How can I check a conditional rule by listing its possible cases?

Treat the condition and the result as two separate statements. Call the condition p and the result q. Each can be either true (T) or false (F), so make one row for each pairing:

p  q
T  T
T  F
F  T
F  F

That gives all four possible cases. Now evaluate the whole conditional, written p → q, on each row. The rule for the standard truth table is simple: it is false only when p is true and q is false. The other three rows are true. [S1]

Think of the false row as the failed promise: the condition happened, but the stated result did not.

Here is an invented example, just for practice:

Rule: “If the library is open, then the study room is available.”

Let p = “The library is open.” Let q = “The study room is available.”

p  q  p → q
T  T    T
T  F    F
F  T    T
F  F    T

Read the rows as possibilities, not as reports about a real library. In the first row, the library is open and the room is available. In the second, the library is open but the room is unavailable; that is the one case that contradicts the conditional. In the third and fourth, the library is not open. Under the standard truth-table definition, both rows count as true for this conditional. [S1]

This table checks the logical form. It does not establish whether a particular library’s rule is accurate or what happens there in practice.

It can feel odd because everyday “if” statements sometimes suggest more than a formal conditional does. In a truth table, p → q makes a specific test: did p happen while q failed? If p is false, that particular failure has not occurred, so the conditional is counted as true by definition. [S1]

For example, if the library is not open, this table does not test whether an open library would have an available room. The table is a way to evaluate the conditional’s logical form, not a prediction, explanation, or guarantee about a real situation. When interpreting a rule in ordinary conversation, check whether the speaker intends only the formal conditional or is also implying something about what causes the result.

Try this invented rule: “If a number is a multiple of 4, then it is even.” First practise the table’s structure. Fill in T or F for the whole conditional in the last column:

p  q  p → q
T  T   ___
T  F   ___
F  T   ___
F  F   ___

Here p means “The number is a multiple of 4,” and q means “The number is even.” The worked table above gives the pattern for the last column: T, F, T, T. To check your method, confirm that you included every pairing of p and q exactly once, then find the row where p is true and q is false. Only that row gets F. [S1]

You can reuse the blank version for any two-part conditional. Write the actual condition and result beside p and q so you do not accidentally switch their roles. The table evaluates the conditional’s truth value for each possible pairing; it does not by itself supply evidence about a real-world example.