How we model the subject area in second-order predicates and do not notice this

Any model has limited accuracy. The more accurate the model needs to be built, the more information will have to be stored for this. If it is possible to collapse an array of data according to some of the criteria, then such a convolution can dramatically reduce the amount of stored information. However, such a convolution is not modeled by regular modeling methods, because it requires the modeling of statements simultaneously with respect to the sets of objects, and not with respect to the objects of these sets. In fact, we need a tool for modeling both first-order predicates and second-order predicates.

I will explain on the most common example. When we write that the machine was released in 1939, and was utilized in 1990, we mean that the machine existed throughout the specified period and at any time interval between the specified dates. An alternative to this statement would be to store information about all possible intervals during which the machine was recognized as existing. But all possible time intervals throughout this period, even with a discretization step per day, is a huge array of data.

Using this dataset is as inconvenient as storing it. Building queries to this data array is also inconvenient. For example, we have a record that the machine machine existed from June 12 to June 17 and was located in this period in the engine room of the HPP. But based on this record, we cannot say anything about the existence and finding of the machine from June 13 to June 15, because with this approach to modeling we need a separate corresponding record to answer this question.

In order not to store each interval separately, we reduce the volume of stored information and the time for processing requests by simulating a set of intervals. We say that in any time interval between 1939 and 1990, the machine tool existed. True, at the same time we lost information about its location, but reduced the amount of stored information and increased the speed of access to this information millions of times. Formally, this information is written as follows: for any element of the set of time intervals from 1939 to 1990, the following is true: the specified machine existed during this time interval. This is modeling in second-order predicates. We recorded our knowledge in the form of knowledge about a multitude of objects: the set of all possible time intervals from 1939 to 1990.

Is it possible to interpret the record that the machine exists from 1939 to 1990 as a statement not about a set, but as a statement about a time interval? Yes you can. But with such an interpretation it is impossible to get an answer to the question: did the machine exist from 1956 to 1958? This modeling method is not used.

We see that the same words have completely different meanings depending on their interpretation. If we write that the machine machine existed from 1939 to 1990, then these words can be interpreted in two different ways:

• as a statement regarding all intervals that can be obtained from the interval from 1939 to 1990.
• as a statement regarding the interval from 1939 to 1990.

The trouble with our language is that it is impossible to separate these two different interpretations of the same words. This makes us think that by indicating the start time and the completion time of some operation, we model the interval, and not the set of all its parts. This leads to the inconvenience we experience when designing systems. Have you noticed that working with dates in queries that we build is different from working with data of another type? Have you ever thought: why? The answer is that, by specifying a time interval, we are modeling a statement with respect to the set of intervals that can be distinguished from this interval, and not with respect to the specified interval. Since it is not possible to simulate second-order predicates in the framework of OOP or other similar modeling standards, it becomes impossible to simulate time interval classes in a regular way.

As soon as we resort to information compression, we deal with statements regarding sets, which requires us to express statements in second-order predicates, which we cannot use for modeling by standard means.

When modeling objects, when we indicate the spatial volume occupied by an object, we, just as in the case of time intervals, can express different thoughts with words alone. For example, it can be said that an object is located inside the volume indicated by us. But there are two different ways to interpret this statement. For example, there are two different ways to interpret a saying a glass of water.

• whether we are talking about the object.
• whether it is a matter.

The difference is the same as with the modeling of the time interval and multiple intervals. In the first case, we simulate the volume of space, in the second - the set of all volumes that can be distinguished from the specified volume. The words are the same, but their meaning is different. In the first case, the statement is constructed in predicates of the first order, in the second - the second. And just as we have difficulties with modeling time intervals, we have difficulties with the modeling of substances using standard methods.

The same problem occurs when we model the state of an object. If you say: with such and such for such a time the light bulb was burning, what do we want to say these? As a rule, we mean that during any period from the specified interval the light bulb is in a state of illumination. That is, what we used to call the state of an object is a set of time intervals during which the object is in a similar state, and not just a single time interval.

If we define the state of an object as the time interval during which the object retains some property, then we need to clarify what is meant not the time interval, but the set of all intervals that can be obtained from the specified interval.
An equivalent alternative to this statement would be the following: at any time from the specified range, the object had the same state. Why I do not like the second interpretation of this thesis? Because it is based on the concept of the moment, which inevitably leads us to the concept of the continuum. My interpretation is based on common sense and does not require the concept of a continuum.

Let's return to the notion of registration. I wrote earlier that by registering the state of an object can be understood:

• one-time registration of the state of the object (stream, environment)
• many registrations of similar states.

However, our consciousness is not aware of a one-time registration. When he is informed that at 12-00 the temperature of the air was 25 degrees, the consciousness can save this information, but cannot imagine. She uses experience, which says that the temperature of the air could not change quickly, consciousness offers its own version, in which the whole story is presented from some point before registration to some point after, for example, from 11-30 to 12-30. Throughout this time interval, the consciousness suggests that the air temperature be constant and equal to 25 degrees. Moreover, not just throughout this interval, but throughout any range from the specified interval! The same happens when we see an old building. We do not simply register the fact that it exists at the time of registration, we imagine its existence for a long time before and some time after. And again - not just during this time interval, but during any of the time ranges within the time interval we have presented!

When registration of a state of an object is understood as a set of registrations, our consciousness extends the hypothesis beyond the boundaries that were built on the basis of one registration. For example, if you report that at 12-00, at 13-00 and at 14-00 the air temperature was the same and equal to 25 degrees, then the limits of the time interval during which we consider the temperature constant, expand from 12-30 to 14 thirty.

We are busy obtaining new data for making predictions with the accuracy we need, building hypotheses, and reconciling reality with these hypotheses.

For this, on the one hand, we are forced to extract new evidence, and on the other, we strive to save the resources needed to store these hypotheses. At the same time, if we try to store the actual data without distortions, then we try to compress the hypotheses as much as possible.

I will give an example. Suppose that we know that Ivanov left point A at 11-00 and arrived at point B at 12-00, moving on foot. In order to imagine this movement, one can imagine a sequence of moments that will convey this movement with a sufficient degree of accuracy. Since the moments are infinite, this method of storing information requires an infinitely large amount of memory. However, as we see, the entire volume of stored information fits into several words and a link to the type of movement. This is enough to imagine any part of the path Ivanova. To do this, we choose any time interval between 11-00 and 12-00 and fill it with a replicated typical movement, which is stored in our memory. Of course, this is only a hypothesis, but it is close enough to predict the consequences we need. For example, after we meet Ivanov, we can ask him: isn’t he tired of walking so much? All information compression methods are based on generalizations of this kind.

Thus, we add to the two interpretations of the term registration a new interpretation. Under the registration of the state of the object in a certain time interval can be understood:

• one-time registration of the state of an object (stream, medium), performed during this time interval.
• a set of registrations of similar states made within this time interval.
• the set of all the ranges lying within the time interval during which the state of the object was considered the same.

If the first two interpretations can be obtained as a result of research, the third interpretation is obtained as a result of a hypothesis that we built on the basis of registration, or a series of registrations. By the way, this means that the continuum of states is also the result of a hypothesis, but not the result of observations.

As well as state registration, the state of an object during a certain time interval can also be interpreted in three different ways:

• as a result of a one-time registration made during this time interval.
• as a result of multiple registrations of similar states made within this time interval.
• as a result of the hypothesis that the state of the object in all ranges lying within this time interval is the same.

When modeling the first thesis, we refer from the time interval to the type of states. When modeling the second thesis, we refer from the time intervals to the type of states. When modeling the third thesis, we refer from the time interval to the type of states, but the meaning of this link is different than in the first case.