I just explained this. When the symbol "4" is used twice in "4+4=8", it must signify a different thing in each of the two instances, or else 4+4 would not equal 8. — Metaphysician Undercover
I understand the general difficulty in having both 4 symbols in "4+4" representing the same object, its indeed not that easy to fathom. I'll try to give here a situation were this can be understood. Its a hypothetical scenario to clarify that this can be the case.
Let's say we live in a country were people live in tribes, now each tribe exactly has 50 men and 50 women, and the progeny of each tribe are separated from their fathers to constitute another tribe, the law dictates that marriages must be fixedly arranged between "tribes" that is if a man of say tribe A marries a woman from tribe B, then all men from tribe A must marry woman from tribe B only, and the same applies for woman, i.e. if a man from tribe A marry a woman from tribe B, then every woman from tribe A must marry a man from tribe B. Now lets fix that when a tribe is married to another tribe, then the result is also 50 girls and 50 boys, and that those would be separated from the parent tribes and so constitute another tribe.
Now the country sets two kinds of descriptions, one is Predicative description, and the other is Functional description.
The predicative description given by the country is the predicate "||" to signify "is married to" and this occurs between TRIBES. While the predicate "m" is used to signify marriage between persons. So the general statement in a laws of that country is:
RULE: For every tribe A for every tribe B (A || B if and only if for every male a of A there is one female b of B such that: a m b, and for every woman a of A there is one male b of B such that: a m b).
Now we have the situation: A || B to mean tribe A is married to tribe B (according to rules above).
Now this is a predicative formulation, why, because A||B is a "proposition", it something that can be true or false, and the symbol || is denoting a "binary relation", so it is a "predicate" symbol.
Notice that we can have the situation were tribe S can marry itself!!! so we can have S || S
Notice that S occurred twice in the proposition "S || S" but still it denotes ONE object, although this object is a totality of many individuals, however that whole of many individuals is considered here as one object. So repeated occurrence of the symbol symbol in an expression doesn't denote different denotation, no here S repeatedly occurred in "S || S" but it still carries the same denotation, namely tribe S.
This also shows that we can have a binary relation between something and itself OTHER than identity, for the expression S || S could have been false? while S is S is always true! Of course this is understood, for example we can have " \not [Sarah hate Sarah]" this is an expression having two occurrence of a symbol that is "Sarah" and yet it refers to the same object, and the binary relation between them that is "hate" can be negated (i.e. its negation is the true statement). And clearly the relation "hate" is not the same as the relation "identity", although it can occur between objects and themselves.
Now the country further uses the following notations to express "is the progeny of tribes", that is:
"P (A || B)", this is read as "The progeny tribe resulting from marriage of A to B"
Notice that expression "P(A||B)" is a "denoting" expression, it denotes a TRIBE. So the expression "P(A||B)" is NOT a predicative expression, since it clearly does NOT constitute a proposition, it is not something that can we can say of being true or false. "P(A||B)" is denotative and not declarative.
But we need a declarative statement "i.e. a proposition, or a predicative expression" about what that denotative expression "P(A||B)" is about? Here were "=" will trip in, to complete the picture and turn it into a proposition. Here the country stipulates:
P(A||B) = C
Now this is a proposition, it is say that the progeny tribe of tribe A married to tribe B , is , tribe C.
Notice here that in that country tribe S is married to itself, and it resulted in tribe Q, so we'll write that as:
P(S||S) = Q
Now we have two distinct occurrence of the symbol S on the left, but still it has the SAME denotational coverage! Both symbols of S denote the same object that is " TRIBE "S" ".
So we can have the same object undergoing some process with ITSELF to resent in other thing, like what happened with S.
The problem with expression 4 + 4 = 8 , is that it in some sense "hides" information, it should have been written as: R(4+4)=8, to mean "the result of adding 4 to 4, is, 8", that would have been more informative. Anyhow mathematicians and logicians shorten that to just 4+4, but what is actually meant is R(4+4). The expression 4 + 4 is deceptive, it gives the impression that "+" is a binary relation occurring between what's denoted by symbol "4" on either side of it, as if it is declaring that "4 is added to 4", which is not what's intended, the foundational mathematicians stipulate "+" as a two place function symbol, and they mention it in the rules of the language, which are often not written explicitly in many contexts, and so it would be considered understood that when they write 4+4 then they mean a denotative expression and not a declarative one, and that 4 + 4 actually means "the result of adding 4 to 4". Anyhow.
Of course you can object to the notion that A,B,S,etc.. here when used to denote "tribes", then they are not actually denoting "individuals", and of course that is correct, they are denoting "multiplicities", but still when B is used it always denote the SAME multiplicity. Whenever we hear B the specific 50 men and 50 woman in that country that were recorded under name "B" would come up into our minds. So all occurrences of B have the same denotational value! or lets say "coverage". IF we accept a totality of multiple individuals as ONE object that is the sum object of all of those, then B would be said to denote ONE object along that understanding.