## What is the logical distinction between “the same” and “equal to?”

8

3

We all understand that Given A = C, and B = C, Then A = B.

However, A is not “the same as” B.

Example:

A is the question, “What animals have feathers and can fly?”

B is the question, “What was the primary subject of artist John James Audubon?”

We can say both A and B = C, and this relationship is bidirectional.

“Birds” ARE “Animals with feathers and can fly”
Also,
“Animals with feathers and can fly” ARE “Birds”

however A is certainly not “the same question” as B. They are not interchangeable in most contexts.

An example of the philosophical dillema this poses came to me on this site in fact. It was regarding the idea of “duplicate questions” where something which had answers "same as" another were therefore deemed to have "same questions." It's obviously false but logically how is this fallacy argued? A commenter rationalized the fact that because it had "the same" answer as another question, they were "the same" questions.

The exact illogical argument is below:

“@VogonPoet - Fair enough. In that case, it is definitely a duplicate. All of the other races mention in 'The Chase' are addressed in the top answer to the duplicate question. –"

Please do not simply restate the problem. This question tries to identify the philosophical difference between two similar expressions which often frustrate good communication, there is a tendency for responses to simply recouch the problem from a different angle or say “it just is.” Respect the quality of the site please. Thank you!

Comments are not for extended discussion; this conversation has been moved to chat.

– Philip Klöcking – 2019-09-27T08:03:22.557

9

## Non sequitur

I'll go off of the example in the comments, namely

“One dollar” = “money” : “Nickel” = “money.” Therefore, “one dollar” = “nickel.”

This is non sequitur - there's no logical reason to assume that Therefore.

Or, alternatively, this could be ambiguity fallacy as this seems to be caused by (intentional?) misapplication of the symbol "=" with different, incompatible meanings. Whatever you mean by this symbol should be properly defined - and in this case you can't use the standard definition of "=" used in math (and thus assume its properties) because that definition of "=" operates on a particular restricted set of objects (e.g. natural numbers or something like that) and doesn't apply to the objects like "one dollar" or "money".

And the thing is, that for most reasonable definitions of what you mean by "=" you'll have to choose which parts of the statement stay valid - either “One dollar” = “money” will be clearly false under that definition, or you won't be able to show that transitivity should apply for "your =" and the therefore part would be without any basis whatsoever - thus the non sequitur fallacy. You could also define "=" in a way where both these things are true, but in that case your "=" would simply mean something like "is vaguely related to the same concept" and in which case "one dollar"="nickel" would be true for that definition of "=" without any problems.

As the other answer states, you can't simply assume that Given A = C, and B = C, Then A = B - transitivity is a property that some well-defined operators have and others don't; it needs to be proven under whatever axioms and operator definitions you choose. If you stay within the bounds of some particular theory such as the math of real numbers or first order logic or whatever, then you can "piggyback" on the various properties that are proven for the symbols used in that theory; but if you make up your own meaning for these symbols (to enable you to say that “One dollar” = “money”) then you're on your own and need to start from scratch to prove what properties hold for symbols in your theory. So there's some ambiguity and vagueness caused by re-purposing "=" to mean something substantially different from the common understanding of "=". Possibly that's a redefinition fallacy, but at least for me the statement stands most clearly as a simple non sequitur.

8Indeed, the "=" sign should probably be substituted with the "element of set" sign there. Both "one dollar" and "one nickel" belong to the set of "things that are money", but two things that belong to the same set aren't required to be the same element of that set. 1 and 2 both belong to the set of natural numbers, but 1 and 2 are not the same natural number. – probably_someone – 2019-09-24T11:48:54.890

3@probably_someone Yes. I was going to say that Set-Theory heavily applies here. In all sets with elements, there is some relationship between those elements, and some operators that can work on those sets. Symbols and terms for such operators may look the same, but have different meaning when the set definition is known. Order of operations is relevant in some sets for particular operators as well. – wolfsshield – 2019-09-24T14:07:55.293

@probably_someone - this may be onto something. It sounds like the solution is to logically disprove the equality of "sameness", or ∃! (N∈{Answers}): P(N'∈{Questions}) - So in this case, ["Birds" ∈ ({What animals have feathers and can fly?} ⋂ {What was John Audubon's main art subject?})], yet it is false that ⋂⇒=. All {Answers} can belong to multiple {Questions}, they have no uniqueness. This is just the problem restated - not a solution. – Vogon Poet – 2019-09-25T20:49:07.863

2@VogonPoet The general class of object that has the important properties of "=" is called an equivalence relation. For a relation ~, we say that ~ is an equivalence relation if it's true that 1) A~A for any A ("~ is reflexive"), 2) A~B implies B~A and vice versa ("~ is symmetric"), and 3) A~B and B~C implies A~C ("~ is transitive"). The subset relation is not symmetric (for example, the even numbers are a subset of the integers, but the integers are not a subset of the even numbers), so it fails criterion 2 of the definition. – probably_someone – 2019-09-25T22:19:50.817

@VogonPoet That said, the version of "sameness" you're going to get out of an equivalence relation can be quite abstract and broad (which is useful for mathematicians, as it gives them a method of linking seemingly unrelated mathematical objects together, but isn't always useful if you're looking for a way to distinguish between two things that are similar). Choosing the right definition of "the same" is key, and in your case, you want to choose the most restrictive possible definition that still adheres to the three criteria (since you're looking for a tool to make fine-grained distinctions). – probably_someone – 2019-09-25T22:25:09.160

Please don't use code formatting for emphasis. A screenreader is going to read out each letter individually, which isn't a great user experience. – TRiG – 2019-09-26T09:35:24.637

6

This is a question in philosophy that deals with the metaphysics of identity. A classic problem in philosophy is the Ship of Theseus and goes back to the pre-Socratics, particularly Heraclitus and his proposition that one cannot stand in the same river twice.

In logic, one often draws a distinction between a name (symbol) and the thing it represents (referent) the relationship between the two being called the thought (reference) as per Ogden and Richard's The Meaning of Meaning though the same ideas are found in ancient Greece. And this is a useful distinction if one presumes a sharp distinction between the outside world and the inside mind like in substance dualism.

In your example, note that you are attempting to apply a property to the propositions that does not hold. A-->C, and B-->C and NB that A-/->B and B-/->A. As such, the transitive property ((A-->B, B-->C)--> (A->C)) does not apply!

This is why it is important to examine not just the symbols in an argument, but the content (of the referents) too, because meaning can be found not only in a proposition, but in the relationship between propositions. Particularly important is recognizing that equality is a bidirectional implication, and that questions and answers aren't related the same way two identical quantities with different labels are.

EDIT

NB: A=B, B=C -> A=C is defined as (A<-->B, B<-->C) --> (A<-->C) because (A-->B, B-->C)-->(A-->C) and (C-->B,B-->A)-->(C-->A) where <--> is defined as --> and <-- true over two symbols.

To address comments below, let's not get caught in a deepity. The questions with the same answer can be seen essentially as two propositions:

1. Birds are animals with feathers that can fly.
2. Birds are the subject of Audubon.

Yes, birds are a common subject, so insofar as these two propositions have the same subject, they have a common attribute. But because they have different predicates, they have other attributes not the same, so the propositions themselves are different. A is not equal to B. In FOPC, A := Fx : x:=b and B := Sx : x:=b and clearly Fb ≠ Sb despite D: of Fb ∩ D: of Sb = b at a minimum.

What is more interesting is having the same proposition in different statements which highlights the difference between syntax and semantics:

1. Birds are animals with feathers that can fly.
2. Animals of flight adorned with feathers are birds.

The difference in statements (syntax) is obvious, and would likely confuse a computer (see Turing Test), and yet most natural language speakers would have little problem in seeing these as equivalent propositions (semantics), that is they really mean the same thing. In FOPC, A := Fx : x:=b and B := Ax : x:=b and clearly Fb = Sb, and thusly A = B.

That makes clear what “the same” is not - it isn’t a bidirectional implication. But what is it then? How do we succinctly describe this fallacy? – Vogon Poet – 2019-09-23T18:49:34.887

1@VogonPoet A and B are simply interrogative forms of two statements both of which feature the same subject C. "Birds are animals that have feathers and fly." and "Birds are the primary subject of the artist John James Audubon." are just two distinct statements. They both belong to a set of propositions that have the subject "bird". There's nothing mysterious here. A is not identical to B. – J D – 2019-09-24T12:49:48.360

@J D - The philosophical question is “why.” Not looking for a simple confirmation. Thanks – Vogon Poet – 2019-09-24T13:23:49.523

2@VogonPoet That's not mysterious either. As symbolic entities, they differ in both syntax and semantics. Sometimes different syntax expresses the same content (synonym), and sometimes the same syntax expresses different content (homonym). In your case, they share some content, but not all. Since they share some, but not all, they are not the same. – J D – 2019-09-24T13:34:02.647

6

To approach this from a slightly different angle, this concept is important in computer programming.

In a lot of languages, the programmer can decide what attributes make an object "equal to" another object.

For example, if you have two "People" objects represented by "first name", "last name" and "address"; you could choose to say that if the first and last names are the same, then the two objects are "equal" (i.e. they represent the same person), despite the differing addresses.

Those two people objects wouldn't be "the same", however, unless you were looking at exactly the same objects in memory - in which case everything about them would be identical, because there would only be one Person object (even if you're looking at it using two different pointers).

As Bill points out in the comments, if two objects are "the same", then a change to one will change the other; whereas if they are just Equal but not the same, then changing one won't affect the other (and may in fact make them not equal any more, depending on the change)

2And if you're looking at two NaNs in the same place, they are "the same" but don't compare equal! – Unrelated String – 2019-09-25T09:37:17.527

1No two NULL are created equal, either! – bishop – 2019-09-25T12:41:11.920

2More importantly if they are "The same" object then changing one will change the other, if they are "Equal" objects they can also be "The Same" but maybe not and changing one will not change the other, it will just make them unequal. – Bill K – 2019-09-25T16:12:50.990

2

@bishop Don't know if you're referencing the recent news story or computer science, but this is a very practical example of the cost of getting this "philosophical" answer wrong: "Having ‘Null’ as a license plate is about as much of a nightmare as you’d expect" https://www.theverge.com/tldr/2019/8/14/20805543/null-license-plate-california-parking-tickets-violations-void-programming-bug

– Jeff Y – 2019-09-25T17:07:03.970

1Wow, @JeffY, I hadn't read that one. Thanks for sharing! – bishop – 2019-09-25T17:35:40.110

4

I do believe you've missed the point of 'duplicate' here. 'Sameness' in this context is a fairly loose and utilitarian construct. Consider: if the temple priestess says she needs a statue of Zeus for entryway, and everyone in the village steps up to sculpt a statue of Zeus, well... the priestess still only needs (and will only use) one of those statues. The statues might be of varying qualities using different materials; they might depict Zeus in different actions or storylines; they might be larger or smaller... The priestess might have a difficult job choosing which one to use (and which ones to toss in the slag heap), but all these statues will be considered 'the same' for the purpose, despite their obvious differences.

We have the same situation here. Whoever voted to close your question did so not because the question was identical to some other question, but because it filled the same functional role on the site. There is a point where one must give up the idiosyncrasies of specific questions and generalize to principle. One wouldn't want a math forum where people were constantly asking for individual sums (what is 2+6? how about 7+1? what about 11-5?); why would we want a philosophy forum where people asked every possible variation on the same basic philosophical question?

Well that’s a great meta post however what about the philosophical question? How does this differ from “equal to” in logic? – Vogon Poet – 2019-09-24T03:13:33.290

Completely different criteria. 'Equal to' in logic is an identity relation; this is simply a functional relation. – Ted Wrigley – 2019-09-24T04:17:38.357

3

Like almost all linguistic conventions, all forms of equality are relative. It is convenient to have numerous synonyms for the different kinds of equality. But they are really interchangeable at some level. What each means is determined by context.

Within the domain of mathematics, one constantly contrives 'equivalence relations' which assign groups of things that are not meant to be separate from the point of view of a given field of study. In the most common modern approach to 'foundations', we define each real number to be a Dedekind Cut: the equivalence class of sets of rational numbers such that if a number is in the set, all numbers less than it are, too. But we have already defined each rational number to be the equivalence class of fractions of integers with no common factors other than 1 between their numerator and denominator. And we have defined the natural numbers to be the equivalence relations of sets of things that can be put into 1-to-1 correspondence. And so forth. That means the number 1 as a real number and as a rational number and as an integer are, by definition, not the same thing. But then we assign embeddings of each more basic kind of number into the more complete sets -- again by stating an equivalence relation...

It is just fine to take this approach outside math, as Wittgenstein does. No two things are ever really identical, and things are 'equal' or 'the same' only in a specific way chosen for a given purpose. Any atom with 6 protons is carbon. They are the same thing. But they also aren't. Some of them have 14 nucleons, and those serve a specific function. They are the same element, but different isotopes. And some of those have a matching complement of 6 electrons, but others don't. Those may be like isotopes but different ions... 'The same' or 'equal' mean nothing without a purpose in mind.

There is no most basic identity relationship, there are just those relationships we choose to serve given purposes. The ones we use most often become confused with some basic intuition of 'real identity', but there is no such thing. You can cling to the notion that your own identity is well defined, and use that as a prototype for other kinds of equality. But are you the same you that you were when you were born? Again, for some purposes yes, and for others no...

2

To Start:

Q1 "What city has the Tower of London?"

A: "London"


Q2 "What city has Big Ben?"

A: "London"


...having the same answer does not always imply equality, or even identity or sameness.

Same/Sameness is related to identity. The inability to distinguish one from the other.

Equal/Equality is about the relationship between elements, and is a comparison - typically of value or worth

"I had tea at lunch" "At lunch, I had tea" <= mean the same thing...meaning is identical, form is different: thus equal not same

1+3=4 2+2=4 : equal, not same 1+3=4 3+1=4 : equal and mathematically same because order of operations doesn't matter however sometimes order does matter but then it wouldn't be equal.

Mathematically: To be the same, you must be equal - in value/result/purpose.

Existence: To be the same, you must be equal - in all metrics (save change over time in the context of "I'm not the same person I was then" - because you are the same identity just you possess different characteristics now).

However you can be equal without being the same.

Sameness and equality also vary by context as to how they play out.

Consider people. You want to be treated equally with everyone, but you don't want to be treated the same as everyone. If Bob likes fish and Joe likes beef, and both want to be treated to an equal meal would you give them the same meal (either fish, beef, or something else that one or both might not like) or do you give them equal meals (Bob gets $10 of fish, Joe$10 beef - assuming equal here is in monetary value rather than mass).

Equal also changes on point of reference. Two people could be equal because they receive the same salary for the same job. But those same two people might not be equal at an amusement park because of some other metric (one's too short or wide for the ride, or too young/old or light/heavy or unhealthy or pregnant or some other limiting factor).

The difference between same and equal is part of the reason it so difficult to nail down fairness. You may treat everyone equal with regard to metric X, but one individual maybe evaluating on metric Y and another on metric Z and neither of them think you've been fair.

Whereas if you treat them the same, again they may not view that as equal or fair. Example: I give out "one-size-fits-all" uniforms, to save money I make them all small, perfect for a a person 5'8" with a particular build. I have personnel that range from thin to fat from 4'6" to 7'2". They all get identical uniforms (nothing observable from the naked eye to discern their differences). Have I treated my people fairly? I gave them the same thing. I didn't give consideration to anything about the individuals I did not discriminate. Yet, while treating them all the same, I have in some metric treated them equally, but with the intent to be that they wear the uniform, I did not treat them fairly, for by most metrics I did not treat them equally.

Your tea lunch example I would argue are the same but not equal, whilst your answer has it backwards. – corsiKa – 2019-09-25T18:32:15.240

1

This answer assumes the question is about logic and not about the closing of another question. If that is the case, then the equality sign (=) is used in first-order logic with identity, but not in propositional logic. If it isn't I will delete this answer.

In first-order logic there is a domain and one may select from that domain a member giving it the name d. One can make a second selection and give that selected member the name e. Then the question arises: Do the two names, d and e, refer to two different objects or one object? If they refer to one object, then d = e. If they refer to two objects, then these names are not identical, ~(d = e).

Consider how forallx describes equality (identity): (page 222)

This does not mean merely that the objects in question are indistinguishable, or that all of the same things are true of them. Rather, it means that the objects in question are the very same object.

Unfortunately this text uses "same" to describe this notion of identity and we want to distinguish "same" from "equal" or "identity", but the rest of the definition offers a way to make that distinction clear. It is worth making this clear, because there are two different concepts going on here.

One object that is merely the "same" as another object might be indistinguishable from the other object. Each may have all the same things true of them. One is in my left hand while the other is in my right hand, but otherwise I can't tell these two objects apart. The critical thing to note is that I have two objects, not just one object. That kind of sameness is not what the authors of forallx mean by identity where an equal sign can be used.

Identity where the equal sign is used in first-order logic means there are two different names, but both names refer to only one object from the domain. Same means there are two objects that happen to be so indistinguishable from each other that for all practical purposes I cannot tell them apart.

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Fall 2019. http://forallx.openlogicproject.org/forallxyyc.pdf

1

I'm assuming that by "equality," you mean the mathematical concept of equality, since that's the only kind of equality denoted by the equals sign in standard writing.

What is the logical distinction between “the same” and “equal to?”

No distinction. They're the same concept. Sameness = equality.

We can say both A and B = C, and this relationship is bidirectional.

No, that's not true. A ≠ C (because the question about flying animals is not the same thing as the concept of birds) and B ≠ C (because the question about Audubon is not the same thing as the concept of birds). And, of course, A ≠ B.

It seems an irrational logic but what is the actual logical distinction between things which have the same answer but are not “the same” - while having equal solutions?

Simply put, it's possible for two questions to have the same answer (the answers are equal), despite not being the same question (the questions are not equal).

In particular, a question is not the same as its answer, and a question is not equal to its answer.

You may think: "But in mathematics, a question is considered equal to its answer, isn't it? After all, we say 2 + 2 = 4, because the question is 2 + 2 and the answer is 4."

That's not true. The equation 2 + 2 = 4 doesn't mean "the answer to the question '2 + 2' is 4"; it means "the number 2 + 2 is the same number as the number 4." It's also completely correct to write 4 = 2 + 2; writing the more complicated expression on the left is merely a stylistic convention.

1

As you can probably glean from all these answers, there is no philosophical "the answer" to your question. From a more practical view, as some answers are pointing out, it depends on which definitions of "same" and "equal" you mean to use, because there is also no "one definition" of each, even in logic. Arguing from linguistics, one could say that they must mean different things, otherwise why have two different words? But they may only have a different range of vernacular meanings, and their respective ranges almost certainly overlap each other, so that a determined or narrow-minded or bad-faith interlocutor could try to force you to accept that they're exact synonyms.

I like your question because it's easy to generate "edge case" examples from. The two that came to my mind are identical (!) twins and Venus.

Colloquially we say that identical twins, especially if they purposely dress alike, are "the same" -- same looks, same clothes, same genetic makeup. But they are not "equal" in that they are physically separate objects, and politically and socially separate entities. On the other hand, it was also colloquial to talk about "The Morning Star" and "The Evening Star" as equal -- in brightness, in appearance -- but not as "the same" object, because they appeared at different times of day. Woops, it turned out to be "the same" physical object, Venus, after all.

Another fun direction of thought is considering mirrors and lenses which produce more than one image of "the same" object. They can be used, together with identical twins, in "magic tricks" and other fooling of the senses for example. And of course our own eyes have lenses as a required component for sight, so...

My philosophical conclusion is that "conceptually the same" and "conceptually equal" are too fraught with human foibles (intellectual and perceptual) to be rationally discussed at this time. Maybe once we humans get our act together intellectually, if ever, we'll be able to return to the question.

+1 for mentioning identical twins. They can be equal in every measurable quality and quantity, but are not the same person. – Monty Harder – 2019-09-25T19:36:34.060

0

First, I will simply get out of the way that this is entirely driven by semantics. Some people will consider "same" and "equal" to be synonyms while others will insist there is a difference. What is important is that you understand what someone means when they say something. In all things, there is a responsibility of the communicatee to ensure they properly received what the communicator is trying to send.

So with that general life lesson out of the way, we get to the topic at hand. In my experience, same tends to indicate that two things have interchangeable effect. Obviously if it is equal, it is same. Thus if I pick up the shirt I wore yesterday from the floor and wear it today, someone may say "He wore the same shirt yesterday!" and they are correct, it is the exact shirt. If however I have a second shirt of the same pattern, they very well may say "He wore the same shirt yesterday!" Now, if it was a cultural faux pas to wear that shirt to the event, then clearly they are correct - the two shirts have had the same effect. If, however, I had spilled coffee on one and not the other, then we begin to point out the differences - clearly it is not the same shirt, even though it appears to be, because the effect of wearing a stained shirt is different than the effect of wearing a clean shirt.

Often when one person tells a story a listener may say "The same thing happened to me!" and tell a completely different story. Maybe in both stories there was a customer service agent who was very rude, but one was in a restaurant and another in a clothing store, so they are not equal stories. But they share similarities, enough that they can be considered the same story. If, for example, they were pitching ideas for TV episodes, the second person might be told "Lucy just pitched that same story. Next!" without caring that there are details that make them unequal.

And this is why there is ambiguity - different people have different thresholds for what makes two things same. There is no ambiguity for equality - either things are equal or they are not. But there is grey area, and people must often debate where something is similar enough to be considered the same. And like so many things, it depends on both context and people's flexibility in accepting sameness. As proper communicators, the onus is on us to understand the framework of communication in which we are participating and follow it.

0

tl;dr- Things are equal in some sense when they're functionally interchangeable in that sense, and they're the same when they're equal in all appreciated senses.

Consider the sets A and B:

A = [ 0, 1 ]
B = [ 1, 0 ]


They're equal in some respects:

1. They're the same basic logical sort of thing, i.e. they're sets of integers.

2. They're of the same length.

3. They contain the same members.

They're potentially unequal in other respects:

1. They have different orderings, if the ordering is meaningful.

• If they're ordered, then the differing orders are a meaningful distinction between them.

• If they're unordered, then the differing orders aren't a meaningful distinction between them.

2. They have different labels, if the labeling is meaningful.

• For example, if this were a computer program, then A and B might imply different locations in memory. However, if the labeling isn't part of their definition, then my choice to label them differently wouldn't reflect a meaningful distinction between them.

If they're equal in all respects, then they're the same.

• A and B are the same if they're the same unordered set that I just happened to write twice with different labels and orderings as semantic happenstance.

• A and B aren't the same if they're ordered sets or/and the different labels imply a meaningful distinction such as existing as different objects in a computer's memory.

In general, things are the same if we can find no meaningful distinction between them. This is a stronger condition than mere equality, as we'll often describe different things as being equal (examples in next section).

### Examples of non-same (different) things that're equal.

This section provides examples to help showcase the distinction between same-ness and equality.

In general, two things are the same thing only if they're indistinguishable in all appreciated respects; this is, things are the same if we literally can't identify an appreciable manner in which they're not he same. However, it's easier for things to be equal; we often consider things to be equal even when they're not the same.

Examples:

1. In math, the expressions 1+1, 2, 3-1, 10/5, etc., are equal despite not being the same expression.

2. In law, Bob and Suzy are equal under the law despite not being the same legal person.

3. In physics, a force and its equal-but-opposite reaction are equal despite not being the same physical action.

4. In C# programming, different object's can still be .Equal().

• For example, this C# program assesses if two object's are the same and if they're equal, finding that they're different-but-equal.

using System;

public class Program
{
public static void Main()
{
var a = "Hello!";
var b = (" " + a).Trim();

var areSameMessage =
"Objects 'a' and 'b' are "
+   (System.Object.ReferenceEquals(a, b) ? "the same" : "different")
+   " objects."
;

var areEqualMessage =
"Objects 'a' and 'b' are "
+   (a.Equals(b) ? "equal" : "not-equal")
+   " objects."
;

Console.WriteLine("a:\t\"" + a + "\"");
Console.WriteLine("b:\t\"" + b + "\"");
Console.WriteLine(areSameMessage);
Console.WriteLine(areEqualMessage);
}
}


which prints

a:    "Hello!"
b:    "Hello!"
Objects 'a' and 'b' are different objects.
Objects 'a' and 'b' are equal objects.

5. In money, 1 Euro is currently equal to about 1.09 US dollars despite these being different amounts of different currencies.

In all of these examples, the point's that we can assess different things as being equal in some sense despite them not being the same thing.

### Technical point: Even identity is subjective, despite being less subjective than equality.

1+1 and 2 are different despite being equal because we can tell them apart; for example, we write and pronounce them differently. Because we can appreciate that these differences exist, 1+1 and 2 are different things despite being equal.

However:

1. What about 1+1 vs. 1 + 1; are they different?
I mean, yeah, technically. For example, this post is stored in digital format on a StackExchange server, and there's a meaningful physical difference between 1+1 and 1 + 1 in the physical world.

2. What about 1+1 vs. 1+1; are they different?
Again, yeah, technically. For example, while editing this answer, if I tried to delete one vs. the other, the result would be observable. So they're different in that they appear in different contexts.

3. What about 1+1 vs. itself a second later?
This is, I'm now referring to the exact same string that appears earlier in this paragraph, stored in the same location on the same computer – but, at slightly different times. So is that different? (Related question.)

Again, yeah, technically... if we care to make that distinction. I mean, we could refer to it as a different thing at different moments in time, if we reallllly wanted to...

4. What about 1+1 and itself at the same moment in time?
Well, now there might, in theory, be some sort of difference, but it's hard to say what that might be. For example, maybe we're living in The Matrix, and when we think about that same thing in different ways, our brains are being altered in ways we can't perceive such that our thoughts map to different concepts beneath our level of perception.

But even if we can't verifiably perceive differing notions of potential distinctions, we can imagine them, then consider the abstraction in which they might exist.

And now we're completely off in Crazytown, right? Like, each step of the way above, we got more-and-more pedantic, with increasingly minor distinctions to the point that we started considering abstraction descriptions that'd only make sense if we seriously regard hypothetical brain-in-vat scenarios.

The point being that we fundamentally can't assert true same-ness; incompleteness doesn't allow it. So, even same-ness is subjective, often a bit fudged from abstract ideality for the sake of simplicity.

In short, my point here is that same-ness doesn't truly require absolute equality in all conceivable senses, but rather in all senses that we care to appreciate.

### Summary: Equality is context-subjective, sameness isn't so much.

Things are equal in some sense when they're interchangeable in that sense.

Things are the same when they're equal in all ways that we care to identify.