Key takeaways
- An argument is valid when its form makes it impossible for all the premises to be true and the conclusion false.
- An argument is sound when it is valid and all its premises are actually true, so a sound argument's conclusion is always true.
- Validity depends only on form; you can judge it without knowing whether any premise is true.
- To test validity with a truth table, look only at the rows where every premise is true; if the conclusion is false in any of them, the argument is invalid.
- A flawed argument does not make its conclusion false: invalid and unsound arguments can still reach true conclusions.
In conversation, "that's a valid point" can mean little more than "that's a fair point". In logic the word is narrower and more useful. It describes the shape of an argument, not whether its claims are true, and it sits alongside a second idea, soundness, that does care about truth.
The two definitions
The Internet Encyclopedia of Philosophy defines both for deductive arguments; in paraphrase:
- An argument is valid when its form rules out true premises with a false conclusion.
- An argument is sound when it is valid and every premise is in fact true.
OpenStax's Contemporary Mathematics splits the work slightly differently, calling an argument sound if all its premises are true and noting that, for a conclusion to be proven true, the argument must be both valid and sound. The two accounts reach the same place: a good argument needs a valid form and true premises.
So validity is a conditional guarantee: if the premises are true, the conclusion is too. Soundness cashes that guarantee in, because the premises really are true.
Four combinations
| Premises all true | Some premise false | |
|---|---|---|
| Valid form | Sound: conclusion guaranteed true | Valid but unsound: conclusion may be false |
| Invalid form | Unsound: conclusion not guaranteed | Unsound |
Examples of each:
- Valid and sound. Every multiple of 6 is even. 18 is a multiple of 6. So 18 is even.
- Valid, unsound, false conclusion. All birds can fly. Penguins are birds. So penguins can fly. The form is fine; the first premise is false.
- Valid, unsound, true conclusion. All cats are reptiles. All reptiles are mammals. So all cats are mammals. Both premises are false and the conclusion happens to be true. Validity was never the problem.
- Invalid, with true premises and a true conclusion. If a number is a multiple of 4, it is even. 12 is even. So 12 is a multiple of 4. Everything stated is true, but the argument affirms the consequent. Run the same form on 10 (10 is even, so 10 is a multiple of 4) and true premises produce a false conclusion.
Example 4 shows the practical meaning of invalidity. An invalid form is one that can lead from true premises to a false conclusion, even if it does not in a particular case. That is why "the conclusion is true, so the argument is fine" is not a defence.
Why validity depends only on form
Replace every content word in example 2 with a letter: all A are B; C is an A; so C is a B. Any argument with that shape is valid, whether it is about birds, numbers or planets. You can judge validity without knowing anything about the subject, which is what makes logic portable. The same is true of the classical forms in our syllogisms guide: each is valid or invalid by its shape alone.
Soundness, by contrast, requires knowing whether the premises are true, which is a question about the world, not about logic.
Testing validity with a truth table
For arguments built from propositional statements, the truth table gives a mechanical test. Levin's open textbook states the principle: an argument is valid provided the conclusion must be true given that the premises are true.
- Translate the premises and the conclusion into symbols.
- Build a truth table with a column for each premise and for the conclusion.
- Mark the critical rows, where every premise is true.
- If the conclusion is true in every critical row, the argument is valid. If it is false in even one, the argument is invalid.
Test: p → q, q → r, so p → r.
| p | q | r | p → q | q → r | p → r |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | T | F | T | F | F |
| T | F | T | F | T | T |
| T | F | F | F | T | F |
| F | T | T | T | T | T |
| F | T | F | T | F | T |
| F | F | T | T | T | T |
| F | F | F | T | T | T |
The critical rows are 1, 5, 7 and 8, the rows where both premises are true. The conclusion is true in all four, so the argument is valid. This is the chain rule, also called hypothetical syllogism, one of the standard rules of inference.
Test: p → q, ¬p, so ¬q (denying the antecedent). Row p = F, q = T makes both premises true and the conclusion ¬q false. One critical row with a false conclusion: invalid.
An equivalent test is to check whether (premise₁ ∧ premise₂ ∧ …) → conclusion is a tautology. The two methods always agree, as explained in tautology, contradiction or contingency?.
Where this matters outside a logic class
- Mathematical proof. A proof is a valid argument whose premises are definitions, axioms and already-proved results. Because those premises are accepted as true, a correct proof is sound, and its conclusion is established.
- Evaluating claims. When you disagree with a conclusion, there are exactly two places to look: a false premise, or an invalid step. Naming which one is the problem makes a disagreement productive.
- Spotting fallacies. Some common reasoning mistakes are invalid forms that resemble valid ones, such as affirming the consequent.
One last point the IEP makes explicitly: an argument being flawed does not show its conclusion is false. Invalid and unsound arguments can have true conclusions. As the IEP puts it, one cannot reject a conclusion simply by finding that a given argument for it is flawed.
Practise on your own arguments
Write two premises and a conclusion, decide whether the form is valid, then decide whether the premises are true. The argument-validity checker in Discrete Math: Study & Quiz helps with the first question; the second is up to you. Our education apps include logic games as well, and the discrete math hub collects the logic series in order.
Good to know
Frequently asked questions
What is the difference between a valid and a sound argument?
If an argument is valid, must its conclusion be true?
Can a sound argument have a false conclusion?
How do I test whether an argument is valid?
Sources
- Validity and Soundness (opens in a new tab)Internet Encyclopedia of Philosophy — accessed
- 2.7 Logical Arguments (Contemporary Mathematics) (opens in a new tab)OpenStax — accessed
- Propositional Logic: Deductions (Discrete Mathematics: An Open Introduction, 3rd edition, Oscar Levin) (opens in a new tab)Open Math Books — accessed
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