We now introduce a method of testing the logical status of wffs, the consistency of wffs, the validity of arguments and the logical equivalence of a pair of wffs, which is more elegant, more convenient and faster than the truth-table method. (Note: Throughout "iff" means "if and only if")

Let's begin by reminding ourselves of the main semantical notions and the connections among these notions.

I. Four Central Concepts of Semantics

A. Definitions

a. validity of an argument: an argument is valid iff there is no row of its truth table such that the premises are all true and the conclusion is false on that row.

b. consistency of a set of wffs: a set of wffs is consistent iff there is at least one row of its truth table such that all the wffs are true; a set of wffs is inconsistent iff there is no row of its truth table such that all the wffs are true.

c. logical status:

1. a wff is a tautology iff it is true on every row of its truth table;

2. a wff is a contradiction iff it is false on every row of its truth table;

3. a wff is a contingency iff it is true on at least one, and false on at least one, row of its truth table.

d. logical equivalence: a pair of wffs is logically equivalent iff they agree in truth value on every row of their truth tables.

B. Connections between these semantical concepts:

a. An argument is valid iff the set of wffs consisting of the premises and the negation of the conclusion is inconsistent.

b. A wff is a tautology iff its denial is an inconsistent set of one wff.

c. Two wffs, p,q, are logically equivalent iff the set of wffs {p, ~q} and the set of wffs {~p,q} are both inconsistent.