VALID AND SOUND ARGUMENTS: EXPLAINED ANSWER KEY

All records and examples are fictional. Classical propositional logic, material conditional, inclusive or. Unknown evidence does not mean false.

Case 1: Valid and sound in record A. Both premises and the conclusion are true.
The form is modus ponens: if p then q; p; therefore q. If both premises are true, p holds and the conditional rules out p being true with q false. The supplied record makes P1 and P2 true, so this particular instance also passes the truth check. This does not establish that any actual dispatch company follows such a rule. The record is a stipulated classroom situation, not a field observation.

Case 2: Valid but unsound in record B. P1 is false, P2 is true and C is false.
The modus ponens form has not changed. Validity asks what follows if all premises are true; it does not certify their truth. Record B contradicts P1 because it supplies a sealed, unscanned parcel. That explains the false conclusion without showing a defect in the inference form. Record B is not a counterexample to validity: a logical counterexample must keep every premise true, and here P1 is false.

Case 3: Valid but unsound in record C. P1 is true under this convention, P2 is false and C is true.
This is the third instance of the same valid form. The supplied record makes P2 false, so soundness fails even though the conclusion happens to match the record. Under the material-conditional convention, “if p then q” is false only when p is true and q false; with p false and q true, it is true. Do not silently read P1 as a causal claim or a claim that scanning happens only after sealing. Neither interpretation is part of this exercise.

Case 4: Invalid and therefore unsound, despite all premises and C being true in record A.
This form affirms the consequent: if p then q; q; therefore p. A counterexample is an unsealed but scanned parcel, as in record C. In that alternative situation P1 and P2 are true while C is false. The conditional gives a sufficient condition for scanning; it does not make sealing necessary. A true instance cannot rescue an invalid form. The counterexample tests the stated argument, not whether the original record was accurately copied.

Case 5: Invalid and unsound. Both premises are true under the stated convention, while C is false.
The form denies the antecedent: if p then q; not p; therefore not q. The conditional permits scanning in the absence of sealing. Record C itself is a counterexample with all premises true and the conclusion false. For this form, the fact that sealing did not happen leaves scanning open. Adding “N is scanned only if N is sealed” would change the premises; it would not vindicate the argument as originally written.

Case 6: Valid and sound in record D. This is modus tollens.
The form is if p then q; not q; therefore not p. If p were true, P1 would require q, contradicting P2. Consequently not p follows. Record D makes both premises true and makes C true. Case 5 denies p and then denies q; this case denies q and then denies p. That difference determines validity. The similar wording does not make the two inferences interchangeable.

Case 7: Valid and sound in the supplied notification record.
This is disjunctive syllogism: p or q; not p; therefore q. The first premise requires at least one of the two alternatives. The second removes p, leaving q. Allowing both alternatives in P1 does not undermine this inference, because P2 already excludes emailing. The record establishes the truth of both premises. The conclusion follows without changing inclusive “or” into exclusive “or.”

Case 8: Invalid and unsound, even though both premises and C are true in this record.
The premises also allow a notice that was both emailed and printed. In that counterexample P1 and P2 stay true while C becomes false. An inclusive “or” does not let you deny one alternative merely by affirming the other. If the record’s “not printed” fact is added as a premise, the conclusion follows trivially from that new premise. It was not a premise in the displayed argument; background information used to check truth must not be smuggled into the validity test.

Case 9: Valid; soundness is not established from the supplied evidence.
Conjunction elimination has the form p and q; therefore p. If the conjunction is true, each part is true, so this form is valid. The truth of P1 is unknown to the learner. That does not make P1 false and does not create a third logical truth value. If P1 is actually true, this argument is sound; if it is false, the argument is unsound. The justified report now is “valid, premise truth unverified,” followed by a request for the actual check and filing evidence.

Case 10: Valid and sound in this record. The conclusion is a conditional, not an assertion that checking occurred.
The form is if p then q; if q then r; therefore if p then r. Assuming p, P1 gives q and P2 gives r. There is no assignment with both premises true and the conditional conclusion false. In this supplied record p is false, q and r true; both premises and the conclusion are true under the stipulated convention. Do not replace C with “it received a receipt because it was checked”: that introduces an occurrence and a causal claim absent from the argument.

References:
https://iep.utm.edu/val-snd/
https://openstax.org/books/introduction-philosophy/pages/5-3-arguments
https://openstax.org/books/introduction-philosophy/pages/5-4-types-of-inferences