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Revision as of 22:16, 16 January 2009
, 22:16, 16 January 2009→1.3.10.5. Articulate Sentences: mathematical markup
The universe of discourse for a sentence, the set whose elements the sentence is interpreted to be about, is not a property of the sentence by itself, but of the sentence in the presence of its interpretation. Independently of how many explicit variables a sentence contains, its value can always be interpreted as depending on any number of implicit variables. For instance, even a sentence with no explicit variable, a constant expression like <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> or <math>^{\backprime\backprime} \underline{1} ^{\prime\prime},</math> can be taken to denote a constant proposition of the form <math>c : X \to \underline\mathbb{B}.</math> Whether or not it has an explicit variable, I always take a sentence as referring to a proposition, one whose values refer to elements of a universe <math>X.\!</math>
The universe of discourse for a sentence, the set whose elements the sentence is interpreted to be about, is not a property of the sentence by itself, but of the sentence in the presence of its interpretation. Independently of how many explicit variables a sentence contains, its value can always be interpreted as depending on any number of implicit variables. For instance, even a sentence with no explicit variable, a constant expression like <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> or <math>^{\backprime\backprime} \underline{1} ^{\prime\prime},</math> can be taken to denote a constant proposition of the form <math>c : X \to \underline\mathbb{B}.</math> Whether or not it has an explicit variable, I always take a sentence as referring to a proposition, one whose values refer to elements of a universe <math>X.\!</math>
Notice that the letters <math>^{\backprime\backprime} p ^{\prime\prime}</math> and <math>^{\backprime\backprime} q ^{\prime\prime},</math> interpreted as signs that denote indicator functions <math>p, q : X \to \underline\mathbb{B},</math> have the character of sentences in relation to propositions, at least, they have the same status in this abstract discussion as genuine sentences have in concrete discussions. This illustrates the relation between sentences and propositions as a special case of the relation between signs and objects.
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To assist the reading of informal examples, I frequently use the letters "s", "t", and "S", "T" to denote sentences. Thus, it is conceivable to have a situation where S = "P" and where P : U �> B. Altogether, this means that the sign "S" denotes the sentence S, that the sentence S is the sentence "P", and that the sentence "P" denotes the proposition or the indicator function P : U �> B. In settings where it is necessary to keep track of a large number of sentences, I use subscripted letters like "e1", ..., "en" to refer to the various expressions.
To assist the reading of informal examples, I frequently use the letters "s", "t", and "S", "T" to denote sentences. Thus, it is conceivable to have a situation where S = "P" and where P : U �> B. Altogether, this means that the sign "S" denotes the sentence S, that the sentence S is the sentence "P", and that the sentence "P" denotes the proposition or the indicator function P : U �> B. In settings where it is necessary to keep track of a large number of sentences, I use subscripted letters like "e1", ..., "en" to refer to the various expressions.