'' arguments are proved utilising the methods used in ''[[qiyas]]'' (reasoning by [[analogy]]).<ref>{{cite book |title=Sharī'a: Theory, Practice, Transformations |last=Hallaq |first=Wael |edition=1st |year=2009 |publisher= Cambridge University Press |location=Cambridge |isbn=978-0521678742 |page=105 }}</ref>
+== Examples ==
-''A fortiori'' arguments are regularly used in [[Halakha|Jewish law]] under the name [[Talmudical hermeneutics#Kal va-chomer (קל וחומר)|kal va-chomer]],<ref>{{cite web|last1=Abramowitz|first1=Jack|title=Torah Methodology #1 – Kal v’Chomer|url=https://www.ou.org/torah/gemara/the-13-rules-of-rabbi-yishmael/methodology1/|website=Orthodox Union|accessdate=20 July 2016}}</ref> literally "mild and severe", the mild case being the one we know about, while trying to infer about the more severe case.
+* If a person is dead (the stronger reason), then one can, with equal or greater certainty, argue ''a fortiori'' that the person is not [[Breathing#Control|breathing]]. "Being dead" trumps other arguments that might be made to show that the person is dead, such as "he is no longer breathing"; therefore, "he is no longer breathing" is an extrapolation from his being dead and is a derivation of this strong argument.<ref>{{Cite book |last=Grabenhorst |first=Thomas Kyrill |url=https://books.google.com/books?id=MrhgAAAACAAJ |title=Das argumentum a fortiori: eine Pilot-Studie anhand der Praxis von Entscheidungsbegründungen |date=1990 |publisher=Lang |isbn=978-3-631-43261-7 |language=de}}</ref>
+*If it is known that a person is dead on a certain date, it may be inferred ''a fortiori'' that he is exempted from the suspect list for a murder that took place on a later date, viz "Allen died on April 22nd, therefore, ''a fortiori'', Allen did not murder Joe on April 23rd."
+* If driving 10&nbsp;mph over the speed limit is punishable by a fine of $50, it can be inferred ''a fortiori'' that driving 20&nbsp;mph over the speed limit is also punishable by a fine of at least $50.
+*If a teacher refuses to add 5 points to a student's grade because the student does not deserve an additional 5 points, it can be inferred ''a fortiori'' that the teacher will also refuse to raise the student's grade by 10 points.
+=== In mathematics ===
+Consider the case where there is a single [[Necessity and sufficiency|necessary and sufficient]] condition required to satisfy some [[axiom]]. Given some theorem with an additional restriction imposed upon this axiom, an "a fortiori" proof will always hold. To demonstrate this, consider the following case:<ref>{{Cite book |last=Kaplansky |first=Irving |title=Set Theory and Metric Spaces |publisher=[[AMS Publishing]] |year=1977 |isbn=978-0-8284-0298-9 |edition=2nd |location=Chelsea, NYC |pages=29 |language=en}}</ref>
+# For any [[Set (mathematics)|set]] A, there does not exist a [[Function (mathematics)|function]] mapping A onto its [[Power set|powerset]] P(A). (Even if A were empty, the powerset would still contain the empty set.)
-In [[Indian logic|ancient Indian logic]] ([[nyaya]]), an inference derived from an ''a fortiori'' postulation is known as ''kaimutika'' or ''kaimutya nyaya'', from the words ''kim uta'' meaning "even more so".{{Citation needed|date=May 2016}}
+# There cannot exist a [[one-to-one correspondence]] between A and P(A).
+Because bijections are a special case of functions, it automatically follows that if (1) holds, then (2) will also hold. Therefore, any [[Mathematical proof|proof]] of (1) also suffices as a proof of (2). Thus, (2) is an "a fortiori" argument.
-In [[Fiqh|Islamic jurisprudence]], ''a fortiori'' arguments are proved utilising the methods used in ''[[qiyas]]'' (reasoning by [[analogy]]).<ref>{{cite book |title=Sharī'a: Theory, Practice, Transformations |last=Hallaq |first=Wael |edition=1st |year=2009 |publisher= Cambridge University Press |location=Cambridge |isbn=0521678749 |page=105 }}</ref>
 ==Types==
-{{unreferenced section|date=March 2017}}
 ===''A maiore ad minus''===
-In [[logic]], '''''a maiore ad minus''''' describes a simple and obvious inference from a claim about a stronger entity, greater quantity, or general class to one about a weaker entity, smaller quantity, or specific member of that class:
+In [[logic]], '''''a maiore ad minus''''' describes a simple and obvious inference from a claim about a stronger entity, greater quantity, or general class to one about a weaker entity, smaller quantity, or specific member of that class:<ref>{{cite book |last1=Fellmeth |first1=Aaron Xavier |last2=Horwitz |first2=Maurice |title=Guide to Latin in international law |date=2009 |publisher=Oxford University Press |location=Oxford |isbn=978-0-19-536938-0 |pages=2–3 |edition=1 |url=http://www.oxfordreference.com/view/10.1093/acref/9780195369380.001.0001/acref-9780195369380 |access-date=21 October 2023}}</ref>
 * From general to particular ("What holds for all X also holds for one particular X")
 * From greater to smaller ("If a door is big enough for a person two metres high, then a shorter person may also come through"; "If a canister may store ten litres of petrol, then it may also store three litres of petrol.")
 * From the whole to the part ("If the law permits a testator to revoke the entirety of a bequest by destroying or altering the document expressing it, then the law also permits a testator to revoke the portion of a bequest contained in a given portion of a document by destroying or altering that portion of the document.")
-* From stronger to weaker ("If one may safely use a rope to tow a truck [in the American usage], one may also use it to tow a car.")
+* From stronger to weaker ("If one may safely use a rope to tow a truck, one may also use it to tow a car.")
 ===''A minore ad maius''===
-The reverse, less known and less frequently applicable argument is '''''a minore ad maius''''', which denotes an inference from smaller to bigger.
+The reverse, less known and less frequently applicable argument is '''''a minore ad maius''''', which denotes an inference from smaller to bigger.<ref>{{cite book |last1=Fellmeth |first1=Aaron Xavier |last2=Horwitz |first2=Maurice |title=Guide to Latin in international law |date=2009 |publisher=Oxford University Press |location=Oxford |isbn=978-0-19-536938-0 |pages=3–4 |edition=1 |url=http://www.oxfordreference.com/view/10.1093/acref/9780195369380.001.0001/acref-9780195369380 |access-date=21 October 2023}}</ref>
 ==In law==
-“Argumentum a maiori ad minus” (from the greater to the smaller) –  works in two ways:
+"Argumentum a maiori ad minus" (from the greater to the smaller) –  works in two ways:
+*  "who may more, all the more so may less" (qui potest plus, potest minus) and relates to the statutory provisions that permit to do something
-*  “who may more, all the more so may less” and relates to the statutory provisions that permit to do something
+* "who is ordered more, all the more so, is ordered less"  and relates to the statutory provisions that order to do something
-* “who is ordered more, all the more so, is ordered less”  and relates to the statutory provisions that order to do something
-An ''a fortiori'' argument is sometimes considered in terms of analogical reasoning – especially in its legal applications. Reasoning ''a fortiori'' posits not mere that a case regulated by precedential or statutory law and an unregulated case should be treated alike since these cases  sufficiently resemble each other, but that the unregulated case deserves to be treated in the same way as the regulated case in a higher degree. The unregulated case is here more similar (analogues) to the regulated case than this case is similar (analogues) to itself.<ref>See Maciej Koszowski,   [http://www.aijcrnet.com/journals/Vol_7_No_1_March_2017/3.pdf The Scope of Application of Analogical Reasoning in Statutory Law]. American International Journal of Contemporary Research no. 1/2017 (v. 7), pp. 28-29.</ref>
+An ''a fortiori'' argument is sometimes considered in terms of analogical reasoning – especially in its legal applications. Reasoning ''a fortiori'' posits not merely that a case regulated by precedential or statutory law and an unregulated case should be treated alike since these cases sufficiently resemble each other, but that the unregulated case deserves to be treated in the same way as the regulated case in a higher degree. The unregulated case is here more similar (analogues) to the regulated case than this case is similar (analogues) to itself.{{citation needed|date=November 2019}}
 ==See also==
 *[[Argumentation theory]]
+*[[Principle of sufficient reason]]
 *[[Rhetoric]]
 ==References==
 {{Reflist}}
+{{Authority control}}
 [[Category:Latin logical phrases]]