Counting as grounded in Assumption of Void

[Eodnhoj7]

Equality, is thus both undefined and subject to equivocation within the law of identity while dually necessitating that identity is not just assumed through recursive Directional forms ...

No idea what you are trying to say but "Equality" is a mathematical concept and "Identity" I guess can be a logical one but you asked if logic needs the equals sign and I told you Propositional Logic does not so the answer to your question is no logic does not need the equals sign. It's not even needed in Predicate Logic and only introduced as FOL + Equality which presumably is due to the logicians wanting to try and formalise some mathematics or other but you'll have to ask someone with a more recent memory of such stuff.

So the principle of identity does not apply to 1=1? and P=P does not necessitate equality as a part of "logic"? How many types of "equalities" are there without it being subject to a fallacy of equivocation?

((P-->P)-->(P-->P))-->(P-->P).... ...

Did you miss it? (P->P)^(P<-P). Still, what you've printed is just another tautology so always true.
Miss it? I was debating with myself to add the other half, but upon recognition a one directional arrow is always a one directional arrow I decided to do a simple one directional string.

But also that identity is contextual recursion considering (P-->P) is fundamentally just an empty loop, thus can be observed as (P) where ( ) is fundamentally empty...like most loops are. ...

Er! All it says is that if there is a P then there is a P and if there is not a P then there is not a P.
False you are assuming it says that, and if that is the case than is justifies the identity property and in intrinsically empty assumption.

Considering all identity is grounded in loops, and loops are purely assumed as self referencing contexts, we are left with identity fundamentally grounded in assumption an void with this loop being the context itself as an inherent form.

Which meaning of "identity" are you using here? But in Logic 'identity' is grounded in the idea that there are things or states of affairs, no things or states of affairs then no Logic.

(P⇄P) is a loop...all "things" and "states of affairs" ranging from the movement of atoms that consitutes and object, the cellestial events, human social interaction, to computing to the various "cycles" which compose even a basic motor are grounded in loops as a context.

[Arising_uk]

So the principle of identity does not apply to 1=1? and P=P does not necessitate equality as a part of "logic"?

The former is a principle of equality in Mathematics, at base it is telling you that the number of objects on the left equals the number of objects on the right. As for the latter, I've already told you that P=P does not exist in Propositional Logic.

How many types of "equalities" are there without it being subject to a fallacy of equivocation?[/color]

No idea, you'd have to talk to someone who remebers their Predicate Logic plus Equality.

Miss it? I was debating with myself to add the other half, but upon recognition a one directional arrow is always a one directional arrow I decided to do a simple one directional string. ...

Except that the direction is different?

False you are assuming it says that, and if that is the case than is justifies the identity property and in intrinsically empty assumption.

Nope, I'm telling you how such an operator is interpreted when supplied with it's variables.

(P⇄P) is a loop... ...

Is it, how interesting. To me it is just meaningless unless you are defining this "⇄" to mean "is a loop" and then it's just pointless as it says nothing.

all "things" and "states of affairs" ranging from the movement of atoms that consitutes and object, the cellestial events, human social interaction, to computing to the various "cycles" which compose even a basic motor are grounded in loops as a context.

Are these just intrinsically empty assumptions?

Still, are you now going to go all loopy on us?

[Eodnhoj7]

So the principle of identity does not apply to 1=1? and P=P does not necessitate equality as a part of "logic"?

The former is a principle of equality in Mathematics, at base it is telling you that the number of objects on the left equals the number of objects on the right. As for the latter, I've already told you that P=P does not exist in Propositional Logic.

How many types of "equalities" are there without it being subject to a fallacy of equivocation?[/color]

No idea, you'd have to talk to someone who remebers their Predicate Logic plus Equality.

Miss it? I was debating with myself to add the other half, but upon recognition a one directional arrow is always a one directional arrow I decided to do a simple one directional string. ...

Except that the direction is different?

False you are assuming it says that, and if that is the case than is justifies the identity property and in intrinsically empty assumption.

Nope, I'm telling you how such an operator is interpreted when supplied with it's variables.

(P⇄P) is a loop... ...

Is it, how interesting. To me it is just meaningless unless you are defining this "⇄" to mean "is a loop" and then it's just pointless as it says nothing.

all "things" and "states of affairs" ranging from the movement of atoms that consitutes and object, the cellestial events, human social interaction, to computing to the various "cycles" which compose even a basic motor are grounded in loops as a context.

Are these just intrinsically empty assumptions?

Still, are you now going to go all loopy on us?

Try to escape a cycle...and you can't...that should answer all of the above.

[Arising_uk]

Try to escape a cycle...and you can't...that should answer all of the above.

Is the above true or is it just another assumption?

If the former can you tell me how you got it?

[Eodnhoj7]

Try to escape a cycle...and you can't...that should answer all of the above.

Is the above true or is it just another assumption?

If the former can you tell me how you got it?

All assumptions, as assumed, are inherently recursive and circular...this "form" is assumed "as it" considering it is an empty context in and of itself.

So yes the argument is true and false....True as self-referential, false due to its potential nature of expansion.

[Arising_uk]

All assumptions, as assumed, are inherently recursive and circular ...

Not sure what you mean by recursion here as that generally has a base case that halts the recursion so not circular at all?

...this "form" is assumed "as it" considering it is an empty context in and of itself.

No idea what this "empty context in and of itself" means at all? Are you talking about Kant's Noumenon? If so all you can say is that due to our idea of causality we could logically assume that there is a 'cause' to Phenomena but that doesn't make it true but just a metaphysical need for some reason or another as all there is are phenomena, so 'it' could be pixie dust, fairies, 'God' or any other psychological need to explain such a 'thing'.

So yes the argument is true and false....True as self-referential, false due to its potential nature of expansion.

An argument cannot be true and false at the same time, let alone due to this "self-referential"(whatever that is supposed to mean?) nor due to this "potential nature of expansion"(whatever this is supposed to mean?)

[Skepdick]

Not sure what you mean by recursion here as that generally has a base case that halts the recursion so not circular at all?

The liar's paradox doesn't have a base case. He is observing the halting problem, but the concept doesn't exist in his vocabulary.

No idea what this "empty context in and of itself" means at all?

When you use logical symbols to represent ontological objects the logical symbol is supposed to represent a noumenon. This is straight out of the law of identity. A rose is a rose is a rose. The word "rose" is supposed to be a noumenon - a thing-in-itself.

We are trapped in this representational/abstract game.

An argument cannot be true and false at the same time, let alone due to this "self-referential"(whatever that is supposed to mean?) nor due to this "potential nature of expansion"(whatever this is supposed to mean?)

It's but language/phrasing getting in the way of substance. The principle of superposition is how an argument is both true and false. Schrödinger's cat is both dead and alive. An undecidable question is both true and false at the same time. He just doesn't have the word "uncertainty" in his vocabulary.

If you were to write a recursive function (self-referential nature) to evaluate the liar's paradox you will soon get a "stack depth exceeded" exception. Out of memory. That's the "nature of expansion".

Again - he doesn't have the "space complexity" concept in his vocabulary.

[Eodnhoj7]

Not sure what you mean by recursion here as that generally has a base case that halts the recursion so not circular at all?

The liar's paradox doesn't have a base case. He is observing the halting problem, but the concept doesn't exist in his vocabulary.

No idea what this "empty context in and of itself" means at all?

When you use logical symbols to represent ontological objects the logical symbol is supposed to represent a noumenon. This is straight out of the law of identity. A rose is a rose is a rose. The word "rose" is supposed to be a noumenon - a thing-in-itself.

We are trapped in this representational/abstract game.

An argument cannot be true and false at the same time, let alone due to this "self-referential"(whatever that is supposed to mean?) nor due to this "potential nature of expansion"(whatever this is supposed to mean?)

It's but language/phrasing getting in the way of substance. The principle of superposition is how an argument is both true and false. Schrödinger's cat is both dead and alive. An undecidable question is both true and false at the same time. He just doesn't have the word "uncertainty" in his vocabulary.

If you were to write a recursive function (self-referential nature) to evaluate the liar's paradox you will soon get a "stack depth exceeded" exception. Out of memory. That's the "nature of expansion".

Again - he doesn't have the "space complexity" concept in his vocabulary.

Labels, as in complex symbols that summate a set of relations, are acts of counting in and of themselves and as such follow the same nature as the thread topic. "Counting" and "Naming" are dualisms of the same nature of "measurement" where one may place a strictly emphasis on quantity, and the other respectively as "quality", the natures are isomorphic.