Changes

The first and last items on this list, namely, the sentence <math>\text{R4a}\!</math> stating <math>x \in Q</math> and the sentence <math>\text{R4e}\!</math> stating <math>\upharpoonleft Q \upharpoonright (x) = \underline{1},</math> are just the pair of sentences from Rule&nbsp;3 whose equivalence for all <math>x \in X</math> is usually taken to define the idea of an indicator function <math>\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}.</math>  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 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 example, the expression <math>^{\backprime\backprime} \downharpoonleft x \in Q \downharpoonright \, ^{\prime\prime}</math> 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 at most be a sign of one.

The first and last items on this list, namely, the sentence <math>\text{R4a}\!</math> stating <math>x \in Q</math> and the sentence <math>\text{R4e}\!</math> stating <math>\upharpoonleft Q \upharpoonright (x) = \underline{1},</math> are just the pair of sentences from Rule&nbsp;3 whose equivalence for all <math>x \in X</math> is usually taken to define the idea of an indicator function <math>\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}.</math>  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 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 example, the expression <math>^{\backprime\backprime} \downharpoonleft x \in Q \downharpoonright \, ^{\prime\prime}</math> 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 at most be a sign of one.

The use of the basic connectives can be expressed in the form of a STR as follows:

The use of the basic logical connectives can be expressed in the form of a STR as follows:

<pre>

<pre>

L0c. [SurcJj Sj]  =  SurjJj [Sj]  =  SurjJj Pj.

L0c. [SurcJj Sj]  =  SurjJj [Sj]  =  SurjJj Pj.

As a general rule, the application of a STR 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 STR 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.

Generally speaking, the application of a rule involves the recognition of an antecedent condition as a case that falls under a clause of the rule.  This means that the antecedent condition is able to be captured in the form, conceived in the guise, expressed in the manner, grasped in the pattern, or recognized in the shape of one of the sentences in a list of equivalents or a chain of implicants.

Generally speaking, the application of a rule involves the recognition of an antecedent condition as a case that falls under a clause of the rule.  This means that the antecedent condition is able to be captured in the form, conceived in the guise, expressed in the manner, grasped in the pattern, or recognized in the shape of one of the sentences in a list of equivalents or a chain of implicants.

A condition is "amenable" to a rule if any of its conceivable expressions formally match any of the expressions that are enumerated by the rule.  Further, it requires the relegation of the other expressions to the production of a result.  Thus, there is the choice of an initial expression that needs to be checked on input for whether it fits the antecedent condition and there are several types of output that are generated as a consequence, only a few of which are usually needed at any given time.

A condition is ''amenable'' to a rule if any of its conceivable expressions formally match any of the expressions that are enumerated by the rule.  Further, it requires the relegation of the other expressions to the production of a result.  Thus, there is the choice of an initial expression that needs to be checked on input for whether it fits the antecedent condition and there are several types of output that are generated as a consequence, only a few of which are usually needed at any given time.

Logical Translation Rule 1

Logical Translation Rule 1

L1b11. [True] = (()) = 1 : U->B.

L1b11. [True] = (()) = 1 : U->B.

Geometric Translation Rule 1

Geometric Translation Rule 1

G1b11. {U} = (()) = 1 : U->B.

G1b11. {U} = (()) = 1 : U->B.

Logical Translation Rule 2

Logical Translation Rule 2

L2b15. [True] = (()) = 1 : U->B.

L2b15. [True] = (()) = 1 : U->B.

Geometric Translation Rule 2

Geometric Translation Rule 2

G2b15. {U} = (()) = 1 : U->B.

G2b15. {U} = (()) = 1 : U->B.

Value Rule 1

Value Rule 1

V1c. ((v , w))

V1c. ((v , w))

Value Rule 1

Value Rule 1

If v, w C B,

If v, w C B,

V1c. (( v , w )).

V1c. (( v , w )).

A rule that allows one to turn equivalent sentences into identical propositions:

A rule that allows one to turn equivalent sentences into identical propositions: