Definition:Compound Statement

Definition

A compound statement is a statement which consists of one or more simple statements (called substatements) joined together with, or modified by, one or more logical connectives.

When expressed symbolically as an entity of symbolic logic, a compound statement can be referred to as a statement form.

Parenthesis

The substatements in a compound statement which are joined by a connective may be compound statements themselves. It is necessary that the interpretation of such a compound statement is unambiguous. A compound statement is ill-formed if it is ambiguous as to how its component substatements are grouped by the action of the connectives.

For example, this compound statement:

$p \lor q \implies \neg \, r \implies p \land q$

could be interpreted in several different ways:

• "If either $p$ or $q$ is true, then it is not the case that the truth of $r$ implies the truth of both $p$ and $q$."
• "Either $p$ is true, or if $q$ is true, then it is not the case that the truth of $r$ implies the truth of both $p$ and $q$."
• ... and so on.

So we need a way, for such a compound statement, to determine which of these interpretations is the one intended.

One way of doing this is to group substatements together so they are treated as one entity. We do this by using the concept of parenthesis, in which brackets are used to identify the substatements of a compound statement that are to be treated as one.

In the example above, the two different interpretations will be written in the style we have chosen as:

$\left({p \lor q}\right) \implies \left({\neg \left({r \implies \left({p \land q}\right)}\right)}\right)$

$p \lor \left({q \implies \left({\neg \left({r \implies \left({p \land q}\right)}\right)}\right)}\right)$

Binding Priorities

However, complicated statements may require a large number of parentheses to write. This can make it difficult to read, whatever style is used to write it. In order to limit the number of brackets that are used, some connectives are usually understood as binding more strongly to its substatements than others.

The binding priority, or precedence, is the convention defining the order of binding strength of the individual connectives.

Tautology and Contradiction

In the study of propositional logic, the actual truth value of a given simple statement is generally not known, or at least, not predecided. The object of the exercise is to see what the truth value of compound statements could be, given that their substatements could be either true or false.

Certain statement forms are such that they are always true, and others are such that they are always false.

• A compound statement which is always true, no matter what the truth value of its component substatements, is a tautology.

• A compound statement which is always false, no matter what the truth value of its component substatements, is a contradiction.

• A compound statement which is neither a tautology nor a contradiction, but whose truth value depends upon the truth value of its component statements is called a contingent statement.

Sources

• Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.1$
• D.J. O'Connor and Betty Powell: Elementary Logic (1980): $\S 1.2$