Statements, predicates and truth

Turn a condition into a precise yes-or-no rule. Combine conditions, check every possible case, and simplify a rule without changing which cases it accepts.

What can be true or false?

When we described sets, a membership condition decided which candidates belonged. Logic studies the structure of those decisions. Start by separating a value, such as $8+5$, from a claim about it, such as $8+5=13$. The expression gives a number; the claim has a truth value.

Expression or sentence	What it is
$8+5=13$	A true statement.
$8+5=12$	A false statement.
$8+5$	A numerical expression, not a statement.
Calculate 8 + 5.	An instruction, not a statement.
Is 8 + 5 equal to 13?	A question, not a statement.

A statement makes a definite claim that is either true or false. A false claim is still a statement. We will use classical two-valued logic: T stands for true, F for false, and each statement has exactly one of these truth values.

A predicate leaves an input to be supplied. Choose $P(n):n>0$, with integer inputs $n\in\mathbb Z$. Substituting 6 produces the true statement $P(6)$; substituting $-4$ produces the false statement $P(-4)$. The domain matters: every permitted input must give a well-defined statement. For instance, a condition involving $1/x$ must exclude $x=0$ unless another meaning at zero has been specified.

Read a divisibility test

For integers a and b with $a\ne0$, $a\mid b$ means “a divides b”: b is an integer multiple of a. For example, $3\mid18$ is true because $18=3\cdot6$. But $4\nmid18$: no integer multiple of 4 equals 18. The bar here means divisibility, not the separator in set-builder notation.

This gives a second predicate, $Q(n):2\mid n$, meaning n is even. Keep the integer domain. Both P and Q give yes-or-no answers, but they ask different questions: positivity and evenness. Next we will decide whether one test, or both tests, must pass.