Anchor Modeling

What I wish source systems would tell us and they hardly ever do. Best laid out as an example, look at this data:

𝟺𝟻𝟽𝟾𝟸𝟷, 𝟹 𝟶𝟶𝟶, 𝟸𝟶𝟸𝟶-𝟶𝟿-𝟸𝟶

This alone does not tell us much, so along with this we need context, commonly in the form of column names:

𝙲𝚄𝚂𝚃𝙾𝙼𝙴𝚁 𝙽𝚄𝙼𝙱𝙴𝚁, 𝙱𝙰𝙻𝙰𝙽𝙲𝙴, 𝚃𝙸𝙼𝙴𝚂𝚃𝙰𝙼𝙿

Fine, this is usually all we get. Now, let’s shake things up a bit by introducing a second line of data. Now we have:

𝟺𝟻𝟽𝟾𝟸𝟷, 𝟷𝟼 𝟶𝟶𝟶, 𝟸𝟶𝟸𝟶-𝟶𝟿-𝟸𝟶
𝟺𝟻𝟽𝟾𝟸𝟷, 𝟹 𝟶𝟶𝟶, 𝟸𝟶𝟸𝟶-𝟶𝟿-𝟸𝟶

Confusing, but this happens. Is the timestamp not granular enough and these were actually in succession? Is one a correction of the other? Can customers have different accounts and we are missing the account number?

Even if you can get all that sorted out, we can shake it up further. Put this in a different context:

𝙿𝙰𝚃𝙸𝙴𝙽𝚃 𝙽𝚄𝙼𝙱𝙴𝚁, 𝚁𝙰𝙳𝙸𝙰𝚃𝙸𝙾𝙽 𝙳𝙾𝚂𝙴, 𝚃𝙸𝙼𝙴𝚂𝚃𝙰𝙼𝙿

Now I feel the need to know more. Are these measurements made by different persons and how certain are they? What is the margin of error? If these were in succession, what were their durations? If only one of them is correct, which one is it?

More sources should communicate data as if it was a matter of life and death. This is what Transitional modeling is all about.

This is a continuation of the articles “She wore a blue dress” and “Rescuing the Excluded Middle“, which introduced crisp imprecision and fuzzy uncertainty. The former being evaluative and the latter both subjective and contextual. The articles discuss, relate, and sometimes further the formalization of transitional modeling, so they are best read with some previous knowledge of this technique. An introduction can be found starting with the article “What needs to be agreed upon” or by reading the scientific paper “Modeling Conflicting, Unreliable, and Varying Information“. In this article I will discuss the effect of a chosen language upon the modeling of posits, with particular homage to the new riddle of induction and Goodman’s predicate ‘grue’.

In order to look at the intricacies of using language to convey information about the real world, we will focus on the statement “She’ll wear a grue dress”. First, this refers to a future event, as opposed to the previously investigated statement “She wore a blue dress”, which obviously happened in the past. There are no issues talking about future events in transitional modeling. Let’s say Donna is holding the dress and is just about to put it on. She would then, with absolute certainty, assert the posit “She’ll wear a grue dress”. It may be the case that the longer time before the dress will be put on, the less certain Donna will be, but not necessarily. If she just after New Year’s Eve is thinking of what to wear at the next, she could still be certain. Donna could have made it a tradition to always wear the same dress.

There is a difference between certainty and probability. If Donna is certain she will wear that dress at the next New Year’s Eve, she is saying her decision has already been made to wear it, should nothing prevent her from doing so. From a probabilistic viewpoint, lots of things can happen between now and New Year preventing that from ever happening. The probability that she will wear the dress at next New Year’s Eve is therefore always less than 1, and will be so for any prediction. Assuming the probability could be determined, it would also be objective. Everyone should be able to come up with the same number. Bella, on the other hand, could be certain that Donna will not wear the dress at the next New Year’s Eve, since she intends to ruin Donna’s moment by destroying the dress. Certainty is subjective and circumstantial. I believe this distinction between certainty and probability is widely overlooked and the concepts confused. “Are you certain? Yes. Is it probable? No” is a completely valid and non-contradictory situation.

With no problems of talking about future events, let’s turn our attention to ‘grue’. Make note of the fact that you would not have reacted in the same way if the statement had been “She’ll wear a blue dress”, unless you happen to be among the minority already familiar with the color grue. If you belong to that minority, having studied philosophy perhaps, then forget for a minute what you know about grue. I will look at the word ‘grue’ from a number of different possibilities, of only the last will be Goodman’s grue.

What is grue?

1. It is a color universally and objectively distinguishable from blue.
2. It is a color selectively and subjectively indistinguishable from blue.
3. It is a synonym of blue.
4. It is an at the current time widely known color.
5. It is an at the current time little known color.
6. It is an at the current time unknown color that will become known.
7. It is an at the current time known color synonymous with blue that at some point in the future will be considered different from blue (Goodman).

In (1) there will likely be no issues whatsoever. Perhaps there is a scientific definition of ‘grue’ as a range of wavelengths in between green and blue. On a side note and right now, the color greige is quite popular and a mix between grey and beige. Using that definition of ‘grue’ anyone should be able to reach the same conclusion whether an actual color can be said to be grue or not. Of course most of us do not possess spectrophotometers or colorimeters, so we will judge the similarity on sight. If enough reach the same conclusion, we may say it’s as close to an objectively determinable color as we will get. This is good, and not much thought has to go into using >grue< in a posit.

In (2) there may be potential issues. Perhaps grue and blue become indistinguishable under certain conditions, such as lighting, or let’s assume that 50% of the population cannot distinguish between grue and blue because of color blindness. Given two otherwise identical dresses of actual different colors, grue and blue, they may assert that she wore or will wear both of these, simultaneously. Such assertions can be made in transitional modeling and possible contradictions found using a formula over sums of certainty (see the scientific paper). To resolve this, non-contradiction either needs to be enforced at write time or periodically analyzed. Unknown types of color blindness could even be discovered this way, through statistically significant contradictory opinions. That being said, one should document already known facts and new findings with respect to effects that may disturb the objectivity of the values used.

In (3) there is a choice or a need for documentation. Either one of ‘blue’ and ‘grue’ is chosen and used consistently as the value or both are used but the fact that they are synonymous is documented. This may be a more common situation than one first may think, since ‘grue’ could be the word for ‘blue’ in a different language. This then raises the question of synonymy. What if there are language-specific differences between the interpretations of ‘grue’ and ‘blue’, so that they nearly but not entirely overlap? If grue allows a bit more bluegreenish tones than blue then they are only close to synonymous. This speaks for keeping values as they were stated, but that values themselves then may need their own model.

With those out of the way, let us look at how well known of a color grue is. In (4) almost everyone has heard of and use grue when describing that color. This is good, both those who are about to assert a posit containing >grue< will know how to evaluate it, and those later consuming information stored in posits will understand what grue is. With (5) difficulties may arise. In the extreme, I have invented the word ‘grue’ myself and nobody else knows about it. However, when interrogated by the police to describe the dress of the woman I saw at the scene of the crime, I insist on it being grue. No other color comes close to the one I actually saw. Rare values, like these, that likely can be explained in more common terms need translation. If done prescriptively the original statement is lost, but if not, it must be done descriptively at the cost of the one consuming posits first digesting translation logic. This is a very common scenario when reading information from some system, in which you almost inevitably find their own coding schemes, like “CR”, “LF”, “TX”, and “RX” turning out to have elaborate meanings.

Now (6) may at first glance seem impossible, but it is not. Let us assume that we believe the dress is blue and the posit temporally more qualified to “She’ll wear a blue dress on the evening of December 31st 2020”. Donna asserts this with 100% certainty the day after the preceding New Year’s Eve. When looking at the dress on December 31st 2020, Donna has learnt that there is a new color named grue, and there is nothing more fitting to describe this dress. Given this new knowledge, that the dress is and always has been grue, she retracts her previous posit, produce a new posit, and asserts this new one instead. The process can be schematically described as:

posit_1     = She'll wear a blue dress on the evening of December 31st 2020

assertion_1 = Donna, posit_1, 100% certainty, sometime on January 1st 2020

assertion_2 = Donna, posit_1, 0% certainty, earlier on December 31st 2020

posit_2     = She'll wear a grue dress on the evening of December 31st 2020

assertion_3 = Donna, posit_2, 100% certainty, earlier on December 31st 2020

Given new knowledge, you may need to correct yourself. This is precisely how corrections are managed in transitional modeling, in a bi-temporal solution, where it is possible to deduce who knew what when. This works for rewriting history as well:

posit_3     = The dress is blue since it was made on August 20th 2018

assertion_4 = Donna, posit_3, 100% certainty, sometime on August 20th 2018

assertion_5 = Donna, posit_3, 0% certainty, earlier on December 31st 2020

posit_4     = The dress is grue since it was made on August 20th 2018

assertion_6 = Donna, posit_4, 100% certainty, earlier on December 31st 2020

The dress is and always has been grue, even if grue was unheard of as a color in 2018. Nowhere do the posits and assertions indicate when grue started to be used though. This would, again be a documentation detail or alternatively warrant explicit modeling of values.

Finally there is (7), in which there is a point in time, t, before which we believe everything blue to be grue and vice versa. Due to some new knowledge, say some yet to be discovered quantum property of light, those things are now split into either blue or grue to some proportions. This is really troublesome. If some asserters were certain “She wore a blue dress” and others were certain “She wore a grue dress”, in assertions made before t, that was not a problem. They were all correct. After that point in time, though, there is no way of knowing if the dress was actually blue or grue from those assertions alone. If we are lucky enough to get hold of the dress and figure out it is blue, things start to look up a bit. We would know which asserters were wrong. Their assertions could be invalidated, while we make new ones in their place. In the less fortunate event that the dress is nowhere to be found, previous assertions could perhaps be downgraded to certainties in accordance with the discovered proportions of blue versus grue.

The overarching issue here, which Goodman eloquently points out, is that this really messes up our ability to infer conclusions from inductive reasoning. How do we know if we are in a blue-is-grue situation soon to become a blue-versus-grue nightmare? To me, the problem seems to be a linguistic one. If blue and grue have been used arbitrarily before t, but after t signify a meaningful difference between measurable properties, then reusing blue and grue is a poor choice. If, on the other hand, blue and grue were actually onto something all along, then this measurable property must have been present and in some way sensed, and many assertions likely to be valid nevertheless. This reasoning is along the lines of philosopher Mark Sainsbury, who stated that:

A generalization that all A’s are B’s is confirmed by instances unless we have good reason to believe that there is some property, O, such that every A-instance is O, and if those A-instances had not been O, they would not have been B.

In other words, some additional property is always hiding behind issue number (7).

With all that said, there are a lot of subtleties concerning values, but most, if not all of them can be sorted out using posits and assertions, with the optional addition of an explicit model of values, together with prescriptive or descriptive measures. That being said, if language is used with proper care and with the seven types of ‘grue’ mentioned above in mind, you will likely save yourself a lot of headaches. We also learnt that people normally think in certainties rather than probabilities.

This is a continuation of “She wore a blue dress“, in which we introduced to the concepts of imprecision and uncertainty. I will now turn the focus back on the imprecise value ‘blue’ and make that imprecision a bit more formal. In the works of Brouwer related to intuitionism an imprecise value can be thought of as a mapping. I will introduce the notation >blue< for such a mapping of the imprecise value ‘blue’. The mapping >blue< would then be:

>blue< : x ⟶ [0,1]

In other words, for any color x it evaluates to either 1 for it being fully considered as blue or 0 if it cannot be considered blue. However, according to Brouwer any value in between is also allowed. It could be 0.5 for half blue, which is also known as a fuzzy impecise value. Allowing these will confuse the with imprecision codependent concept of uncertainty. I will therefore restrict imprecise values, such as blue to:

>blue< : x ⟶ {true, false}

The reasoning is that subjectivity enters already in the evaluation of this mapping. In the terminology of transitional modeling, it is when asserting the statement “She wore a blue dress” that the asserter evaluates the actual color x of the dress against the value ‘blue’. As such, the posit will be crisp from the asserter’s point of view. Given that the dress was acceptably ‘blue’ enough, the asserter can determine their certainty towards the posit. Values can therefore be said to be crisp imprecise values, but only relative a subject.

If we assume that the occasion when she wore a dress took place on the 1st of April 2020 and this is used as the appearance time in the posit, then it is also an imprecise value. Most of us will take this as the precise interval from midnight to midnight on the following day. At some point in that crisp interval, the dress was put on. Even so, putting on a dress is not an instantaneous event and time cannot be measured with infinite precision, so regardless of how precisely that time is presented, appearance time will remain imprecise.

With finer detail, the appearance time could, for example have been expressed as at two minutes to midnight on the 1st of April 2020. But, here we start to see the fallacy of taking some time range for granted though. With the same reasoning as before we would assume that to refer to the interval between two minutes and one minute to midnight. However, there is no way of knowing that a subject will always interpret it this way. So, we need the mapping once again:

>two minutes to midnight on the 1st of April 2020< : x ⟶ {true, false}

It seems as if the evaluation of this mapping is not only subjective, but also contextual. If we know that it could have taken more than a minute to put on the dress in question, then maybe this allows for both tree and one minute to midnight evaluating to true. Even when such a range is possible to specify it is almost never available in the information we consume, so we often have to deal with evaluations like these. We have, however, become so used to evaluating the imprecision that we do so more or less subconsciously.

But, didn’t we lose a whole field of applicability in the restriction of Brouwer’s mapping? That fuzziness is actually not all lost. I believe that what assertions do in transitional modeling is to fill that gap, while paying respect to subjectivity and contextuality. It is not possible to capture the exact reasoning behind the assertion, but we can at least capture its result. Recall that an assertion is someone expressing a degree of certainty towards a posit, here exemplified by “She wore a blue dress”. An example of an assertion is: “Archie thinks it likely that she wore a blue dress”. With time involved this becomes: “On the 2nd of April Archie thinks it likely that she wore a blue dress two minutes to midnight on the 1st of April”. Even more precisely and closer to a formal assertion: “Since the >2nd of April< the value >likely< appears for (Archie, certainty) in relation to ‘since the >1st of April< the value >blue< appears for (she, dress color)'”.

As can be seen, assertions can themselves be formulated as posits. Given the example assertion, it’s value is also imprecise, with a mapping:

>likely< : x ⟶ {true, false}

We have however, in transitional modeling, decided that certainty is better expressed using a numerical value. Certainty is taken from the range [-1, 1], with 1 being 100% certain, -1 being 100% certain of the opposite, and 0 for complete uncertainty. Certainties in between represent beliefs to some degree. We have to ask Archie, when you say ‘likely’, how certain is that given as a percentage? Let’s assume it is 80%. That means the corresponding mapping becomes:

>0.8< : x ⟶ {true, false}

Certainty is just another crisp imprecise value, but relative a subject who has performed a contextual evaluation of the imprecise values present in a posit with the purpose of judging their certainty towards it. An asserter (the subject) made an assertion (the evaluation and judgement), in transitional modeling terminology.

The interesting aspect of crisp imprecise values are that they respect “tertium non datur”, which is Latin for “no third is given”, more commonly known as the law of the excluded middle. In propositional logic it can be written as (P ∨ ¬P), basically saying that no statement can be both true and not true. An asserter making an assertion, evaluating whether the actual color of the dress can be said to be blue, obeys this law. It can either be said to be blue or it cannot. This law does not hold for fuzzy imprecise values. If something can be half blue, then neither “the dress was blue” nor “the dress was not blue” is fully true.

Fuzziness is not lost in transitional modeling though. Since certainty is expressed in the interval [-1, 1], it encompasses that of fuzzy values. The difference is that fuzziness comes from uncertainty and not from imprecision. Uncertainty is subjective and contextual, whereas fuzzy imprecise values are assumed objective and universal. I believe that this makes for a richer and truer to life, albeit more complex, foundation. It also rescues the excluded middle. Statements are either true or false with respect to crispness, but it is possible to express subjective doubt. Thanks to the subjectivity of doubt, contradicting opinions can be expressed, but that is the story of my previous articles, starting with “What needs to be agreed upon“.

As a consequence of the reasoning above, a posit is open for evaluation with respect to its imprecisions. Such imprecisions are evaluated in the act of performing an assertion, but an assertion is also a posit. In other words, the assertion is open for evaluation with respect to its imprecisions (the >certainty< and >since when< this certainty was stated). This can be remedied by someone asserting the assertion, but then those assertions will remain open, so someone has to assert the new assertions asserting the first assertions. But then those remain open, so someone has to assert the third level assertions asserting the second level assertions asserting the first level assertions, and so on…

Rather than having “turtles all the way down“, in transitional modeling there are posits all the way down, but for practical purposes it’s likely impossible to capture more than a few levels. The law of the excluded middle holds, within a posit and even if imprecise, but only in the light of subjective asserters performing contextual evaluations resulting in their judgments of certainty. To some extent, the excluded middle has been rescued!

Since we have started to recognize “keys” in our information modeling tool (from version 0.99.4) I will have this timely discussion on identification and identity. Looking at my previously published articles and papers, I have repeatedly stated that identification is a search process by which circumstances are matched against available data, ending in one of two outcomes: an identity is established or not. What these circumstances are and which available data you have may vary wildly, even if the intent of the search is the same. Think of a detective who needs to find the perpetrator of a crime. There may have been strange blotches of a blue substance at the crime scene, but no available register to match blue blotches of unknown origin to. We have circumstances but little available data, yet often detectives put someone behind bars nevertheless.

On the other hand, think of a data integrator working with a data warehouse. The circumstance is a customer number and you have a neat and tidy Customer concept with all available data in your data warehouse. The difference to the detective is the closeness of agreement between different runs of the identification process. The process will look very much the same for the next customer number, and the next, and the next. So much so that the circumstance itself may warrant its own classification, namely being a “key” circumstance. In other words, a “key” is when circumstances exist that every time produce an identical search process against well defined and readily available data. As such, a “key” does not in any way imply that it is the only way to identify something, that it is independent of which time frame you are looking at it, or that it cannot be replaced at some point.

These are the reasons why, in Anchor and Transitional modeling, no importance has been given to keys. Keys cannot affect a model, because if they did, the model itself would reflect a single point of view, be bound to a time frame, and run the risk of becoming obsolete. That being said, if a process is close to perfectly reproducible, it would be stupid not to take advantage of that fact and help automate it. This is where the concept of a “key” is useful, even in Anchor and Transitional modeling, which is why we are now adding it as an informational visualization with the intent of also creating some convenient functionality surrounding them. Even so, regardless of which keys you add to the model, the model is always unaffected by these, precisely for the reasons discussed above.

I hope this clarifies my stance on keys. They are convenient for automation purposes, since they help the identification process, but shall never affect the model upon which they work in any way.

Visualization and editing of keys has been added in version 0.99.4 (test) of the free online Anchor modeling tool. This is so far only for informational purposes, but is of great help when creating your own automation scripts. Note that a key in an Anchor model behaves like a bus route, stopping on certain items in the graph. In order to create a key, select an anchor and at least one attribute (shift-clicking lets you do multiple select). To edit a created key, click on its grey route to highlight it red. You can then add or remove items or change it’s name. Click again to leave key editing mode. Along with this come some improvements to the metadata views in the database, and among them the new _Key view.

As you intuitively know, there is only one time. Yet in the domain of information modeling we speak of “valid time”, “transaction time”, “user defined time”, “system time”, “application time”, “happening time”, “changing time”, “speech act time”, “inscription time”, “appearance time”, “assertion time”, “decision time” and so on, as not being the same. In fact, I will boldly say that the only one of these coming close to true time is happening time, defined in Anchor modeling as ‘the moment or interval at which an event took place’, even if it goes on to define other types of time. However, if we assume that only happening times exist, then all other types of time should be able to be represented on the form:

[Event, Timepoint].

In Transitional modeling we have the concept of a posit, with its appearance time defined as ‘the time when some value can be said to have appeared (or will appear) for some thing or a collection of things’. However, it also has assertion time, defined as ‘the time when someone is expressing an opinion about their certainty toward a posit’. To exemplify, “Archie and Bella will be ‘married’ on the 1st of April” and “Charlie is expressing that he is almost certain of this on the 31st of March”. In this case the 1st of April is the appearance time and 31st of March the assertion time.

Given the previous assumption on how to represent everything as events and time points, we can rewrite the previous example as [The value ‘married’ appears for Archie and Bella, 1st of April] and [The value ‘almost certain’ is given to a posit by Charlie, 31st of March]. Now they are on the same form, indicating that there is indeed only one true time. Why, then, do we feel the need to distinguish between appearance and assertion time? Why not just have a single “event time”? Well, as it turns out, there is a crucial difference between the two, but it has less to do with time and more to do with the actual events taking place.

Some events are temporally orthogonal to each other. Charlie can change his mind about how certain he is of the posit, independently of the posit itself. The posit “Archie and Bella will be ‘married’ on the 1st of April” remains the same, even if Charlie changes his mind and [The value ‘quite uncertain’ is given to a posit by Charlie, 1st of April]. Maybe Charlie realized that he may have been the subject of an elaborate prank, strengthened by the fact that he was only given one day’s notice of the wedding and that pranks are quite frequent on this particular day. To summarize, for one given point in appearance time there are now two points in assertion time. This means assertion time runs orthogonal to appearance time. They are two different “dimensions” of time.

But, really, there is only one time. We choose to view these as two dimensions, not because there are different times, but because there are different types of events. Plotting these on a plane just makes it easier for us to illustrate this fact. In Transitional modeling, assertion time is not only orthogonal to appearance time, it is also relative the one making the assertion. Let’s introduce Donna and [The value ‘absolute prank’ is given to a posit by Donna, 31st of March]. In other words, Donna knew from the start that the wedding was a prank. Now, for one given point in appearance time there are three points in assertion time, two belonging to Charlie and one belonging to Donna, with one also coinciding. Even so, there is only one time, but different and subjective events.

If these temporally orthogonal events are abundant, even the list of types of time presented in the beginning of the article may seem few. The problem is that we are used to seeing only one objective assertion time coinciding with the appearance time. Looking at the representation on a (helpfully constructed) plane, this would be the 45 degree line on which a (helpfully positioned) bitemporal timepoint has the same value for both its coordinates: (tx, ty) with tx = ty. Most information, likely wrongfully, is represented on this line. We have lost the nuance of distinguishing between the actual information and the opinion of the one stating it. This makes it easy to fall into the trap thinking that there is no need to distinguish between orthogonal events. I have, unfortunately, seen many a database in which attempts have been made to crush orthogonal events into a single column with less than desirable results and the negative impact discovered in irrecoverable retrospect.

I believe that every new modeling technique, and any modeler dealing with time in existing techniques, must decide on which events it wants to recognize as important and if they are temporally orthogonal or not. If not, they will never be able to represent information close to how information behaves in reality. Orthogonal events will need different timelines and different timelines need to be managed separately, such as being stored in different columns or tables in a database. I think there are many orthogonal events of interest, some quite generally applicable and some very specific to certain use cases. While we could get away with a single “event time” we often choose not to. The reasoning is that by making orthogonal events integral to a modeling technique allows for it to provide their theory, stringency, consistency and optimization.

Recognizing orthogonal events can therefore be a smart move. The events of interest in Transitional modeling is “the appearance of a value” and “having an opinion”. The events of interest in the works of Richard T. Snodgrass are “making a database fact valid in the modeled reality”, “making a database fact true in the database”. The events of interest in the works of Tom Johnston are “entering a certain state”, “utterances about enterings of states”, and “the inscription of utterances”, and so goes on for all modeling techniques. We have all probably added to the confusion, but if we can start to recognize a common ground and that we only slightly differ in the events we recognize, this terminological mess can be untangled. With all the notions afloat, the question that begs answers is what events you recognize and if any of them are subjective? Feel free to share in the comments below!

I do think it’s time for all of us to abide by the thought of one true time, dissociated by temporally orthogonal events, and be careful when using the misleading ‘dimensions of time’ notion.

This is an article about imprecision and uncertainty, two in general poorly understood and often mixed up concepts. It’s also about information, which I will define as saying something about something else¹. Information is the medium we use to convey and invoke a sense of that else; sharing our perception of it. The funny thing is, when we say something about something else, many things about the else will always get lost in translation. Information is, therefore, always imprecise and uncertain to some degree. What is perplexing, and less funny, is how we often tend to forget this and treat information as facts.

I think we have a desire to believe that information is precise and certain. The stronger the desire, the greater the willingness to interpret it as facts. Take Günther Schabowski as an example. When he, although uncertain, quite precisely stated that “As far as I know [the new regulations are] effective immediately, without delay.” Those new regulations were intended to be temporary travel regulations with relaxed requirements, limited to a select number of East Germans. This later on the same day led to the fall of the Berlin wall and eventually contributed to the end of the cold war, if we are to believe Wikipedia. Even small words from the right mouths can have large consequences.

Now, in order to get a better understanding of imprecision and uncertainty, let us look at the statement 𝕊𝕙𝕖 𝕨𝕠𝕣𝕖 𝕒 𝕓𝕝𝕦𝕖 𝕕𝕣𝕖𝕤𝕤 in conjunction with the following photo.

First, we assume that whoever 𝕊𝕙𝕖 is referring to is agreed upon by everyone reading the statement. Let’s say it’s the woman in the center with the halterneck dress. Then 𝕨𝕠𝕣𝕖 is in the preterite tense, indicating that the occasion on which she wore the dress has come to pass. In its current form, this is highly imprecise, since all we can deduce is that it has happened, sometime in the past.

Her dress looks 𝕓𝕝𝕦𝕖, but so do many of the other dresses. If they are also 𝕓𝕝𝕦𝕖 we must conclude that 𝕓𝕝𝕦𝕖 is imprecise enough to cover different variations. One may also ask if her dress will remain the same colour forever? I am probably not the only one to have found a disastrous red sock in the (once) white wash. No, the imprecise colour 𝕓𝕝𝕦𝕖 is bound to that imprecise moment the statement is referring to. To make things worse, no piece of clothing is perfectly evenly coloured, but this dress is at least in general 𝕓𝕝𝕦𝕖.

Finally, it’s a 𝕕𝕣𝕖𝕤𝕤, but there are an infinite number of ways to make a 𝕕𝕣𝕖𝕤𝕤. Regardless of how well the manufacturing runs, no two dresses come out exactly the same. The 𝕕𝕣𝕖𝕤𝕤 she wore is a unique instance, but then it also wears and tears. Maybe she has taken it to a tailor since, and it is now a completely different type of garment. In other words, what it means to be a 𝕕𝕣𝕖𝕤𝕤 is imprecise and what the 𝕕𝕣𝕖𝕤𝕤 actually looked like is imprecisely bound in time by the statement.

In fact, 𝕊𝕙𝕖 𝕨𝕠𝕣𝕖 𝕒 𝕓𝕝𝕦𝕖 𝕕𝕣𝕖𝕤𝕤 would have worked just as well in conjunction with any of the women in the photo². Me picking one for the sake of argument had you focusing on her, but in reality, the statement is so imprecise it could apply just as well to anyone. Imprecise information is such that it applies to a range of things. 𝕊𝕙𝕖 ranges over all females, 𝕨𝕠𝕣𝕖 ranges from now into the past, 𝕓𝕝𝕦𝕖 ranges over a spectrum of colours, 𝕕𝕣𝕖𝕤𝕤 ranges over a plethora of garments. 𝕊𝕙𝕖 𝕨𝕠𝕣𝕖 𝕒 𝕓𝕝𝕦𝕖 𝕕𝕣𝕖𝕤𝕤, taken combined increases the precision, since not every woman in the world has worn a blue dress. Together with context, such as the photo, the precision can even be drastically increased.

With a better understanding of imprecision, let’s look at the statement anew and how: 𝗔𝗿𝗰𝗵𝗶𝗲 𝘁𝗵𝗶𝗻𝗸𝘀 𝕊𝕙𝕖 𝕨𝕠𝕣𝕖 𝕒 𝕓𝕝𝕦𝕖 𝕕𝕣𝕖𝕤𝕤. Regardless of its imprecision, 𝗔𝗿𝗰𝗵𝗶𝗲 is not certain that the statement is true. The word 𝘁𝗵𝗶𝗻𝗸𝘀 quantifies his uncertainty, which is less sure than 𝗰𝗲𝗿𝘁𝗮𝗶𝗻, as in: 𝗗𝗼𝗻𝗻𝗮 𝗶𝘀 𝗰𝗲𝗿𝘁𝗮𝗶𝗻 𝕊𝕙𝕖 𝕨𝕠𝕣𝕖 𝕒 𝕓𝕝𝕦𝕖 𝕕𝕣𝕖𝕤𝕤. Maybe 𝗗𝗼𝗻𝗻𝗮 wore the dress herself, which is why her opinion is different. Actually, 𝗔𝗿𝗰𝗵𝗶𝗲 𝘁𝗵𝗶𝗻𝗸𝘀 𝕊𝕙𝕖 𝕨𝕠𝕣𝕖 𝕒 𝕓𝕝𝕦𝕖 𝕕𝕣𝕖𝕤𝕤, 𝗯𝘂𝘁 𝗶𝘁 𝗺𝗮𝘆 𝗵𝗮𝘃𝗲 𝗯𝗲𝗲𝗻 𝘁𝗵𝗲 𝗰𝗮𝘀𝗲 𝘁𝗵𝗮𝘁 𝕊𝕙𝕖 𝕨𝕠𝕣𝕖 𝕒 𝕡𝕚𝕟𝕜 𝕕𝕣𝕖𝕤𝕤. From this, we can see that uncertainty is both subjective and relative a particular statement, since 𝗔𝗿𝗰𝗵𝗶𝗲 now has opinions about two possible, but mutually exclusive, statements. These are, however, only mutually exclusive if we assume that he is talking about the same occasion, which we cannot know for sure.

Somewhat more formally, uncertainty consists of subjective probabilistic opinions about imprecise statements. Paradoxically, increasing the precision may make someone less certain, such as in: 𝗔𝗿𝗰𝗵𝗶𝗲 𝗶𝘀 𝗻𝗼𝘁 𝘀𝗼 𝘀𝘂𝗿𝗲 𝘁𝗵𝗮𝘁 𝔻𝕠𝕟𝕟𝕒 𝕨𝕠𝕣𝕖 𝕒 𝕟𝕒𝕧𝕪 𝕓𝕝𝕦𝕖 𝕙𝕒𝕝𝕥𝕖𝕣𝕟𝕖𝕔𝕜 𝕕𝕣𝕖𝕤𝕤 𝕥𝕠 𝕙𝕖𝕣 𝕡𝕣𝕠𝕞. This hints that there may be a need for some imprecision in order to maintain an acceptable level of certainty towards the statements we make. It is almost as if this is an information theoretical analog to the uncertainty principle in quantum mechanics.

But is this important? Well, let me tell you that there are a number of companies out there that claim to use statistical methods, machine learning, or some other fancy artificial intelligence³, in order to provide you with must-have business-leading thingamajigs. Trust me that a large portion of them are selling you the production of 𝕊𝕙𝕖 𝕨𝕠𝕣𝕖 𝕒 𝕓𝕝𝕦𝕖 𝕕𝕣𝕖𝕤𝕤-type of statements rather than fact-machines. Imprecise results, towards which uncertainty can be held. Such companies fall into four categories:

• Those that do not know they aren’t selling facts.
  [stupid]
• Those that know they aren’t selling facts, but say they do anyway.
  [deceptive]
• Those that say they aren’t selling facts, but cannot say why.
  [honest]
• Those that say they aren’t selling facts, and tell you exactly why.
  [smart]

Unfortunately I’ve met very few smart companies. Thankfully, there are some honest companies, but there is also an abundance of stupid and deceptive companies. Next time, put them to the test. Never buy anything that doesn’t come with a specified margin of error, a confusion matrix, or some other measure indicating the imprecision. If the thingamajig is predicting something, make sure it tells you how certain it is of those predictions, then evaluate these against actual outcomes and form your own opinion as well.

Above all, do not take information for granted. Always apply critical thinking and evaluate its imprecision and the certainty with which and by whom it is stated.

¹ 𝘐𝘯𝘧𝘰𝘳𝘮𝘢𝘵𝘪𝘰𝘯 𝘵𝘩𝘢𝘵 𝘵𝘢𝘭𝘬𝘴 𝘢𝘣𝘰𝘶𝘵 𝘪𝘵𝘴𝘦𝘭𝘧 𝘪𝘴 𝘶𝘴𝘶𝘢𝘭𝘭𝘺 𝘤𝘢𝘭𝘭𝘦𝘥 𝘮𝘦𝘵𝘢-𝘪𝘯𝘧𝘰𝘳𝘮𝘢𝘵𝘪𝘰𝘯.

² 𝘈𝘵 𝘭𝘦𝘢𝘴𝘵 𝘧𝘰𝘳 𝘴𝘰𝘮𝘦𝘰𝘯𝘦 𝘸𝘪𝘵𝘩 𝘮𝘺 𝘭𝘦𝘷𝘦𝘭 𝘰𝘧 𝘬𝘯𝘰𝘸𝘭𝘦𝘥𝘨𝘦 𝘢𝘣𝘰𝘶𝘵 𝘨𝘢𝘳𝘮𝘦𝘯𝘵𝘴.

³ 𝘙𝘰𝘣𝘣𝘦𝘥 𝘰𝘧 𝘪𝘵𝘴 𝘰𝘳𝘪𝘨𝘪𝘯𝘢𝘭 𝘮𝘦𝘢𝘯𝘪𝘯𝘨, 𝘴𝘪𝘯𝘤𝘦 𝘸𝘦 𝘢𝘳𝘦 𝘧𝘢𝘳 𝘧𝘳𝘰𝘮 𝘩𝘢𝘷𝘪𝘯𝘨 𝘤𝘰𝘯𝘴𝘤𝘪𝘰𝘶𝘴 𝘮𝘢𝘤𝘩𝘪𝘯𝘦𝘴.