506: Rules of Structured Descriptions

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The description of rules of structured descriptions

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Topics

About: notation

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Motive for Structured Descriptions
• 2: The Rules of Structured Descriptions of Propositions
• 3: The Rules of Structured Descriptions of Definitions
• 4: It is Work in Progress

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Starting Context

• Nothing

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Target Context

• The reader will have the description of the rules of structured descriptions.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Motive for Structured Descriptions

When we read Description (or Natural Language Description) of a proposition in this series, we often feel the format not optimal for comprehending the contents as promptly as possible, and we began to think that there should be a format that is more efficient for reading.

The most important point is that descriptions are expressed in the same structure, then, we will know where to look at for what information.

Certainly, some people will prefer different structures than ours, but even if they dislike our structure, still, they will know where to look at for what information.

An idea is that all the entities in the proposition are listed in the 1st block and the statements are listed in the 2nd block. Then, we will easily comprehend what entities are involved in the proposition.

Each block should be visually kind. For example, the entities should be listed line by line, which will help the reader to grasp the entities promptly or to look up an entity while looking at the statements; any 2 statements should not be in the same line and the logical connection ('$\wedge$' or '$\vee$') between them should be a line by itself, and the supposition part and the conclusion part of each statement should not be in the same line and the implication direction mark ('$⟹$' or '$⟺$') between them should be a line by itself, which will help the reader to grasp the logical construction of the statements promptly.

Another idea is that notations are concise and consistent. For example, should we write "$r\in \mathbb{R}$" or "$r$ is any real number"? We will take the former. In fact, "~ is any ~" should be able to be expressed as $\text{ ~ }\in \{\text{ the ~ s }\}$. Certainly, when the collection, $\{\text{ the ~ s }\}$, is like $\mathbb{R}$, it is concise, but what if the collection does not have such a standard notation? Well, we will nevertheless adopt an expression like $T\in \{\text{ the topological spaces }\}$ for the sake of consistency (consistency over conciseness). Besides, the expression, $\text{ ~ }\in \{\text{ the ~ s }\}$, seems to have the benefit of forcing the author to be more precise: for example, one is often tempted to write "$U$ is any open subset." but feels to be compelled to write $U\in \{\text{ the open subsets of }T\}$: sometimes, the space on which the subset is open becomes unambiguous.

Although at first one was thinking of only Structured Descriptions of propositions, but one has extended it also to Structured Descriptions of definitions. Certainly, many simple definitions do not particularly necessitate Structured Descriptions, but some complicated definitions could use it.

Structured Description is logically complete, which means that no information is lacking compared with Natural Language Description.

So, Structured Description is self-sufficient without Natural Language Description, and while we may show both for a little while, Natural Language Description will be dismissed eventually, probably.

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2: The Rules of Structured Descriptions of Propositions

As the 1st rule, the author has the discretion to break some rules in order to realize better readability, because the whole purpose is better readability but not observing the rules.

This is an example.

Entities:
$T^{\prime }$: $\in \{\text{ the topological spaces }\}$
$A$: $\in \{\text{ the possibly uncountable index sets }\}$
$B^{\prime }$: $\in \{\text{ the bases of }{T}^{\prime }\}$, $=\{B_{\alpha }^{\prime }\subseteq T^{\prime }|\alpha \in A\}$
$T$: $\subseteq T^{\prime }$, with the subspace topology
$B$: $\{B_{\alpha }^{\prime }\cap T|\alpha \in A\}$
//

Statements:
$B\in \{\text{ the bases of }T\}$.
//

The entities are listed line by line.

Sometimes, one wonders whether one should list an entity in the Entities block: for example, should $A$ above be listed independently or can it be just mentioned in the line for $B^{\prime }$? Well, that depends on the case: one has listed it independently, because it is used also for $B$. And should an entity like the topology of $T^{\prime }$ not be listed explicitly? Well, that depends on whether it is explicitly mentioned in Proof, and even if it is, one may choose to mention it in the $T^{\prime }$ line instead of listing it independently. Although one does not perceive any necessity to put any 2 major entities into a line, there can be some subtleties in real cases and the 1st rule applies there.

"$\in \{\text{ the ~ s }\}$" is used instead of "is any ~". Note that that means "any ~" but not "a ~". "is a ~" is expressed as "$\mathrm{\exists }\in \{\text{ the ~ s }\}$".

"$\{\text{ ~ }\}$" there does not necessarily mean a "set" but a collection (see this on the difference between 'set' and 'collection'), so, do not worry about that "$\{\text{ the topological spaces }\}$" is not really any set or something.

The "//" marks denote the ends of the blocks, which may seem to be unnecessary if nothing is inserted between the blocks and something after the Statements block is guaranteed to not be confused to be a part of the Statements block, but as one expects some cases like a comment is inserted between the blocks, the marks stay.

Although the Statements block above has only 1 statement, when it has multiple statements some of which are conditional ones, it will be like this.

Statements:
(
~
$⟹$
~
)
$\wedge$
~
//

The implication direction mark, "$⟹$" or "$⟺$", is in a line by itself in principle, which will be visually divide the supposition part and the conclusion part, but one does not prohibit some minor implication direction marks embedded in some lines, according to the 1st rule.

Any number of parentheses can be used in order to avoid any ambiguity. One always uses "$()$" but not "$$" or "$[]$" for that purpose.

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3: The Rules of Structured Descriptions of Definitions

The rules of Structured Descriptions of definitions are almost the same with the ones of Structured Descriptions of propositions, but there are some differences.

Structured Description of any definition has the Conditions block instead of Statements block.

This is an example.

Entities:
$T_1$: $\in \{\text{ the topological spaces }\}$
$T_2$: $\in \{\text{ the topological spaces }\}$
$p$: $\in T_1$
$\ast f$: $:T_1\to T_2$
//

Conditions:
$\mathrm{\forall }{U}_{f(p)}\subseteq T_2$ such that $U_{f(p)}\in \{\text{ the neighborhoods of }f(p)\}$, $\mathrm{\exists }{U}_p\subseteq T_1$ such that $U_p\in \{\text{ the neighborhoods of }p\}$ $\wedge$ $f(U_p)\subseteq U_{f(p)}$.
//

While there can be multiple concerned entities, only one of them is the entity defined in the definition, and the entity is indicated by "*".

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4: It is Work in Progress

This is an attempt to make descriptions of definitions and propositions to be comprehended more promptly, and one knows that the rules will be improved as one writes and reads descriptions.

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References

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