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Operator/Bigelow/Pargetter: an operator is something that does not occur in the normal language (everyday language).
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Everyday language.
In contrast, it occurs in equations.
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Equations.
For example, the plus sign:
Plus/Plus sign/+/Bigelow/Pargetter: if it stands between two number names (!), (e.g."3" and "5", it makes a new term out of it, namely "(3 + 5)" which in turn refers to a number.
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Names, >
Reference, >
Numbers.
That is, plus makes a new name from two old names.
Everyday language: it is not absolutly necessary to add operators to it. You can also have their tasks done by predicates. For example, instead of + we can accept a predicate R.
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Predicates.
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Operator/Predicate/Bigelow/Pargetter: in general: for an n-digit operator, there will be a (n+1)-digit predicate:
O (x1,... xn) = x (n+1),
instead we can say:
Ro(x1,…xn, x(n+1)).
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Operator/Bigelow/Pargetter: For example, plus, plus sign,"+": is applied to number names and provides new number names.
Operator: corresponds semantically to a relation.
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Relations.
Relation/Bigelow/Pargetter: is used differently in semantics: instead of using a relation to determine the semantic value of a sentence, we use it to obtain the referent of a compound referential term.
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Semantics, >
Semantic value.
Relation/syntactic rule:
Ro(x1,… x(n+1)).
Is true iff (x1,... x (n+1) stand in the required relation.
On the other hand,
Operator/syntactic rule:
O (x1,... xn)
refers to x(n+1) iff x1,…x(n+1) stand in the required relation
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Operator: an n-digit operator could be replaced by a predicate for a relation for a (n+1)-digit relation - but there are different syntactic rules: statement about relation true/false.
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Truth values.
Statement with operator: refers to - With probability: e.g. P (p v q) = P(p) + P(q) - P(p u q) is the compound expression and not a name, but a sentence.
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Sentences.
This converts it into a referring statement (this in turn is a name).
+Plus sign: two-names-one-name-operator - P: sentence-to-name-operator.
Operator/Probability/Bigelow/Pargetter: here "P" is an operator that can be applied to composite terms (possibly with "+").
This compound term (example P(p v q) = P(p) + P(q) - P (p u q)
is not a name,...
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...but a sentence.
Operator/probability: thus P converts a sentence into a referring expression. (This is again a name.)
Probability/Possibility/Necessity/Modality/Bigelow/Pargetter: "necessary" and would be > would be can be reinterpreted as probability operators.
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Probability, >
Possibility, >
Necessity, >
Modalities.
For example, "it is likely to grade n that...".
Such operators make sentences out of sentences.
For example, operator P: for each sentence p
P(p)
will refer to the number n, iff
it is likely to grade n that p
is true.
