Proportion and Proportionality

The Greek word [analogia] means “proportionality”, as the Latin proportio.

1. Proportion

A relation is a stable connection between two things:

shell : fish                       old age : life
glove : hand                    pilot : ship

A proportion is an analogy between at least two relations (not between individuals, as in categorical analogy); it implies at least four terms. Two pairs of beings are in a relation of proportion if, in their respective fields, they are bound by the same, or a similar relation.

shell : fish ~ feather : bird                   — cover the body of
glove : hand ~ shoe : foot                   — protect the
leader : group ~ captain : ship             — guide the
old age : life ~ evening : day                — last moment of the

The analogy of proportions is expressed through parallel syntactic structures:

(Since) a ship needs a pilot, any group needs a leader!

In arithmetic, a proportion is defined as the relation between two numbers, such as ‘17 / 27’. The same proportional relation binds two pairs of numbers a/b and d/d if they obey the following rule:

3/2 = 9/6, same proportion 1.5
a/b = c/d <=> ad = bc (a = bc/d, etc.)

The analogy between proportions corresponds to the linear equation with one unknown, that is to say, to the “rule of three”:

a / b = c / x where ax = bc, ax-bc = 0; and x = bc / a

In geometry, two similar figures have the same shape and different dimensions. Two congruent triangles have equal angles and proportional sides.

The process of understanding is the same in the case of mathematics as it is in argumentation. The reasoning by which the value of ‘x’ is mathematically extracted from the arithmetical proportion is the same as the argument which extracts the necessity of a leader from the analogy of proportion between a ship’s crew and a group of people more generally.

The analogy of proportion is at the basis of a specific kind of metaphor:

old age, evening of life.

The analogy of proportion is open to ironic self-refutation:

A woman without a man is like a fish without a bicycle.

2. Proportionality

Lat. ad modum argument, Lat. modus, “measure, “just measure”
NB: Besides “moderation”, the Lat. temperentia can mean “just measure, fair proportion”.

The argument of proportionality justifies a provision or an action by claiming that it is well proportioned to the facts, gradual. It is invoked a contrario in routine press releases such as:

(The association, the trade union, the government…) X condemns the disproportionate use of force used against

Let us consider a situation of unrest, described by the current government as a seditious uprising, led by a handful of terrorists. The government organizes a large military presence to “show strength not to have to use it”. This strategy of psychological war may have perverse effects. In reality, the argument of proportionality allows calculations that defeat the desired effect:

The deployment of strength, far from minimizing the enemy, made it stronger. (Pierre Miquel, [The Algeria War], 1993[1])

This conclusion is based on the topos, “one does not use cannons to shoot flies”. A strong refutation of a (declared) weak position entails the same kind of paradox.

The argument of proportionality is a form of argument on the right measure, which can also be defined as the intermediate measure, S. Moderation.

The proportionality strategy can be used to avoid the risks posed by the escalation strategy.

[1] Pierre Miquel, La Guerre d’Algérie. Paris: Fayard, 1993, p. 190.