§5. Truth-Value Analysis

The "truth functions of given components are all the compounds constructed from them by means exclusively of conjunction and negation (and the dispensable further connectives 'v', '→', '↔')" (p. 33, hyperlinks mine). "P" is also counted as a truth function of "p".

Schemata

Truth functions containing literals are not statements, as each literal is semantically empty. Literals and truth functions are therefore called "schemata", and each is a singular "schema".

The "interpretation" of a schema is either 1. its truth value, or 2. the statement it represents.

"⊤" ("tee", or even "true") stands for "true", and "⊥" (there is no established pronunciation for the "Up Tack", but the monosyllabic "false" suffices) stands for "false" ("F" is needed for different purposes later).

Resolution of Schemata

"Resolution" is the process whereby schemata may be reduced logically. Obviously "~⊥" resolves simply to "⊤". More complex schemata can resolve according to at least nine rules, which Quine lays out on page 34 as

(i) Delete ⊤ as a component of conjunction...
(ii) Delete ⊥ as a component of alternation...
(iii) Reduce a conjunction with ⊥ as a component to ⊥.
(iv) Reduce an alternation with ⊤ as a component to ⊤.
(v) Drop ⊤ as antecedent of a conditional...
(vi) Reduce a conditional with ⊥ as antecedent, or ⊤ as consequent, to ⊤...
(vii) If a conditional has ⊥ as consequent, reduce the whole to the negation of the antecedent.
(viii) Drop ⊤ as component of a biconditional...
(ix) Drop ⊥ as component of a biconditional and negate the other side...
(emphases and quotation marks omitted by me)

Truth Value Analysis

Evaluating schemata for each possible interpretation of its literals is called truth-value analysis. "We make a grand dichotomy of cases by putting first '⊤' and then '⊥' for some chosen letter..." (p.37). "It is better to choose the letter which has the most repetitions, if repetitions there be, and to adhere to this plan also at each later stage... This strategy tends to hasten the disappearance of letters, and thus to minimize work." (p. 38).

[Note: At this point in my studying I have chosen to significantly downsize my annotation ambition, as well as to come back to link more of the text in each post to other posts for the sake of speeding up the process here and now.]