Valid Argument Forms

definitions
discrete-math
logic
valid-arguments

An argument is a sequence of statements: one or more premises, followed by a conclusion. An argument is valid if whenever all the premises are true, the conclusion is also true — equivalently, if

\[(\text{Premise}_1 \land \text{Premise}_2 \land \cdots \land \text{Premise}_n) \to \text{Conclusion}\]

is a tautology.

Validity is about the argument’s form, not the truth of its statements: an argument can be valid with a false premise or a false conclusion. It can also have true premises and a true conclusion and still be invalid, if the truth of the conclusion doesn’t actually follow from the premises.

Provided in the exam. The named forms below are printed on the MATH1061/MATH7861 Examination Formula Page, alongside the logical-equivalence-laws — so recognising which form an argument matches, and naming it, is the assessed skill rather than recalling the list.

Why is an argument automatically valid if it has a false premise? If any premise is false, the conjunction of the premises is false — and a false statement implies anything vacuously (\(\mathbf{c} \to X\) is always T). So \((\text{Premise}_1 \land \cdots \land \text{Premise}_n) \to \text{Conclusion}\) is trivially a tautology regardless of the conclusion.

Determining validity

  • Truth table method: build a truth table for the premises and the conclusion. The argument is valid iff every row where all premises are T also has the conclusion T.
  • Rules of inference method: chain the laws of logical equivalence (see logical-equivalence-laws) and the valid argument forms below to derive the conclusion from the premises.
  • Invalidity method: assume all the premises are true and the conclusion is false, then reason out what that assumption forces each variable to be. Reaching a contradiction (some variable forced to be both T and F) means the assumption was impossible — the argument is valid. Reaching a consistent assignment instead gives a counterexample — the argument is invalid. (This directed search is usually faster than building the full truth table once there are 3+ variables.)

Valid argument forms

Name Form(s)
Modus Ponens \(p \to q,\ p\ \therefore q\)
Modus Tollens \(p \to q,\ \sim q\ \therefore \sim p\)
Generalization \(p\ \therefore p \lor q\); or \(q\ \therefore p \lor q\)
Specialization \(p \land q\ \therefore p\); or \(p \land q\ \therefore q\)
Conjunction \(p,\ q\ \therefore p \land q\)
Elimination \(p \lor q,\ \sim q\ \therefore p\); or \(p \lor q,\ \sim p\ \therefore q\)
Transitivity \(p \to q,\ q \to r\ \therefore p \to r\)
Proof by division into cases \(p \lor q,\ p \to r,\ q \to r\ \therefore r\)
Contradiction rule \(\sim p \to \mathbf{c}\ \therefore p\)

These don’t need to be memorised — they’re a reference to check your reasoning against, not a checklist to recite.

See also