Figure 1. Five continuants extended in time, one non-continuant (the grey line) and two reference intervals.
Latest version: http://www.ihmc.us/users/phayes/IKL/INTEROP/INTEROPERABILITY.html
One central presumption underlying the design of IKL is that when sentences are transmitted across a network or archived for later use, their meaning should be the same at the point of use as it was at the point of origin, no matter how remote these are from one another. This is the central defining characteristic of a 'network logic': transmission of content commutes with entailment. If A entails B when A is stored or transmitted, then A should still entail B when it is retrieved at another time or received at another node of the network; and vice versa. This has several significant consequences, but the two most important are that IKL must be non-indexical and panoptic.
An indexical expression is one in whose meaning is affected by the context of its use. Words like 'here', 'now', 'you' and so on - words whose meaning depends systematically on their immediate context of use - make English an indexical language. To see intuitively why an interchange language cannot be indexical, imagine recording facts about the present time using a word like 'now', and storing them for later use in this form, so that the fact that it is raining at some time would be recorded literally as 'raining now'; and then this stored fact is retrieved from memory at some future date. Clearly it has become meaningless, since 'now' at the time of retrieval does not refer to the same time as 'now' when the information was archived. Temporal data cannot be stored using temporal indexicals. Tensed language suffers the same fate, since references to the future will become references to the past in some future. Similarly, geographically sensitive information cannot be transmitted to another location using spatial indexicals such as 'here'. A similar case can be made for other, less physical, kinds of contextuality. For example, if it is known that someone has a false belief about identity – say, that Michael Eisner is really Osama bin Laden – then the appropriate way to represent this in IKL is to refer to that person's beliefs explicitly, rather than simply record this belief as a proposition in IKL. The correct way to encode such information into IKL is to replace the indexicals with identifying information which uses some kind of global temporal, spatial or other contextual reference framework whose meaning is universally agreed and will remain stable: to de-indexicalize it.
This may not always be possible, and there are appropriate work-arounds. First, a truly 'global' coordinate system may not be achievable, and so an identification of the particular coordinate system (of, for example, dates or times, or GPS coordinates) should be recorded as part of the coordinate information itself, perhaps as meta-data. This provides for archived information to be transcribed into other coordinate systems in use elsewhere or at later times. Many existing 'datatyping' schemes use this principle. Second, it may be impossible for lack of information: one may not know the 'when' and 'where' which were the now and here of some piece of indexically expressed information. In cases like this, IKL content should express the doubt explicitly by using existential assertions, e.g. "at some time in 2004" might be expressed by a construction such as
(exists ((t isTimeInterval))(and (subInterval t 2004)... ))
IKL is panoptic (footnote, not a standard logical term) in the sense that its universe of discourse contains all things which are in all other universes of discourse. This is a consequence of its being non-indexical, since if it were not panoptic, then the meaning of a quantified sentence would either change when the sentence is moved across the network, or else would not satisfy the local meaning of the quantifiers. The bare quantifier 'exists' in IKL does not mean 'really exists' or 'is actual', but something more like 'can be possibly referred to by someone'. Since beliefs may be false, this has to be allowed to include fictional entities, so that for example it is quite coherent to say in IKL
(exists ((x Person)) (and (not (isActual x)) ... ))
which might be loosely expressed, using 'exist' in its more normal sense, as 'there are some imaginary people who ... '. Any name which can be used by anyone to refer to anything, real or imaginary, or any thing which anyone considers to 'exist' in any sense, or which has existed in the past or will exist in the future, or even is simply something whose existence is being considered hypothetically; these all count as things which exist in the IKL universe. It is all-inclusive. This means that an unrestricted quantifier 'forall' is a much stronger assertion in IKL than might be expected, and a simple unrestricted exists is similarly weaker; and in fact, bare quantifiers are rarely used in IKL sentences, except in cases where the body of the sentence already implies some kind of restriction on what is being said. To achieve the meaning of a normal quantifier, it is usually necessary in IKL to use a restricted quantifier: rather than saying 'forall...' and intending (because one has never thought of anything else, or because this is locally understood, or implicit in the context) that this means, say, 'for all people...', one must usually in IKL be explicit about what one is quantifying over, typically by writing the restriction into the quantifier:
(forall ((x isPerson)) ... )
IKL content is therefore often reliant upon some framework of classification of things into categories or classes, whose primary use is to provide appropriate quantifier restrictions. Such a framework of categories is often referred to as a system of types or sorts, and many logics and notations are designed to conform to them, with special mechanisms for handling sortal reasoning or even allowing type checking to be done at parse time. IKL is not a typed logic in this sense. When translating content into IKL from such a typed or sorted notation, any sortal restrictions on existence, and all relationships between sorts or types, must be stated explicitly in IKL. (More on the topic of sorts and types below.)
The non-indexicality of IKL mean that translations from other languages may need to be handled with care, since names and quantifiers in those other languages may require modification when content is rendered into IKL. In particular, information in tensed languages will usually need to be translated into a different form when put into IKL, since the IKL quantifiers are not restricted to things which exist at any particular time. As this case is so common, we review it here in detail.
Consider TIKL, a (hypothetical) tensed version of IKL in which sentences are asserted "at a time", which we will call the reference interval, and are understood to be about "the world at the time" in which they are asserted, so that 'exists' means 'exists now', i.e. "exists in the reference interval". We will write TIKL content like this, to distinguish TIKL content from the IKL into which it is translated. Things that exist during a time-interval may or may not exist outside it, and may or may not have the same identity conditions outside it. For example, consider three people, Mary, Joe and David, and the description Committee Chair.
Figure 1. Five continuants extended in time, one non-continuant (the grey line) and two reference intervals.
During the interval I marked by the box in figure 1, Mary is the Committee Chair, Joe exists, but David, sadly, has recently died and no longer exists. Note that this way of talking treats 'exists' as referring to things that exist during the interval, a different way of interpreting it than the much more inclusive meaning intended for the IKL quantifier, described above. To translate the TIKL exists quantifier into IKL, therefore, we must restrict the IKL exists quantifier to things that exist in the reference interval, and in order to be able to do this we need an IKL name for this class or category of things. We will assume that this need is satisfied by a relation called existsIn between things and intervals, so that (existsIn I x) is true when x is a thing that exists in I: in our example, (existsIn I Mary) but not (existsIn I David).
While the first of these is equivalent to asserting (exists (x)(= x Mary)) in I, the second is not handled by asserting (not (exists (x)(= x David))) in J, since this second sentence is a logical contradiction. Simply using a name such as "David" in a logical sentence implicitly asserts that some thing exists with that name: not only can we speak no ill of the dead, in such a logic, we cannot refer to them at all. For this reason many practical indexical languages, such as that used by Cycorp [ref Cycl, ref Charles Klein personal], often adopt a different convention whereby quantifiers range over all entities, as in IKL, not just those which are present during the reference interval. The result is an indexical panoptic logic, a kind of hybrid between IKL and a conventional tense logic. This requires that the indexical language use a predicate like existsIn, or perhaps an indexical version such as existsNow, in order to distinguish 'living' from 'dead' things. (Other conventions are also possible, such as quantifiers which range only over things existing in the present or past, but excluding the future. We will not review all the possibilities here.)
Note that to say that Mary, for example, existsIn I, is not to say that the interval wholly contains Mary. Mary also exists outside that interval: in contrast for example to Event13, whose entire lifetime is inside the reference interval. Entities such as Mary in this example, which are intuitively the "same thing" throughout their lifetimes, are called continuants. For IKL purposes, and speaking informally, any object or event or process which lasts for a time and is referred to in the same way throughout its life is a continuant. In contrast, a role or function, such as CommitteeChair in our example, which may be filled by different continuants at different times, is not a continuant. The distinction is important, as continuants should be translated into IKL differently from non-continuants. A continuant is the 'same thing' at different times, even though its properties and relationships may change, so we can use the simple name to refer to the continuant both in a tensed language and in a panoptic logic such as IKL. In contrast, non-continuant names or descriptions, such as CommitteeChair, PresidentOfTheUSA or MyCar'sBattery, must be somehow temporally decontextualized when translated into IKL. Various options for this can be used, but the main point here is that some kind of decontextualizing translation is required for names of non-continuants. Using a simple name to refer to a non-continuant in a panoptic language leads quickly to inconsistency. While in TIKL it might be acceptable to treat CommitteeChair as a name and assert an equation
?? (= Mary CommitteeChair) ??
in IKL this would immediately produce errors, since we could conclude for example that Mary was equal to Joe because they had both been Chair at some time. (Even in a tensed logic, difficulties arise if the non-continuant changes identity during the reference interval, as in the above example with interval J. A better way to encode such facts in TIKL is to treat the non-continuant role as a property of the continuant which occupies that role:
(CommitteeChair Mary) [during I ]
which allows both (CommitteeChair David) and (CommitteeChair Mary) to be true in J without leading to the conclusion that David equals Mary. This translates naturally into IKL by treating the property as a fluent [McCarthy 1882] , i.e. by adding the time-interval as an extra argument, translating the property into a relation between the continuant and the time-interval. It is conventional to insert the time-interval as the last argument of a fluent:
(CommitteeChair Mary I)
An alternative translation 'attaches' the interval argument to the continuant name, forming a term which denotes a new entity which might be paraphrased as Mary-during-I:
(CommitteeChair (tsl Mary I))
We will call this a time-slice of the continuant, and will use the function tsl consistently to form time-slices. (This function is not strictly required in IKL, since we could simply use the time-interval as the slicing function directly, but the resulting formulae are often rather unintuitive, so we retain the 'dummy' slicing function.)
Since in IKL properties and relations are treated as individuals, we could also express this as time-slicing on the property, giving the rendering
((tsl CommitteeChair I) Mary)
which – once one gets used to the unusual notation – has the merit of 'slicing' the non-continuant part of the original assertion, which corresponds intuitively to the idea that Mary is the same person whether or not she is chairing the committee. One way to understand this intuitively is to think of tsl as a function on IKL relations which selects those parts of the relational extension which have temporal segments, determined by I, between which the corresponding relation would hold in TIKL, with that same reference interval.
Finally, yet another alternative is to reify the TIKL expression as a proposition in IKL, and then assert that this proposition 'holds true in' the time-interval:
(holdsIn I (that (CommitteeChair Mary)))
but in order to be fully related to other IKL sentences, this would also requires that holdsIn be adequately axiomatized.
We could express the equivalence of these various translations by writing suitable axioms, for example:
(forall ((c continuant) (I timeInterval) x)(iff
(x c I)
(x (tsl c I))
))
(forall ((c continuant) (I timeInterval) x)(iff
((tsl x I) c)
(x (tsl c I))
))
(forall ((c continuant) (I timeInterval) x)(iff
(holdsIn I (that (x c)))
(x (tsl c I))
))
More complex axioms would be required for handling more elaborate sentential forms, of course. In general, these translation techniques apply to any atomic assertion involving a relation and an argument sequence, by following a suitable time-slicing rule and mapping continuant names into themselves.
[Footnote. It would be possible to treat CommitteeChair as a continuant, and personal names as non-continuants. This would amount to thinking of the role itself as a more important determiner of what is 'real' than personal identity. A secret serviceman whose job is to bodyguard the President, regardless of who happens to be the incumbent at the time, might reasonably think this way. But the main logical point here is that one cannot consistently treat both Mary and Committee Chair as continuants, since their identities diverge as time goes by.]
@@@ This needs rewriting, its taking too long. define an actual translation function from TIKL to IKL and be done with it. First show why naive map doesnt work, to motivate point reduction.
Also give translations for panoptic case ala CYC and for temporal case, to emphasise similarities and differences. Conveyor example. @@@
So far we have considered techniques for translating atomic sentences from TIKL into IKL. All other sentences can be viewed as built up from atomic sentences using the connectives and and not, and the universal quantifier forall. Translating from non-atomic sentences of TIKL into IKL is not completely straighforward, since the temporal quantifiers must be property rendered into IKL timeless quantifiers. The translation given here assumes that truth or existence in a time-interval means truth or existence throughout the time-interval, i.e. intuitively, at all points of the time-interval, and therefore throughout all sub-intervals of the interval.
By 'point' here we mean simply the smallest size of interval recognized by the temporal ontology in use: this might be a 'unit interval' or 'clock-tick' for a discrete time ontology, or a mathematical point for an ontology which treats times as real numbers. The essential properties of a point, in the sense used here, are:
(1) every proposition is either true or false at a point, and every thing either exists there or does not;
(2) something is true throughout an interval exactly when it is true at all points of the interval; and
(3) intervals overlap exactly when they share at least one point.
See [ref ContextMereo] for axioms defining some general conditions for such points to exist, and a discussion of the conditions under which they are plausible.
Given the above properties, truth at a point is transparent to conjunction and negation, and hence to all the other propositional connectives. That is, when I is a single timepoint, P [at I] is false just when (not P) [at I] is true, and (and P Q) [at I] is true just when P [at I] and Q [at I] are both true, ie when (and . As noted, the universal quantifier in
This means that we can transcribe boolean combinations directly from TIKL to IKL, provided that we are talking at a timepoint. However, we can translate any TIKL which is asserted at a non-point interval as being an implicit quantification over all the points in the interval.
Because continuants can change their properties without changing their identity, care is needed when stating facts involving continuants. In general, a property (or relation) should be simply asserted of a continuant in IKL only when it is true of the continuant throughout its lifetime, or true of the continuant inherently, regardless of its circumstances. This is sometimes called a rigid property or relation [ref OntoClean]. A common technical definition of rigidity is that a property is rigid when if it is true of a continuant, then the only way for it to become false is for the continuant to cease to exist. This way of speaking is natural for a tensed language like TIKL, but the same point is made in an IKL panoptic framework by saying that a rigid property must apply to something throughout its temporal extent; it cannot be true of it at one time but not at another. So for example being human is generally taken to be rigid, whereas, say, being a committee chairman is clearly not. The consequence is that a rigid property can be simply predicated of a continuant in IKL directly, without needing to relativize the property to a time or a time-slice.
It is not always completely obvious whether a property should be taken to be rigid, and the choice has consequences for what can be reasonably said about the things the property applies to. If one removes a wheelchair motor and uses it to power a robot [ref battlebot], is it still a wheelchair motor? If someone uses a rusty old car engine block as a decorative garden planter [ref engineplanter], is it still an engine block? There can be rational disagreement about the answers to such questions; but the key point here is that the notions of a continuant, and of a rigid property, should 'go together' in a panoptic logic. So for example, if being an engine block is rigid, then the engine block must stay being an engine block for as long as it exists. This has some strong consequences: for example, if we wish to say that the rusty old planter is not an engine block any longer, then would be obliged to say that at some time the engine block ceased to exist and a new, different, entity - an ex-engine-block - came into existence at the same time, made from the same materials. This is not a mere change of status of a continuant, but two distinct, different continuants, which coincide exactly at the instant of one's birth and the other's death. This is artificial and would violate many other natural assumptions about endurance, causality, sharing a common substance, and so on. If one wishes to contemplate re-using engine blocks as garden planters, it would be better to either say that it is still an engine block even though being used as a planter, or else to characterize the property of being an engine block as a non-rigid property, thereby allowing the single metal continuant to survive its transition from engine block to mere rusty planter.
Rigid properties are often understood to be the essential characteristics of a continuant, the 'handles' by which it is classified into some taxonomy of types, sorts or species. So for example, one person might decide that 'wheelchair motor' refers to a kind of motor, and that this classification is retained throughout its existence, independently of any actual use to which it might be put. With this understanding, being a wheelchair motor is a rigid property of a continuant, but not all wheelchair motors are motors in wheelchairs. In contrast, non-rigid properties are often identified with the roles that a thing might play in some larger scheme of things. If we consider 'wheelchair motor' to mean 'motor used to power a wheelchair', then being a wheelchair motor is a role which a thing might satisfy or not at different times; so is not rigid, and hence not a characteristic description of a continuant.
Notice that rigidity of properties is not affected by the common discussions about retaining identity when parts are replaced. My truck has a battery as a part, and it is the same truck when the battery is replaced by another. Perhaps, with enough perserverance, I can replace all the parts of the truck, and it still be the same truck. None of this is in conflict with the claim that the truck is a continuant, and hence that being that truck is a rigid property. One way to describe such examples, in fact, is to think of the larger whole as an assembly of parts, each playing a role, and for the identity conditions of the whole continuant to be defined more by the roles which constitute the assembly having suitable fillers than by the particular continuants which play those roles at any given moment. So, on this view, as long as my truck has all the parts it needs to be a truck then it is still my truck, even if some parts are replaced. If you take away some of the parts, it is still my truck, but needs repair. But if you take away all the parts then it ceases to be a truck; and then at that time it ceases to exist - comes to the end of its lifetime - because being a truck is rigid, so a truck cannot become anything other than a truck.
We use the term 'continuant' simply to mean anything with an extended lifetime, whose properties may change. Continuant is often contrasted with occurrent, and sometimes this distinction is elevated into an absolute ontological distinction between metaphysically incompatible categories [ref Simons, snap/span, Dolce]. (A closely similar, perhaps identical, distinction is sometimes made using the terminology 'perdurant'/'endurant'. [ref ??] ) The claimed basis for this distinction is that continuants - roughly, physical objects - cannot have temporal parts, while occurrents - roughly, events - can, and that this necessitates a fundamentally different way of determining identity conditions. IKL ontologies are not required to take this seriously. Although distinctions like these can be made within IKL, they need not be, and no important consequences follow if this distinction is simply ignored.
Accepting this division leads to artificial ontological distinctions, such as those between a living person and that person's life, which complicate formal ontologies while providing no apparent extra utility in reasoning. Imposing this distinction as absolute and binary also makes it very hard to properly account for the many stable (hence object-like) but dynamic (hence event-like) entities - such as ocean waves, flames and thunderstorms - which can apparently be fully described only by using both perspectives. The philosophical basis for the distinction is at best controversial, but even if the distinction is admitted on philosophical grounds, this does not constrain the utility of the time-slicing technique (contrary to claims made in several 'foundational' ontology guides [ref Dolce]), since all we have to do, in order to satisfy the philosophical niceties, is to refrain from describing a time-slice of a continuant as a "temporal part". For all these reasons, IKL does not require that any particular logical or ontological distinction be made between processes and things, between continuants or occurrents, or between perdurant vs. endurant entities; or that these various categories, if present in an ontology, should be handled any differently when translating indexical logic to IKL. The various techniques described above for associating temporal interval names to assertions apply with equal correctness to anything which exists in time and lasts for a time-interval; which is all that is meant by "continuant" in the sense used here. As an example, we might use similar logical forms to describe both For the first ten minutes, the football match was boring and As a child, I was rather shy. Both of them say something about a temporal slice of a thing:
(Boring (tsl FootballMatch I))
(RatherShy (tsl PatHayes J))
where I is the ten-minute interval at the beginning of the match and J is a rather longer interval from 1949 to approximately ten years later. That the first refers to an event which happened, usually classified as an occurrent, but the second to a person who lived, usually taken to be the archetype of a continuant, may be important for other reasons – such as mappings to and from natural language – but is not, in itself, indicative of any differences in how IKL can describe them formally.
@@@@still to do@@@@
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