| Line 6: |
Line 6: |
| | | | |
| | =====1.3.12.1. Syntactic Transformation Rules===== | | =====1.3.12.1. Syntactic Transformation Rules===== |
| − |
| |
| − | A sentence that is written in a context where it represents a value of 1 or 0 as a function of things in the universe U, where it stands for a value of "true" or "false", depending on how the signs that constitute its proper syntactic arguments are interpreted as denoting objects in U, in other words, where it is bound to lead its interpreter to view its own truth or falsity as determined by a choice of objects in U, is a sentence that might as well be written in the context "[ ... ]", whether or not this frame is explicitly marked around it.
| |
| − |
| |
| − | More often than not, the context of interpretation fixes the denotations of most of the signs that make up a sentence, and so it is safe to adopt the convention that only those signs whose objects are not already fixed are free to vary in their denotations. Thus, only the signs that remain in default of prior specification are subject to treatment as variables, with a decree of functional abstraction hanging over all of their heads.
| |
| − |
| |
| − | : [u C X] = Lambda (u, C, X).(u C X).
| |
| − |
| |
| − | As it is presently stated, Rule 1 lists a couple of manifest sentences, and it authorizes one to make exchanges in either direction between the syntactic items that have these two forms. But a sentence is any sign that denotes a proposition, and thus there are a number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged. Consider the sense of equivalence among sentences that is recorded in Rule 4.
| |
| − |
| |
| − | <pre>
| |
| − | Rule 4
| |
| − |
| |
| − | If X c U is fixed
| |
| − |
| |
| − | and u C U is varied,
| |
| − |
| |
| − | then the following are equivalent:
| |
| − |
| |
| − | R4a. u C X.
| |
| − |
| |
| − | R4b. [u C X].
| |
| − |
| |
| − | R4c. [u C X](u).
| |
| − |
| |
| − | R4d. {X}(u).
| |
| − |
| |
| − | R4e. {X}(u) = 1.
| |
| − | </pre>
| |
| − |
| |
| − | The first and last items on this list, namely, the sentences "u C X" and "{X}(u) = 1" that are annotated as "R4a" and "R4e", respectively, are just the pair of sentences from Rule 3 whose equivalence for all u C U is usually taken to define the idea of an indicator function {X} : U -> B. At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their own ostensible types and the ruling type of a sentence. On reflection, and taken in context, these problems are not as serious as they initially seem. For instance, the expression "[u C X]" ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence. As a general rule, if one can see it on the page, then it cannot be a proposition, but can be, at best, a sign of one.
| |
| − |
| |
| − | The use of the basic connectives can be expressed in the form of a ROST as follows:
| |
| − |
| |
| − | <pre>
| |
| − | Logical Translation Rule 0
| |
| − |
| |
| − | If Sj is a sentence
| |
| − |
| |
| − | about things in the universe U
| |
| − |
| |
| − | and Pj is a proposition
| |
| − |
| |
| − | about things in the universe U
| |
| − |
| |
| − | such that:
| |
| − |
| |
| − | L0a. [Sj] = Pj, for all j C J,
| |
| − |
| |
| − | then the following equations are true:
| |
| − |
| |
| − | L0b. [ConcJj Sj] = ConjJj [Sj] = ConjJj Pj.
| |
| − |
| |
| − | L0c. [SurcJj Sj] = SurjJj [Sj] = SurjJj Pj.
| |
| − | </pre>
| |
| | | | |
| | As a general rule, the application of a ROST involves the recognition of an antecedent condition and the facilitation of a consequent condition. The antecedent condition is a state whose initial expression presents a match, in a formal sense, to one of the sentences that are listed in the STR, and the consequent condition is achieved by taking its suggestions seriously, in other words, by following its sequence of equivalents and implicants to some other link in its chain. | | As a general rule, the application of a ROST involves the recognition of an antecedent condition and the facilitation of a consequent condition. The antecedent condition is a state whose initial expression presents a match, in a formal sense, to one of the sentences that are listed in the STR, and the consequent condition is achieved by taking its suggestions seriously, in other words, by following its sequence of equivalents and implicants to some other link in its chain. |