Abstract Objects in Logic

Question

I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, it is true. Yet, when we consider x(x+1) and X^2 + X, we can say that the x is the same for = 1. However, we call this intensional since the two expressions are true for the same value. This I understand. However, what I am having difficulty with is the claim that numbers are by their very nature abstract objects. So, how is it that there exists any truth values for mathematical statements? I know this seems like a general question but I am having difficulty in wrapping my head around the fact since a proposition about an abstract object by its very nature is intensional. Why then is the number 1 fixed. Is it simply because we agree that 1 is 1 and nothing else? And, does mathematical logic itself establish the meaning of 1?

Answer 1

There are a lot of questions here. Some insights:

1. Objects are not "concrete" or "abstract" in isolation. There is a hierarchy of abstraction. It has no bottom and probably has no top.

2. There is a traditional distinction between what we call "constants" and "variables". $0$ and $1$ are numbers, sure. And we can think of them as constants. But as ink stains on a page, they are exactly the same sort of symbol as $x$. It is up to a logic to give them meaning. And logics can differ on what they mean. So, in a context where you're talking about the numbers, $0$ and $1$ are numbers, because numbers are what you're talking about. But in a context where you're talking about matrices, for example, $0$ and $1$ are not numbers. They're matrices.

3. We call a definition intensional when it is a listing of the properties an object must satisfy. So, if $X$ is defined by $X(X+1) = X^2 + X$, there are only so many things $X$ could be. $X$ has to belong to an algebra with addition and multiplication, for example.

4. If you list out all of the things that $X$ could be, you can alternatively define $X$ in terms of that listing. That is the extensional definition.

Answer 2

See Extensionality.

The distinction concerning concepts traces back to "traditional" logic : comprehension or intension vs. extension.

A modem reference for the distinction is the Logique de Port-Royal (1662), one of the most widely read treatises of the 17th-century:

In these universal ideas, it is very important to correctly distinguish the comprehension and the extension.

By the comprehension of the idea we understand the attributes which it involves and which cannot be withdrawn without destroying the idea, as the comprehension of the idea of triangle involves extension, figure, three lines, three angles, equality of those angles summed up to two right angles, etc.

By the extension of the idea we understand the subjects to which the idea applies, and which are also known as the inferiors of a general term which, in relation to them, is called superior; as the general idea of a triangle extends to all the different species of triangles.

This distinction is clear also in George Boole's The Mathematical Analysis of Logic (1847) :

That which renders logic possible, is the existence in our minds of general notions, - our ability to conceive of a class, and to designate its individual members by a common name.

Thus, concepts and their extensional counterpart, classes, are the very foundation of logic.

We can follow these ideas until Gottlob Frege, which articulated explicitly the principle of comprehension in his two-volume work of 1893/1903, Grundgesetze der Arithmetik; this principle says that, given a well-defined concept, there is the class of all objects that satisfy the concept.

---

Having said that, we call "extensional" a concept or a definition; I do not think that it is correct to say that an object - abstract or not - is "extensional".

According to the traditional view, the concept of number has some attributes which form its intension. The numbers form the extension of the number concept.

Thus, on what assumptions you are asserting :

an abstract object by its very nature is intensional... ?

In the history of logic and philosophy (end 19th- beginning 20th-centuries) we find a distiction between extensional contexts : a "standard" assertoric statement, and intensional ones.

The paradigmatic example of intensional context is "I believe that Walter Scott is the author of Waverley". The truth-conditions for this statement depends on my "knowledge": for sure I'm able to assert the "tautology : "I believe that Walter Scott is Walter Scott", but if I'm not an expert of english literature, I'm not able to assert the first one.

In modern logic, the "standard" approach is truth-functional, i.e. extensional. But of course there are now attempt at a formalization of intensional discourses , from Alonzo Church's Logic of Sense and Denotation (1951), to Jaakko Hintikka's Knowledge and Belief (1962).