The overlooked part of Russell's paradox

[RJG]

In order to 'X<X' to be meaningful at all, X must be defined as something that has the property that it can be greater or less than something else, i.e. it must be a number of some kind.

Thom, you are talking about math here, not logic. In logic:

• "X" can represent anything. It does not have to be a number.
  "<" can represent "before/after", "less than/greater than", "inside/outside", etc etc

***********
The Axioms of Simple Logic:

• X=X is true
  X=~X is logically impossible (e.g. something can’t be what it is not)
  X<X is logically impossible (e.g. something can’t exist before it exists)

A 'set' is not a number, so the falsehood of 'X<X' would not logically apply to it.

Logically, a "set" (or "list", or "box", or anything real/imaginary) could not be external to itself to then contain itself or come before itself or...etc etc).

To better understand the impossibility of X<X, try to put a box into itself, or pick up a stone and tap anything you want, now try to tap itself. It can't be done because we can't do (or even imagine!) that which is logically impossible.

But I think you're being too hard on them - paradoxes are useful for understanding and revealing things about our logical systems. (In my experience, they can also fun and instructive thought exercises.)

Yes, paradoxes are like riddles. But it seems that some people take these paradoxes to imply a greater (and more ignorant) meaning, such as so-called proof that "the system of logic is flawed; or always ends in contradictions or paradoxes". I do take issue to, and resent this type of thinking, as it implies that man is immune (infallible) to error himself. This is just "let's blame the system" for the error, but never our own intelligence (or lack thereof).

The human error in applying logic (resulting in paradoxes/contradictions) does NOT mean there is an error in logic! (...the error is with the human's misapplication of logic!)

[Thomyum2]

Thom, you are talking about math here, not logic. In logic:

• "X" can represent anything. It does not have to be a number.
  "<" can represent "before/after", "less than/greater than", "inside/outside", etc etc

***********
The Axioms of Simple Logic:

• X=X is true
  X=~X is logically impossible (e.g. something can’t be what it is not)
  X<X is logically impossible (e.g. something can’t exist before it exists)

But you are using mathematical symbols in your non-mathematical arguments. Even non-mathematical logic requires clear definition. (I seem to remember from a past post that you work in the legal field? If so, I'm surprised you're not more rigorous in your definitions.) '<' conventionally means 'less than'. If you're going to use it to mean one of those other things, then of course it needs to be defined as such, and then that will require 'X' to be further defined as well if it's going to be meaningful. For example: Using the definition 'before', if X refers to 'My cat', then the statement 'My cat' < 'My cat' is non-sensical - neither true nor false. If you redefine "<" to mean 'existed before', then "X<X" could even be said to result in true statement because my cat existed yesterday and exists today, so my cat was existing yesterday before my cat was existing today. You would have to define 'X' to mean 'the entire time period during which my cat existed' for the 'before' definition of "<" to be useful in producing a clear impossibility, and even then it's an extremely awkward way to reason. Logic doesn't work well if you can arbitrarily change the definitions of the symbols or terms you're using.
[/quote]

To better understand the impossibility of X<X, try to put a box into itself, or pick up a stone and tap anything you want, now try to tap itself. It can't be done because we can't do (or even imagine!) that which is logically impossible.

Yes, one can't imagine a box doing this, but as other posts have pointed out, a set is not a box - it's a collection of distinct elements. In set theory, two sets are said to be equal if they have the same number of members. You can have a set that is a member of itself that also has the same number of members as itself. So it's sort of analagous to saying that (X) = (X + 0). 'X' and 'X + 0' are two different things, but because they represent the same number they're said to be equal.

But I think you're being too hard on them - paradoxes are useful for understanding and revealing things about our logical systems. (In my experience, they can also fun and instructive thought exercises.)

Yes, paradoxes are like riddles. But it seems that some people take these paradoxes to imply a greater (and more ignorant) meaning, such as so-called proof that "the system of logic is flawed; or always ends in contradictions or paradoxes". I do take issue to, and resent this type of thinking, as it implies that man is immune (infallible) to error himself. This is just "let's blame the system" for the error, but never our own intelligence (or lack thereof).

The human error in applying logic (resulting in paradoxes/contradictions) does NOT mean there is an error in logic! (...the error is with the human's misapplication of logic!)

I agree that a paradox does not mean that logic itself is flawed, and that they do often occur due to fallacious reasoning. But not always. What Godel's work showed that there will be paradoxes and contradictions even when the logic is sound, not because logic is flawed, but because it is limited. Any axiomatic system is 'incomplete', to use Godel's word, and so not everything can be proven to be true or false with logic.

[Steve3007]

Again, it's not just a matter of whether this list is in the house or not, it's also about whether this list lists itself or not.

Yes, I realize that. But I see now what you meant by "choice". My mistake in not seeing that. If the list I labelled 'J' in this post is defined as:

"J = A list, in the house, which lists all lists in the house which list themselves."

then there would be no contradiction in it either listing itself or not listing itself. Both J = {C, F, I} and J = {C, F, I, J} are logically consistent with that definition of list J. If it lists itself, then it lists itself. If it doesn't, it doesn't.

Where I still don't get your point is where you said "You cannot take something as an ALL, and then add to it whilst keeping the context the same...". So I'll look back at that below.

Tell me where there is a contradiction in the following:

I form a list of the three self-listing lists in this house. This list does not list itself, therefore, this list does not list itself because it does not satisfy the red part of B (nor could it without running into contradictions as highlighted in the OP and my last two or three replies to you).

I don't see any contradictions anywhere, except in Russell's paradox itself (which we're not discussing here because Russell's paradox is about lists of lists that don't list themselves), but I'll go back to where you made the point originally and look at it again:

First you established ALL the self-listing lists in the house (of which there were 3), then you made a list of ALL the self-listing lists in the house (of which there were 3). So the ALL had already been exhausted. It is contradictory for this list to list itself because the list will not longer semantically qualify as ALL the self-listing lists in the house. You cannot have ALL plus one.

No, I still see no contradictions. In both cases it qualifies as all the self-listing lists in the house. You seem to be talking about the situation before and after list J was compiled. You seem to be making a rule regarding the ordering of events. You seem to be saying something along the lines of: "We cannot add J as a member of itself until J is a member of itself". But the instant that J is added to itself, it is a member of itself. And the instant that J is removed as a member of itself, it isn't a member of itself. Hence J = {C, F, I} and J = {C, F, I, J} are both logically self-consistent.

---

I think the difference between the situation we're discussing and Russell's paradox is this:

The list of all lists that list themselves can either list itself or not list itself without contradiction. If it lists itself then it lists itself. If it doesn't, then it doesn't. (Not RP).

This list of all lists that don't list themselves can neither list itself nor not list itself without contradiction. If it lists itself then it doesn't list itself. If it doesn't, then it does. (RP).

The first is a choice, and the second is a self-contradiction (aka a paradox).

[Steve3007]

Philosopher19, I've re-read through your subsequent posts on this. It does indeed seem to me that you're talking about something to do with the temporal ordering of events. You seem to be arbitrarily making a rule along the lines of:

"Once you've decided how many self-listing lists there are you can't change that."

So you seem to be saying that, if there are initially 3 self-listing lists in that house, adding J as another self-listing list is not allowed. Fair enough. If you want to add that rule, then J = {C, F, I} is the only option for J, because, according to this rule, j cannot self-list. But remember, this is just an abstract logical game to which you've added an arbitrary (not logically necessary) rule. There's no logical necessity for that rule to be added. Without it, J can still be {C, F, I, J} without any contradictions.

And if you do want to add a rule like that, how did C, F and I ever get created? The rule would have to be something like "once I've added 3 self-listing lists to the house, no more shall be added."

[RJG]

The Axioms of Simple Logic:

• X=X is true
  X=~X is logically impossible (e.g. something can’t be what it is not)
  X<X is logically impossible (e.g. something can’t exist before it exists)

"X" can represent anything. It does not have to be a number.
"<" can represent "before/after", "less than/greater than", "inside/outside", etc etc.

*******************

To better understand the impossibility of X<X, try to put a box into itself…

Yes, one can't imagine a box doing this…

So then you agree that it is logically impossible for a box to contain itself? ...yes?

...but as other posts have pointed out, a set is not a box - it's a collection of distinct elements.

Logic doesn't care what X represents. X could represent a box, a list, a set, or anything real or imaginary, and the logic still holds true. Logic doesn't stop being logic just because we are now talking about "sets" instead of "boxes".

You can have a set that is a member of itself…

Not so. This is bad math/logic. A set cannot logically or mathematically be a "member of itself" (contain itself). Not only is this logically and mathematically impossible, but it can't even be humanly imagined!

Thom, humor me here a bit, and just try to visually imagine this set X (a particular collection of distinct elements) containing itself. It can't be done! -- It is as impossible to imagine as imagining a "square circle" (X=~X) or "married bachelor" (X=~X) or a "box containing itself" (X<X).

So it's sort of analagous to saying that (X) = (X + 0). 'X' and 'X + 0' are two different things, but because they represent the same number they're said to be equal.

Logically (and mathematically), X plus nothing (X+0) is still X.
X=X+0 = X=X = logically TRUE

What Godel's work showed that there will be paradoxes and contradictions even when the logic is sound, not because logic is flawed, but because it is limited.

To me, this is self-contradictory; i.e. "pure nonsense". It is like saying - if we do math correctly we will end up with an error in math.

Any axiomatic system is 'incomplete', to use Godel's word, and so not everything can be proven to be true or false with logic.

I agree that not everything can be proven true or false. But we can certainly weed out all the non-truths (the "logical impossibilities") from our contaminated pool of knowledge by applying the axioms of Simple Logic.

[philosopher19]

Philosopher19, I've re-read through your subsequent posts on this. It does indeed seem to me that you're talking about something to do with the temporal ordering of events. You seem to be arbitrarily making a rule along the lines of:

"Once you've decided how many self-listing lists there are you can't change that."

I like how you've highlighted the temporal ordering of events, and whilst it is not wholly irrelevant to the example I presented, it is not the be-all and end-all of the OP and my replies to you. The focal point is the following:

Absolute consistency with regards to the semantic of 'All'. Compare the following:

X) Suppose you have all possible lists in your house that don't list themselves. You want to make a list of all the lists that don't list themselves in your house. Can you make such a list? No because it is impossible for you to make another list that doesn't list itself precisely because of that which is underlined in red. If you've already got all possible lists that don't list themselves, then by definition, you cannot have another list like this. However, this does not mean that a list listing all such lists doesn't exist in your house. If your house has a list of all lists, then it lists all such lists in your house, plus itself (no contradictions). See?

Y) Suppose you have all possible lists in your house that do list themselves. You want to make a list of all the lists that do list themselves in your house. Can you make such a list? No because it is impossible for you to make another list that does list itself precisely because of that which is underlined in red. If you've already got all possible lists that do list themselves, then by definition, you cannot have another list like this. However, this does not mean that a list listing all such lists doesn't exist in your house. If your house has a list of all lists, then it lists all such lists in your house, plus itself (no contradictions). See?

Mainstream logic or maths would have your reject the possibility of you forming a list of absolutely all lists that do not list themselves. This is correct because you cannot form such a list (as highlighted in X). But this does not mean that such a list cannot exist (as highlighted in X). Rejection of there being a list of all lists, or a set of all sets, is blatantly contradictory, and it is rooted in not being aware of or recognising the OP.

Do you see/understand both X and Y? I strongly recommend that you have a read of the following to reinforce the above:

http://philosophyneedsgods.com/2021/05/ ... -infinity/

[Steve3007]

...The focal point is the following:

Absolute consistency with regards to the semantic of 'All'...

OK. With my usage of the word "semantic", that means consistency in the meaning of the word "all" as we're using it. If J is "the list of all lists that list themselves" and C, F and I are lists that list themselves, then both J = {C, F, I} and J = {C, F, I, J} are consistent. In both cases, J is the list of all lists that list themselves. No inconsistency in the meaning of the word "all" there.

...Compare the following:

X) Suppose you have all possible lists in your house that don't list themselves...

OK. So if by "all possible" you mean all lists that it is possible for us to dream up without logical contradiction, then we're now talking about an infinite quantity of lists.

... You want to make a list of all the lists that don't list themselves in your house. Can you make such a list?

This is just the standard Russell's paradox again, because we're talking about a list of lists that don't list themselves. You can start to make such a list until it comes to deciding, without contradiction, whether that list lists itself. If it does it doesn't. If it doesn't it does. As previously discussed.

No because it is impossible for you to make another list that doesn't list itself precisely because of that which is underlined in red. If you've already got all possible lists that don't list themselves, then by definition, you cannot have another list like this.

No, this isn't a reason why you can't make that list. If you make the statement "I have all possible lists of some particular type" and then you add another list of that type all you're doing is demonstrating that you were not telling the truth when you said "I have all possible lists of some particular type".

However, this does not mean that a list listing all such lists doesn't exist in your house. If your house has a list of all lists, then it lists all such lists in your house, plus itself (no contradictions). See?

The house doesn't have a list of all lists. You said my house contains "all possible lists that don't list themselves". You stated that as a premise.

Y) Suppose you have all possible lists in your house that do list themselves.

Ok, so this time the premise is that we have all the lists that it would be possible to dream up which list themselves. Again, an infinite quantity.

You want to make a list of all the lists that do list themselves in your house. Can you make such a list?

Yes. But if I create that list in my house, all I'm doing is showing myself to have been incorrect to state that my house contained all possible lists that list themselves.

No because it is impossible for you to make another list that does list itself precisely because of that which is underlined in red. If you've already got all possible lists that do list themselves, then by definition, you cannot have another list like this.

All you've shown is that if I say "I have all possible widgets in my house" and then I add another widget, I wasn't telling the truth.

However, this does not mean that a list listing all such lists doesn't exist in your house. If your house has a list of all lists, then it lists all such lists in your house, plus itself (no contradictions). See?

No. I'm afraid I don't see your point, unless that point is simply to say "if I claim to have all possible somethings in my house and then I add another something, I wasn't telling the truth when I said I already had them all".

Mainstream logic or maths would have your reject the possibility of you forming a list of absolutely all lists that do not list themselves.

If by "mainstream logic" you mean Russell's paradox, then, as mentioned, the paradox is specifically about whether that list lists itself. Yes?

Do you see/understand both X and Y? I strongly recommend that you have a read of the following to reinforce the above:

I'll read it, but unless it says something different to what you've said so far, I don't see it changing anything.

[Steve3007]

...I strongly recommend that you have a read of the following to reinforce the above:

http://philosophyneedsgods.com/2021/05/ ... -infinity/

I've had a look at it. At first glance it seems to say the same things that you've said in various topics in this forum. But, although I may be repeating things I've said before when reading your comments on this forum, I'll make a few initial comments on it.

The following quotes are from the philosophyneedsgods.com site to which you linked.

If x, y, and z are sets that are not members of themselves, and I form a set of these three sets, to represent this, I can write something like: p = {x, y, z}. I cannot write x = {x, y, z} because x is in x, which makes x a member of itself.

I agree that if you write x = {x, y, z} then you are contradicting the condition "x is not a member of itself".

If x, y, and z are sets that are members of themselves, and I form a set of these three sets, to represent this, I can write something like: p = {x, y, z}. Consistency with the previous paragraph dictates that I cannot write x = {x, y, z} because x is in x, which makes x a member of itself twice (which is contradictory as nothing is a member of itself twice, or nothing is itself twice).

Since the condition in this paragraph is different, there's no reason for it to be consistent with the previous paragraph. You're perfectly fine writing x = {x, y, z}. That is consistent with the condition "x is a member of itself". It doesn't make x a member of itself twice and, even if it did, it's not obvious why that would be a problem. If I list the same item twice on a shopping list, that's not a logical problem as far as I can see. Worst case: I buy two of them. As we know (and as we've seen RJG ignoring!), we're talking about references here, and referencing something twice is not the same as that thing "being itself twice". I can create as many references to the same thing as I like.

[Thomyum2]

To better understand the impossibility of X<X, try to put a box into itself…

Yes, one can't imagine a box doing this…

So then you agree that it is logically impossible for a box to contain itself? ...yes?

...but as other posts have pointed out, a set is not a box - it's a collection of distinct elements.

Logic doesn't care what X represents. X could represent a box, a list, a set, or anything real or imaginary, and the logic still holds true. Logic doesn't stop being logic just because we are now talking about "sets" instead of "boxes".

You can have a set that is a member of itself…

Not so. This is bad math/logic. A set cannot logically or mathematically be a "member of itself" (contain itself). Not only is this logically and mathematically impossible, but it can't even be humanly imagined!

Thom, humor me here a bit, and just try to visually imagine this set X (a particular collection of distinct elements) containing itself. It can't be done! -- It is as impossible to imagine as imagining a "square circle" (X=~X) or "married bachelor" (X=~X) or a "box containing itself" (X<X).

I'll humor and do this and agree that I cannot visually imagine a set containing itself. But that I can't imagine something doesn't make it true of false. What you're using here is argument and not logic. You're trying to persuade, but logic requires not that you persuade but that you prove - that you show or demonstrate that it is true or false.

So it's sort of analagous to saying that (X) = (X + 0). 'X' and 'X + 0' are two different things, but because they represent the same number they're said to be equal.

Logically (and mathematically), X plus nothing (X+0) is still X.
X=X+0 = X=X = logically TRUE

I think we're getting off track because of language here - terms like 'contain' or 'list' or 'is a member of', commonly reference physical objects, and these would have to be much more rigorously defined in order to a make a logical proof out of the statement.

Let's try a different approach. For purposes of logic as it applies to set theory (which is a branch of mathematics) let's instead use the phrase 'define in terms of'.

As you've agreed, we can define 'X' to be anything. So as a number, we can define 'X' in terms of itself, and once we do this, we can arrive at logical conclusions as to the truth or falsehood of the statement. For example:

• Let X = X + 0 --> true for all numbers

• Let X = X + 1 --> false for all numbers

• Let X = 2 * X --> true only when X=0, false for all other numbers

• Let X = X squared --> true only when X=0 or X=1, false for all other numbers.

These ways of defining 'X' result result in a statement that is true or false, or a statement that is true for some number(s) and false for all other numbers. The self-referential definitions here aren't a problem and there aren't any paradoxes.

If 'X' represents a set, we can also define in terms of itself, as shown in other posts: Let X = {X,1,2}. Doing this for a set shouldn't be a problem, just as it wasn't a problem for a number above - we should be able to say that this is either true for all sets, or never true, or true for only certain sets. But what's different is that as a set it can result in a paradox. Certain ways of defining sets result in a statement that appears to be neither true nor false, or both true and false.

What Godel's work showed that there will be paradoxes and contradictions even when the logic is sound, not because logic is flawed, but because it is limited.

To me, this is self-contradictory; i.e. "pure nonsense". It is like saying - if we do math correctly we will end up with an error in math.

Any axiomatic system is 'incomplete', to use Godel's word, and so not everything can be proven to be true or false with logic.

I agree that not everything can be proven true or false. But we can certainly weed out all the non-truths (the "logical impossibilities") from our contaminated pool of knowledge by applying the axioms of Simple Logic.

Russell's solution to the paradox was just what you're proposing - applying a set of axioms to 'weed out' these contradictions. But no matter how many axioms he (and others) tried to add, there were still contradictions. And Godel showed the futility of that project - that this is necessarily the case for any axiomatic system. It's not a problem or error in logic or math - it's just the nature of it. After all, Godel's proofs were made using logic too and as far as I know, no one in the 90 years since has identified a logical error in his proofs.

[RJG]

Thom, humor me here a bit, and just try to visually imagine this set X (a particular collection of distinct elements) containing itself. It can't be done! -- It is as impossible to imagine as imagining a "square circle" (X=~X) or "married bachelor" (X=~X) or a "box containing itself" (X<X).

I'll humor and do this and agree that I cannot visually imagine a set containing itself. But that I can't imagine something doesn't make it true of false.

You can't imagine it because it is logically impossible to imagine! -- X<X is logically impossible no matter what X represents.

What you're using here is argument and not logic. You're trying to persuade, but logic requires not that you persuade but that you prove - that you show or demonstrate that it is true or false.

So are you trying to say that X<X 'is' actually logically possible???

[philosopher19]

Yes. But if I create that list in my house, all I'm doing is showing myself to have been incorrect to state that my house contained all possible lists that list themselves.

Right, and this is precisely my point. So how can you form a set of all sets that are members of themselves? Suppose you take all the sets that are members of themselves, and then group them into a new set. Is this new set a member of itself? If it is a member of itself, then you did not form a set of all sets that are members of themselves (precisely because it did not include this new set). It is, as you rightly recognise, you showing yourself that you were incorrect to state that you had all possible sets that are members of themselves. The same problem applies when you try to form a set of all sets that are not members of themselves.

Do you see my point? Of course, if one just accepts that the set of all sets encompasses all sets besides itself as well as itself (which is just another way of saying the set of all sets encompasses all sets, including itself), then the matter would be over. But to my understanding, a lack of awareness of the overlooked part of Russell's paradox (as highlighted in the OP), has lead mainstream set theorists to conclude that the notion of a set of all sets is inconsistent.

OK. With my usage of the word "semantic", that means consistency in the meaning of the word "all" as we're using it. If J is "the list of all lists that list themselves" and C, F and I are lists that list themselves, then both J = {C, F, I} and J = {C, F, I, J} are consistent. In both cases, J is the list of all lists that list themselves. No inconsistency in the meaning of the word "all" there.

But you cannot say J = {C, F, I} and J = {C, F, I, J} mean the same thing. The latter clearly implies that J is a member of itself. The former implies no such thing. It does not suffice that you say J is a member of itself and then demonstrate this via writing J = {C, F, I}. How has this self-membership been demonstrated here in this notation?

[Thomyum2]

Thom, humor me here a bit, and just try to visually imagine this set X (a particular collection of distinct elements) containing itself. It can't be done! -- It is as impossible to imagine as imagining a "square circle" (X=~X) or "married bachelor" (X=~X) or a "box containing itself" (X<X).

I'll humor and do this and agree that I cannot visually imagine a set containing itself. But that I can't imagine something doesn't make it true of false.

You can't imagine it because it is logically impossible to imagine! -- X<X is logically impossible no matter what X represents.

What you're using here is argument and not logic. You're trying to persuade, but logic requires not that you persuade but that you prove - that you show or demonstrate that it is true or false.

So are you trying to say that X<X 'is' actually logically possible???

I'm not saying that is possible; I'm saying that you haven't shown that it's not possible. To which I'll add: Did you not read the rest of my post?

[RJG]

The Axioms of Simple Logic:

• X=X is true
  X=~X is logically impossible
  X<X is logically impossible

"X" can represent anything. It does not have to be a number.
"<" can represent "before/after", "less than/greater than", "inside/outside", etc etc.

*************

So are you trying to say that X<X 'is' actually logically possible???

I'm not saying that is possible; I'm saying that you haven't shown that it's not possible.

If you agree that X=X is logically true, then you also agree that X<X is not possible! ...as X<X logically contradicts the truthfulness of X=X. ...unless you are trying to say that X=X 'is' NOT True???


*************

To which I'll add: Did you not read the rest of my post?

Thom, if we can't agree on the logical impossibility of X<X, then I see no reason to discuss anything further.

[Thomyum2]

The Axioms of Simple Logic:

• X=X is true
  X=~X is logically impossible
  X<X is logically impossible

"X" can represent anything. It does not have to be a number.
"<" can represent "before/after", "less than/greater than", "inside/outside", etc etc.

*************

So are you trying to say that X<X 'is' actually logically possible???

I'm not saying that is possible; I'm saying that you haven't shown that it's not possible.

If you agree that X=X is logically true, then you also agree that X<X is not possible! ...as X<X logically contradicts the truthfulness of X=X. ...unless you are trying to say that X=X 'is' NOT True???
*************

To which I'll add: Did you not read the rest of my post?

Thom, if we can't agree on the logical impossibility of X<X, then I see no reason to discuss anything further.

I agree that for any number X, 'X=X' (meaning 'X is equal to X') is true and 'X<X' (meaning X is less than X) is false.

I do not agree that this will always remain the case if the definitions of 'X' and '<' and '=' are arbitrarily changed or expanded to other meanings. Changing the definitions of symbols and using non-numerical values for variables introduces a great deal ambiguity into these statements and clarification would be necessary if you wish do this (I gave some examples of this a few posts back).

'X=X' may seem self-evident but if you're not talking about numbers, I don't understand how you can know what it means. What does 'cat=cat' even mean? The word 'cat', or an actual cat? Which cat? At what time? Equal how? In weight, size, age? Surely you see what I mean. Would you agree to the truth or falsehood of a statement without knowing the intended definitions of the terms used in those statements?

It's OK if you'd rather not discuss it further - we're off topic now anyway. It will likely come up again on another thread at some point and we can pick up where we left off.

[RJG]

'X=X' may seem self-evident but if you're not talking about numbers, I don't understand how you can know what it means.

Firstly, math is logic. If the variables (e.g. X) represent numbers then we typically call this math, otherwise it is called logic. ...or if you prefer to look at it the other way, then - logic is math in words. Essentially they are the same with different names, typically dependent on variable type.

What does 'cat=cat' even mean? The word 'cat', or an actual cat?...

If X represents the word "cat", then X represents the word (not the animal), and vice versa. In logic, X represents whatever it is you are talking about.

...Which cat? At what time? Equal how? In weight, size, age? Surely you see what I mean.

You are making it much more complicated than it really is. If X represents this cat (specifically identified) or that cat, or the weight, size, or age, then that is what X represents! ...X represents whatever it is you are talking about.

For example, If X represents a list, then it does NOT represent the reference-to-this-list; it ONLY represents the list itself. The list and the reference-to-the-list are two different animals. Hence the trickery in paradoxes similar to Russell's paradox. Having two different assignments (meanings) for a single variable (e.g. X) is what creates the deception/trick (perceived paradox).