Lat. a fortiori ratione, “for a stronger reason”.
Ratio, “reason”; fortis, “strong”, fortior “stronger”.
The argument a fortiori applies in two directions:
(1) “From bigger to smaller” (Lat. a maiori ad minus). This formula allows inferences from more to less:
The hook can hold a load of up to 20kg, so it can support 10kg.
If he is capable of killing someone, he is capable of striking someone.
Other expressions to the same effect: “for stronger reason”, “all the more reason to/for”, “those who can do hard things can readily do easy ones”, etc.
(2) “From smaller to bigger” (Lat. a minori ad maius). This formula rejects inferences from less to more:
The hook cannot hold a load of more than 20kg, so it certainly cannot support a 30 kg burden.
If one has no right to strike, one has no right to kill.
Other expression to the same effect: “still / much / even less”, etc.
This scheme can be specified in a discursive domain, for example as a consolation discourse:
The idea that “death should spare young people” is more acceptable (more normal) than “death should spare the elderly”. And you know that around you many younger people have died. Therefore, accept death.
This form underlies the statement “others died much younger”, said to comfort the living for the death of an elderly relative.
1. A fortiori, a transcultural topos
The a fortiori argument scheme is a clear example of a cross-cultural interpretative – argumentative rule.
1.1 Greco-Latin tradition
In the Greco-Latin tradition all collections of argument schemes throughout the history of Western argumentation mention the a fortiori rule. Aristotle illustrates this rule via the following examples:
If even the gods are not omniscient, human beings are certainly not. (Rhet, II, 23, 1397b15, RR, p. 359)
A man who strikes his father also strikes his neighbors […] for a man is less likely to strike his father than to strike his neighbors (ibid.).
The second argument can be used in the following situation. Somebody was assaulted. Who is guilty? We know that someone in the victim’s neighborhood committed violence against his own father. The a fortiori line casts suspicion upon he who has already committed more strongly prohibited forms of violence. The conclusion is that the police should question him.
1.1 Muslim legal argumentation
In Muslim legal argumentation, the bi-l-awla argument corresponds exactly to the a fortiori argument. The problem is discussed in the Koran (Surah 17, verse 24), dealing with the respect that a child owes to his parents:
Do not make “pffft!” to them!
The prohibition refers to a minimal impolite retort of a child shrugging off the words of his parents, or obeying them reluctantly, puffing out a sigh of exasperation. The a fortiori principle extends the prohibition to all disrespectful behavior: “since it is forbidden even to say “pff!” to one’s parents, it is all the more forbidden to say harsh words to them, to bully or to hit them”.
The prohibition takes its support on the lowest point on the scale, the epsilon of disrespect. Commentators have noticed that a fortiori argument can be a case of semantic deduction (Khallâf , p. 216).
1.3 Talmudic exegesis
The rules of Talmudic exegesis have been established by various authors since Hillel (1st century CE). The entry “Hermeneutics” of the Encyclopædia Judaïca, enumerates the thirteen interpretation rules of Rabbi Ishmael.
The first one is the rule qal va-homer, “how much more”, going a fortiori from the “minor” (qal) to the “major” (homer). (Jacobs &Derovan 2007, p. 25).
This rule helps to determine what is lawful and what is not, for example the conditions under which the Easter sacrifice, Pesach, should be offered. The Bible asks that Pesach be offered at Easter. Some actions are forbidden on the Sabbath, so what is one to do when Pesach coincides with the Sabbath? The calculation a fortiori gives the answer: the sacrifice Olat Tamid (“daily burnt-offering”) is offered every day, including Shabbat. Pesach is more important than Tamid (proof: if one does not respect Tamid, one does not incur penalties; if one does not respect Pesach, the sanctions are severe). Since not to celebrate Pesach is more serious than not to celebrate Tamid, and Tamid is lawful when Easter falls on the day of Shabbat, it is therefore a fortiori lawful to proceed to sacrifice Pesach when Easter falls on the day of Sabbath.
The reasoning can be expressed as a rhetorical syllogism:
Problem: the Pesach sacrifice must be offered on Passover.
Some actions are forbidden on Shabbat
Question: What should we do when Passover coincides with Shabbat?
Data: We know that 1) Tamid must be celebrated on Shabbat 2) Not celebrating Pesach is worse than not celebrating Tamid
Argumentation: Topos of the opposites on (2):
Celebrating Pesach is more important than celebrating Tamid.
This, combined with (1), leads to the conclusion:
Conclusion: Pesach can be celebrated when Easter coincides with Shabbat.
1.4 Chinese tradition
Confucius, The Analects. Bk 11, §12. Trans. Robert Eno 
Ji Lu asked about serving the spirits. The Master said, “While you are yet not able to serve men, how could you be able to serve the spirits?”
“May I ask about death?”
“When you do not yet understand life, how could you understand death?”
Han Fei Tzu, “Precautions within the palace”. Trans. Burton Watson 
Thus, the actor Shih aided Lady Li to bring about the death of Shen-sheng and to set Hsi-ch’i on the throne.1 Now, if someone as close to the ruler as his own consort, and as dear to him as his own son, still cannot be trusted, then obviously no one else is to be trusted either.
1 Lady Li and Hsi-ch’i “forced Shen-sheng to commit suicide in 656 BC”. “Hsi-ch’i (…) succeeded to the throne in 651 BC” (Burton Watson’s note to the text)
A fortiori can therefore be considered a good candidate for universality.
2. Nature of gradation
The application of the a fortiori rule presupposes both that the facts put in relation fall within a certain category and that they are hierarchically positioned within this category. This gradation may follow very different principles:
— Objective gradation: “he can hardly go from his bed to the window, and you would like to take him shopping downtown?”
— Socio-semantic gradation: “even grandparents sometimes make big mistakes, so their grandchildren…”
— Gradation based on the authority of the sacred book: “the Pesach sacrifice is more important than the Tamid sacrifice”.
In the “Argumentation within Language” theory (Ducrot 1973) the concept of a graduated category is represented as an argumentative scale, the a fortiori rule being an argumentative operator on such scales.
3. A fortiori in paragon scales
Some of these scales are topped by an ultimate individual, the paragon, the most excellent specimen of the category. The absolute degree in the category is established in terms of comparability with the paragon: “sly as a fox”.
These paragon scales are effective in rejecting a complaint: “You say that what happens to you is unjust. That’s true. But consider that Christ is the Innocent par excellence. Now, you are not Christ, and Christ accepted an unjust death. You must therefore accept this injustice.”
An episode of the Spanish Civil War (1936-1939). Paco, a somewhat turbulent villager, turns himself in after the war, upon the request of Mosén Millán, a priest. Mosén Millán assures him that he would be convicted but that he would have his life saved. Paco surrendered, and now he is to be shot with his companions.
— Why do you want to kill me? What did I do? We didn’t kill anyone. Tell them I’ve done nothing wrong. You know very well that I’m innocent, that we’re all innocent.
—Yes, my son. You are all innocent. But what can I do?
— They want to kill me because I fought back at Pardinas; OK, but the other two did nothing wrong.
Pedro clung to the cassock of Mosén Millán, and repeated: “They did nothing, and they are going to kill them. They did nothing.” Moved to tears, Mosén Millán said to him:
— Sometimes, my son, God allows the death of an innocent. He allowed it for his own son, who was more innocent than you three.
On hearing these words, Paco remained paralyzed and mute. The priest said nothing either.
Ramón J. Sender [Requiem for a Spanish Peasant] 
 Ramón J. Sender  = (1981). Requiem por un campesino Español. Barcelona: Destinolibro, 7th ed.. P. 100-101. Confucius, The Analects. An Online Teaching Translation. R. Eno 2015 (Version 2.21) http://www.indiana.edu/~p374/Analects_of_Confucius_(Eno-2015).pdf  Han Fei Tzu. Basic Writings. Section 17, “Precautions within the palace”. Translated by Burton Watson. New York, London, Columbia University Press, 1964. P. 84-85.