What are some logically equivalent formulations of “uniqueness”?

Question

A monoid is a mathematical structure with an associative law of composition and an identity element. It can be proven that if an element of a monoid has an inverse, then the inverse is unique:

Assume y and z are two inverse elements of x, and e is the identity element of the monoid.

y = y • e = y • (x • z) = (y • x) • z = e • z = z

This is an algebraic proof that there cannot be more than one inverse element of x.

We considered two variables, y and z, and assumed they were both inverses to x.

It followed, by substitution of definitions, and applications of monoid axioms, that y = z.

That is, all things claiming to be an inverse to x are equal to each other.

---

I understand this proof logically, but I find it to be more of a symbolic “trick” than a deep way to represent what uniqueness means.

When you study different formal systems which are logically equivalent to one another, you find that two sentences which technically express the same fact nonetheless don’t appear to be “saying the same thing”, because they portray the world from a different perspective. (As an aside, it would be fascinating if we could formalize the difference between those two things.)

My question is about to what extent does the form of logic - the way we have defined and structured a logical system - predispose us to choose that “idiom” as a way to prove “uniqueness”, when there could be other ways to conceptualize what uniqueness is.

The only insight I can offer so far is this.

It is very common to interpret “P(c)”, where P is predicate “red”, and c is a constant denoting a particular cup, as “Cup c is red.”

My problem might just be that I haven’t studied the meaning of variables in logic enough yet.

If a variable is not quantified over, it is called free.

An interpretation can satisfy a formula, or not satisfy it. Whether it does or not, depends exclusively on the choice of a structure, and the assignment of the free variables.

I personally find the following sentence to be a useful example to analyze:

Trisha is the only doctor.

Which conventionally would be proven by a sentence of the form,

Trisha is a doctor, and for anyone else who is a doctor, that person is also Trisha.

I agree with Mauro that this might be a misinterpretation of what “Doctor(Trisha) and for all x, Doctor(x) implies x = Trisha” is saying - but I think this implies that a common way of explaining the meaning logical sentences is inaccurate, and we need to be more precise.

x is a symbol called a “variable”. An assignment allows us to choose what element in a domain x is mapped to.

There is this aspect of first order logic I haven’t seen discussed much yet which is arbitrary assignments. I feel like this needs to be made more precise. Is an arbitrary assignment an assignment, the details of which haven’t been disclosed to the human reading the sentence? Or is an arbitrary assignment the lack of an assignment?

I will come back and add more. But now the question is more about how to formally evaluate the proposition “Trisha is the only doctor”.

For example, a model satisfies a formula when the formula is true for an arbitrary assignment.

I think what I was getting at is, to say “for all x such that x is a doctor, x is Trisha” is awkward when you think about how this is formally evaluated: in principle, variable x must pass over every element in the domain, and it is asserted that for none of them is Doctor() true, except Trisha.

I realize one mistake I made was thinking of “Trisha” more as a predicate than strictly an identifier for an element in the domain; but I still think the monoid example gives us food for thought, and I am confident there are other presentations of “uniqueness” than the common one I mentioned.

---

“Uniqueness” sounds like a meta-property. Perhaps second-order logic has a way to express “P(c) uniquely”, such as “Unique(P, c)”. In fact, I know there is a “unique existential quantifier” like: “∃!x P(x)”.

Answer 1

You are reading the definition in a way that is not quite right. There is no logical term for "other". A better reading is "x is P and anything that is P is x". This seems to reasonably summarize the intuition of uniqueness that most people have.

And the logical definition should not be construed as any sort of metaphysical or conceptual definition (although there are probably philosophers who think that sort of definition is the right metaphysical or conceptual definition). All the logical definition does is give a suitable definition of uniqueness based only on a relatively minimal set of concepts. You could just as well add uniqueness as a primitive concept, but then you would need to add axioms to let you prove that "x is the unique P" if and only if "x is P and anything that is P is x". So you have complicated the logic for no real gain.

Answer 2

To me, an intuitive way to express uniqueness would be something like:

Francis is Pope and anybody who is not Francis is not Pope.

But, "Anybody who is not Francis is not Pope," is the contrapositive of, "Anybody who is Pope is Francis". And this is exactly what the conventional way of writing uniqueness yields. In symbolic form:

Pope(francis) ∧ (∀x)(Pope(x) → (x = francis))

To express that something satisfies a predicate uniquely without naming it, we just apply existential generalisation:

(∃y)[ Pope(y) ∧ (∀x)(Pope(x) → (x = y)) ]

So, it's not really that odd after all.

As to your proposal with inverses, I don't see how it will help. Why should it be that for any individual there exists an inverse that when composed with it yields a unique identity element? You have assumed uniqueness by speaking of "the identity element". And why should the composition relation be commutative and associative? With mathematical objects you may assume what you like, but I don't see how these assumptions would generalise.

As to your proposal about names being unique. In formal logic, we always assume that a name refers to a unique individual. But the assumption that an individual has a unique name is not a standard assumption, because it makes the rules for instantiation and generalisation more complicated. It is however a fairly common assumption in AI systems. But this doesn't really help to express uniqueness. If there happens to be one and only one Pope, and Pope(francis) is true, it doesn't follow that Pope(conifold) leads to a contradiction, it just means that Pope(conifold) is false under the standard interpretation of those terms as they relate to the actual world. It wouldn't be convenient to have to express that Francis is the only Pope by enumerating all the people who are not.

Answer 3

Uniqueness means different things for different mathematical objects, all related to how we view equality between objects.

This is observed in your post, where you notice that two sets being equal, and two well-formed formula being "equal" mean very different things.

Since, two sets are equal if they have the same elements, and two well-formed formula are "equal" if they logically imply each other.

So uniqueness looks different, whenever equality looks different. And depending on the notion of equality, the "sameness" of the objects may not be obvious at all. Topologists provide some famous examples of this.

The following is not an actual mathematical convention ( as far as I am aware), but I am presenting it in the spirit of the OP's question.

In general, we can consider different notions of Uniqueness by considering an equivalence Relation E over our domain of Discourse U, let x be an element in U

Consider the equivalence class [x] = {y∈U|yEx}

Definition: x is E-unique ( unique up to E) with respect to some property P iff for all z, P(z) ⟺ z∈[x]

Here is another formulation of uniqueness.

x is unique with respect to some property P iff |Y = {z|P(z)}| = 1 and x∈Y.

i.e. The set of all objects which satisfy P, has cardinality 1 and x is an element of the set.

Answer 4

As far as I have seen, the ubiquitous way of proving “x has property P uniquely” in mathematics is by showing “if there was some other y which also had property P, it would follow necessarily that y is actually still just x”.

This isn't correct. You need the meanings of arbitrary constant and specific constant.

Definition.

C is a specific constant iff ∃!x [x=C]

Definition.

C is an arbitrary constant iff ∀x [x=C] ∨ ∃!x [x=C]

Suppose x1 is some specific constant such that what x1 denotes has property/attribute P. We symbolize that by writing P(x1). Now, let α be an arbitrary constant. IF[ If P(α) then α = x1 ] THEN x1 denotes the unique thing which has property P.

We do not conclude α is x1, we conclude any thing which α denotes is the thing x1 denotes. It seems to me, that you are confusing the referrer with the referent. Since α is an arbitrary constant, it is possible it denotes any of all things. We start out assuming α is an arbitrary constant, and conclude it's a specific constant. Therefore, there is a unique thing with property P.

What are some logically equivalent formulations of "uniqueness"?

I have already expressed my favorite.

∃!x[P(x)] = P(x_n) ∧ P(α) → x_n = α, where x_n is any specific individual constant, and α is any arbitrary individual constant.

So you start off assuming some individual constant is arbitrary, and denotes a thing with property P. If later you conclude it's a specific constant, then the thing denoted by it is the unique thing with property P.

P(α) → x_n = α iff ∀x[P(x) → x_n = x]

So another way to express uniqueness is

∃!x[P(x)]= P(x_n) ∧ ∀x[P(x) → x_n=x]

Through Existential Instantiation and Existential Generalization the previous statement is logically equivalent to

∃!x[P(x)]= ∃y [P(y) ∧ ∀x[P(x) → y = x]]

These three statements are all logically equivalent ways to formulate "uniqueness".

You mentioned the statement Trisha is the only doctor.

Formally that means

∃x[D(x)] ∧ ∀x[D(x) → Trisha = x]

This doesn't have precisely what is needed for uniqueness, unless I interpret your statement as

Trisha is a doctor AND Trisha is the only doctor.

I know that's what you mean.

That does have the form

D(Trisha) ∧ ∀x[D(x) → Trisha = x ]

Which is one of the three logically equivalent ways of formulating "uniqueness".

This does not reflect human intuition too well, in my opinion. We aren’t wont to say, “Trisha is the strongest person in the world, and all the other strongest people are also Trisha!”

The meaning of 'other' isn't explicitly used in the logical definition of uniqueness. Logically what is meant is, "Trisha is the strongest person in the world, and any person who is the strongest person in the world is Trisha.'

Consider a formal system where the symbols are the “main characters”, so to speak. It would be false to say “a = b” ever, because the symbol a is not the symbol b. The only thing a equals is a. Maybe you can think of this as a world of entities without a symbolic layer of denotation above them.

The symbols a,b are referrers, and are clearly different symbols. But if a=b, then they symbolize the same referent.

It seems to me your confusion can be eliminated by learning the difference between referrer and referent. You must have, in your words, "a symbolic layer of denotation above them [the symbols of your formal language].

Also, you said ,"consider a formal system where the symbols are the "main characters"." This raises the question, "what does the word 'symbol' mean?"

Symbols are terms that symbolize, i.e. stand for something. The definition of symbol is

Definition.

C is a symbol iff ∃!x [x = C]

In words, C is a symbol if and only if there is a unique thing denoted by C.

Not all terms symbolize a unique thing. Consider the expression

x^2 - 1 = 0

Here the term x symbolizes 1 or -1, so it's not a symbol. You can call it a symbol, but you won't know what you mean. It's a mark, or a term, which in this case symbolizes exactly two things.