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Facts

A Meaning is a specification that includes:

Here, a list of things in the collection as the Subjects of a Preposition, is a list that has as many entries as the number of / Capitals in the Preposition, so that when the Preposition occurs in a Sentence, there is one entry in the list for each Subject of the Preposition.

With reference to an Adverb, two Meanings are Related, if they are identical, except possibly for the things in the collection that they specify for the Noun that Agrees with that Adverb.

A Sentence is determined to be true or false, for a given Meaning, by the following rules:

From these rules, a Sentence of the form ) - A B, where A and B are Sentences, means that A implies B, for it is true provided that B is true if A is true.

The thing that corresponds to a Phrase, for a given Meaning, is the thing that would correspond to that Phrase by the above rules, if it was an Object Phrase of a larger Sentence.

With reference to a Definition Paragraph, two Meanings are Similar if they are identical, except possibly for the things in the collection that they specify for Nouns that are Subjects of the Abbreviation in that Definition Paragraph.

A Meaning is Allowed by a Definition Paragraph if, for every Meaning that is Similar to that Meaning, with reference to that Definition Paragraph:

A Statement in a Development is a Fact if it is true for every Meaning that is Allowed by all the Definition Paragraphs that are Active for that Statement. A Sentence is an Absolute Fact if it is true for every Meaning.

The purpose of the 2nd, 3rd, and 4th requirements on a Regular Development is to ensure that Definition Paragraphs do not restrict Meanings beyond the requirement that the specifications of true or false, and if appropriate, also of things, for Abbreviations, are as determined by the corresponding Definitions. If the 2nd requirement was omitted, there could be two simultaneously Active Definition Paragraphs such as:

+Df +A ( +B +C
+Df +A - ( +B +C

No Meaning is Allowed by both of these. If the 3rd requirement was omitted, there could be a Definition Paragraph such as:

+Df +A - +A

No Meaning is Allowed by this. If the 4th requirement was omitted, there could be two simultaneously Active Definition Paragraphs such as:

+Df +A = +B +C
+Df +B - = +A +C

No Meaning is Allowed by both of these, as can be checked by considering the 2 × 2 × 2 = 8 possible specifications of true and false, independently, for +A, +B, and +C.

For any Paragraph in a Regular Development, any partly specified Meaning, such that all the specifications have been given, except that for Nouns and Prepositions that are Defined by Definition Paragraphs, Active for that Paragraph, no specifications have been given, where the Definition is a Phrase, and only the specifications of things in the collection have been given, where the Definition is not a Phrase, can be uniquely extended to a Meaning that is Allowed by all the Definition Paragraphs Active for that Paragraph, by first completing the specifications for the Noun or Preposition Defined by the first of those Definition Paragraphs, as required to make the Meaning Allowed by that Paragraph, then doing the same for the second of those Definition Paragraphs, and so on. Every Meaning that is Allowed by all those Definition Paragraphs is obtained in this way.

Furthermore, in a Regular Development, if a Statement A is a Fact, and among the Definition Paragraphs Active for A there is a Definition Paragraph B, such that the Nouns or Prepositions Defined by B, and any Definition Paragraphs Active for A that come after B, have no Relevant occurrences in A, then A is true for all Meanings that are Allowed by the Definition Paragraphs Active for A that come before B, even if those Meanings are not Allowed by B, or by Definition Paragraphs Active for A that come after B. For given a Meaning that is Allowed by the Definition Paragraphs Active for A that come before B, we can change the specifications for the Noun or Preposition Defined by B, so as to make the Meaning Allowed by B, and this has no effect on whether A is true or false for the Meaning. We can then do the same for any Definition Paragraphs Active for A that come after B, dealing with these Definition Paragraphs one by one, in the order in which they come, and at each step, the changes to the specifications have no effect on whether A is true or false for the Meaning. At the end of this, we have obtained a modified Meaning that is Allowed by all the Definition Paragraphs Active for A, and for which A is true, if and only if A is true for the original Meaning. But A is a Fact, so A is true for the modified Meaning, so A is true for the original Meaning.

As a result of this, in a Regular Development, the 5th requirement means that if a Statement Paragraph is the first Paragraph after a Method Application, and the Statement in that Statement Paragraph is a Fact, then that Statement is true for all Meanings that are Allowed by all the Definition Paragraphs Active for that Statement that come before that Method Application, even if those Meanings are not Allowed by one or more Definition Paragraphs, Active for that Statement, within that Method Application. Thus in a Regular Development, if a Statement Paragraph A is Active for another Paragraph B, and the Statement in A is a Fact, then that Statement is true for every Meaning that is Allowed by all the Definition Paragraphs that are Active for B, for the Definition Paragraphs Active for A are also Active for B, unless they are within a Method Application, such that A is the first Paragraph after that Method Application.

For every Statement A in a Regular Development, there exists a Sentence B, such that the Sentence B does not contain a Relevant occurrence of any Word that is Defined by a Definition Paragraph that is Active for the Statement A, and the Sentence = A B would be a Fact if it occurred as a Statement at the same point in the Development as the Statement A. Then if the Statement A is a Fact, the Sentence B is an Absolute Fact.

Sometimes a suitable Sentence B, and a proof that = A B is a Fact, can be obtained by starting with the Sentence = A A, which is an Absolute Fact, and using Abbreviate to remove, from the second Subject of =, all Relevant occurrences of the Noun or Preposition that is Defined by the last Definition Paragraph that is Active for the Statement A, then doing the same for the second last Definition Paragraph that is Active for A, and so on.

Sometimes this can be obstructed, because a Relevant occurrence of a Defined Preposition, that we want to remove, has among its Subjects a Noun that would become Governed by an Adverb in the Definition of that Preposition, so that the direct removal of the Defined Preposition, by Abbreviate, would not be Legitimate. For example, if one of the Definition Paragraphs Active for the Statement A is:

+Df +sm// +a +b [c = +ft// +c +a +ft// +c +b

and in the course of forming the Sentence B, we need to replace a Verb occurrence of the Phrase +sm// +c +d, we cannot directly use Abbreviate, because of the [c in the Definition. When that happens, if the Definition is E, we can choose a Noun, say +f, that has no occurrences in the Statement A or the Definition Paragraphs Active for A, and does not Agree with any Adverb in A or in any of those Definition Paragraphs, and form a new Sentence G, that is identical to E, except that each occurrence in E of an Adverb that Agrees with the Noun for which the problem occurred, (here +c), is replaced by the corresponding Adverb that Agrees with the alternative Noun, (here +f), and each occurrence of the problem Noun that is Governed by one of those Adverbs is replaced by the alternative Noun. Then the Sentence = E G is an Absolute Fact, and we can use G instead of E in the formation of B. An example of how to prove a Statement such as = E G is given near the start of the Knowledge Base.

Once a suitable Sentence B has been constructed, and = A B has been proved to be a Fact, B can be proved to be a Fact by Tabulate, if A is a Fact. Then since B has no Relevant occurrences of any Noun or Preposition that is Defined by a Definition Paragraph that is Active for the Statement A, the result above implies that B is an Absolute Fact.

A Development is Logical if it is Regular, and every Statement is a Fact. The Knowledge Base on this website is an example of a Logical Development.

The English and Verish text on this website is licensed for use under the Free Software Foundation Free Documentation License, and the software is licensed for use under the Free Software Foundation General Public License.

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