Propositional logic and truth tables: QCE Philosophy and Reason
“Translate and symbolise propositions using logical operators, and use truth tables to test propositional arguments for validity”
Propositional logic symbolises statements with letters and five operators: negation, conjunction (and), disjunction (inclusive or), the conditional (if... then) and the biconditional (if and only if). Each operator is defined by a truth table; the conditional is false only when a true antecedent meets a false consequent. To test an argument, build a full truth table with a column for every premise and the conclusion, then look for a row where all premises are true and the conclusion false. If no such row exists the argument is valid; a single such row makes it invalid and is the counterexample.
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What this dot point is asking
QCAA wants you to move from ordinary English into the symbols of propositional logic (also called sentential logic) and then use truth tables as a mechanical test of validity. You need the five operators, their truth tables, and the method of checking every row for a line where the premises are all true and the conclusion false. This is core IA1 and external-exam content.
The answer
Symbolising propositions
A proposition is a statement that is true or false. We assign capital letters (P, Q, R) to simple propositions and combine them with logical operators (connectives):
- Negation (not): the tilde, written as not-P or with the symbol applied to P.
- Conjunction (and): P and Q.
- Disjunction (or, inclusive): P or Q.
- Conditional (if... then): if P then Q, where P is the antecedent and Q the consequent.
- Biconditional (if and only if): P if and only if Q.
Translation tips: "but", "yet" and "however" usually signal conjunction; "unless" usually translates as a disjunction or a conditional; "only if" puts the condition in the consequent (P only if Q means if P then Q). Watch scope: "not P and Q" is ambiguous, so use brackets.
The truth tables for each operator
Each operator is defined entirely by how it maps the truth values of its parts. With T for true and F for false:
- Negation: not-P is T when P is F, and F when P is T.
- Conjunction (P and Q): T only when both P and Q are T; otherwise F.
- Disjunction (P or Q): F only when both are F; otherwise T. (This is inclusive "or".)
- Conditional (if P then Q): F only when P is T and Q is F; otherwise T. So a conditional with a false antecedent is automatically T.
- Biconditional (P if and only if Q): T when P and Q have the same value; F when they differ.
The conditional is the one students find counter-intuitive: "if the moon is cheese then 2 plus 2 is 5" is true, because the antecedent is false. The conditional only makes a false claim when a true antecedent leads to a false consequent.
Building a full truth table
For an argument with two simple propositions there are 2 to the power 2 = 4 rows; with three there are 8 rows. List every combination of T and F for the basic letters, then compute each compound column step by step, working from the smallest sub-formulas outward.
Testing an argument for validity
An argument is valid if there is no row where every premise is true and the conclusion is false. Method:
- Symbolise each premise and the conclusion.
- Build a truth table with a column for each premise and the conclusion.
- Scan for any row in which all premises are T and the conclusion is F.
- If such a row exists, the argument is invalid (that row is the counterexample). If none exists, it is valid.
Worked example: modus ponens
Argument: "If P then Q; P; therefore Q." Premises: (if P then Q) and P; conclusion: Q.
| P | Q | if P then Q | P | Q |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | T | F | T |
| F | F | T | F | F |
Only row 1 has both premises (the conditional and P) true. In that row the conclusion Q is also true. There is no row with true premises and a false conclusion, so modus ponens is valid.
Argument: "If P then Q; Q; therefore P" (affirming the consequent). Premises: (if P then Q) and Q; conclusion: P.
Step 1: Symbolise the premises and conclusion
Premise 1 is the conditional "if P then Q". Premise 2 is Q. The conclusion is P. We need a column for each.
Step 2: List every truth-value combination
With two letters there are rows: (T, T), (T, F), (F, T), (F, F) for P and Q.
Step 3: Compute the compound columns
| P | Q | if P then Q | Q (premise) | P (conclusion) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | F | F |
Step 4: Scan for a counterexample row
Look for a row where both premises are true and the conclusion is false. Row 3 (P false, Q true) has the conditional true and Q true (both premises true) yet the conclusion P false. That single row is a counterexample, so the argument is invalid: affirming the consequent fails the truth-table test.
Tautologies, contradictions and contingencies
A formula true in every row is a tautology (for example "P or not-P", the law of excluded middle). One false in every row is a contradiction (for example "P and not-P"). One that is sometimes true and sometimes false is contingent. An argument is valid exactly when the conditional "(premises) therefore (conclusion)" is a tautology.
- Reading "or" as exclusive
- In standard propositional logic "or" is inclusive: P or Q is true when both are true. Only treat it as exclusive if the question explicitly says so.
- Mishandling the conditional
- "If P then Q" is false in only one case: P true and Q false. A false antecedent makes the whole conditional true.
- Declaring validity from one convenient row
- You must check every row. Validity requires that no row has all premises true and the conclusion false.
- Confusing "only if" and "if"
- "P only if Q" means if P then Q (Q is necessary). "P if Q" means if Q then P. They reverse the conditional.
Try this
Q1. Give the truth table for the conditional "if P then Q" and explain when it is false. [3 marks]
- Cue. False only when P is true and Q is false; true in the other three rows.
Q2. Symbolise: "You will pass only if you study." [2 marks]
- Cue. "Only if" puts the condition in the consequent: if you pass then you studied (pass implies study).
Q3. Use a truth table to test "P or Q; not-P; therefore Q" (disjunctive syllogism). [4 marks]
- Cue. The only row with both premises true is P false, Q true; there Q is true, so the form is valid.
Exam-style questions
Questions in the style of QCAA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.
Practice5 marksSymbolise the argument and use a truth table to test it for validity: "If the alarm sounds, the building is evacuated. The building was not evacuated. Therefore the alarm did not sound."Show worked answer →
A 5 mark response symbolises, builds the table, and reads off validity.
Symbolisation. Let P = "the alarm sounds" and Q = "the building is evacuated". Premises: (if P then Q) and not-Q. Conclusion: not-P. This is modus tollens.
Truth table (rows P, Q): when P=T, Q=T the conditional is T but not-Q is F; when P=T, Q=F the conditional is F; when P=F, Q=T the conditional is T but not-Q is F; when P=F, Q=F the conditional is T, not-Q is T, and the conclusion not-P is T.
Reading it. The only row where both premises are true is P=F, Q=F, and there the conclusion not-P is also true. No row has true premises and a false conclusion, so the argument is valid.
Markers reward correct symbolisation, a complete table, and the conclusion that modus tollens is valid because no counterexample row exists.
Practice4 marksExplain why "if P then Q" is true whenever P is false, and why this matters for testing arguments with truth tables.Show worked answer →
A 4 mark response explains the material conditional and its testing role.
The rule. The conditional "if P then Q" is defined as false in exactly one case: a true antecedent (P) with a false consequent (Q). In every other case, including both rows where P is false, it counts as true (vacuously true).
Why it matters. When you scan a truth table for validity, you only need rows where all premises are true. Conditional premises are automatically true wherever their antecedent is false, so those rows are still candidates for a counterexample. Mishandling this (wrongly marking a false-antecedent conditional as false) would corrupt the validity test.
Markers reward the material-conditional rule and a clear link to checking rows in the validity method.