translate and symbolise propositions using logical operators, and use truth tables to test propositional arguments for validity

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:

1. Symbolise each premise and the conclusion.
2. Build a truth table with a column for each premise and the conclusion.
3. Scan for any row in which all premises are T and the conclusion is F.
4. 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.

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.

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.

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.