Valid vs sound arguments: definitions, examples and a truth-table test for validity

By Muaz Notiar, founder of Tangibool Published 7 November 2026 10 min read

An argument is valid when its conclusion must be true whenever all its premises are true; validity is about form. An argument is sound when it is valid and its premises really are true, so a sound argument always has a true conclusion.

This guide defines both, sorts examples into a 2 × 2 grid of valid or invalid with true or false premises, and shows the truth-table test for validity step by step. That test is a mechanical check of form, which a free tool can run; whether the premises are true is a question about the world, which no truth table can answer.

Definitions

An argument is a list of premises and a conclusion that is claimed to follow from them.

• Valid. There is no possible situation in which all the premises are true and the conclusion is false. If the premises were true, the conclusion would have to be true. See valid argument.
• Invalid. There is at least one possible situation in which all the premises are true and the conclusion is false. That situation is a counterexample.
• Sound. Valid, and all the premises are actually true. See sound argument.
• Unsound. Invalid, or valid with at least one false premise.

Validity does not care whether the premises are true. It asks a hypothetical question: if they were true, could the conclusion be false? Soundness adds the factual question: are they true?

The 2 × 2 grid

One example in each cell:

1. Valid, premises true: sound. If an animal is a mammal, it is warm-blooded. A whale is a mammal. So a whale is warm-blooded. The form is modus ponens, which is valid, and both premises are true.
2. Valid, a premise false: unsound. If an animal is a fish, it can fly. A salmon is a fish. So a salmon can fly. The form is the same, so it is just as valid, but the first premise is false, and so is the conclusion.
3. Invalid, premises true: unsound. If a city is in France, it is in Europe. Paris is in Europe. So Paris is in France. Every statement here is true, conclusion included, but the argument is not valid: its form is affirming the consequent, and the counterexample is a city in Europe that is not in France, such as Madrid. The true conclusion is luck, not logic.
4. Invalid, a premise false: unsound. If 2 + 2 = 5, then 3 is even. 3 is even. So 2 + 2 = 5. Same invalid form, and a false premise as well.

Three facts follow from the definitions and are worth learning as they stand:

• A valid argument with true premises always has a true conclusion. That is what validity is for.
• A valid argument can have a false conclusion, but only if at least one premise is false.
• An invalid argument can have true premises and a true conclusion. Example 3 is one.

Testing validity with a truth table, step by step

For arguments built from statements joined by “and”, “or”, “not” and “if … then”, validity can be tested mechanically.

The argument: if the battery is charged, the car starts. If the car starts, I drive to work. The battery is charged. So I drive to work.

Step 1. Give each simple statement a letter. b: the battery is charged. s: the car starts. w: I drive to work.

Step 2. Write the premises and the conclusion in symbols. Premises b → s, s → w and b. Conclusion w.

Step 3. Make the truth table. Three letters give 2³ = 8 rows. One column per premise and one for the conclusion. Row 0 has every letter false:

Computed by the Tangibool engine

Step 4. Find the rows where every premise is true. Here there is one: row 7.

Step 5. Check the conclusion on those rows. On row 7 the conclusion w is true. No row has true premises and a false conclusion, so the argument is valid. The argument validity checker does the same test and reports that the conclusion follows, and the conditional “premises imply conclusion” is a tautology, true on every row.

An invalid argument, tested the same way

The argument: if the store is open, the lights are on. If the lights are on, someone is inside. Someone is inside. So the store is open. With o, l and i for the three statements:

Computed by the Tangibool engine

All three premises are true on rows 1, 3 and 7. On rows 1 and 3 the conclusion is false, so the argument is invalid, and either row is a counterexample. Row 1 reads: the store is closed, the lights are off, and someone is inside, a security guard on a night round. The argument validity checker gives that same row, and the conditional behind the argument is not a tautology.

A row like this is the strongest way to show an argument invalid: it describes a situation the arguer cannot rule out.

The checker tests form; you test the facts

The validity checker treats each letter as a statement that could be true or false and looks at every combination. It knows nothing about batteries, whales or Paris. So it can tell you whether an argument is valid, but never whether it is sound. To judge soundness you need to check each premise against the world, which is the work of evidence, not of logic.

That division is useful. When someone disagrees with a conclusion, the argument has to fail in one of two ways: the form is invalid, which a checker can show, or a premise is false, which only evidence can show. Knowing which saves a lot of arguing past each other.

Valid argument forms worth knowing

The invalid look-alikes are affirming the consequent (if p then q; q; so p), denying the antecedent (if p then q; not p; so not q) and affirming a disjunct (p or q; p; so not q). The worked examples for modus ponens, modus tollens, affirming the consequent and denying the antecedent show each table.

Two strange cases of validity

Two kinds of argument are valid for reasons that have nothing to do with their subject:

• Inconsistent premises. If the premises cannot all be true together, there is no row where they are all true, so there is no counterexample, and the argument is valid whatever the conclusion. From “p” and “not p”, any q follows. Such an argument is valid but can never be sound. See inconsistent premises.
• A conclusion that is always true. If the conclusion is a tautology, such as “q or not q”, no row can make it false, so it follows from anything.

Both are correct by the definition, and both are reasons why a good checker also reports whether the premises can be true together. Tangibool's argument checker shows that second verdict beside the first.

Validity, soundness and other kinds of argument

Validity and soundness belong to deductive arguments, where the conclusion is claimed to follow with certainty. Inductive arguments, such as “every swan I have seen is white, so all swans are white”, claim only that the conclusion is probable. They are called strong or weak, and a strong inductive argument with true premises is called cogent. Inductive arguments are never valid in the deductive sense: the black swans of Australia are the counterexample.

Arguments about “all” and “some”, such as “all mammals are warm-blooded”, belong to predicate logic. A truth table can still test their propositional shape, as in example 1, by talking about one individual: “if the whale is a mammal, it is warm-blooded”.

Practice: valid or invalid, sound or unsound?

1. If it snows, school is closed. It is snowing. So school is closed. Valid (modus ponens). Sound if, on that day, the rule really holds and it really is snowing.
2. All birds can fly. Penguins are birds. So penguins can fly. Valid, unsound: the first premise is false, and so is the conclusion.
3. If you are in London, you are in England. You are in England. So you are in London. Invalid (affirming the consequent): you could be in Manchester. Unsound whatever the facts.
4. Either the meeting is on Monday or on Tuesday. It is not on Monday. So it is on Tuesday. Valid (disjunctive syllogism). Sound if those really were the only two days.
5. If a number is divisible by 4, it is even. 10 is not divisible by 4. So 10 is not even. Invalid (denying the antecedent), and the conclusion is false: 10 is even.
6. If the alarm is set, the light is red. The light is not red. So the alarm is not set. Valid (modus tollens).

Common mistakes

1. Calling an argument valid because its conclusion is true. Example 3 has a true conclusion and an invalid form.
2. Calling an argument invalid because a premise is false. Example 2 is valid with a false premise; it is unsound, not invalid.
3. Saying a premise or a conclusion is valid. Statements are true or false; arguments are valid or invalid.
4. Treating “sound” as “convincing”. A sound argument can still be unconvincing to someone who doubts a premise, and an unsound one can be persuasive.
5. Testing only the rows you can think of. Validity is about every possible combination; the truth table, or the checker, makes sure none is missed.

Test an argument yourself

1. Give each simple statement a letter and write the premises and the conclusion with -> for “if … then”, and AND, OR and NOT.
2. Open the argument validity checker, put one premise on each line and the conclusion in its own box.
3. If it says the conclusion does not follow, read the counterexample back into words: it is the situation that breaks the argument. Then decide, separately, whether the premises are true.

Frequently asked questions

What is a valid argument?

An argument in which the conclusion must be true whenever all the premises are true. Validity depends on the form of the argument, not on whether its premises are actually true.

What is a sound argument?

A valid argument whose premises are all true. Its conclusion is therefore true.

What is the difference between valid and sound?

Validity is about the connection between premises and conclusion; soundness also requires the premises to be true. Every sound argument is valid, but a valid argument with a false premise is unsound.

Can a valid argument have a false conclusion?

Yes, but only if at least one premise is false. With all premises true, a valid argument cannot have a false conclusion.

Can an invalid argument have a true conclusion?

Yes. “If a city is in France, it is in Europe; Paris is in Europe; so Paris is in France” has true premises and a true conclusion, but the form is invalid.

What makes an argument valid?

The absence of any counterexample: no possible situation makes all the premises true and the conclusion false. A truth table checks every situation for arguments built from “and”, “or”, “not” and “if”.

How do you test whether an argument is valid?

Translate it into symbols, make a truth table with a column for each premise and the conclusion, find the rows where every premise is true, and check that the conclusion is true on all of them.

Can an argument be sound but not valid?

No. Soundness includes validity: an argument is sound only if it is valid and its premises are true.

Related pages

• Guides: rules of inference
• Glossary: valid argument, sound argument, premise, conclusion, counterexample
• Lesson 8: arguments and validity and Lesson 9: counterexamples
• The argument validity checker
• Practice: argument validity and logic exercises for philosophy courses, with answers

Written by Muaz Notiar. The Tangibool engine (semantics tangibool-semantics/1, engine 1.0.0) computed the 2 truth tables marked “Computed by the Tangibool engine”, and the 11 tool results the guide links to were re-run through the tools' own code in the build of 2026-11-07 and match what the text says. © 2026 Foreranger LTD.