Propositional Logic · step 5 of 13

Logical consequence & valid arguments

We write Γ ⊨ A to say that A is a logical consequence of the premises Γ: every valuation that makes all of Γ true also makes A true. An argument is valid exactly when its conclusion is a consequence of its premises — and validity says nothing about whether the premises are actually true. A valid argument with false premises is still valid; a true conclusion reached from irrelevant premises is still invalid.

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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.

Write the argument symbolicallyList the premises P₁ … Pₙ and the conclusion C using consistent variables. Most marks lost here come from inconsistent letters between premises.
Look for a counter-valuation firstTry to make every premise true and the conclusion false. Success means the argument is invalid and you have a complete refutation.
If that fails, show it cannot be doneArgue that the demands are contradictory, or equivalently that P₁ ∧ … ∧ Pₙ → C is a tautology.
State the verdict in wordsSay "valid" or "invalid", and for invalid give the full assignment. A counter-valuation with an unassigned variable does not refute anything.

Worked example

Is the argument p → q, q ⊨ p valid?

Try to make both premises true and the conclusion false: we need p = F.
Set q = T. Then premise q holds.
With p = F, the premise p → q is F → T, which is true.
So both premises are true while the conclusion p is false.

Answer. Invalid — the counter-valuation p = F, q = T makes the premises true and the conclusion false. This is the classic fallacy of affirming the consequent.

Where marks get dropped

These are the specific errors that cost credit on logical consequence & valid arguments questions — QED's rubric penalises each of them separately.

Confusing ⊨ with →. The arrow is a connective inside a formula; ⊨ is a claim about valuations made in the metalanguage. Writing "p ⊨ q ∧ r" inside a formula is a syntax error.
Judging validity by whether the conclusion sounds true. Validity depends only on the form — check the valuations, not the plausibility.
Confirming validity by checking a handful of rows. A single unchecked row can hide the counter-valuation.

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Answer in real notation

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Logical consequence & valid arguments — frequently asked questions

What is the difference between valid and sound?

Valid means the conclusion follows from the premises. Sound means valid AND the premises are actually true. Propositional logic exams almost always ask about validity only.

Does Γ ⊨ A mean I can prove A from Γ?

Yes for propositional logic, because the standard proof systems are both sound and complete: ⊨ and ⊢ pick out exactly the same pairs. That equivalence is a theorem, not a definition.

How do I show an argument with three premises is invalid?

One valuation suffices: give the truth value of every variable, show each premise evaluates to true, and show the conclusion evaluates to false.

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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