Changes

If a sign, as accepted and interpreted in a particular setting, has an ''existentially unique'' denotation, that is, if there exists a unique object that the sign denotes under the operative sign relation, then the sign is said to possess a ''EU-denotation'', or to have a ''EU-object''.  When this is so, the sign is said to be ''eudenotational'', otherwise it is said to be ''dysdenotational''.

If a sign, as accepted and interpreted in a particular setting, has an ''existentially unique'' denotation, that is, if there exists a unique object that the sign denotes under the operative sign relation, then the sign is said to possess a ''EU-denotation'', or to have a ''EU-object''.  When this is so, the sign is said to be ''eudenotational'', otherwise it is said to be ''dysdenotational''.

Using the distinction accorded to eudenotational signs, the issue about the ontological status of variables can be illustrated as turning on two different ''acceptations'' of the list <math>X = \{ x_1, \ldots, x_n \}.\!</math>

Using the distinction accorded to eudenotational signs, the issue about the ontological status of variables can be illustrated as turning on two different acceptations of the list X = {x1, ..., xn}.

1. The natural (or naive) acceptation is for a reader to interpret the list as referring to a set of objects, in effect, to pass without hesitation from impressions of the characters "x1", ..., "xn" to thoughts of their respective EU objects x1, ..., xn, all taken for granted to exist uniquely.  The whole set of interpretive assumptions that go into this acceptation will be referred to as the "object convention".

# The natural (or naive) acceptation is for a reader to interpret the list as referring to a set of objects, in effect, to pass without hesitation from impressions of the characters <math>{}^{\backprime\backprime} x_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} x_n {}^{\prime\prime}\!</math> to thoughts of their respective EU-objects <math>x_1, \ldots, x_n,\!</math> all taken for granted to exist uniquely.  The whole set of interpretive assumptions that go into this acceptation will be referred to as the ''object convention''.

 

# The reflective (or critical) acceptation is to see the list before all else as a list of signs, each of which may or may not have a EU-object.  This is the attitude that must be taken in formal language theory and in any setting where computational constraints on interpretation are being contemplated.  In these contexts it cannot be assumed without question that every sign, whose participation in a denotation relation would have to be indicated by a recursive function and implemented by an effective program, does in fact have an existential denotation, much less a unique object.  The entire body of implicit assumptions that go to make up this acceptation, although they operate more like interpretive suspicions than automatic dispositions, will be referred to as the ''sign convention''.

2. The reflective (or critical) acceptation is to see the list before all else as a list of signs, each of which may or may not have a EU object.  This is the attitude that must be taken in formal language theory and in any setting where computational constraints on interpretation are being contemplated.  In these contexts it cannot be assumed without question that every sign, whose participation in a denotation relation would have to be indicated by a recursive function and implemented by an effective program, does in fact have an existential denotation, much less a unique object.  The entire body of implicit assumptions that go to make up this acceptation, although they operate more like interpretive suspicions than automatic dispositions, will be referred to as the "sign convention".

In the present context, I can answer questions about the ontology of a "variable" by saying that each variable xi is a kind of a sign, in the boolean case capable of denoting an element in B = {0, 1} as its object, with the actual value depending on the interpretation of the moment.  Note that xi is a sign, and that "xi" is another sign that denotes it.  This acceptation of the list X = {xi} corresponds to what was just called the "sign convention".

In the present context, I can answer questions about the ontology of a "variable" by saying that each variable xi is a kind of a sign, in the boolean case capable of denoting an element in B = {0, 1} as its object, with the actual value depending on the interpretation of the moment.  Note that xi is a sign, and that "xi" is another sign that denotes it.  This acceptation of the list X = {xi} corresponds to what was just called the "sign convention".