Self-Assessment Quiz: Rules of Inference

Twenty questions to check your grasp of validity, the inference rules, the fallacies, the quantifier rules, and derivations. Answer from memory before opening the key. Aim for 16+.

Question 1

An argument is valid exactly when:

A) its conclusion is true B) all its premises are true C) no truth assignment makes every premise true while the conclusion is false D) it uses at least one named inference rule

Question 2

An argument is sound when it is:

A) valid B) valid and all premises are actually true C) valid or all premises are true D) made only of modus ponens steps

Question 3

"The conclusion of this argument is true, so the argument is valid." This reasoning is:

A) correct — a true conclusion certifies validity B) incorrect — validity is about the premise-conclusion connection, not the outcome C) correct only if the premises are also true D) incorrect only for quantified arguments

Question 4

Which argument form is modus tollens?

A) $p \rightarrow q,\ p \ \therefore\ q$ B) $p \rightarrow q,\ \neg q \ \therefore\ \neg p$ C) $p \rightarrow q,\ q \ \therefore\ p$ D) $p \rightarrow q,\ \neg p \ \therefore\ \neg q$

Question 5

"If the code is correct, the tests pass. The tests passed. Therefore the code is correct." This is:

A) valid (modus ponens) B) valid (modus tollens) C) invalid — affirming the consequent D) invalid — denying the antecedent

Question 6

From $p \lor q$ and $\neg p$, which rule lets you conclude $q$?

A) addition B) simplification C) disjunctive syllogism D) conjunction

Question 7

From $p \rightarrow q$ and $q \rightarrow r$, hypothetical syllogism concludes:

A) $q$ B) $r$ C) $p \rightarrow r$ D) $p \land r$

Question 8

"From $p$, conclude $p \lor q$ for any $q$ whatsoever." This rule is:

A) invalid — you can't introduce an arbitrary $q$ B) valid — it is addition C) valid only when $q$ is true D) the same as simplification

Question 9 (True/False, justify)

True or false: A valid argument can have all true premises and a false conclusion. Justify in one sentence.

Question 10

Denying the antecedent has the form:

A) $p \rightarrow q,\ \neg q \ \therefore\ \neg p$ B) $p \rightarrow q,\ \neg p \ \therefore\ \neg q$ C) $p \rightarrow q,\ q \ \therefore\ p$ D) $p \rightarrow q,\ p \ \therefore\ q$

Question 11

Both fallacies of this chapter (affirming the consequent, denying the antecedent) share the same counterexample row. It is:

A) $p=$T, $q=$T B) $p=$T, $q=$F C) $p=$F, $q=$T D) $p=$F, $q=$F

Question 12

The resolution rule, $p \lor q,\ \neg p \lor r \ \Rightarrow\ q \lor r$, is the engine behind:

A) truth tables B) Prolog and many SAT solvers C) the contrapositive D) universal generalization

Question 13 (True/False, justify)

True or false: A fallacy can sometimes also be a valid argument form. Justify in one sentence.

Question 14

Universal instantiation (UI) lets you go from:

A) $P(c)$ to $\forall x\,P(x)$ B) $\forall x\,P(x)$ to $P(c)$ for any element $c$ C) $\exists x\,P(x)$ to $P(c)$ for a fresh $c$ D) $P(c)$ to $\exists x\,P(x)$

Question 15

The side condition on universal generalization (UG) is that the element $c$ must be:

A) a fresh name used nowhere else B) equal to $0$ C) genuinely arbitrary — no special assumption was made about it D) an element you can compute explicitly

Question 16

Why must existential instantiation (EI) use a fresh name?

A) to keep the proof short B) because you don't get to choose which element satisfies $P$, so the name must carry no prior commitments C) because $\exists$ ranges over an infinite domain D) it doesn't — any name works

Question 17 (Short answer)

In one sentence, explain the difference between an argument that is invalid and one that is valid but unsound, and how the fix differs.

Question 18

In the method for building a derivation (§3.5), you should:

A) work forward from the premises and hope to reach the goal B) work backward from the goal to choose rules and sub-goals, then write the proof forward C) build the full truth table first D) cite later lines when convenient

Question 19

A derivation line reads "7. $r$ — Modus ponens, 5, 6," where line 5 is $r \rightarrow s$ and line 6 is $r$. Is line 7 justified?

A) Yes — that's exactly modus ponens B) No — modus ponens from $r \rightarrow s$ and $r$ yields $s$, not $r$ C) Yes, but only if $s$ is also true D) No — modus ponens needs three inputs

Question 20 (Short answer)

The chapter says a derivation is "hard to find but easy to check." State, in one sentence, what makes checking easy, and name the later chapter where this find/check asymmetry returns as a major theme.

Answer Key

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