Changes

I am going to revert to my customarily sloppy workshop manners and refer to propositions and proposition expressions on rough analogy with functions and function expressions, which implies that a proposition will be regarded as the chief formal object of discussion, enjoying many proposition expressions, formulas, or sentences that express it, but worst of all I will probably just go ahead and use any and all of these terms as loosely as I see fit, taking a bit of extra care only when I see the need.

I am going to revert to my customarily sloppy workshop manners and refer to propositions and proposition expressions on rough analogy with functions and function expressions, which implies that a proposition will be regarded as the chief formal object of discussion, enjoying many proposition expressions, formulas, or sentences that express it, but worst of all I will probably just go ahead and use any and all of these terms as loosely as I see fit, taking a bit of extra care only when I see the need.

Let <math>Q\!</math> be a propositional expression with an unspecified, but context-appropriate number of variables, say, none, or <math>x,\!</math> or <math>x_1, \ldots, x_k,\!</math> as the case may be.

Let <math>Q\!</math> be a propositional expression with an unspecified, but context-appropriate number of variables, say, none, or <math>x\!</math>, or <math>x_1, \ldots, x_k\!</math>, as the case may be.

:* Strings and graphs that have no labels are called ''bare''.

:* Strings and graphs that have no labels are called ''bare''.

:* A bare terminal node, symbolized by a small circle <math>\circ</math> in text, is known as a ''stone''.

:* A bare terminal node, symbolized by a small circle <math>{}^{\backprime\backprime} \circ {}^{\prime\prime}</math> in text, is known as a ''stone''.

:* A bare terminal edge, symbolized by a vertical bar <math>\vert</math> in text, is known as a ''stick''.

:* A bare terminal edge, symbolized by a vertical bar <math>{}^{\backprime\backprime} \vert {}^{\prime\prime}</math> in text, is known as a ''stick''.

Let the ''replacement expression'' of the form <math>Q[\circ /x]</math> denote the proposition that results from <math>Q\!</math> by replacing every token of the variable <math>x\!</math> with a blank, that is to say, by erasing <math>x.\!</math>

Let the ''replacement expression'' of the form <math>Q[\circ /x]</math> denote the proposition that results from <math>Q\!</math> by replacing every token of the variable <math>x\!</math> with a blank, that is to say, by erasing <math>x\!</math>.

Let the ''replacement expression'' of the form <math>Q[\,\vert /x]</math>  denote the proposition that results from <math>Q\!</math> by replacing every token of the variable <math>x\!</math> with a stick stemming from the site of <math>x.\!</math>

Let the ''replacement expression'' of the form <math>Q[\,\vert /x]</math>  denote the proposition that results from <math>Q\!</math> by replacing every token of the variable <math>x\!</math> with a stick stemming from the site of <math>x\!</math>.

In the case of a propositional expression <math>Q\!</math> that has no token of the designated variable <math>x,\!</math> let it be stipulated that <math>Q[\circ /x] = Q = Q[\,\vert /x].</math>

In the case of a propositional expression <math>Q\!</math> that has no token of the designated variable <math>x\!</math>, let it be stipulated that <math>Q[\circ /x] = Q = Q[\,\vert /x]</math>.

I think that I am at long last ready to state the following:

I think that I am at long last ready to state the following: