Changes

In the present context, I can answer questions about the ontology of a &ldquo;variable&rdquo; by saying that each variable <math>x_i\!</math> is a kind of a sign, in the boolean case capable of denoting an element of <math>\mathbb{B} = \{ 0, 1 \}\!</math> as its object, with the actual value depending on the interpretation of the moment.  Note that <math>x_i\!</math> is a sign, and that <math>{}^{\backprime\backprime} x_i {}^{\prime\prime}\!</math> is another sign that denotes it.  This acceptation of the list <math>X = \{ x_i \}\!</math> corresponds to what was just called the ''sign convention''.

In the present context, I can answer questions about the ontology of a &ldquo;variable&rdquo; by saying that each variable <math>x_i\!</math> is a kind of a sign, in the boolean case capable of denoting an element of <math>\mathbb{B} = \{ 0, 1 \}\!</math> as its object, with the actual value depending on the interpretation of the moment.  Note that <math>x_i\!</math> is a sign, and that <math>{}^{\backprime\backprime} x_i {}^{\prime\prime}\!</math> is another sign that denotes it.  This acceptation of the list <math>X = \{ x_i \}\!</math> corresponds to what was just called the ''sign convention''.

In a context where all the signs that ought to have EU-objects are in fact safely assured to do so, then it is usually less bothersome to assume the object convention.  Otherwise, discussion must resort to the less natural but more careful sign convention.  This convention is only &ldquo;artificial&rdquo; in the sense that it recalls the artifactual nature and the instrumental purpose of signs, and does nothing more out of the way than to call an implement &ldquo;an implement&rdquo;.

In a context where all the signs that ought to have EU objects are in fact safely assured to do so, then it is usually less bothersome to assume the object convention.  Otherwise, discussion must resort to the less natural but more careful sign convention.  This convention is only "artificial" in the sense that it recalls the artifactual nature and the instrumental purpose of signs, and does nothing more out of the way than to call an implement "an implement".

I make one more remark to emphasize the importance of this issue, and then return to the main discussion.  Even though there is no great difficulty in conceiving the sign "xi" to be interpreted as denoting different types of objects in different contexts, it is more of a problem to imagine that the same object xi can literally be both a value (in B) and a function (from Bn to B).

I make one more remark to emphasize the importance of this issue, and then return to the main discussion.  Even though there is no great difficulty in conceiving the sign <math>{}^{\backprime\backprime} x_i {}^{\prime\prime}\!</math> to be interpreted as denoting different types of objects in different contexts, it is more of a problem to imagine that the same object <math>x_i\!</math> can literally be both a value (in <math>\mathbb{B}\!</math>) and a function (from <math>\mathbb{B}^n\!</math> to <math>\mathbb{B}\!</math>).

In the customary fashion, the name "xi" of the variable xi is flexibly interpreted to serve two additional roles.  In algebraic and geometric contexts "xi" is taken to name the ith "coordinate function" xi : Bn >B.  In logical contexts "xi" serves to name the ith "basic property" or "simple proposition", also called "xi", that goes into the construction of a propositional universe of discourse, in effect, becoming one of the "sentence letters" of a truth table and being used to label one of the "simple enclosures" of a venn diagram.

In the customary fashion, the name "xi" of the variable xi is flexibly interpreted to serve two additional roles.  In algebraic and geometric contexts "xi" is taken to name the ith "coordinate function" xi : Bn >B.  In logical contexts "xi" serves to name the ith "basic property" or "simple proposition", also called "xi", that goes into the construction of a propositional universe of discourse, in effect, becoming one of the "sentence letters" of a truth table and being used to label one of the "simple enclosures" of a venn diagram.