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There are several features of this definition that need be understood. Indeed, there are problems involved in this whole style of definition that need to be discussed, and this requires a slight digression.

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Before I move on I will need to go back and pick up a collection of basic definitions from the beginning of the Subsection. Also, cumbersome as it may be, I will need to use the form "-( )-" for the pair of "struck-through parentheses" that I normally use for logical negation.

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~~=====1.3.10.3. Propositions and Sentences=====~~

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The foregoing "definitions" are the bare essentials that are needed to get the rest of this discussion going, but they have to be regarded as almost purely informal in character, at least, at this stage of the game. In particular, these definitions all invoke the undefined notion of what a "sentence" is, they all rely on the reader's native intuition of what a "set" is, and they all derive their coherence and their meaning from the common understanding, but the equally casual use and unreflective acquaintance, that just about everybody has of the logical connectives "not", "and", "or", as these are expressed in natural language terms.

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As formative definitions, these initial postulations neither acquire the privileged status of untouchable axioms and infallible intuitions nor do they deserve any special suspicion, at least, nothing over and above the reflective critique that one ought to apply to all important definitions. These dim beginnings of anything approaching genuine definitions also serve to accustom the mind's eye to a particular style of observation, that of seeing informal concepts presented in a formal frame, in a way that almost demands their increasing clarification. In this style of examination, the frame of the set-builder expression "{x in X : ... }" functions like the "eye of the needle" through which one is trying to transport a suitably rich import of mathematics.

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Part the task of the remaining discussion is gradually to formalize the promissory notes that are represented by these terms and stipulations and to see whether their casual comprehension can be converted into an explicit subject matter, one that depends on grasping the corresponding collection of almost wholly, if still partially, formalized conceptions. To this I now turn.

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Jon Awbrey

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