Validity and soundness · printable packet

Modus ponens, modus tollens, disjunctive and hypothetical syllogism, the three fallacies they are confused with, and why a form applies only to whole formulas.

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.

1. What you will learn

By the end of this lesson you will be able to name the standard valid forms and their schemas, recognize affirming the consequent, denying the antecedent and affirming a disjunct, say which row breaks each of them, write the conclusion a named form produces, and decide whether an argument is an instance of a form at all.

2. What you already have

You can test an argument for validity and build a countermodel when it fails. This lesson collects the handful of patterns that come up constantly, so that the test need not be rerun every time, and names the near-misses that are the reason it sometimes must be.

3. Terms to use precisely

4. Named forms and formal fallacies

Term	What it means
Valid	No assignment makes every premise true and the conclusion false.
Sound	Valid and with all premises actually true.
Affirming the consequent	The invalid move from P -> Q and Q to P.
Denying the antecedent	The invalid move from P -> Q and ~P to ~Q.

Four patterns cover most of what is used. Modus ponens: from $\phi \to \psi$ and $\phi$, conclude $\psi$. Modus tollens: from $\phi \to \psi$ and $\neg \psi$, conclude $\neg \phi$. Disjunctive syllogism: from $\phi \vee \psi$ and $\neg \phi$, conclude $\psi$. Hypothetical syllogism: from $\phi \to \psi$ and $\psi \to \chi$, conclude $\phi \to \chi$. Each has a near-neighbor that fails. Affirming the consequent — from $\phi \to \psi$ and $\psi$, conclude $\phi$ — and denying the antecedent — from $\phi \to \psi$ and $\neg \phi$, conclude $\neg \psi$ — are both broken by the single row where the antecedent is false and the consequent true, and both amount to reading $\to$ as $\leftrightarrow$. Affirming a disjunct fails because $\vee$ allows both. A form applies only to instances: the letters stand for whole formulas, so affirming one conjunct of a compound antecedent is not affirming the antecedent.

Another way: steps

Write the first premise as a conditional or a disjunction and name its parts.
Ask what the second premise does: affirm or deny, and which part.
Name the form, and recall whether it is valid.
If in doubt, test it: the four-row table settles it in a minute.

Another way: example

From $(P \vee Q) \to R$ and $\neg R$, conclude $\neg (P \vee Q)$. The consequent is denied and the whole antecedent is what gets negated, so this is modus tollens, and it is valid.

5. Two separate questions about a reason

Validity asks whether the conclusion follows from the premises. Soundness also asks whether those premises are true. An argument can therefore be valid but unsound. For example, 'If the room is empty, the lights are off; the room is empty; therefore the lights are off' has valid form even if an inspection reveals that somebody is in the room. A false premise prevents soundness without changing the form.

An invalid argument can happen to have true premises and a true conclusion. Imagine a light that is in fact on: 'If the switch is on, the light is on; the light is on; therefore the switch is on.' Even if the switch is also on, a separate power source shows why the premises do not guarantee it. Actual truth and necessary support are different tests.

The named forms in this lesson record patterns you can check with truth tables. They apply to whole formulas, including compound formulas, rather than to isolated letters inside them. A valid form plus verified true premises guarantees a true conclusion. A true conclusion alone does not tell you whether its offered argument is good. When evidence about a premise is missing, say that soundness is not established; do not invent the missing fact.

6. Evaluate the inference before evaluating its evidence

Validity concerns whether the premises guarantee the conclusion. It is assessed by considering every assignment that makes all premises true. Soundness concerns a valid argument whose premises are also actually true. These are different properties, so a careful evaluation records both. You can settle the form from the stated formulas while remaining uncertain about whether a factual premise accurately describes the world.

Suppose an argument says P -> Q, P, therefore Q. Its form is valid because the conditional cannot remain true with P true and Q false. Now suppose an inspection establishes P false in the actual case. That discovery makes this instance unsound without changing its form. The same inference pattern still preserves truth whenever its premises are true; this instance simply lacks all-true premises.

If both premises of a valid argument are actually true, its conclusion must be true. This is the benefit of soundness. There cannot be a sound argument with a false conclusion in the stated classical framework. An alleged example must contain a false premise, an invalid inference, or a change in meaning between the formal representation and the described situation.

The reverse diagnosis needs care. A true conclusion does not show that the argument is sound. It could arise from false premises through a valid form, or from true premises through an invalid form. A conclusion might be correct for reasons entirely absent from the argument offered. Evaluate the support itself rather than giving it credit merely because you agree with the result.

Keep uncertainty distinct from established falsehood. If a premise has not been checked, you may know the form is valid while soundness remains unestablished. Lack of evidence does not itself make the premise false. In a written review, distinguish 'unsound because this premise is false' from 'soundness has not yet been established because this premise needs evidence'. That distinction prevents turning an honest evidence gap into an unsupported accusation.

7. Recognize the valid rule and its invalid lookalikes

Modus ponens moves from a conditional and its antecedent to its consequent. Modus tollens moves from a conditional and the denial of its consequent to the denial of its antecedent. Both can be checked by demanding their conclusions false and observing that a premise would then fail. The rules operate on whole formulas, not only on single statement letters.

Affirming the consequent is the tempting reverse move: P -> Q, Q, therefore P. A row with P false and Q true refutes it. The result Q may have a different sufficient condition, or may hold without P for some other reason. A premise that makes P sufficient for Q does not make P necessary. To justify the reverse, an additional premise such as Q -> P would be needed and would itself require support.

Denying the antecedent is another invalid pattern: P -> Q, ~P, therefore ~Q. The same row P false, Q true is a countermodel. It satisfies the conditional and the denial of P while falsifying the denial of Q. Ruling out one sufficient route does not rule out every route to the result. The invalidity lies in that missing restriction, not in the mere presence of negations.

Compound substitution preserves a valid pattern when each repeated component is unchanged. From (P | Q) -> R and P | Q, modus ponens yields R. From P -> (Q & R) and ~(Q & R), modus tollens yields ~P. But replacing the complete denied consequent by ~Q alone changes the displayed rule inputs. You would need an additional justified inference connecting the available formulas rather than pretending they already match.

When classifying an argument, do not rely only on a familiar name. Write its exact premises and target, then test the pattern. An argument can contain a conditional without instantiating either MP or MT. It can also be valid through a longer combination of rules. The definition of validity is the foundation; named forms are convenient, previously justified cases of that definition.

8. Keep a transparent two-part audit

An inference audit should explain whether a countermodel exists or why it cannot exist. A premise audit should identify what would establish each premise's truth in the actual case. For a room-status argument, the latter might involve a current closure notice, a precise definition of unavailable, and an observation about which date the notice covers. A truth table supplies none of those records by itself.

If a premise is a general rule, check its scope. 'Every approved record has a signature' differs from 'most approved records have signatures' and from 'records approved after July require signatures'. Translating all three as the same unrestricted conditional erases important qualifications. A formally valid derivation from an inaccurate translation does not establish a sound argument about the original situation.

If a premise reports an observation, check its time, object, and reliability. The light being on yesterday is not the same atomic claim as the light being on now. Two reports about different rooms do not supply a contradiction about one room. A stable atom key connects the formal work to the factual audit and prevents apparent validity from depending on a silent change of subject.

Validity also differs from how persuasive an argument feels. A vivid example may make an invalid inference seem convincing; an unfamiliar symbolic proof may be valid despite being hard to follow. Good communication should make the valid dependency clear, but confidence, popularity, and rhetorical force are not substitutes for the all-premise-true test.

Finally, a sound argument can still be unhelpful to an audience if its premises are harder to establish than its conclusion or if it merely repeats the conclusion among the premises. Soundness is an important logical property, not a complete account of informative explanation, relevance, or responsible inquiry. Other courses examine those additional standards. Here the central achievement is to keep the guarantee of the inference distinct from the actual truth and evidence of its starting points.

9. Auditing a room-status argument

An organizer argues: if the building is closed, the meeting room is unavailable; the building is closed; therefore the meeting room is unavailable. Let C mean building closed and U mean room unavailable. The form C -> U, C, therefore U is valid. A false conclusion would require U false; with C true that would make the conditional premise false, so no countermodel exists.

The organizer still needs evidence for both premises. Suppose the building is actually open. The second premise is false, and this particular argument is unsound. The room might nevertheless be unavailable because of repairs. A true conclusion would not repair the false premise or make the argument sound.

Now reverse the reasoning: if the building is closed, the room is unavailable; the room is unavailable; therefore the building is closed. C false and U true give a countermodel. Repairs supply a possible concrete interpretation of that assignment. This argument is invalid even on a day when the building really is closed.

A useful audit therefore has two columns: inference check and premise evidence. Inference checking can be completed from the declared formulas. Premise evidence requires the relevant building notice and room status record. Neither column replaces the other. If a notice is missing, record that uncertainty rather than treating a valid arrow pattern as proof of the actual building's status. The logical model is intentionally small, with two atoms, so that this distinction remains visible.

10. Where this goes wrong

The commonest error is reading a conditional as a biconditional, which is what turns modus ponens into affirming the consequent and modus tollens into denying the antecedent. The second is matching a form to part of a formula: affirming $P$ when the antecedent is $P \wedge Q$ matches nothing. The third is treating a name as evidence — an argument is valid because no row breaks it, and the name is a record of that, not a substitute for it.

11. A valid form with a false actual premise

12. A true conclusion from an invalid inference

13. A sound modus tollens argument in a stipulated model

14. Missing evidence is not a false premise

15. Guided practice

Argument: P -> Q, P; therefore Q. Actual assignment is P=T,Q=F. Write the validity symbol (V for valid, I for invalid), the number of actually false premises, and the soundness symbol (S for sound, U for unsound).

16. Guided practice

For P -> Q,P; therefore Q, assume actual P=F,Q=T. Count false premises, then count countermodels to the inference form across all assignments.

17. Guided practice

Test P -> (Q | R), P; therefore Q | R. Decide validity and give a countermodel only if invalid.

18. Practice

Argument: P -> Q, Q; therefore P. Actual assignment is P=T,Q=T. Write the validity symbol (V for valid, I for invalid), the number of actually false premises, and the soundness symbol (S for sound, U for unsound).

19. Practice

C means the building is closed and U means the room is unavailable. Formalize the premise: if the building is closed, the room is unavailable.

20. Somewhere new

Argument: P -> Q, ~P; therefore ~Q. Actual assignment is P=F,Q=F. Write the validity symbol (V for valid, I for invalid), the number of actually false premises, and the soundness symbol (S for sound, U for unsound).

21. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

22. Test question

Argument: P -> Q, ~Q; therefore ~P. Actual assignment is P=F,Q=F. Write the validity symbol (V for valid, I for invalid), the number of actually false premises, and the soundness symbol (S for sound, U for unsound).

23. What you can do now

You can name the form of an argument, say whether it is valid, and produce the row that breaks it when it is not. Say in your own words why affirming the consequent and denying the antecedent are broken by the same valuation.

Working for the steps left to you

Soundness is not established

An unchecked premise is not thereby proven false.