Willow often says that things are "defined in themselves". I am not exactly sure what is meant by that, but I assume that it is an attempt to express the principal of identity; which is that a thing is identical with itself, and with no other.

Another possibility is that it comes from Willow's understanding of Spinoza, who says that a thing is either conceived through itself or through another. Only substance is, according to Spinoza, conceived through itself. Modes are conceived through their relations to other modes and, ultimately, through substance. So existence is of the essence of substance, but existence is not of the essence of modes. So, for example I can be conceived as not existing, but substance (God) cannot be so conceived. So my essence does not involve existence but Gods' essence does. Anyway this is getting away from the topic of the thread. I am thinking about starting a thread on Spinoza's understanding of substance and modes, but I'm not sure I will have the time.

I note you never got around to talking about the fate of the point but instead wandered off to talk about something else.

So you didn't realise that tautologies exclude semantics so are no use when it comes to making actual sense?

Only substance is, according to Spinoza, conceived through itself. Modes are conceived through their relations to other modes and, ultimately, through substance. So existence is of the essence of substance, but existence is not of the essence of modes. — John

This would be what Peirce's secondness challenges. Uniqueness would still be defined relatively. Inidividuation or identity is a difference that makes a difference. So - following Aristotle - the substantial is has some particular matter and some particular form. That is, it stands in relative contrast to the absolutely vague and the absolutely general.

This secondness or substantiality then shows itself in the sharp possibility of a reaction. One thing can react with definiteness to another thing. We have the dyad of some relation. We have a difference that is distinctive as part of a context and so can go on to be remembered as changing its developing history. We have the uniqueness of some difference that actually made a difference to the whole.

What remains unclear to me is what it means to say that the principle of identity does not apply to something. Zalamea helpfully formalizes the principles of vagueness and generality on page 21 of his paper; he describes them as failures of distribution of the principles of contradiction and excluded middle, respectively. Is there an analogous way to formalize the principle of identity and/or its failure, which would show what you have in mind here? — aletheist

So 2ns is dyadic reaction. Actuality is being defined in terms of a difference that makes a difference. This is quite in contrast to a tautology where the actual is simply a difference. It is not about what the world can see and remember as a concrete happening - a unique event that produce some further change. It is simply presuming the existence of some thing as that which is "the same as itself" - absolutely secure in its difference from other things without further determination. Nothing has to be actually shown or remembered by way of the demonstration of some reaction.

Thus it is not hard to see secondness contradicting identity in a big way. Actuality is about some relative change that is definite due to a context. A thing must be reacting to at least one other thing.

And then when does such an interaction ever exhaust all properties. If I bump into a car in the pitch dark, I have some idea perhaps of an encounter with something metal and solid. But is it a Porsche or a Fiat. Might it just be a lamppost?

So identity is only approached in the limit by the dyad of actual interaction. And Newtonianism said at the end of the day, the limit itself dissolved into the purely relative. In space, am I drifting away from the rocket, or is the rocket moving away from me? The mechanics of the situation are fundamentally reversible or symmetric. We can't any longer use local differences as the guide to what is actually the case. The identity of individuals can't be arrived via the exhaustion of their observables, even if it can be approached with arbitrary closeness in principle.

So 2ns switches things in flipping it so that actuality is not real in the sense of a limit state having been concretely achieved - the usual classical notion of identity. Instead a limit is a limit - the place that "exists" in the apophatic sense of never actually being arrived at. So 2ns is substantiality approaching its exhaustive limit, not substantiality in and of itself, nothing further needing securing.

I think I get what you mean when you say that the principle of identity does not apply to the actual - it is a limit that existing things can approach, but never fully achieve. But in what sense, then, is this distinctive of 2ns, in the same way that the inapplicability of the principles of contradiction and excluded middle are distinctive of vageness/1ns and generality/3ns, respectively? Again, can you state and/or formalize exactly what you mean by the principle of identity in this context?

Any thoughts on the excerpts that I posted as possible clues to why Zalamea claims that the synthetic continuum is recovered fully by category theory via synthetic differential geometry or smooth infinitesimal analysis?

Yes, Spinoza's concept of substance is contradictory to Aristotle's concept. Spinoza denies that there can be many finite substances and contends that there can be only one infinite substance.

But in what sense, then, is this distinctive of 2ns, in the same way that the inapplicability of the principles of contradiction and excluded middle are distinctive of vageness/1ns and generality/3ns, respectively? — aletheist

Vagueness and generality are defined as not being constrained in two of the ways that actual particulars are constrained. So materiality can be vague and not substantial. Form can be general and likewise not substantial.

Although given 3ns, these categorical distinctions are themselves all just aspects of the single triadic, irreducibly complex, sign relation. So that is why forms or universals can be real but not actual. And more unusually, the same is being said of the material principle. Materiality (as the vagueness of pure tychic spontaneity) is also a real potential, but not actually substantial 2ns (as it lacks yet the regularisation of habitual form or 3ns).

You see here that I go back and forth between hylomorphism and semiotics as of course the two are essentially the same metaphysical scheme with semiotics doing the better job of explaining the "how" by its foregrounding the mechanical role played by the sign relation in producing a world of suitably "deadened" substance.

Anyway 2ns would stand in relation to the law of identity as this same kind of protest - I am not constrained by that constraint which is said to be required to produce the brutely particular.

So 2ns instead talks about the deeper process that produces the brute particular. It points to the materiality and the formality, the vagueness and the generality, that have to be in interaction to produce actual substantial events, or differences that make a difference. 2ns treats actuality as what you get in the limit (with full 3ns). So actuality does apply to 2ns ... in the limit. But then 2ns is thus not actuality as brutely conceived by the law of identity. It is completely contextual once you step back to see the full 3ns scheme of things.

And this becomes more acceptable if we choose our intuition pumps more carefully and stop imagining reality in already presumptively Newtonian terms - like billiard balls rattling about on green baize.

What happens when two clouds collide? Where does any one cloud stop and start? What is the definite shape of any cloud? What is the physical logic of cloudy objects?

Clouds surely have actuality - we talk about them enough. But really, the law of identity fails to apply in a big way. And we can now specify the nature of that failure in the language of fractal maths. The contextuality of identity stands completely exposed these days.

I mean even Cantor was on the right track without really understanding it.

Have you checked - https://en.m.wikipedia.org/wiki/Cantor_set

Yes, Spinoza's concept of substance is contradictory to Aristotle's concept. Spinoza denies that there can be many finite substances and contends that there can be only one infinite substance. — John

Yep. The problem with Spinoza is that he was right about there having to be a "One", but wrong in conceiving of that basic materiality as a singular substance rather than as the vagueness of unbounded action. So it is material cause ... in its most insubstantial form. So action utterly lacking in form or purpose. An everythingness that is a singular being only because we call its fundamental disunity or lack of direction a single property or characteristic.

Vagueness is the canonical many. And when the question is asked of how many manys there are, the answer that comes back is "I am only counting the one".

I note you never got around to talking about the fate of the point but instead wandered off to talk about something else. — apokrisis

I said:

Take the rationals, and make a cut at <2. 2 stays on on side, all the numbers less than 2 on the other. One side contains 2, while the other might approach 2, but by the very fact of the cut, never reaches it. — Banno

Anyway 2ns would stand in relation to the law of identity as this same kind of protest - I am not constrained by that constraint which is said to be required to produce the brutely particular. — apokrisis

The law of identity expressed in what way, either verbally or formally (or both)? What exactly is this constraint with which 2ns "refuses" to comply? Would "contextuality" be a good descriptive term for this characteristic, as the second member of a trichotomy with vagueness and generality? What about "substance" to go along with matter and form?

The more I think about it, the more I really like "contextual" as a candidate for the 2ns counterpart to "vague" for 1ns and "general" for 3ns. If I am tracking with you properly, it expresses the specific kind of indeterminacy that is characteristic of the actual, especially as manifested in Peirce's semeiotic concept of the index. Consider this passage:

A sign is objectively general, in so far as, leaving its effective interpretation indeterminate, it surrenders to the interpreter the right of completing the determination for himself. "Man is mortal." "What man?" "Any man you like." A sign is objectively vague, in so far as, leaving its interpretation more or less indeterminate, it reserves for some other possible sign or experience the function of completing the determination. "This month," says the almanac-oracle, "a great event is to happen." "What event?" "Oh, we shall see. The almanac doesn't tell that." — CP 5.505, c. 1905

As a first cut: A sign is objectively contextual, in so far as, leaving its interpretation indeterminate, it relies on some aspect of the actual situation to complete the determination. "That house is on fire." "What house?" "That one over there."

Peirce used a very similar example to illustrate the indexical nature of pronouns, which "call upon the hearer to use his powers of observation, and so establish a real connection between his mind and the object; and if the demonstrative pronoun does that - without which its meaning is not understood - it goes to establish such a connection; and so is an index. The relative pronouns, who and which, demand observational activity in much the same way, only with them the observation has to be directed to the words that have gone before" (CP 2.287, c. 1893). Proper names are also indices, and rely entirely on the interpreter already being familiar with whom or what they reference.

Now back to the other passage:

The general might be defined as that to which the principle of excluded middle does not apply. A triangle in general is not isosceles nor equilateral; nor is a triangle in general scalene. The vague might be defined as that to which the principle of contradiction does not apply. For it is false neither that an animal (in a vague sense) is male, nor that an animal is female. — CP 5.505, c. 1905

As a first cut: The contextual might be defined as that to which the principle of identity does not apply. This object from one point of view, or at one time and place, is not the same as this object from another point of view, or at another time and place.

As I noted before, Zalamea - right after quoting the same passage - formalizes generality and vagueness as "failures of distribution" of the principles of excluded middle and contradiction, respectively. (Robert Lane provides a helpful explanation of the important differences between these principles, as defined and used by Peirce, and the modern laws of excluded middle and non-contradiction.) He associates the general with the universal quantifier and the vague with the existential quantifier, so it seems like the contextual should be associated with a singular proposition. After wrestling with his notation for a while - I am still not sure that I am interpreting it correctly - I came up with these formulations:

• If x is general, then it is not necessarily true that for any predicate P, ∀xP ∨ ∀x¬P
• If x is vague, then it is not necessarily true that for any predicate P, ¬(∃xP ∧ ∃x¬P)

As a first cut: If x is contextual, then it is not necessarily true that under all circumstances, x = x.

What do you think?

I said — Banno

Yeah. But anyone can wiki the set theoretic definition. Keep up.

Would "contextuality" be a good descriptive term for this characteristic, as the second member of a trichotomy with vagueness and generality? What about "substance" to go along with matter and form? — aletheist

I don't think it is essential to arrive at one perfect word. Peirce called them one, two, and three precisely because the same basic triadic relation could have its many manifestations.

But if vagueness is the best term for 1ns, and generality the best for 3ns, then another term for 2ns (after hierarchy theory) would be specificity.

Now the issue is that you think that 2ns needs to be contradicted by (or there be a failure of distribution concerning) the law of identity. So somehow 2ns itself should mean the opposite of the individual, the specific, the determinate,

But the Peircean triad actually wants to give the particular its real place in the scheme of things. So we don't need to contradict identity itself to contradict the principle of identity.

I mean 2ns looks the most like the regular reductionist notion of the atomistically and mechanically determinate - in simply being Newtonian action and reaction.

That is why I said the contradiction lies in the genesis of specificity. Peirceanism says it is a contextual deal. The laws of thought say it is brutely tautological. So the opposition is there between the holism and atomism, but Peirceanism would still call 2ns "actuality" or one of its synonyms, like particular, local, substantial, specific, determinate, individual, etc. The difference is that what the laws of thought presume as the brute foundations - nominalistic identity - Peirceanism shows to be the emergent final product.

As a first cut: If x is contextual, then it is not necessarily true that under all circumstances, x = x. — aletheist

Or x = not not-x is true. That employs the context to derive the specificity via a dichotomy.

Check out the Spencer-Brown's laws of form. Or Kaufmann's note on Peirce's sign of illation - http://homepages.math.uic.edu/~kauffman/Peirce.pdf

Yes, Spinoza's concept of substance is contradictory to Aristotle's concept. Spinoza denies that there can be many finite substances and contends that there can be only one infinite substance. — John

No, the two concepts aren't contradictory in any way. They are actually compatible. It is true that Aristotle means something different by Substance than Spinoza, however, the two concepts (their meanings) are not contradictory, but complementary. Substance in Spinoza is that which cannot be conceived as not existing, and which must be conceived through itself. There is only one element of Aristotle's metaphysics which fits this description - and there is only ONE of them - the Prime Mover. So Substance in Spinoza is NOT Substance in Aristotle, but rather Prime Mover. Hence the two definitions of Substance aren't even incompatible to begin with.

I don't think it is essential to arrive at one perfect word. — apokrisis

I agree, but I think that it will be helpful to clarify the distinctions that we are trying to draw if we can assign a term to 2ns that goes along with "vague" for 1ns and "general" for 3ns.

But if vagueness is the best term for 1ns, and generality the best for 3ns, then another term for 2ns (after hierarchy theory) would be specificity. — apokrisis

I am not very familiar with hierarchy theory, but I know that you refer to it a lot, and I might do some reading about it once I finish my current exploration of category theory and smooth infinitesimal analysis. Although I can see how "specific" is an antonym for both "vague" and "general," it strikes me as too much of a synonym for "determinate," such that the principle of identity would apply. Furthermore, we can "specify" something by description, rather than requiring an index to pick it out within a particular context.

I mean 2ns looks the most like the regular reductionist notion of the atomistically and mechanically determinate - in simply being Newtonian action and reaction. — apokrisis

I agree, especially since part of Peirce's point in describing vagueness/1ns as the inapplicability of PC and generality/3ns as the inapplicability of PEM is to define 2ns as that to which both PC and PEM do apply. However, he also wrote those two definitions of "individual" that I quoted a while back (CP 3.611-613, 1911). The first requires determinacy with respect to every general character, and thus - as he wrote elsewhere (see below) - can only be an ideal limit; while the second makes individuality a matter of reaction, and therefore existence. Both effectively deny the identity of indiscernibles, the first by virtue of the different "hecceities" that two distinct individuals must have, and the second because no two reacting things can have the same spatial (or, I would add, temporal) relations.

The latter is what I had in mind when I suggested as an example of contextuality, "This object from one point of view, or at one time and place, is not the same as this object from another point of view, or at another time and place." I was also thinking of this passage:

The logical atom, or term not capable of logical division, must be one of which every predicate may be universally affirmed or denied ... Such a term can be realized neither in thought nor in sense ... In thought, an absolutely determinate term cannot be realized, because, not being given by sense, such a concept would have to be formed by synthesis, and there would be no end to the synthesis because there is no limit to the number of possible predicates. A logical atom, then, like a point in space, would involve for its precise determination an endless process. We can only say, in a general way, that a term, however determinate, may be made more determinate still, but not that it can be made absolutely determinate. Such a term as "the second Philip of Macedon" is still capable of logical division - into Philip drunk and Philip sober, for example; but we call it individual because that which is denoted by it is in only one place at one time. It is a term not absolutely indivisible, but indivisible as long as we neglect differences of time and the differences which accompany them. Such differences we habitually disregard in the logical division of substances. In the division of relations, etc., we do not, of course, disregard these differences, but we disregard some others. There is nothing to prevent almost any sort of difference from being conventionally neglected in some discourse ... This distinction between the absolutely indivisible and that which is one in number from a particular point of view is shadowed forth in the two words individual {to atomon} and singular (to kath' hekaston); but as those who have used the word individual have not been aware that absolute individuality is merely ideal, it has come to be used in a more general sense. — CP 3.63, 1870

This is the basis on which I have elsewhere suggested "extreme realism" as the view that reality consists entirely of generals, or at least that everything real is general to some degree. In other words, there are no absolute individuals/singulars that are determinate in every conceivable respect. Hence "general" here encompasses both the "positive generality" of 3ns as "conditional necessity" and the "negative generality" (i.e., vagueness) of 1ns as "the merely potential," since they both constitute forms of indeterminacy.

I remain intrigued by the prospect of identifying a third kind that pertains uniquely to 2ns, and still think that "contextuality" is the most promising candidate. It is that which precludes any individual from being absolutely singular, because it cannot strictly satisfy the principle of identity while occupying different places at different times and/or being referenced from different points of view.

"That is why I said the contradiction lies in the genesis of specificity. Peirceanism says it is a contextual deal. The laws of thought say it is brutely tautological. — "apokrisis

I guess you have been talking mainly about how the genesis of individuality/determinacy/identity is contextual, rather than brute, because it involves the ongoing interaction between vague freedoms and general constraints. I have been talking mainly about how the nature of individuality/determinacy/identity is contextual, rather than absolute, because nothing is exactly the same as anything else, including itself.

Or x = not not-x is true. That employs the context to derive the specificity via a dichotomy. — apokrisis

I am not sure what you mean here. Could you please elaborate?

As a second cut: If x is contextual, then it is not necessarily true that under all circumstances, x = ¬¬x.

This version appeals to me because it seems to parallel better the "failures of distribution" of the other two principles for the vague and the general. Peirce stated within his first definition of "individual" that "the principles of contradiction and excluded middle may be regarded as together constituting the definition of the relation expressed by 'not'" (CP 3.612, 1911). Hence if one or the other does not apply, negation is left undefined. The same is true if this formulation of the principle of identity does not apply, and it also eliminates the (classical) logical equivalence of the other two principles.

Another way to put it is that if generality and vagueness are real yet not actual, then the actual would be the not real. Or if the first two are the ideal limits, then actuality is the thus limited.

That sounds paradoxical but makes the point that actuality - as the 2ns of substantial events - is the emergent outcome of the two real causes of being, the material potential of 1ns and the formal constraint of 3ns. So actuality is not real in being merely an effect, not a cause.

Zalamea p23 highlights that points are the limits on actuality - and so actual actuality is an unreal possibility in being completely or Platonically ideal. Existence can approach but not reach the perfection of discontinuous actualisation that the principle of identity demands.

Indeed, the actual, the given, the present, the instant, are no more than ideal limits: limits of possibility neighbourhoods which contain those actuality marks, those points impossible to be drawn, those fleeting presents, those impalpable instants.

Zalamea also highlights the irreducible mutuality that is thus at the heart of spatiotemporal existence when he goes on to talk about Peirce foreshadowing a modern desire for possibilitia surgery techniques.

For me - coming at this from a more physical perspective where energy or action must be made properly part of any world geometry model - you can understand energy in terms of spatiotemporal curvature. So now we can understand the continuum - that blackboard that is already determinate in being definitely dimensional - in terms of its own more primal pre-geometry.

Briefly, you have the two things going on as a reaction. We start with the unboundedness of disconnected curvature - a roil of hot spacetime indeterminate fluctuation. A chaos of directions all erasing each other. That is, an infinity of scraps of hyperbolic curvature. Space as energetic action is buckled maximally at every point and so curves apart from itself to lack all actual connection. The only continuity is this sea of rupture. 2ns exists only as the reactions that are immediate hyperbolic divergence - fluctuation dyads that are breaking apart as soon they connect, leaving behind no history or memory, no 3ns of some context of continuity.

But in the very fact of chaotic or locally hyperbolic curvature, you then have the latent possibility of a constraint to flat and simply connected Euclidean space. If only 1ns could be cooled and its wild curvature could start to join up to share a common story - each point or rupture be flattened just enough for a history of ongoing relations to start to form. Ie: the birth of 3ns as now a telos. Euclidean flatness could become the thermal goal. If definite dimensionality begins to form - like the three dimensions of the universe in which it concretely expands - then you can establish the feedback loop that drives the primal chaos towards the flat connectedness of a true continuum state.

So a world gets born by starting with unbounded freedom. From every possibility of a point or locale there is the possibility of a momentum or curvature. You just get these two complementary things together in the pregeometry as a necessity. If there is a locale as spontaneous fluctuation of pure possibility or 1ns, then it come automatically with the equally phantom possibility of its motion or action. And given no restrictions or bounds as yet on that other possibility, it would have to be as unrestricted as possible - hence hyperbolic curvature, or Planck scale divergence.

But then given also we are presuming there must in fact also be interaction or constraint going on between these extremophile locales, these unbounded fluctuations (a reasonable conclusion as it must be the case as otherwise we could not be here to question its existence) then it only takes a little bit of interacting to provide a little more persistent for the fluctuations, the ruptures, to start to cool and start to line up in flatter fashion.

Given such an interaction - a state of 3ns - constraints would provide a generalised flattening force, while the vagueness with its unbounded curvature would provide the energy to be disposed of as a developing extent of spacetime. We start with a lump blob of energy - a wodge of fluctuations going off in disconnected directions. Then like a ball of pastry, it gets rolled flat and spread very thin. It tells the story of a Big Banf becoming a Heat Death via an asymptotic story of self-equilibrating cooling and expanding. At the Heat Death, the fluctations or local curvature is almost completely dissipated, leaving just a Euclidean actuality of a maximally cold, dark, even, and perfectly connected void.

So if we are to get deeply physical about the mathematical continuum, we have to wind the story back even pre-dimensionally or pre-geometry. Peirce's blackboard analogy talks of an infinity of flat dimensions. But even vaguer would be an infinity of hyperbolically curved fluctuations that lack all connection or communication, and so flatness can become part of their telos if flatness is also a latent possibility of that pre-geometric beginning.

And it must be because any curvature at all is already speaking to the otherness that would be flatness and connectedness instead.

No, the two concepts aren't contradictory in any way. — Agustino

This is a nonsensical objection.

The two concepts are contradictory because Spinoza conceives substance as one and infinite and Aristotle conceives substance as plural and finite. Aristotle does not conceive the prime mover as substance at all, as you admit, nor as having any attributes other than being prime mover, so the two conceptions of Spinoza's substance and Aristotle's prime mover are also obviously incompatible.

Another way to put it is that if generality and vagueness are real yet not actual, then the actual would be the not real. — apokrisis

Peirce's definition of "real" is that which has characters regardless of what anyone thinks about it. He came to realize by about 1896 that all three categories are real in this sense, which is why Max Fisch characterized him as a "three-category realist" from that point until the end of his life. Consider also this passage:

Existence, then, is a special mode of reality, which, whatever other characteristics it possesses, has that of being absolutely determinate. Reality, in its turn, is a special mode of being, the characteristic of which is that things that are real are whatever they really are, independently of any assertion about them. — CP 6.349, 1902

Of course, here we once again encounter the notion that existing things are "absolutely determinate," which - as we have been discussing - is really (pun intended) just an ideal limit. Strictly speaking, nothing "exists" in this sense. Is that what you are getting at by suggesting that the actual is the not real?

Existence can approach but not reach the perfection of discontinuous actualisation that the principle of identity demands. — apokrisis

Again, please elaborate. How does the principle of identity demand discontinuous actualization?

The first requires determinacy with respect to every general character, and thus - as he wrote elsewhere (see below) - can only be an ideal limit; while the second makes individuality a matter of reaction, and therefore existence. — aletheist

I already covered this where you first made your objections about my mention of an unresolved "tension".

If 3ns is the constrained totality, then 1ns and 2ns stand constrained by it as first, the 1ns of in fact unconstrained possibilty, and then second, the 2ns of now constrained or determinate possibility.

So actuality from this standpoint is simply regularity of spontaneity. Energy or fluctuation has become so ordered by global law or habit as to be fixed in its dimensionality and thus completely determinate and countable as a physical degree of freedom.

This is where we actually are in quantum cosmology. We can count the total number of physical degrees of freedom in the visible universe at its Heat Death - there are 10^122.

So in including energy in the physical picture closes it, turns the apparently open or infinite into a tale of the inherently finite.

This means the continuum is "grainy" under a quantum gravity "theory of everything" view of the Universe. But grainy doesn't mean definite or determinate discontinuity - as in the points of a line, or the pixels of an image. It means that the necessary duality in terms of the forces of integration and differentiation - the constraining generality of global 3ns and constructing actuality of local 2ns - are already present in germinative fashion in the vague potentiality of 1ns.

So again, 1ns is unconstrained possibility. 2ns is constrained possibility (local actual constructive freedom). And 3ns is the constraint of both kinds of possibility.

Peirce's definition of "real" is that which has characters regardless of what anyone thinks about it. — aletheist

So then the actual only has character to the degree that reality has thought about it. That fits.

What materialists call the actual is only that which physical pan-semiosis has secured as some persisting mark of being.

If 3ns is real, it is real because it can't be wished away. Likewise the ultimate tychism of 1ns is real in the same fashion.

But 2ns is only real in that it is the emergent result of the other foundational reals - the actual causes of actual being. So it is not itself really real in being the product of pansemiotic "thought", or the universal growth of reasonableness. Matter is effete mind as they say. Or in other words, any material event could have been thought otherwise. Newtonian mechanics was always about inserting ourselves and our desires into reality. We want to discover the "hard facts" of atomistic events so as to then be able to rearrange the machinery of existence for our own convenience.

There is much to ponder here, but you still have not explained - at least, not in a way that "clicks" for me - what you mean when you say that the principle of identity (x = x, or perhaps x = not not-x) demands discontinuous actualization by employing the context to derive specificity via a dichotomy. If you could spell this out, I would be grateful.

Both effectively deny the identity of indiscernibles, the first by virtue of the different "hecceities" that two distinct individuals must have, and the second because no two reacting things can have the same spatial (or, I would add, temporal) relations.

The latter is what I had in mind when I suggested as an example of contextuality, "This object from one point of view, or at one time and place, is not the same as this object from another point of view, or at another time and place." — aletheist

Again, as already said, semiosis goes further. It defines indiscernability as a pragmatic issue - the principle of indifference that underlines probability theory.

So indiscernability is not ontic, but epistemic. If the Universe has a purpose, then that in itself creates a boundary, an event horizon, where it will cease to sweat the detail. It meets that purpose and then everything beyond that is a matter of generalised indifference.

So maths is hung up on radical openness. Counting seems something that extends to infinity because the very definiteness of any first step seems to already to guarantee the openness of that. And then the radicalness of that openness leads to a desperation to also produce a matching closure. Philosophy of maths ties itself into knots to discover global bounds on unbounded construction - the 2ns of already determinate degrees of freedom.

But semiosis already comes with the closure to match the openness. If there is openness due to their being a purpose, then there is a closeness in the way that also creates the possibility of its own satisfaction.

Thus at the level of 1ns, the continuum is neither an open or closed set. It is a clopen set - https://en.wikipedia.org/wiki/Clopen_set

Or as I said earlier, a scalefree situation like a fractal. If you are seeking either points or their contextual neighbourhoods, they exist with perfect evenness across every possible scale of being (and so, like a fractal, there is radical openness). And yet at the same time, the perfection of that evenness is a scale symmetry or the definiteness of actual closure. You can use a single number to capture the exact (symmetry broken) dimensionality of the resulting structure. The Cantor set for instance has an "angle" of ln(2)/ln(3) or ≈ 0.631.

There is nothing to prevent almost any sort of difference from being conventionally neglected in some discourse ... — CP 3.63, 1870

So note Peirce argues here by implication for the pragmatic principle of indifference. If we are now talking cosmology, it is the Universe that is indifferent to any difference that doesn't make a difference in being beyond the needs of its genenralised purpose.

Of course the theist will run into real difficulties here because beyond the Universe is some further 3ns of a god with a mind. Peircean semiotics was heading so nicely in the direction of pure self-manifestation on the logical grounds that "nothingness is impossible". And then the theist has to do a sharp 180 when this dangerous final answer eventually does show itself. 1ns can't be the true initial conditions of existence as Peirce's own logic makes necessary. And from that balking at the final hurdle, the whole point of the metaphysics falls apart. Transcendentalism re-enters to claim its false dominion.

Anyways...

This distinction between the absolutely indivisible and that which is one in number from a particular point of view is shadowed forth in the two words individual {to atomon} and singular (to kath' hekaston); but as those who have used the word individual have not been aware that absolute individuality is merely ideal, it has come to be used in a more general sense. — CP 3.63, 1870

So this is dichotomistic reasoning. The general and the particular are defined via their mutual opposition. And so they both only exist in a relative fashion. Nothing ends up real - except to the degree its own reality is secured by a sufficient distancing from its apophatically reality-erasing "other".

Thus you have not fully surrendered to the essence of semiotics. Everything is ultimately emergent, immanent, a matter of relative development. Even the universals that are real under semiotics are only historically realised - ideal only relative to their own developed degree of expression.

So any law is not exceptionless. It may approach the Platonic ideal, but it can't arrive there. That makes even laws or 3ns emergent and not really real (in that wishes can change things still - as we humans demonstrate in defying laws like universal gravitation with wings and aeroplane engines.)

Thus when we are talking about the mirror that are the three laws of thought and the semiotic triad, it is this kind of immanent mutuality we would seek on each level. And in your language, you keep wanting to talk about generality or vagueness in unsuitably specific terms. You are treating the categories as hard and existent, not developmental and relative to the business in hand.

So as I say, there are three laws of thought that constrain naked possibility so that only brute substantial actuality appears left. The world is a nominalistic totality of facts, a cause-transcending state of affairs. We only then need a mechanical mode of reasoning - the atomism of predicate logic - to make complete sense of existence.

But then Peirceanism is the counter to that - the counter that doesn't destroy the particular, but instead flips the perspective to show how it is the irreducibly emergent and not the brutely foundational.

So that then is how generality and vagueness emerge as the "other" of the LEM and PNC.

Generality denies the LEM by saying that particularity is always contexted by a purpose. So middles have to be excluded to the general satisfaction of some constraint. There has to be a semantic act of judgement, an act of measurement.

But then that purpose can be satisfied often quite easily in practice. So constraint is self-limiting in its essence. There is always a point beyond which even the universal law doesn't in fact give a bugger. So indiscernability arises for that self-interested reason.

The 1ns of vagueness then stands against the "2ns" of the PNC. With generality vs the LEM, it was 3ns against 3ns - top level against top level. But now we have an odd mixing of levels that seems confusing.

However it seems to work out right. Where the PNC talks in terms of crisp possibility, vagueness says no, crispness is only relative to its "other" of the vague. So what the laws of thought treat as having simple existence - bivalence - semiotic shows to be an emergent property that has to erupt via a fluctuation of vague suggestiveness, followed up by the solidification of established 3ns, or universalised habit.

Then we arrive at the 1ns of the laws of thought - the principle of identity. This then is made to stand in dichotomous contrast to what? It stands in apophatic relation to 2ns/actuality ... whatever that really means in Peirce-speak.

So in fact we already find this a well traversed issue in Leibniz's doctrine of indiscernibles. But where the reductionist thinks that the differences that make a difference are atomistically unbounded - there is no reason why we could ever in principle cease the pursuit of further detail, chase down the last decimal of the expansion of pi until we are exhausted - the Peircean system offers principled relief. We can stop when the differences cease to matter to our over-riding purpose. And the same goes for the Universe (whose primary telos is thermalisation as far as we can discern).

So the laws of thought presume the brute existence of the indiscernible difference that secures the principle of identity. And Peirceanism flips this to say indiscernability kicks in at the point where some 3ns ceases to have a reason to care, and so 1ns is left undisturbed.

Thus indiscernability describes some prevailing state of equilibrium where there is enough 3ns to create generalised order, and enough 1ns to generate freedoms to be regulated. 2ns as actuality is the fractal balance where we can put a definite number on the real actions of integration vs differentiation (flattening and curving) going on.

So again, 2ns in Peirceanism is about the emergence of crisp possibility or determinate degrees of freedom. And this then stands "other" to the 1ns of the laws of thought in denying their assertion that crisp possibility or determinate degrees of freedom are instead the brute foundational facts in nature.

The laws of thought in turn try to dismiss vagueness as merely semantic, not ontic. Any vagueness in logic is due to informal issues like measurement error ... or indifference to the finer facts, an insufficiency in taking care.

So quite neatly, the laws of thought also employ the same dichotomistic othering. They just take for granted what semiotics demands be tracked via some developmental history reflecting a "growth of reasonableness".

This is the basis on which I have elsewhere suggested "extreme realism" as the view that reality consists entirely of generals, or at least that everything real is general to some degree. — aletheist

This of course is what I deny. There is only relativity, never the absolute.

However, vagueness can't be absolute either. And generality can approach its own limit asymptotically. So certainly the Universe can be expected to wind up as totally generic as possible - the shorthand description of its ultimate Heat Death.

[edit:] So where I agree is that it is "constraints all the way down". There is no foundation of actual material being, just always a constraint acting to suppress free variety. Reality is thus always contextual (while remaining also irreducibly tychic). And if that's what you mean here, then I guess that is extreme realism in deny non-emergent or transcendent reality to brute particulars.

As a second cut: If x is contextual, then it is not necessarily true that under all circumstances, x = ¬¬x. — aletheist

Huh? X is being made its own context. That is the tautology here. The assertion is being made that the context is crisply existent too.

The upshot then is that the statement is true only to the degree that either term is true. But also each is true in direct proportion to the truth of its "other". So as least we do have a definite form of relative truth in play. Each term is as true as the other, even if neither might never be a perfect truth ... and that also is stated in the reciprocality of the relation. If either one was perfectly true, it would negate its other - erase it from existence and so lose the other as that to which anything was being related. So it is a limit statement - with each limit having to have the positive existence of its other (in familiar yin-yang fashion).

So yes, in the end, it is a statement of contextuality. But contextuality "absolutely divided". So that is why 2ns is about emergent particularity or individuation and the 1ns of identity is about the brute facticity of the same. The triadic scheme makes the middle the last thing to emerge from an organic process of constraint on absolute possibility. Monadic reductionism presumes it is the first brute fact to get the mechanics of construction started.

Hence if one or the other does not apply, negation is left undefined. — aletheist

Yep. I have said that there must be this coupling of constraints and unbounded freedom that then leaves the third thing of determinate possibility - material degrees of freedom. So it is a pincer movement to arrive at where reductionist ontology wants to start itself.

Again, much to ponder. Thanks for taking the time. For the moment, I can offer just a few initial responses.

If we are now talking cosmology, it is the Universe that is indifferent to any difference that doesn't make a difference in being beyond the needs of its genenralised purpose. — apokrisis

Which is what, in your view? According to Peirce, "the universe is a vast representamen, a great symbol of God's purpose, working out its conclusions in living realities … The Universe as an argument is necessarily a great work of art, a great poem - for every fine argument is a poem and a symphony - just as every true poem is a sound argument" (CP 5.119, 1903, emphasis added). The dynamic object of this sign is God Himself, and its immediate object is His purpose, the development of reason - i.e., the growth of our knowledge about God and His creation. As an argument, the interpretant is its conclusion, the living realities that the universe is constantly working out.

1ns can't be the true initial conditions of existence as Peirce's own logic makes necessary. — apokrisis

My interpretation is that 3ns is the true initial condition of reality - which is prior to both possibility (1ns) and existence (2ns) - as Peirce's own cosmology makes necessary (Ens necessarium).

So again, 2ns in Peirceanism is about the emergence of crisp possibility or determinate degrees of freedom. — apokrisis

This is one point at which I am having consistent trouble tracking with you. I understand 2ns in Peirceanism to be about brute reaction/resistance, the absence of freedom (1ns) and reason/purpose (3ns).

This of course is what I deny. There is only relativity, never the absolute. — apokrisis

How is this inconsistent with my suggestion that everything is general (i.e., indeterminate) to some degree? Perhaps I just need to clarify that this is generality in the broad sense, both negative (vagueness) and positive. It is a corollary of the thesis that all three categories are present and irreducible in every actual phenomenon.

X is being made its own context. That is the tautology here. The assertion is being made that the context is crisply existent too - thus bringing out that which Peirceanism would seek to deny. — apokrisis

By "here," do you mean the standard application of the principle of identity as x=x and/or x=not-not-x? Who is making the assertion "that the context is crisply existent too"?

The upshot then is that the statement is true only to the degree that either term is true. — apokrisis

Which statement? Which terms? I want to make sure that I clearly understand what you are saying here. Also, how would you fill in the blank with some formalized version of the principle of identity?

If x is contextual, then it is not necessarily true that _____.

While I am at it, do you agree or disagree with my other "first cut" definitions of "contextual" that parallel what Peirce wrote about "vague" and "general"?

A sign is objectively contextual, in so far as, leaving its interpretation indeterminate, it relies on some aspect of the actual situation to complete the determination. "That house is on fire." "What house?" "That one over there."

The contextual might be defined as that to which the principle of identity does not apply. This object from one point of view, or at one time and place, is not the same as this object from another point of view, or at another time and place.

But where the reductionist thinks that the differences that make a difference are atomistically unbounded - there is no reason why we could ever in principle cease the pursuit of further detail, chase down the last decimal of the expansion of pi until we are exhausted - the Peircean system offers principled relief. We can stop when the differences cease to matter to our over-riding purpose. — apokrisis

When we seek the truth, differences never cease to matter. A difference, by its very nature, as a difference, is a difference, and therefore it must be treated as a difference. If one adopts the perspective that a difference may be so minute, or irrelevant, that it doesn't matter, and therefore doesn't qualify as a difference, then that person allows contradiction within one's own principles ( a difference which is not a difference), and the result will be nothing other than confusion.

Which statement? Which terms? I want to make sure that I clearly understand what you are saying here. Also, how would you fill in the blank with some formalized version of the principle of identity? — aletheist

If x is not not-x, then it just seems straightforward that it is claiming its identity apophatically. I don't get that there can be any difficulty.

I am what I am because I am not what I am not.

But the difference is that now I have made the natural relativity of the question of identity explicit. I can suggest some feature as a characteristic that is "me" - like I am male. Because I'm not female. Then I can start to measure my maleness in terms of its,distance from femininity. Like perhaps I'm really hairy, so more truly male. Or perhaps I've got shapely legs, so that seems more ambiguous.

A silly example but it illustrates the principle. To state I am male is not very helpful. To state that I'm not female is to anchor the statement in a Universe of relative measurement. The context is now made fully part of the deal.

While I am at it, do you agree or disagree with my other "first cut" definitions of "contextual" that parallel what Peirce wrote about "vague" and "general"? — aletheist

But contextuality leaves it open whether the further possibility is 1ns or 2ns. It could a future condtional (the coming battle with the Persian fleet) or it could be some event already fixed by a determination (what will I discover when I finally check my ticket for the lottery drawn last week?).

So contextuality is simply 3ns or the generality of a constraint. And possibility divides into two kinds - that already determined by the past yet simply unknown or unmeasured, and then the true spontaneity of an undetermined future. Newtonianism talks about the first. Quantum theory is our best model for handling the latter.

Which is what, in your view? — aletheist

Following the second law of thermodynamics, I put it quite simply - to dissipate vagueness.

This is of course a very weak kind of telos from a theist point of view. But there you are. The world exists because vagueness proved to be intrinsically unstable. To the degree it existed, it already contained the possibility of regulation that could organise it to turn it into a crisp nothingness - a Heat Death void.

And throw all the Peirce quotes at me that you like. Semiosis starts with 1ns and so is radically at odds with any conventional transcendent monotheism - any 3ns notion of a higher purpose or creating mind or pre-existent harmony.

This is one point at which I am having consistent trouble tracking with you. I understand 2ns in Peirceanism to be about brute reaction/resistance, the absence of freedom (1ns) and reason/purpose (3ns). — aletheist

Why should any reaction be determinate in itself? Two things may collide and bounce. But how do you know which hit which, or who came off worst? You need some kind of fixed backdrop to close the story - give it a context against which the elements of the reaction can be measured. So that was what Newton's three laws of motion were about (and the triadicity was hardly an accident).

As I say, once we talk about 3ns or generality as constraint or purpose or law, then 2ns becomes the constraint of 1ns and hence the determinate thing of constrained possibility ... or a material degree of freedom.

Any point particle has six degrees of freedom - three directions of translational symmetry or straight line inertial motion, and three directions of rotational symmetry, or inertial spin. So Newton captured this fact that the fixing of a Euclidean flat backdrop then left these irreducible degrees of freedom. Constraint could stop everything but these last, now crisply definite, forms of local symmetry breaking. A rolling ball or spinning top - in a frictionless world - would remain in motion without change forever.

So Newtonianism is about a set of absolute freedoms. And thus also the corollary of absolute constraints.

Of course then along came relativity to demonstrate all this classical definiteness was relativistically contextual and quantumly indeterminate. That is why Peirce gets credit for foreseeing the physical revolutions about to come.

If it matters to you, then you have a reason to enquire. So where is the difficulty.

Mind you, if you claim that everything actually does matter to you, excuse me if I think that is patent bullshit. Does it make any difference to you if I wear a red or blue shirt tomorrow? Do you need that to be another determinate fact ... or do you believe in free will in contradiction to your what you just posted?