# What is the meaning of "comprehension" in logic?

I was reading about the axioms of Zermelo-Fraenkel set theory and the axiom of restricted comprehension. This led me to find out what the meaning of this word is and why it's called this. Then I saw that it has something to do with intension vs extension (although maybe I'm wrong). Then I googled the word comprehension and this is what Wikipedia says:

In logic, the comprehension of an object is the totality of intensions, that is, attributes, characters, marks, properties, or qualities, that the object possesses, or else the totality of intensions that are pertinent to the context of a given discussion.

I'm not a native speaker and I should note that I'm not very good at logic and I want to know what the meaning of this word is and how it's defined. Is it like the ability to understand something or does it have another meaning?

• "the significance of general ideas has two aspects: the comprehension [la comprehension] and the extension [l'étendue]. The comprehension consists in the set of attributes essential to the idea. For example, the comprehension of the idea ‘triangle’ includes the attributes extension, shape, three lines, and three angles. The extension of the idea consists in the inferiors or subjects to which the term applies, which for Port-Royal includes “all the different species of triangles”. but I still don't the meaning of this word in this context. Apr 6, 2019 at 20:47
• I mean its any difference between the comprehension we use in everyday language and the comprehension that is used in this context? Apr 6, 2019 at 20:48
• Yes, the philosophical meaning of "comprehension" is a lot more technical than the usual common language definition. Basically, it seems to distinguish between comprehension (basic understanding), and extension (broader understanding). First, they contribute to the history of semantics by distinguishing the comprehension (or intension) of a general term from its extension (denotation)...In recognizing these two modes of signification—the comprehension and the extension—Port-Royal imports the distinction between incomplete and complete entities into the signification of general terms. Apr 6, 2019 at 21:16
• It just means "set formation." Forget any philosophical associations. In this context comprehension just means set formation. If you can form a set based on a predicate, that's unrestricted comprehension. If you are required to start with a known set and then reduce it by a predicate to form another set, that's restricted comprehension. Whether the word originally had philosophical associations are not, it's best to just translate "comprehension" as "set formation" and don't get hung up on these other issues IMO. Apr 6, 2019 at 23:52

An object can be defined in at least two ways, with an intentional definition or an extensional definition. This is how Wikipedia describes the two:

In logic and mathematics, an intensional definition gives the meaning of a term by specifying necessary and sufficient conditions for when the term should be used.

An extensional definition of a concept or term formulates its meaning by specifying its extension, that is, every object that falls under the definition of the concept or term in question.

So an intentional definition provides the necessary and sufficient conditions for the term. For example, an even natural number would be a natural number that is divisible by 2. We could not provide an extensional definition of even natural number since that would require specifying each even natural numbers and there are infinitely many of them.

Based on the Wikipedia article cited by the OP, "the comprehension of an object is the totality of intensions". It contains all the necessary and sufficient conditions that could be used to define the object.

Now consider the "axiom of restricted comprehension". Wikipedia notes:

In many popular versions of axiomatic set theory the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.

Some mathematicians call it the axiom schema of comprehension, although others use that term for unrestricted comprehension, discussed below.

Because restricting comprehension solved Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory.

Note that there is some ambiguity on how this term is used and that there are other terms in use for the same idea. Even if one is a native speaker one has to be careful with technical terms. However, the key point is this axiom "says that any definable subclass of a set is a set".

Also note that the reason this is restricted comprehension is to avoid Russell's paradox that arises from an axiom of unrestricted comprehension.

Wikipedia contributors. (2019, March 3). Axiom schema of specification. In Wikipedia, The Free Encyclopedia. Retrieved 23:45, April 6, 2019, from https://en.wikipedia.org/w/index.php?title=Axiom_schema_of_specification&oldid=885934953

Wikipedia contributors. (2019, January 6). Extensional and intensional definitions. In Wikipedia, The Free Encyclopedia. Retrieved 23:29, April 6, 2019, from https://en.wikipedia.org/w/index.php?title=Extensional_and_intensional_definitions&oldid=877075627

The philosophical conceprt of "comprehension" and the so-called principle of comprehension has its source in traditional logic.

See Port Royal Logic:

[for] Port-Royal [...] the significance of general ideas has two aspects: the comprehension [la comprehension] and the extension [l'étendue]. The comprehension consists in the set of attributes essential to the idea. For example, the comprehension of the idea ‘triangle’ includes the attributes extension, shape, three lines, and three angles. The extension of the idea consists in the inferiors or subjects to which the term applies, which for Port-Royal includes “all the different species of triangles”.

See: Antoine Arnauld et Pierre Nicole, La logique ou l'art de penser (3eme ed, 1668), page 69.

Following the Scottish philosopher William Hamilton, in his Logic, page 59, the distiction has been reformulated as that between intension and extension:

the Internal Quantity of a notion, its Intension or Comprehension, [and] the External Quantity of a notion or its Extension.

Consider the well-kown traditional example : "every Human is Mortal".

We read it today in terms of classes (or sets), i.e. estensionally, as meaning that the set of Humans is a subset of the set of Mortals.

But we can also read it intensionally as asserting that attribute of Mortality is contained into the concept of Humanity.

Thus, a concept is "made of" attributes : its intension (or comprehension), and to a concept corresponds the class of objects to which the concept applies (the objects "falling under" the concept) : its extension.

The comprehension principle is the principle asserting that we can use the attributes defining a concept (its intension) in order to carve out from the universe of objects the collection of all and only those objects to which the concept applies (its extension).

In order to understand your misconception, you have to go back to the French origins of the word, comparing two different meanings of the same term :

Compréhension : la compréhension d'un concept est la donnée des concepts plus généraux qui peuvent en être prédiqués, et peuvent entrer dans sa définition; là où l' extension d'un concept est la classe (l'ensemble) des individus qui répondent à ce concept.)

with :

Compréhension : est un processus psychologique lié à un objet physique ou abstrait tel que la personne, la situation ou le message selon lequel une personne est capable de réfléchir et d’utiliser des concepts pour traiter cet objet de manière appropriée.

Both derive from Latin comprehensiō, comprehensiōnem, from comprehendere (“to grasp”), from the prefix com- + prehendere (“to seize”).