Need a template to render logical symbols and formatting.
Write the essentials for truth functional schemata, Boolean term schemata, Boolean existence schemata, quantification—monadic and general or polyadic, substitution, deduction, completeness, decidability, and soundness.
Topics from Part 4 of Quine’s book: singular terms, identity, descriptions, elimination of singular terms, elimination of variables, classes, number, axiomatic set theory.
When the section on implication is written place in it Quine’s example which is now in the section on use and mention.

THE OUTLINE

Propositional or truth functional calculus

The general truth function

Statements and statement letters

Statements, whose characteristic in logic, it is to be true (T) or false (F or ^) are represented by letters p, q, r… represent letters.

A truth function of a number of statements is one whose inputs are the truth values of the statements and whose value, for each combination of input values, is a truth value—i.e., one of T and F. It is convenient to work with letters ‘p’, ‘q’, ‘r’… representing statements. A truth functional interpretation of a letter is statement that gives the letter a definite truth value or a definite truth value.

Truth functions can have any number of inputs. We consider some common one place and two place functions (unary and binary), develop their properties, and show all truth functions can be built from them.

Then the truth functional combinations of letters are called truth functional schemata (singular: schema).

Letters S, S', S''… or S₁, S₂, S₃… stand for schemas. A truth functional interpretation of a schema is an assignment of statements with definite truth values or definite truth values to the letters.

Use and mention

The following true statement mentions two countries.

America is larger than Australia.

The following true statement uses the names of the countries.

‘America’ is shorter than ‘Australia’.

The following explains how this distinction between use and mention is significant in talking of linguistic objects, particularly logical ones.

That a statement has a truth value T is syntactic and we might say:

The truth value of p is T.

On the other hand that p is true is semantic—i.e. of the meaning of p or, equivalently, of the relation between p and the world. And to say that a statement is true we would write:

‘p’ is true.

Quine gives the following example that concerns the difference between implication and the conditional ‘®’.

The conditional ‘®’ is a connective written ‘p ® q’, said ‘if p then q’ but not ‘p implies q’, and is false only in the case that ‘p’ is true and ‘q’ is false.

On the other hand a truth functional schema S is said to imply another schema S' if there is no way of interpreting the letters (T or F) so as to make the first (S) true and the second (S') false. What is the relation between implication and the conditional? This may be given in terms of validity: A truth functional schema is valid if it comes out true under every interpretation of its letters. Then: Implication is validity of the conditional. Thus we might say:

‘S₁’ implies ‘S₂’

but

‘S₁ ® S₂’.

Identity

I(p), the identity function is a one place function whose truth value is the same as p's. Since ‘I(p)’ is equivalent to ‘p’ the truth functional calculus need never work with it. It is superfluous.

Negation

The negation of S, a function of a single schema, is written – S; its truth value is the ‘opposite’ of the truth value of S. When the negation is of a single letter we write S or p (the common symbolization with a top bar as in ā is not conveniently available on Word). – – p is true if and only if p is true so there is no need to ever write the double negation

Conjunction

Conjunction is a two place function or connective: it is the logical ‘and’. Conjunction here uses juxtaposition as its implicit symbol, it is said ‘p and q’ or just as it is written ‘pq’; it is true when both ‘p’ and ‘q’ are true and otherwise false. Conjunction is obviously symmetric—the order of p and q does not affect the truth value of conjunction; and it is associative because ‘p(qr)’ is true if and only if ‘p’ and ‘qr’ are true, i.e. if and only if all three are true; but this is just the condition for truth of ‘(pq)r’. ‘pqr…w’ is similarly indifferent to grouping (and order) and is true if and only if all of ‘p’, ‘q’, ‘r’… and ‘w’ are true.

The general truth function

We will show that the general truth function is a combination of negations and conjunctions.

An n place truth function has n inputs; let them be ‘p₁’ … ‘p_n’. Each letter has two input truth values, so the number of combinations of input values is 2ⁿ. Of these combinations the result will be true in some cases and false in others. An example of a case in which the result is false might be ‘p₁’, ‘p₂’, ‘p₃’, … ‘p_n-1’, ‘p_n’; their conjunction is true when the result is false; collect all such cases and form the conjunction of their negations; provided that there are some cases when the n place function is false, the compound is false when at least one factor is true but otherwise true and is thus equivalent to the given n place truth function. If the n place function is always true write ‘– (p₁p₁p₂p₃ … p_n-1p_n)’ which is always true since the expression in brackets is always false. Note, by the way, the superfluity of identity from the equivalence of ‘I(p)’ to ‘– – p’.

Alternation

The non-exclusive (truth function) or uses the symbol Ú or vel from Latin; it is a two place function, is said ‘p or q’ and written ‘p Ú q’; its value is true when at least one of p and q is true and otherwise false. Now ‘pq’ is precisely when both ‘p’ and ‘q’ are false; so ‘– (pq)’ is true (precisely) when not both are false—i.e. precisely when ‘p Ú q’ is true. That is, ‘p Ú q’ is equivalent to ‘– (pq)’ (the truth functions are identical). So, alternation can be written in terms of negation and conjunction; this shows, as we also know from consideration of the general truth function; alternation is superfluous. But similarly, conjunction can be written in terms of alternation for ‘pq’ is equivalent to ‘– (p Ú q)’. So the general truth function can be written in terms of negation and alternation; either alternation or conjunction can be taken as primitive and the other superfluous; but it is convenient to retain both for their use and convenience.

This is a convenient place to introduce

De Morgan’s laws

Expression of conjunction and alternation in terms of one another generalize to the following equivalences:

‘– (p Ú q Ú … Ú s)’ to ‘(pqr … s)’

and

– ‘(pqr … s)’ to ‘(p Ú q Ú … Ú s)’.

Sheffer’s stroke

This connective written ‘p | q’ is true if and only if ‘p’ and ‘q’ are not both true—i.e., ‘– (pq)’. ‘– p’ is equivalent to ‘p | p’ and ‘pq’ to ‘(p | q) | (q | p)’ so all truth functions can be written in terms of the stroke. This connective was introduced in 1913 by H.M. Sheffer; he also introduced another ‘¯’, with ‘p ¯ q’ equivalent to pq; all truth functions can be expressed in terms of it: ‘p’ as ‘p ¯ p’ and ‘pq’ as ‘p ¯ q’, and hence the rest.