# What's the difference between a second-order relation and a relation between objects?

I was reading an article in philosophy and found this:

Some philosophers have denied that there is such a relation as identity. Thus Ludwig Wittgenstein writes (Tractatus 5.5301): "That identity is not a relation between objects is obvious." At 5.5303 he elaborates: "Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing." Bertrand Russell had earlier voiced a worry that seems to be motivating Wittgenstein's point (The Principles of Mathematics §64): "[I]dentity, an objector may urge, cannot be anything at all: two terms plainly are not identical, and one term cannot be, for what is it identical with?" Even before Russell, Gottlob Frege, at the beginning of "On Sense and Reference," expressed a worry with regard to identity as a relation: "Equality gives rise to challenging questions which are not altogether easy to answer. Is it a relation?" More recently, C. J. F. Williams has suggested that identity should be viewed as a second-order relation, rather than a relation between objects, and Kai Wehmeier has argued that appealing to a binary relation that every object bears to itself, and to no others, is both logically unnecessary and metaphysically suspect.

So what's the difference between a second-order relation and a relation between objects? Is it related to second-order logic and could you explain so that a layman can understand?

• A relation between objects is e.g "x is less then y" between numbers or "x is taller than y" between humans. Sep 27, 2019 at 16:35
• A second-order relation is a relation hodling between relations between objects. Consider e.g. two relations R and S between numbers and consider the relations holding between R and S if they have the same domain. Sep 27, 2019 at 16:37
• I don’t know anything about logic. But for you to get the history of some of this, it may help to go back to at least Hegel’s Phenomenology. I can’t remember the section. Also, see this announcement from Dresden and read over the matters to be discussed. tu-dresden.de/gsw/phil/iphil/das-institut/news/… Sep 27, 2019 at 16:42
• “Anticipation of Wittgensteinian motives in Hegel’s philosophy” Univ Dres Topics. ; there is an anticipation of a certain area of W’s Tractatus in a certain section of H’s Phenomenology. Sep 27, 2019 at 16:49

Here are the questions:

So what's the difference between a second-order relation and a relation between objects? Is it related to second-order logic and could you explain so that a layman can understand?

A first-order logic has a domain, such as the natural numbers. Relations of first-order logic relate one or more objects of this domain. If there is only one object the relation can be thought of as describing a property or a predicate of the object. For example, the term R(a) means that the object with the name a has the property (or predicate) R. If R is interpreted as "is an even number" and a is interpreted as "2", then R(a) means the proposition "2 is an even number".

There are also ways to quantify the objects of the domain using the logical symbols "there exists" (∃) and "for all" (∀). These quantifiers range over all objects in the domain and only those objects. They are called "logical symbols" because a model does not assign specific meaning to these two symbols as it would for a domain or a relation.

The quoted paragraph references identity (=), so let's consider what that means as an example.

In first-order logic with identity the logical symbol, =, is a special binary relation. Just like ∃ and ∀ the model does not have to define what this logical symbol means. The identity relation works like this for all models: Suppose a is the name of an object from the domain and b is the name of an object from the domain. If a = b then the name a refers to the very same object that the name b refers to. If that is not the case, that is, if the names a and b refer to two different objects, then a ≠ b.

A second-order logic has all that a first-order logic has (except perhaps identity), but it also allows one to quantify over the relations like R even though R is not a member of the domain.

Having the ability to quantify over all relations (or properties or predicates) allows one to eliminate the need for the identity logical symbol and use the "identity of indiscernibles" instead. Here is how Wikipedia describes that concept:

The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa; to suppose two things indiscernible is to suppose the same thing under two names.

This is what the identity symbol meant in first-order logic: two names refer to the very same object. In second order logic one can quantify over all those properties or relations as well as the objects in the domain. That is, a = b can be replaced by ∀R(R(a) ↔ R(b)). For all predicates R, the object represented by a possesses R if and only if the object represented by b possesses R.

Wikipedia contributors. (2019, September 22). Identity of indiscernibles. In Wikipedia, The Free Encyclopedia. Retrieved 18:16, September 27, 2019, from https://en.wikipedia.org/w/index.php?title=Identity_of_indiscernibles&oldid=917052463