A **compound** proposition consists of other propositions combined in such a way
that the truth value of the compound can be unambiguously determined if the truth
values of its component propositions are known. The ways in which propositions can
be compounded are known as **logical connectives**. I call a logical connective L
**genuine** if the truth value of L(p, q, r, ...) depends on the truth values of all
of its components. A proposition that is true whatever the truth values of its components,
or is false whatever the truth values of its components, is known as a **tautology**.
A tautology is **ungenuine**.

The four possible logical connectives on a single proposition p can be shown in
**truth table** form as follows:

p | T(p) | P(p) | N(p) | F(p) | |

1 | 1 | 1 | 0 | 0 | |

0 | 1 | 0 | 1 | 0 |

On two propositions, p and q, there are 16 possible logical operations: Of these 16
there are 8 that are true and 8 that are false when p and q are both true,
we will call these **positive** and **negative** operations respectively.
The truth tables for the negative operations are found by replacing 0 by 1 and 1 by 0
in the truth tables for the corresponding positive operations.

Of the 8 positive operations on two components three are not genuine operations
(one is true regardless of the values of p and q, one has the same truth values as p,
and one has the same truth values as q). The 5 genuine positive operations are conveniently
denoted by the vowels A, E, I, O, U. The 5 genuine negative operations can be correspondingly
denoted by the same letters with hats: Â, Ê, Î, Ô, Û (or bars
but I could't find how to do those in HTML.) Their values can be exhibited in the following
**truth tables**:

p | q | pAq | pEq | pIq | pOq | pUq | pÂq | pÊq | pÎq | pÔq | pÛq | ||

1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | ||

1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | ||

0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | ||

0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |

These operations can be translated into corresponding ordinary English words, as
indicated below, although care is of course needed in the converse operation of translating
English arguments into the logical notation, since there are many variations of phrasing
and ambiguities in the normal English usage.

pAq = ‘p and q’ (*conjunction*)

pEq = ‘p is logically equivalent to q’ or ‘p if and only if q’ (*biconditional*).

pIq = ‘p implies q’ or ‘if p then q’ (*implication* or *conditional*)

pOq = ‘p or q or both’ (*disjunction* or the *inclusive or*).

pUq = ‘p is implied by q’ (*converse implication*)

pÂq = ‘not both p and q’ (*nand*).

pÊq = ‘p or q but not both’ (the *exclusive or*).

pÎq = ‘not p implies not q’ (*inverse implication*).

pÔq = ‘neither p nor q’ or ‘not p and not q’.

pÛq = ‘p is false unless q is true’ (*contrapositive implication*).

Some more long-winded phrases are sometimes used to express the above propositions. For example pIq = ‘p is a sufficient condition for q’, pUq = ‘p is a necessary condition for q’, and pEq = ‘p is a necessary and sufficient condition for q’.