Changes

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.

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.

To assist the reading of informal examples, I frequently use the letters <math>^{\backprime\backprime} s ^{\prime\prime}</math> and <math>^{\backprime\backprime} t ^{\prime\prime},</math> to denote sentences.  Thus, it is conceivable to have a situation where <math>s ~=~ ^{\backprime\backprime} p ^{\prime\prime}</math> and where <math>p : X \to \underline\mathbb{B}.</math>  Altogether, this means that the sign <math>^{\backprime\backprime} s ^{\prime\prime}</math> denotes the sentence <math>s,\!</math> that the sentence <math>s\!</math> is the sentence <math>^{\backprime\backprime} p ^{\prime\prime},</math> and that the sentence <math>^{\backprime\backprime} p ^{\prime\prime}</math> denotes the proposition or the indicator function <math>p : X \to \underline\mathbb{B}.</math>  In settings where it is necessary to keep track of a large number of sentences, I use subscripted letters like <math>{}^{\backprime\backprime} e_1 {}^{\prime\prime}, \, \ldots, \, {}^{\backprime\backprime} e_n {}^{\prime\prime}</math> 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.

A "sentential connective" is a sign, a coordinated sequence of signs, a significant pattern of arrangement, or any other syntactic device that can be used to connect a number of sentences together in order to form a single sentence.  If k is the number of sentences that are connected, then the connective is said to be of order k.  If the sentences acquire a logical relationship by this means, and are not just strung together by this mechanism, then the connective is called a "logical connective".  If the value of the constructed sentence depends on the values of the component sentences in such a way that the value of the whole is a boolean function of the values of the parts, then the connective is called a "propositional connective".

A ''sentential connective'' is a sign, a coordinated sequence of signs, a significant pattern of arrangement, or any other syntactic device that can be used to connect a number of sentences together in order to form a single sentence.  If <math>k\!</math> is the number of sentences that are connected, then the connective is said to be of order <math>k.\!</math> If the sentences acquire a logical relationship by this means, and are not just strung together by this mechanism, then the connective is called a ''logical connective''.  If the value of the constructed sentence depends on the values of the component sentences in such a way that the value of the whole is a boolean function of the values of the parts, then the connective is called a ''propositional connective''.

=====1.3.10.6.  Stretching Principles=====

=====1.3.10.6.  Stretching Principles=====