An analysis of identity

This article by Dan Polansky deals with the concept of identity or sameness and related concepts. It deals with some of the usual philosophical questions, but it also contains two practical applications: Pattern identity for mind uploading and Children as quasi-identical to parents.

Introduction
Identity can be quickly approximately defined by example, yet it can raise multiple interesting questions. What is going to be defined first by means of example is what is known as numerical identity as opposed to qualitative identity. Let us consider the shell game, which employs three shells. These are numerically distinct/non-identical: they are three shells, not one shell. Yet they are indistinguishable and thus qualitatively identical. By contrast, a particular shell at one point in time is numerically identical to that same shell at a later point, at which it possibly has a different location. Thus, numerically identical can be defined as "one and the same", whereas qualitatively identical can be defined as "exactly alike" or "indistinguishable".

Another case of numerical identity can be linked to a person who has radically changed. The person has borrowed a lot of money, visits the creditor and says: "I have radically changed. I am a completely new person. The person, who borrowed money from you, does not exist any more." The creditor is not convinced, and nor is the judge. Whatever quandaries there are of personal identity over time, an individual human body is usually taken to correspond to one person, and the person is the same or "self-same" entity during the lifetime regardless of changes of personality.

Another case is of a mosaic window, whose pieces have been replaced one at a time. The replacement of each piece is relatively small relative to the whole, yet after each piece is replaced, the whole mass of the window has been replaced. One may consider the result to be the same entity as the original, or one may not. If one does not, the question can be asked at which point did the original window lose its numerical identity. The case is somewhat reminiscent of replacement of cells in a human body during its lifetime. Another somewhat similar case is of a desktop computer in which one replaces the CPU in the motherboard with a 4 times faster one while other parts remain the same. On some level of analysis, the much faster computer does feel like a new item although only a physically small part has been exchanged.

Another case is of a cathedral that has been burned, but a new one has been built that looks exactly the same, at the same location. The new cathedral may be considered to be numerically non-identical, but for many a practical purpose, it does not make much of a difference, and if one treats it as numerically identical, not much has to be lost in way of modeling or representation ability.

Identity per type of entity
Different entity types lead to different identity considerations. Thus, one may well distinguish e.g.: Most of these are currently not treated in this article as dedicated subjects. For instance, when one considers a particular flock of birds and its change over time, one can see that it is a rather different kind of unit than non-metabolic body (a cup) and even metabolic body (a cat); the metabolic body outwardly appears to be something like the non-metabolic body, whereas the parts of the flock--the birds--have no apparent material ties and their spatial relationships keep changing a lot during the flight.
 * physical mass bodies without metabolism, e.g. a cup or an atom
 * physical mass bodies with metabolism, e.g. a horse or a tree
 * their development over time, e.g. from sapling to an old tree
 * metamorphosis from caterpillar to butterfly
 * minds
 * mental objects, e.g. propositional beliefs
 * processes, e.g. a procession of humans
 * bodies or quasi-bodies in flow, e.g. rivers, waterfalls and flames
 * behavioral patterns
 * qualities, e.g. blueness
 * mathematical immutables: numbers, geometric shapes, mathematical functions, etc.
 * texts of books and their development through time
 * languages and their development through time, including forking
 * human organizations, including states and companies
 * animal collectives such as flocks (of birds) and herds

Identity over time
Identity over time seems to present a set of problems or questions. One such question is this: At first impression, this formulation seems to suggest some kind of philosophical confusion more than any real problem; since, if the temperature of a particular cup located in view on a desk changes, surely the cup is not same as for temperature but it is the same (or self-same) cup as it was before its temperature change.
 * How can something change and remain itself at the same time? Since, if it has changed, it is no longer the same thing.

Another way of approaching this puzzle is to construe the time dimension as an analogue of the space dimensions and thus construe, say, a particular cup as an immutable object in space-time. What we would usually consider to be changes or mutations of the cup are just movements of some kind of mental pointer along time axis, within the same 4-dimensional space-time cup object. Here we can generally ask what makes different parts, with different time coordinate, belong to the same cup, but we can equally well ask what makes different parts, with different space coordinates but not time coordinates, belong to the same cup. This argument is inspired by Quine's Word and Object.

To the above space-time argument, one can object that the time dimension is nothing like the space dimensions even if one can represent them all together in one immobile 4-dimensional time-space. Popper pointed out that while the time dimension or axis is clearly objectively given, the space dimensions are not clearly objectively given but rather are picked somewhat arbitrarily, against some frame of reference. Philosophers of physics discuss the subject of the arrow of time (direction of time), one proposal relating to entropy increase along the time axis (arguably dubious), but there does not seem to be any discussion about the arrow of space dimensions. One can suspect that the problems of identity of an object along the time axis are quite different from those along the space axes.

Be it as it may, changes of an object over time do in fact present a set of interesting problems, but not formulated as a purely logical puzzle along the above lines. One such problem was exemplified in the introduction as a mosaic window whose pieces have been replaced one at a time.

Further reading:
 * Identity Over Time, Stanford Encyclopedia of Philosophy

Identity over time of individual animals
Identity over time of a cup can be approximated as invariance on the set of atoms. This is a mere first approximation, as a cup can lose a handle without ceasing to be the same individual cup, but is a workable first idea. However, that works very poorly with animals given their metabolism, exchange of substance with the environment.

This applies to the animal of humans as well. The set of atoms belonging to an individual body is far from constant, and so is the set of cells.

This allows various deliberations to the effect that the identity of an individual human is far from materially constituted (after all, many cells get replaced in a period of multiple years). It may lead to considerations in sections Children as quasi-identical to parents and Pattern identity.

A special case worth mentioning is that is biological metamorphosis, from caterpillar to butterfly. The butterfly seems to be numerically identical to the caterpillar it was before, but the anatomy is very different. Thus, when one compares the two time-occurrences of the individual animal, they seem to be qualitatively non-identical.

Identity over time of languages
A language is an interesting case to investigate as for numerical identity. It is not a mass extended object, whether metabolic or non-metabolic, yet it is an empirical entity undergoing change in time. It seems reasonably well described as a set of empirically instantiated patterns.

Let us take the case of Old English (before ca. 1100), Middle English (ca. 1100-1500 AD; Norman conquest dates to 1066) and modern English (ca. 1500-now). They are tracked in information systems as distinct entities, but following the criterion of continuity, they are part of a single entity. They are sometimes ranked as "stages". That single entity they are part of can to a limited extent be likened to a river which has one look or form up the stream and quite a different look or form down the stream. The slicing up into stages raises the question: what demarcates them as separate entities and what demarcates their boundaries other than convention?

The entity of language shows the somewhat arbitrary act of classification, here into a language vs. dialect. For instance, Slovak, mutually intelligible with Czech, was considered by some thinkers to be a dialect of Czech or Czechoslovak. Then, depending on the approach, one either has one entity or two entities, at least in the domain of discourse as opposed to the real world, which, figuratively speaking, does not care about humans implying things by means of language. (And yet, in so far as linguistic theories impact language codification efforts, they do impact the real world: once language and its spelling is codified, the language use tends to undergo something like normalization, reduction of scatter or diversity in the use, especially the formal use.)

The case of the arguably hybrid language English (Romance-Germanic, as for vocabulary) points to as if different flows being joined to create a single flow, as if two rivers becoming one at the confluence point. Since, the English vocabulary results from incorporation of both Old English vocabulary and Romance vocabulary. The idea of tracing the flow backwards as if a single river flows, following the continuity principle, breaks down, unless one is willing to consider Latin to be as if another ancestral branch of modern English.

The above shows that the question when is A one and the same thing as B needs to contrasted to the question when is A part of one and the same thing as B.

Identity over time and mergers and splits
Mergers and splits are worth a separate treatment. One may start the analysis with those pertaining to corporations/firms, and have the case of rivers merging in mind as a certain analogy. When a large firm buys a very small one, the resulting firm is naturally the large firm. But when two similarly large firms merge, the result can often be considered to be a new separate entity. How the identity is considered before law is probably a technical matter, requiring knowledge of the law mechanics. It seems possible for a smaller firm to perform a hostile takeover of a larger firm with the use of borrowed money, and for legal purposes, the result can be legally considered identical to the pre-merger smaller firm. Whether it is true or not, the legal customs can bring in something like a conventional aspect of what the law considers to be identical as opposed to what really is identical. A complication is that numerical identity over time does not require qualitative identity of the time-occurrences of the entities considered for numerical identity.

As another example, we can consider an apple. We split the apple into approximately same halves using a knife. There is no sense in saying that any of the parts is identical to the original apple. However, if, by contrast, we only slice away a tiny slice from the apple, it makes sense to consider what remained of the apple--almost complete apple--to be the same apple. The amount to be sliced can be varied continuously, and there would have to be an arbitrary threshold after which we cease considering the larger part to be identical to the original apple.

The above is reminiscent of a cup retaining its identity even after its handle gets broken away, as if split away. But maybe this is just our pragmatic interpretation: a cup without a handle can still well be used as a non-leaking drinking vessel. This would be less the case if a small hole was made near the bottom of the cup, without the cup losing its handle; it would still seem to be the same cup, albeit leaking one. By contrast, if the cup is shattered to pieces, we may think of having a shattered cup, or rather no cup at all.

Disassembly and reassembly
Concrete objects seem to keep their identity after a disassembly and reassembly. Thus, we can consider a chair whose parts are tied together using fasteners (rather than e.g. glue). We loosen the fasteners and obtain parts instead of the chair. At that point, the chair does not exist as a chair. We tie the fasteners again, and obtain a usable chair. Arguably, the resulting chair is the self-same chair as the one before the disassembly. And yet, it seems strange for an individual object to be able to cease to exist and then start existing again as the same object.

Ship of Theseus
Similar to disassembly and reassembly and to incremental replacement of parts in a mosaic window is the ship of Theseus paradox. One imagines a ship has its parts exchanged step at a time, and someone collects the parts, then building a structurally identical ship from the removed parts. One would think the reassembled second ship is numerically identical to the original, but also numerically identical to the ship that resulted from part-per-part exchange. And this cannot be, numerically. In fact, arguably, upon very strict identity interpretation, neither of two processes produce genuinely numerically identical ship.

Following Hobbes, one of the ships is identical per the same "form", whereas the other one per "matter". What Hobbes calls a "form", the modern mind could call pattern, arrangement of parts, manner of organization of parts, configuration of parts or the like. We may ask which of the two ships is more numerically identical to the original. The part-per-part-exchange ship is both formally identical and both has continuity over time; it is not identical as for building material. The removed-parts-reassebled ship is formally identical, but its continuity over time is very weak, but it is identical as for building material. Thus, three considerations have been identified: formal, material, and continuity.

Further reading:
 * Ship of Theseus
 * Ship of Theseus, britannica.com

Pragmatism
Some considerations such as the apple in section Identity over time and mergers and splits suggest that identity may be more arbitrary and "mere model" than one would wish. In various applications, instead of worrying about whether things really are identical, we may instead use the concept of considering them or treating them as identical.

Let us have a chair that we use and let someone swap the chair for another one that looks and works exactly the same, without our knowledge. When coming again to the chair and sitting on it again, we are likely to fall into the error of thinking this is the same chair we had before. This error about the numerical identity has no consequences for us as long as the new chair has the same qualities and uses. The new chair is not numerically identical, but no use gets harmed by our considering it to be numerically identical.

A key pragmatic element relating to numerical identity is knowledge of the object. When a device has reliably served us over long time, its being swapped for nominally qualitatively identical device (same type, model, etc.) decreases our knowledge about reliability, and may reduce our psychological sense of comfort and confidence.

Identity of rivers
Rivers are another interesting case for numerical identity analysis. They are subject to the famous quote by Heraclitus. The putative problem that quote suggests seems to be treated in a fairly satisfactory manner by Quine. Per Quine, if one does not worry about quandaries of the extension of a river in space, it is not clear why one should worry about quandaries of the extension of the river in time.

Another possible problem is how to deal with branching and tributaries. Let us take the case of rivers Elbe and Vltava in Czechia. There is a confluence of Elbe and Vltava near Czech town Mělník, where Vltava is considered to be a tributary to Elbe. However, at the point of the confluence, the volume of water coming from Vltava is higher than that coming from Elbe. Thus, one could argue that Vltava is Elbe and that the upper part of Elbe, above Mělník, is a different river. To completely eliminate this kind of quandary, one could replace the concept of a river with the concept of the river-branching-tree covering all the watercourses in the river basin. However, this would create strange utterances, such as that one would claim to live near the river where in fact they would only be living near a small stream entering the river. Therefore, the river-branching-tree solution is impractical.

Qualitative identity
Two things are qualitatively identical if they share the same qualities. But what does it mean exactly? Returning to the shell game, let us consider two shells that are indistinguishable by the naked eye. But there can be some microscopic crack or non-homogeneity in one or the other so that an image under electron microscope would reveal a difference. Are such two micro-distinguishable shells still considered qualitatively identical? For some practical applications, too detailed concept of qualitative identity would cause a harm: thus, two coins of the same value are qualitatively identical in the sense of being perfectly interchangeable, and there is no concern with whether any even easily observable detail about the coins would help distinguish them.

However, the above was about something like absolute qualitative identity. We can consider relative qualitative identity, by which either a) the qualitative identity is considered with respect to a given set of qualities rather than all potential qualities a mind can discern, or b) the qualitative identity is considered as given by sharing a narrower or broader class, and thus, two things can be more qualitatively identical than other two things. The case b) seems to match the treatment by Stanford Encyclopedia of Philosophy, per which: "Things with qualitative identity share properties, so things can be more or less qualitatively identical. Poodles and Great Danes are qualitatively identical because they share the property of being a dog, and such properties as go along with that, but two poodles will (very likely) have greater qualitative identity." By contrast, oxfordreference.com definition is "Two things are qualitatively identical if they share all their properties [...]", which points to absolute qualitative identity.

Further reading:
 * qualitative identity, oxfordreference.com

Personal identity
The term "personal identity" seems to be used in way that goes in a direction different from what so far has been presented. Thus, it would be part of one's identity whether one is a woman or a teacher. However, these are not questions of sameness or selfsameness, over time or otherwise, but rather of being an instance of a class, or having a certain kind of being, e.g. being a teacher.

However, the concept of identity as discussed so far does apply to persons. It is in that persons, entities that reside in human bodies, can undergo a change so big that one may want to say that the result is a new entity, a new person, non-identical to the original. This can bring about the subject of conditions of identity, that is, under which conditions are two nominally distinct things in fact the same thing. One such applicable condition can be one of continuity, that is, that if something undergoes a continuous change, that is, a change through a progression of states such that the successive states are close to one another, the result is considered to be the same or self-same entity. In that sense, even a person whose personality changes very much is the numerically same person as a result of the change being a result of accumulation of small changes.

Further reading:
 * Personal Identity, Stanford Encyclopedia of Philosophy
 * Personal Identity, Internet Encyclopedia of Philosophy
 * , wikipedia.org

Pattern identity
Relating to personal identity is what roboticist Hans Moravec in his book Mind Children calls a pattern identity. By his conception, a person who has been "uploaded" into a human-like robot (android) and has lost the biological body is still the same person via being the same pattern. The idea is that a person is not a material body but a pattern that continues to maintain itself. This idea has a deep practical application in so far as some seriously believe the idea is correct and that see their work in robotics or artificial intelligence as a contribution to mind uploading in the future.

However, the concept of pattern is rather broad and unclear. Let us illustrate the concepts on the bit pattern 01101. The same pattern is seen in letter sequences ABBAB and CDDCD. It is also seen in letter sequence CDdcD when one disregards the capitalization. It is equally well seen in a sequence of photographs of cats and dogs appropriately ordered, even if no two animals in the photographs look alike. This emphasizes the substrate independence of the concept of pattern. A configuration of a chessboard would be such a pattern, independent of the particular physical chessboard, and being the same pattern/configuration even in an electronic computer, where one cannot ask what is the mass of the chess figures making out the pattern/configuration.

Expanding on the above, we can conceive of various patterns of patterns. For instance, we may treat blocks of bits as single bits using a mapping, e.g. 00 → 0, 01 → 1, 10 → 1, 11 → 0. Or we can take three bits, and set the result to 1 if the number of 1s is odd, to 0 otherwise. Further mapping can be considered in a nested fashion, that is, based on the 1-derived pattern, we may obtain 2-derived pattern. This is a sketch illustrating the pattern-of-pattern-of-pattern and similar higher order ideas.

Patterns as explained above are immutable objects. However, Hans Moravec as a mind is a mutable object, one undergoing change. What, then, does Hans Moravec mean?

The kind of pattern identity involved in a mind is not a static one but rather a dynamic one. A dynamic pattern can be seen in a digital computer. Let us, say, consider an 8-bit computer with 16 KB of memory. At each point in time, the memory defines a static, fixed pattern. However, this pattern is self-modifying (given a host hardware), resulting in the behavior of the system being a sequence of patterns, or, figuratively, a pattern in motion. It is on this dynamic level that a moving pattern of the mind could be pattern-identical to a moving pattern in a human-like robot. (The pattern in the 16 KB is self-modifying in the sense that part of it is, at each point in time, interpreted by the host hardware as instructions for the modification of this pattern.)

Identity crisis
In psychology, the term "identity crisis" seems to refer to intense examinations of questions like, what kind of person do I want to become, or perhaps, what kind of person am I really, deep within? It does not seem concerned with questions of identity in philosophy; it seems the term "identity" has here been borrowed for a different purpose. On the other hand, one may wonder whether a deep change in personality does in fact warrant an allocation of a new entity at least in the modeling of the person. One might alternatively want to call it a "being crisis", as in whether one should acquire or develop "being an entrepreneur", "being a monk" and other beings.

Further reading:
 * , wikipedia.org
 * topic identity crisis, britannica.com
 * Identity Crisis, encyclopedia.com

Identity of information objects in computing
As for identity over time, information objects in computing such as memory objects and files present a whole different world, very different from physical objects.

Let us consider a text file and ask what constitutes its identity over time. The text file has a file name and its content changes over time. A backup copy of the file can be created. If the file gets destroyed, it can be restored from the backup copy. But then, on some level of analysis, the restored file is a distinct file, even if it has the same name and the same content. Moreover, one can open the file in the text editor and replace its content with a wholly different content, breaking the content continuity over time. By contrast, when one expands the file and makes minor editing to it, there is a content continuity.

For practical purposes, it makes sense to consider a file restored from backup to be the same (self-same) file as the one there was before it was destroyed.

Similar considerations apply to pre-computing text objects, their identities and versions. One does not need a computer to expand a text object on a sheet of paper, edit it by striking parts out and as if inserting parts by writing them on the margin and connecting them via a line with the location of insertion, to create a copy (even by writing) and use the copy instead of a lost original, etc. What computers bring about is the rapid ease and speed with which the operations of expansion, editing, backing up, restoration, and copying can be done, as well as the range of the types of information objects, whether texts, images, audio recordings, etc.

Identity of variable referents in programming
One can perhaps liken numerical identity and qualitative identity to certain features of programming languages. Thus, in Python, one could map the numerical identity to "is" and the qualitative identity to "="; this would be rather natural in so far as "is" tests whether the expressions compared point to the same object whereas "=" invokes the appropriate equality comparison function. However, since a class can redefine "=" to do whatever it wants, violating the natural intuitive semantics, this presents a limitation.

However, this consideration seems to tell us nothing about the interesting considerations of identity of physical objects and minds.

Conditions of identity
One can discuss conditions of numerical identity, that is, under which conditions are things that are nominally two in fact one thing. One such condition is sameness of atoms: the set of atoms belonging to a cup does not much change during its lifetime nor can a given atom be member of multiple cups. However, that works poorly with living things as for identity over time, since they have metabolism. Another condition is continuity, which can be investigated in the context of text works being edited and expanded, computer files being so treated, political entities, languages, etc. Yet another condition is something like pattern identity.

Applications
The philosophical questions about identity seem to be primarily of interest to philosophers only. But there can be practical applications as well. For instance, one can ask whether one can in some sense consider oneself identical to one's children. This proposition seems weird at first, but if it could be established that one's children are at least weakly identical or quasi-identical to oneself, this could have a major impact on one's overall life plan or design.

In the field of personal identity, one can wonder whether one can bring about changes to one's personality so deep that one can really meaningfully consider oneself to be a new person in some sense, or whether it is just a figure of speech.

Another application is the discussion of pattern identity by Hans Moravec in his book Mind Children. If one is convinced by his analysis, one would accept to be "uploaded" into a human-like robot (android) while losing one's biological body; one would consider oneself to be identical enough with what some consider to be a mere simulacrum.

The questions of identity can possibly contribute to design considerations of information systems. This is especially the question, when are two nominally distinct entities in fact the same? In a geographic information system, one can wonder which of merging rivers is identical to the result of the merger, if any at all and why. In an information system about works of art, one may wonder whether a cathedral that has been destroyed and a faithful replica has been build in its place should have a new entity/item in the information system for the replica.

Children as quasi-identical to parents
One application of investigations of identity is to see children as quasi-identical to their parents. At first, this may seem rather counterintuitive or implausible. Surely a child is not numerically identical to the parent since they are two, and if one takes both parents into account, three. Moreover, if a child were somehow numerically identical to the parents, it is not clear why the child would not be numerically identical to the siblings, by transitivity of numerical identity. However, let us consider a human parent capable of cloning so that the child is genetically identical to the parent. When the parent is alive, they are indeed two, but when the parent dies, there is one person with the genetic identity of the previous parent, the child (bar identical siblings of the parent and their ancestors). Thus, if one happens to only takes into account the frames where there is a childless parent and lone child in existence and when one notices the genetic identity of the two person occurrences, one may think of this as quasi-numerical identity over time. That this is not a genuine or full numerical identity is revealed by the intermediate frame where both parent and child are alive.

One may object that human cloning is currently impossible and that a human is constituted by continuity of memories, plans and desires, not merely by the biological sameness of the body in different time frames. Even so, if one wishes to as if live forever, living via one's children may be as close as one can get. To make this idea more plausible, one may note that one's body does not have substance invariance over time due to metabolism and does not have cellular invariance over time due to cells dying and being replaced, and new additional cells appearing during childhood. Thus, one may realize that the kind of numerical identity one sees over time in a cup does not apply to individual humans (or plants in fact), and that there is something that continues to exist in an individual human that is distinct from the set of atoms or the set of cells. One candidate thing constituting the individual identity over time is certain patterns that persist during lifetime of a human. As for biology, many such patterns persist between a parent and a child even if in a modified form. As for cultural patterns (especially language but also certain norms), a parent can make sure that as many of the cultural patterns as possible replicate into the child, or at least that the most constitutive patterns replicate.

Following the above deliberation, the father would consider a son to be more of the quasi-identical copy than a daughter, and similarly for mother and daughter; since, on the chromosomal level, there would be a greater affinity, as well as on the anatomical level and genetically driven behavioral level.

There is another way to think of the continuing identity of a human individual, the purely genetic level. Thus, the individual human can see themselves as a representative of a certain alliance of genes. As is typical of alliances, alliance members have more of a stability or unity than the alliance itself. From that perspective, given absence of cloning, the alliance as an entity cannot be preserved, but one can do as much as one can to serve the alliance members, the genes. And indeed, a portion of the genes gets replicated in a child. Since different children have different portion of the genes replicated (bar identical twins), having more than one child serves the alliance as a whole better. (It serves better anyway since the world is full of dangers.)

Thus, a parent can think that their children are in a strange sense quasi-identical to the parent. Maybe the parent is wrong about it, but the above analysis of the concept of identity suggests the idea is not entirely wrong. Some parents may be lead by similar considerations to think that having at least one child is a good idea even if they would be disinclined to have children otherwise. And thus, what appeared to be an abstract impractical philosophical deliberation turns out to have a very material application, one of the greatest significance it can have on a personal level.

If one thinks the prefix "quasi-" is not detracting enough from the word it modifies (here "identity" and "identical"), one can think "pseudo-" or even "quasi-quasi-", as uncustomary as it is.

Identity across possible worlds
Identity across possible worlds is required for the rigid-designator Kripkean treatment of modal logics to work. Thus, if we say "Walter Scott could have not written Ivanhoe", what this means is that the terms Walter Scott and Ivanhoe, by implication, succeed in referring to something not only in the present world but also in possible worlds. A rigid designator is defined as one that refers to the same object across possible worlds, regardless of whether the object meets the same descriptions in these worlds as in this world.

It is not entirely clear what constitutes an individual identity across possible worlds. If one conceives of a possible world as something independent, not forked from the past of this world, one would need to rely on something like qualitative identity. Since, what if not a bundle of qualities would identify the individual object? We cannot rely on continuity over time. If, by contrast, we consider a possible world that is a time development of a forked past state of the present world, we can track a present object to its time-occurrence in the forking time frame, and from there continue the tracking in the other world, arriving at the same object in the present in the other world. (If we use this method to track the identity of a human between this world and a particular possible world, and if the human has a child born after the world forking event, arguably, the child is not necessarily numerically identical across these two worlds since, assuming that the genetic recombination results from chance events, it is not even genetically identical.)

If we consider a possible world that is non-forked and if we decide to track the identity of concrete individuals via their quality bundles (which is not entirely like numerical identity, per indistinguishable shells in shell game), we need to establish the identity of qualities across the possible worlds. However, some qualities do not need to exist across possible world, e.g. color, since the possible world can lack electromagnetic waves. This needs a deeper consideration.

If we consider this world and the possible worlds to be deterministic, there cannot be any genuine forking, and then, the modalities of necessity and contingency lose any true meaning. However, even if this world is deterministic, our knowledge of it does not suffice to model it as such, and then, it may make sense to talk and reason as if the world were non-deterministic, where the apparent non-determinism is generated e.g. by deterministic chaos. Then we may want to wish to maintain the illusion that the modalities of necessity vs. contingent make sense; they make sense with respect to the non-deterministic model of the world that we have in our minds.

Further reading:
 * 6. Identity across possible worlds in Identity, Stanford Encyclopedia of Philosophy
 * Transworld Identity, Stanford Encyclopedia of Philosophy

The identity of indiscernibles
The identity of indiscernibles is a principle that states that two objects that are indiscernible are identical, that is, are one object. To what extent this is plausible depends on what we consider to be properties one can use to discern objects. Thus, in a shell game, the shells appear indistinguishable to human eyes, but if one takes the property of being at location L at time T to be a property used for discernment, the shells are in fact discernible. With the shells, one may argue that even if one discounts different locations as properties, there would be some crack or irregularity within the shells that makes them discernible in principle. However, it is not clear what kind of irregularities would be found e.g. in individual atoms, that is, what distinguishes, say, two distinct atoms of hydrogen other than their location and relation to other atoms. This would be even more salient for the philosophical atoms, which are not the physicist's atoms but rather, perhaps, the physicist's quarks.

As something of an aside from the point of view of formal symbolic logic: We can study the principle is relation to a particular language and associated axioms of first-order predicate logic. Let us have a very simple language, consisting of a single one-place predicate: F(x). And let us have a single axiom: there exists x: F(x). Then, if we have the model consisting of, say, 0 and 1, and such that F(0) and non-F(1), there is indeed identity of indiscernibles since all elements in the domain of discourse can be differentiated using F. But as soon as we have a domain of 0, 1, and 2, in another model, then regardless of for which items in the model F holds true, it is unable to distinguish all three items. (We do not assume any constant symbols "0", "1" and "2" referring to these items in the domain; the language consists of F(x) and nothing else.)

Further reading:
 * The Identity of Indiscernibles, Stanford Encyclopedia of Philosophy
 * Identity of indiscernibles, britannica.com

Identity and language
The concept of identity permeates language; the great majority of uses of language seems to invoke the concept. It is so even if we disregard the words "same", "self-same", "identical", "distinct" and "different". Thus, we can say "Peter woke up. He realized he was late." There, the second sentence's "he" refers to the same Peter to which "Peter" refers in the first sentence, and what we deal with is identity over time. In fact, the sentence "Peter woke up" alone implies numerical identity over time, of Peter before waking up and Peter after waking up.

Numerical identity is one of the meanings of the verb "to be", together with is-a relationship and instance-of relationship. Thus, in the sentence "The winner of the lottery is Joe Hoe", the "is" indicates numerical identity rather than is-a or is-instance-of.

We have treated of identity as a two-place predicate. However, in the phrase "the identity of the perpetrator", it is implied to be a property of an object. The statement "we do not know the identity of the perpetrator" means that we do not know who the perpetrator is. In this connection, one might speak of multiple identities a person has, e.g. if the person presents themselves under different names and different facial appearances (hair length, presence of mustache, etc.). The connection of this to the concept of numerical identity is somewhat puzzling. One might think that X is an identity of Y if X is Y. Thus, "Joe Hoe" would be one identity of Y, where Y is "the perpetrator of the bad act under discussion". This requires further elaboration, possibly with the use of the sense-referent or intension-extension distinctions.

Identity and cognition
The concept of identity permeates cognition even without reference to language. The human visual system sees objects as persisting over time, e.g. an apple, a stone, a tree or a river. This ability to see objects as persistent (maintaining their identity over time) is not restricted to humans but is present e.g. in a dog or a chimpanzee as well.

Identity in quantum theory
There is a peculiar situation concerning identity of particles in quantum theory. This subject is not covered here; this section is here as a placeholder for later expansion, and to link to Stanford Encyclopedia of Philosophy.

Further reading:
 * Identity and Individuality in Quantum Theory, Stanford Encyclopedia of Philosophy

Predicate logic and identity
First-order predicate logic with identity contains the symbol "=" for the purpose, and associated inference rules. One may ask whether this is necessary, that is, whether one could not instead introduce a two-place predicate I(x, y) and then codify the identity using formulas as follows: And a schema, for all one-place predicates F: This seems insufficient per the following article: This article seems, given model M, to construct model M*, which makes sure I is no longer guaranteed to assert identity in M*. If this understanding of the article correct, this motivates the introduction of symbol "=" as something different from a predicate symbol.
 * For all x: I(x, x)
 * For all x, y: I(x, y) => (F(x) => F(y))
 * Absolute Identity and Absolute Generality by Timothy Williamson, media.philosophy.ox.ac.uk

We can specify I(x, y) as follows: For all x: I(x, x), and I holds of no other pairs. It seems the statement "I holds of no other pairs" cannot be expressed in first-order logic without identity.

The language of first-order logic fundamentally depends on the concept of identity. Each occurrence of a symbol is an occurrence of the same symbol. Thus, in "For all x: F(x)", both occurrences of x are of the same x rather than of implied subscripted x1 and x2. Similarly for different occurrences of predicate symbols, e.g. F.

Further reading:
 * First-order logic
 * Type–token distinction

Relative identity
The subject of "relative identity" is treated by various sources. From the name, one can think of qualitative identity relative to a set of predicates capable of distinguishing things. This section needs to be expanded; the content is currently delegated to further reading.

Further reading:
 * Relative Identity, Stanford Encyclopedia of Philosophy
 * Identity (section Relative identity) by Phillip Bricker, 1996, encyclopedia.com

Set theory
In the mathematical set theory, two sets are considered to be identical, that is, one and the same, if they have the same elements. This would not need to be the case; one could think of something like internal hidden property or key of a set that would distinguish it from other sets with the same elements. However, it is unclear what theoretical advantage this would bring. In any case, in a programming language such as Python, both mutable and immutable sets can have the same elements yet be distinct objects.

National identity
National identity is another subject identified using the word "identity". One would speak of a person's "national identity".

However, one can think of national identity as the identity of a nation, especially over time. This brings the question "what is a nation", a search for definition, criteria or characterization. Since a key element of identification of a nation is a shared language, one may think the deliberations on national identity will show some overlap with the subject of Identity over time of languages. This subject is currently delegated to further reading.

Further reading:
 * , wikipedia.org
 * , wikipedia.org
 * National identity, encyclopedia.com

Identity theft
The subject of "identity theft" is another one named using the word "identity". But what kind of identity is being stolen? Surely, no one can possibly steal the two-place relation of numerical identity. Nor can they steal the two-place relation of qualitative identity. However, if an identity is also the facial appearance and manner of verbal and other behavior, such an identity could indeed be stolen by, say, an operative of a secret service pretending to be someone else. Thus, X steals the identity of distinct Y if X convinces Z that X is Y, for a nefarious purpose. This can be done e.g. with the help of person name, social security number and credit card number. The fraudster can use that information to quasi-prove he is some one else. These person identifying pieces of information are rather distinct from personal identity as the sense of who one is.

Further reading:
 * , wikipedia.org
 * Identity theft, britannica.com
 * Identity Theft, encyclopedia.com

David Hume
David Hume is a great philosopher with significant things to say on identity. The content is currently delegated to further reading.

Note: SEP's David Hume entry does not mention the word "identity".

Further reading:
 * Hume on identity over time and persons by Jeff Speaks, 2006, nd.edu
 * e. Personal Identity in Hume, David, Internet Encyclopedia of Philosophy
 * Hume on Personal Identity, rintintin.colorado.edu

John Locke
John Locke's treatment of personal identity has a dedicated SEP entry, worth perusing.

Further reading:
 * Locke on Personal Identity, Stanford Encyclopedia of Philosophy

Thomas Hobbes
Thomas Hobbes treats of identity in The first grounds of philosophy, mentioning the ship of Theseus paradox.

Further reading:
 * CHAP. XI. Of Identity and Difference. in THE FIRST GROVNDS OF PHILOSOPHY., quod.lib.umich.edu

Gottlob Frege
Frege's Sense and reference (Sinn und Bedeutung) starts by pointing out that the identity relation (under the head of "equality") raises some puzzling questions, such as whether it is a relation between referents or between referent-bearing signs. If the former, identity is a relation I such that for all x, I(x, x) and I is true of nothing else; but its cognitive value is near zero. By contrast, 2 + 3 = 5 brings a revelation, just like "Hespherus is Phosphorus". One can read more at Wikisource.

Further reading:
 * On Sense and Reference

Willard Van Orman Quine
Quine has paragraph 24 Identity, page 114, in his Word and Object. Sameness of river is discussed in paragraph 36 Time, page 171. Search online does not seem to find a good reading, so I'll leave it with the book identification for those interested in Quine.

Stuart Fullerton
Stuart Fullerton wrote an interesting book on sameness and identity.

Further reading:
 * On Sameness and Identity. by Stuart Fullerton, gutenberg.org

Erik Erikson
Erik Erikson, German-born American psychoanalyst, seems to have a special concept of identity, in the field of personal identity. This section is to be expanded.

Further reading:
 * Erikson's stages of psychosocial development
 * Wikidata: Erik Erikson.

Identity politics
The meaning of "identity" in the term "identity politics" seems to be rather different from the one covered in this article. This section is here to support further reading and concept clarification. A speculative hypothesis about this use of the word "identity" is that it was thought to be any answer to the question "Who are you?" But that is not so; if the answer is "Mark Twain", that is indeed a statement of identity, as in "I am Mark Twain", that is, the referents of "I" and "Mark Twain" are one and the same. By contrast, if the answer is "African American", the result is "I am African American" and is not a statement of identity but rather of instance-of relationship. If this hypothesis is correct, one could try to figure out an alternative name for the answer to an instance-of question; "class" comes to mind, not as in sociology but as in ontology.

Further reading:
 * , wikipedia.org
 * Identity Politics, Stanford Encyclopedia of Philosophy
 * Identity politics, britannica.com