Propositional Logic · step 1 of 13

Truth tables & connectives

A truth table lists every possible assignment of truth values to the variables in a formula and computes the formula’s value under each one. With n variables there are 2ⁿ rows, so a three-variable formula needs eight. The only connective students reliably get wrong is implication: p → q is false in exactly one case, when p is true and q is false — so "if 2 + 2 = 5 then I am the Pope" is a true statement.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Count the rowsn distinct variables give 2ⁿ rows. Write the variables in a fixed order and fill the columns in the standard TTFF / TFTF pattern so you cannot skip a case.
Add a column per subformulaWork outward from the innermost brackets. Each column should apply exactly one connective to columns you have already computed.
Apply the connective definitions¬p flips; p ∧ q is true only when both are; p ∨ q is false only when both are false; p → q is false only for T → F; p ↔ q is true when the two sides match.
Read off the final columnAll T means tautology, all F means contradiction, a mix means contingency. If you were asked to compare two formulas, they are equivalent exactly when their final columns are identical row for row.

Worked example

Build the truth table for (p → q) ∧ ¬q and say when the formula is true.

Four rows: (p,q) = (T,T), (T,F), (F,T), (F,F).
Column p → q: T, F, T, T.
Column ¬q: F, T, F, T.
Conjunction of the two columns: F, F, F, T.

Answer. The formula is true only in the last row, p = F and q = F. (Together with modus tollens, this is why (p → q) ∧ ¬q entails ¬p.)

Where marks get dropped

These are the specific errors that cost credit on truth tables & connectives questions — QED's rubric penalises each of them separately.

Treating p → q as false whenever p is false. A false antecedent makes the implication vacuously TRUE — this single error accounts for most lost truth-table marks.
Reading ∨ as exclusive. In logic p ∨ q is inclusive: it is true when both disjuncts hold. Exclusive or is written ⊕ and has a different table.
Dropping rows. Eight rows for three variables, sixteen for four — an incomplete table cannot justify a claim of tautology.

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Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.

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Truth tables & connectives — frequently asked questions

How many rows does a truth table with 4 variables have?

Sixteen — 2⁴. Each extra variable doubles the table, which is why exam questions rarely go beyond three or four variables and why truth tables become impractical as a proof method for large formulas.

Why is p → q true when p is false?

Because the implication only claims something about the case where p holds. If p never happens, the promise is never broken, so the statement is counted as true. This is called a vacuous truth.

Do I have to show every intermediate column?

In an exam, yes — the intermediate columns are where the method marks live. QED marks them as separate rubric criteria, so a correct final column with no working still loses points.

The rest of Propositional Logic

Connectives, truth tables, equivalences and normal forms. Each subtopic below has its own method, worked example and mark-losing traps.

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