''Argumentum a fortiori'' (literally "argument from the stronger
eason Eason is a surname.
The name comes from Aythe where the first recorded spelling of the family name is that of Aythe Filius Thome which was dated circa 1630, in the "Baillie of Stratherne". Aythe ''filius'' Thome received a charter of the lands of F ...
) (, ) is a form of
argumentation
Argumentation theory, or argumentation, is the interdisciplinary study of how conclusions can be supported or undermined by premises through logical reasoning. With historical origins in logic, dialectic, and rhetoric, argumentation theory, incl ...
that draws upon existing
confidence
Confidence is a state of being clear-headed either that a hypothesis or prediction is correct or that a chosen course of action is the best or most effective. Confidence comes from a Latin word 'fidere' which means "to trust"; therefore, having ...
in a
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
to argue in favor of a second proposition that is held to be
implicit
Implicit may refer to:
Mathematics
* Implicit function
* Implicit function theorem
* Implicit curve
* Implicit surface
* Implicit differential equation
Other uses
* Implicit assumption, in logic
* Implicit-association test, in social psycholog ...
in, and even more certain than, the first.
Usage
American usage
In ''
Garner's Modern American Usage
''Garner's Modern English Usage'' (''GMEU''), written by Bryan A. Garner and published by Oxford University Press, is a usage dictionary and style guide (or ' prescriptive dictionary') for contemporary Modern English. It was first published in 1 ...
'', Garner says writers sometimes use ''a fortiori'' as an
adjective
In linguistics, an adjective ( abbreviated ) is a word that generally modifies a noun or noun phrase or describes its referent. Its semantic role is to change information given by the noun.
Traditionally, adjectives were considered one of the ...
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'' ead ''even more compelling''one."
Jewish usage
''A fortiori'' arguments are regularly used in
Jewish law
''Halakha'' (; he, הֲלָכָה, ), also transliterated as ''halacha'', ''halakhah'', and ''halocho'' ( ), is the collective body of Jewish religious laws which is derived from the written and Oral Torah. Halakha is based on biblical comman ...
under the name kal va-chomer, literally "mild and severe", the mild case being the one we know about, while trying to infer about the more severe case.
Islamic jurisprudence
''Fiqh'' (; ar, فقه ) is Islamic jurisprudence. Muhammad-> Companions-> Followers-> Fiqh.
The commands and prohibitions chosen by God were revealed through the agency of the Prophet in both the Quran and the Sunnah (words, deeds, and ex ...
, ''a fortiori'' arguments are proved utilising the methods used in ''
qiyas
In Islamic jurisprudence, qiyas ( ar, قياس , " analogy") is the process of deductive analogy in which the teachings of the hadith are compared and contrasted with those of the Quran, in order to apply a known injunction ('' nass'') to a ...
'' (reasoning by
analogy
Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject ...
).
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
Breathing (or ventilation) is the process of moving air into and from the lungs to facilitate gas exchange with the internal environment, mostly to flush out carbon dioxide and bring in oxygen.
All aerobic creatures need oxygen for cellu ...
. "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 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, 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.
In mathematics
Consider the case where there is a single
necessary and sufficient
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth o ...
condition required to satisfy some
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
. Given some theorem with an additional restriction imposed upon this axiom, an "a fortiori" proof will always hold. In order to demonstrate this, consider the following case:
# For any
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
A, there does not exist a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
mapping A onto its
powerset
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
P(A).
# There cannot exist a
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a c ...
of (1) also suffices as a proof of (2). Thus, (2) is an "a fortiori" argument.
Types
''A maiore ad minus''
In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, ''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:
* 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
n the American usage
N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''.
History
...
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.
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.
See also
*
Argumentation theory
Argumentation theory, or argumentation, is the interdisciplinary study of how conclusions can be supported or undermined by premises through logical reasoning. With historical origins in logic, dialectic, and rhetoric, argumentation theory, incl ...
*
Rhetoric
Rhetoric () is the art of persuasion, which along with grammar and logic (or dialectic), is one of the three ancient arts of discourse. Rhetoric aims to study the techniques writers or speakers utilize to inform, persuade, or motivate par ...