When dealing with simple arguments, the first two columns of a truth table are guide columns in which the variables, or letters, of the argument are listed, followed by a column for each premise and then a column for the conclusion. In this case, though, the guide columns happen to be identical to the premise columns, so we won’t repeat them.

The symbol ∴ indicates that a conclusion follows (it’s often translated as therefore).

(Note: The rules themselves are set out in lowercase letters; capital letters, though, are used to symbolize statements in a proof.)

Notice that the rules are not used jointly but are applied one at a time and once per line. They can, however, be applied multiple times in the same proof, either sequentially or occasionally.

statements (A, E, I, and O) are related in interesting—and useful—ways. They found that standard-form statements with the same subject and predicate but different quantity or quality will have truth values that are predictably correlated. By tradition, this connection is described as one of opposition, and it is formalized and illustrated in the square of opposition.

These are known as singular statements. Each one asserts something about a single person or thing, including objects, places, and times. Each subject term is a noun (including proper names), pronoun, or noun phrase referring to an individual, particular item. In a way, the predicate terms specify classes, but alas, the subject terms don’t.

In categorical reasoning, the statements, or claims, of interest are categorical statements

Categorical logic is different. The basic unit of concern in categorical logic is the statement component.

A negation is the denial of a statement, which we indicate with the word not or a term that means the same thing.

Two simple statements joined by a connective to form a compound statement are known as a conjunction.

Remember that a statement (or claim) is an assertion that something is or is not the case. In other words, it is the kind of thing that can be either true or false. A simple statement is one that doesn’t contain any other statements as constituents. A compound statement is composed of at least two simple statements.

Here, we take up an exploration of propositional logic (or truth-functional logic)—the branch of deductive reasoning that deals with the logical relationships among statements.