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An argument is valid when no valuation makes every premise true and the conclusion false, and that is the same as its corresponding conditional being a tautology.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to state what validity is, test an argument by crossing out the rows where a premise fails and checking the conclusion on what survives, give the column of an argument's corresponding conditional, distinguish validity from soundness and from the truth of the conclusion, and say why an argument with jointly unsatisfiable premises is valid.
You can build a truth table and decide whether a formula is a tautology. Validity is that machinery pointed at a group of formulas: some premises and a conclusion, tested together on every row at once.
| Term | What it means |
|---|---|
| Premise | A statement granted for the purpose of testing an argument. |
| Conclusion | The statement claimed to follow from the premises. |
| Entailment | Every assignment satisfying all premises also satisfies the conclusion. |
| Countermodel | One assignment making every premise true and the conclusion false. |
An argument is valid when there is no valuation on which every premise is true and the conclusion is false. That is the whole definition, and three things follow from reading it literally. First, validity is about the form: the subject matter of the atoms never enters, so an argument is valid or not whatever its letters stand for. Second, it is a conditional claim — it says what holds if the premises do — so a valid argument may have false premises, and adding the claim that they are true is soundness, a different and stronger thing. Third, only the rows on which every premise survives can matter, and the rest of the table can be ignored. The same fact can be written as a single formula: $\Gamma \models \phi$ exactly when the corresponding conditional, the premises conjoined arrowing to the conclusion, is a tautology. So validity, tautology and unsatisfiability are three views of one test.
Another way: steps
Another way: example
From $P \to Q$ and $\neg Q$ conclude $\neg P$. The second premise leaves the two rows with $Q$ false; the first then removes the one with $P$ true. On the single surviving row $\neg P$ holds, so the argument is valid.
To test {P -> Q, P} entails Q, first select rows where both premises hold. P must be true. On that half of the table, P -> Q requires Q true. No true-premise row leaves Q false, so the entailment holds. A row with P false and Q false is irrelevant: it falsifies a premise and therefore is not a countermodel.
Now test {P -> Q, Q} entails P. Choosing P false and Q true makes both premises true and the conclusion false. That one row ends the test with a negative result. It does not matter whether P happens to be true in the actual case; entailment concerns what the premises guarantee.
The same idea can be expressed as a conditional: conjoin the premises and put the conclusion after an arrow. The argument is valid exactly when this corresponding conditional is a tautology. This is a test for consequence, not a new premise to assume. Keep the original premises visible so that an incorrect assignment can be diagnosed rather than merely labeled wrong.
Entailment is a relationship between a collection of premises and a conclusion. The premises entail the conclusion when every assignment that makes all premises true also makes the conclusion true. The definition does not ask whether the premises and conclusion have identical columns. A conclusion can be weaker than the premises: P & Q entails P even though P can be true while the conjunction is false.
Start an exhaustive test by identifying the common models of the premises. These are the rows on which every premise is true together. Then inspect the conclusion only on those rows for a failure. A row falsifying any premise is not a countermodel, even if it also falsifies the conclusion. It represents a situation outside the guarantee being tested. Keeping an explicit all-premises column helps prevent counting the wrong rows.
For premises P | Q and ~P, the only common model over P,Q is P false, Q true. The conclusion Q holds there, so it is entailed. If you change the conclusion to P, the same common model falsifies it, refuting entailment. The premise set has not changed; the result differs because consequence concerns a particular target formula, not a general impression that the premises are strong.
A single countermodel is sufficient to refute the entailment claim. Ten thousand confirming assignments could not compensate for one genuine exception to an every-assignment guarantee. Conversely, failure to find a countermodel after a few attempts is not enough to establish entailment. You need an exhaustive table, a complete constraint argument, or an appropriate proof whose rules preserve truth.
The vocabulary of validity expresses the same formal test for an argument: an argument is valid exactly when its premises entail its conclusion. The term soundness adds another requirement, actual truth of the premises. This lesson focuses on the guarantee within the formal representation. Evidence about actual records, policies, or observations remains a separate input to evaluating a sound argument.
A direct countermodel search begins by demanding the conclusion false and every premise true. Translate those demands into requirements on atoms and compound parts. If they can all be satisfied together, you have a countermodel. If every possible way of satisfying them leads to conflict, the entailment holds. This is the same test as a truth table organized around the particular failure you are seeking.
Suppose the premises are P -> Q and Q -> R, and the conclusion is P -> R. To falsify the conclusion, P must be true and R false. The first premise then forces Q true. But Q true with R false falsifies the second premise. There is no other way to falsify the conclusion, so the countermodel search is exhausted. The two premises entail the conditional target even without an additional premise asserting P.
Notice that last distinction. Deriving the conditional P -> R does not amount to asserting P or R. If P is false, both chain premises may still hold while R is false. The entailment guarantees the relationship expressed by the conditional, not either side independently. Checking the whole conclusion prevents an argument from gaining a stronger result than its premises support.
When a demand has alternatives, preserve them. To make P & Q false, at least one conjunct must be false; there are several assignments meeting that requirement. To make P | Q false, both must be false. Treating these demands as interchangeable can produce a false validity result by overlooking a surviving candidate. Record branches or use an exhaustive table when the alternatives are difficult to track.
A useful alternate test conjoins the premises and makes that conjunction the antecedent of a conditional whose consequent is the conclusion. This associated conditional is a tautology exactly when entailment holds. Its only false case would be all premises true and conclusion false. The construction does not add a new assumption to the original argument; it packages the same failure condition into one formula for classification.
Adding extra premises preserves an entailment already established in classical propositional logic. Any assignment satisfying the enlarged premise set also satisfies the original set, so its conclusion still holds. This does not mean the new premises are true or wise to add. It means that narrowing the set of candidate assignments cannot create a countermodel that was absent before.
Removing premises is different because it may introduce new candidate assignments. From P & Q, the conclusion Q follows. If the premise is weakened to P, Q need not follow: P true and Q false is now allowed. Whenever a premise is deleted or weakened, retest the consequence instead of carrying the old verdict over without justification.
If the premises are inconsistent, there are no all-premise-true assignments. Consequently there cannot be an assignment with all premises true and conclusion false, whatever conclusion is proposed. In classical logic an inconsistent set therefore entails every formula. This is sometimes called vacuous entailment. It does not establish that every conclusion is actually true, because the premise set cannot all be actually true under the model.
A tautological conclusion is entailed by any premise set because it is true on every assignment, including all common premise models. These two boundary cases arise for different reasons: inconsistent premises leave no candidate rows, while a tautological conclusion succeeds on every candidate row. Distinguishing them helps explain why formal validity alone does not measure an argument's usefulness or the reliability of its evidence.
Report your result with the premise set and target explicit. If entailment fails, supply a full countermodel and verify each premise. If it holds, identify why every candidate satisfies the target or why none can falsify it. Then keep any practical recommendation separate. The logical guarantee is precise and valuable, but it does not by itself select good premises, establish relevance to a decision, or verify the facts a decision relies on.
A fictional delivery office states: if a parcel has been dispatched, a tracking record exists. It also states that this parcel has been dispatched. Let D mean dispatched and K mean tracking record exists. The premises are D -> K and D, and the proposed conclusion is K. For a countermodel we would need K false. Since D is a premise, D must be true. But D true with K false falsifies D -> K. No countermodel remains, so the premises entail K.
Change the second premise to 'a tracking record exists' and propose 'the parcel has been dispatched'. The premises become D -> K and K, with D as conclusion. Now D false and K true satisfy both premises while falsifying the conclusion. A tracking record might have been prepared before dispatch. The reverse inference is invalid.
The numbers of premises and atoms are the same in the two arguments: two premises and two atoms. Their different results come from the direction of support, not their length. A checklist that merely records that an arrow and two letters are present would miss the difference.
Finally, validity does not check the office's database. If dispatch records are unreliable, the valid first argument may start from a false premise. To rely on its conclusion about an actual parcel, check the factual premise and the stated rule separately. Formal consequence tells you what follows if those inputs are true; it does not supply their evidence.
The commonest error is judging an argument by its conclusion: an argument with a true conclusion can be invalid, and an argument with a false one can be valid. The second is treating validity as a claim that the premises hold; it never is, which is why soundness is a separate word. The third is stopping the hunt after a few rows — a countermodel found settles invalidity at once, but validity is a claim about every row and needs all of them checked or an argument that covers them all.
State the target argument.
P | Q, ~P; therefore Q
The test holds both premises fixed.
Apply the denied premise.
P=F
A common model must satisfy ~P.
Apply the disjunction premise.
Q=T
With P false, the disjunction requires Q true.
Inspect the conclusion there.
Q=T on the only common model
No premise-true row falsifies the target.
State the consequence.
Valid: {P | Q,~P} entails Q
The premises guarantee the conclusion on every common model.
State the proposed inference.
P -> Q, Q; therefore P
A countermodel must retain both given premises.
Falsify the conclusion.
P=F
This is required to refute entailment.
Preserve the atomic premise.
Q=T
Q is explicitly asserted.
Check the conditional premise.
P -> Q=T
Its antecedent is false.
Verify the whole witness.
Premises T,T; conclusion F
All values belong to the same assignment.
State the consequence failure.
Invalid; countermodel P=F,Q=T
One true-premise false-conclusion assignment refutes entailment.
State the premises and target.
P -> Q, Q -> R; target P -> R
The entire conditional is the proposed conclusion.
Demand the target false.
P=T,R=F
These are its only falsifying input values.
Keep the first premise true.
Q=T
The true P forces Q through the first conditional.
Evaluate the second premise.
Q -> R=F
Q true with R false violates it.
Reject that candidate.
A premise fails
A countermodel cannot falsify any premise.
Check that the search is exhaustive.
Every falsification of P -> R requires the rejected values
No alternate target-false case remains.
State the valid conclusion.
{P -> Q,Q -> R} entails P -> R
The conclusion is a relationship, not an independent assertion of P or R.
State the proposed test row.
P=F,Q=F for premises P,P -> Q and conclusion Q
The conclusion is false on this assignment.
Check every premise.
P=F; P -> Q=T
The atomic premise fails.
Reject it as a countermodel.
Do the premises P & (Q | R) entail Q | R? Decide validity and construct a countermodel if they do not.
P & (Q | R)
∴ Q | R
valid invalid — countermodel:
Complete the row filter for premises P | Q and ~P, with conclusion Q.
Filter for all premises true.
Only FT survives: models model
~P excludes P-true rows, and the disjunction then requires Q.
Inspect Q on the surviving row.
Q=T; countermodels countermodels
The only candidate satisfies the conclusion.
State the entailment result.
valid
No true-premise row falsifies Q.
Do the premises P entail P & Q? Decide validity and construct a countermodel if they do not.
P
∴ P & Q
valid invalid — countermodel:
Do the premises P -> Q, ~Q entail ~P? Decide validity and construct a countermodel if they do not.
P -> Q
~Q
∴ ~P
valid invalid — countermodel:
D means dispatched and K means a tracking record exists. Formalize: whenever a parcel is dispatched, a tracking record exists. Do not assert the reverse.
Answer:
Do the premises P | Q, ~P, ~Q entail R? Decide validity and construct a countermodel if they do not.
P | Q
~P
~Q
∴ R
valid invalid — countermodel:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Do the premises P -> R, Q -> R, P | Q entail R? Decide validity and construct a countermodel if they do not.
P -> R
Q -> R
P | Q
∴ R
valid invalid — countermodel:
You can test an argument for validity and say what the verdict does and does not claim. Say in your own words why an argument can be valid and yet establish nothing.
14. An irrelevant false-premise row, step 3
Not a countermodel
Falsifying the conclusion is insufficient unless all premises remain true.