A network engineer observes: 'If the server is overloaded (P), then response times increase (Q).' They see that response times have increased (Q is true). What can they definitively conclude?
AThe server is overloaded — Q being true confirms P
BNothing definitive — Q being true does not guarantee P is true
CThe server is not overloaded — increased response times rule out overload
DP → Q is no longer a reliable rule, since Q occurred without confirmed P
Inferring P from Q and P→Q is the fallacy of affirming the consequent. Many things could cause increased response times (a DDoS attack, a memory leak, heavy legitimate traffic) — the conditional P→Q does not say Q can only happen if P happens. The only valid inference from Q and P→Q is that you cannot yet conclude ¬P. Contrast with modus tollens: if you observe ¬Q (normal response times), then you CAN conclude ¬P (server is not overloaded).
Question 2 Multiple Choice
Which of the following inference patterns is logically valid?
AFrom P→Q and Q→R, conclude P→R (hypothetical syllogism)
BFrom P→Q and Q, conclude P (affirming the consequent)
CFrom P→Q and ¬P, conclude ¬Q (denying the antecedent)
DFrom P∨Q and P, conclude ¬Q (disjunctive elimination)
Hypothetical syllogism (option A) is valid: if P implies Q and Q implies R, then P implies R by transitivity of implication. This is the chain rule of logic and underlies multi-step proofs. Options B and C are both classic invalid forms — common fallacies. Option D is wrong because P∨Q being true with P true tells you nothing about Q; Q could also be true.
Question 3 True / False
A valid argument with true premises must have a true conclusion.
TTrue
FFalse
Answer: True
This is the definition of validity combined with the truth of premises — it describes a *sound* argument. Validity guarantees that IF the premises are true, THEN the conclusion is true. When the premises are additionally stipulated to be true, the conclusion is forced to be true. This is precisely why valid inference rules are so powerful in mathematics: the axioms are taken as true, so valid deductions from them produce true theorems.
Question 4 True / False
An argument is valid if and mainly if its conclusion is true.
TTrue
FFalse
Answer: False
Validity is a structural property of the argument form, completely independent of whether the conclusion is actually true. 'All fish can fly; salmon are fish; therefore salmon can fly' is a valid argument (the conclusion follows necessarily from the premises) even though the conclusion is false (because a premise is false). Conversely, a conclusion can be true while the argument that reaches it is invalid. Validity concerns the logical relationship between premises and conclusion, not the truth values of either.
Question 5 Short Answer
What is the difference between a valid argument and a sound argument? Give an example of a valid but unsound argument.
Think about your answer, then reveal below.
Model answer: A valid argument has a form that guarantees the conclusion is true whenever all premises are true — the conclusion follows necessarily from the premises. A sound argument is valid AND has true premises. A valid but unsound argument: 'All mammals can breathe underwater; dolphins are mammals; therefore dolphins can breathe underwater.' The form (all A are B; x is A; therefore x is B) is valid, but the first premise is false, making the argument unsound even though the conclusion happens to be true.
The distinction matters for proof-checking: you must verify both that your logical steps are valid (the structure is correct) and that your starting assumptions are true (the premises hold). In mathematics, axioms supply the true premises; logic rules supply validity. An error in either dimension produces a flawed proof — but they are distinct types of error.