Deductive reasoning: a primer

• A. The argument is invalid because whales live in the ocean.
• B. The argument is valid because the conclusion necessarily follows from the premises. ✓
• C. The argument is invalid because the second premise is a generalization.
• D. The argument is valid only if whales are fish.

Why: This is a classic valid syllogism. If all mammals are warm-blooded and whales are mammals, the conclusion that whales are warm-blooded must be true. Validity concerns the logical structure, and here the conclusion necessarily follows from the premises.

• A. Modus tollens
• B. Affirming the consequent
• C. Modus ponens ✓
• D. Denying the antecedent

Why: This is modus ponens: from 'If P, then Q' and 'P', we conclude 'Q'. It is one of the most fundamental valid forms of deductive reasoning.

• A. It is valid because it follows modus ponens.
• B. It is valid because all even numbers are divisible by 4.
• C. It is invalid because it affirms the consequent. ✓
• D. It is invalid because 10 is not an even number.

Why: The argument takes 'If P, then Q' and 'Q', and concludes 'P'. This is affirming the consequent, a logical fallacy. The premise that 10 is even does not establish that 10 is divisible by 4; for example, 10 is not divisible by 4.

• A. Its conclusion is true, regardless of its premises.
• B. It is valid and all of its premises are true. ✓
• C. Its premises are persuasive to most readers.
• D. It has at least two premises and one conclusion.

Why: Soundness requires both validity and true premises. A valid argument with a false premise is unsound, and an argument with a true conclusion can still be unsound if its reasoning is invalid.

• A. A valid argument must have true premises, while a truthful argument must follow a valid form.
• B. Validity concerns whether the conclusion must follow from the premises; truth concerns whether the premises and conclusion actually match reality. ✓
• C. Validity and truth are the same thing, measured in different ways.
• D. Truth applies only to conclusions, while validity applies only to premises.

Why: Validity is a property of the argument's structure: if the premises were true, the conclusion would have to be true. Truth is a property of individual statements matching reality. An argument can be valid with false premises, and a true statement can appear in an invalid argument.

• A. Close the door.
• B. The sky is blue. ✓
• C. What time is it?
• D. Wow, that is amazing!

Why: A proposition is a declarative statement that is either true or false. 'The sky is blue' is declarative and has a definite truth value, while commands, questions, and exclamations do not have truth values.

• A. P ∧ Q
• B. P ∨ Q
• C. P → Q ✓
• D. ¬P → Q

Why: The conditional 'If P, then Q' is symbolized as P → Q. The conjunction (∧) means 'and,' the disjunction (∨) means 'or,' and the negation in D changes the meaning entirely.

• A. Only P is true
• B. Only Q is true
• C. Both P and Q are true ✓
• D. At least one of P or Q is true

Why: A conjunction is true only when both of its component propositions are true. If either part is false, the whole conjunction is false.

• A. No birds can fly.
• B. Some birds cannot fly. ✓
• C. All birds cannot fly.
• D. Some birds can fly.

Why: The negation of 'All birds can fly' is 'It is not the case that all birds can fly,' which is equivalent to 'Some birds cannot fly.' Saying 'No birds can fly' goes too far, since it asserts something stronger than the negation requires.

• A. Conjunction (∧)
• B. Inclusive disjunction (∨) ✓
• C. Conditional (→)
• D. Biconditional (↔)

Why: An inclusive disjunction 'P ∨ Q' is true whenever at least one part is true, so it is false only when both P and Q are false. The conjunction is false whenever either part is false, and the conditional is false only when the antecedent is true and the consequent is false.

• A. Its premises are all true.
• B. If its premises are true, its conclusion must be true. ✓
• C. Its conclusion is true, regardless of the premises.
• D. It has at least one premise that is true.

Why: Validity concerns the logical relationship between premises and conclusion: if the premises were true, the conclusion could not be false. Whether the premises actually are true is a separate question, addressed by soundness.

• A. Valid and has at least one true premise.
• B. Invalid but has a true conclusion.
• C. Valid and all of its premises are true. ✓
• D. Persuasive to most listeners.

Why: Soundness requires both validity and true premises. A valid argument with a false premise is unsound, and an invalid argument is never sound, even if its premises and conclusion happen to be true.

• A. The argument is sound because the conclusion is logically derived.
• B. The argument is valid but unsound, because a premise is false. ✓
• C. The argument is invalid because penguins are not birds.
• D. The argument is sound because it follows a familiar pattern.

Why: The form of the argument is valid: if both premises were true, the conclusion would follow. However, the first premise is false (penguins cannot fly), so the argument is unsound.

• A. If P then Q; Q; therefore P.
• B. If P then Q; P; therefore Q. ✓
• C. Some P are Q; some Q are R; therefore all P are R.
• D. P or Q; P; therefore not Q.

Why: This is modus ponens, a classic valid form. Option A commits the fallacy of affirming the consequent, option C draws an unwarranted universal conclusion, and option D is an incorrect use of disjunction that would require it to be exclusive and does not show a valid pattern here.

• A. Yes, if at least one premise is false. ✓
• B. No, a valid argument always has a true conclusion.
• C. Yes, but only if the argument is also unsound.
• D. No, because validity guarantees the truth of the conclusion.

Why: Validity only guarantees that the conclusion is true if the premises are true. If a premise is false, a valid argument can have a false conclusion. Such an argument is valid but unsound.

• A. All warm-blooded animals are whales.
• B. All whales are warm-blooded. ✓
• C. Some whales are not warm-blooded.
• D. No mammals are whales.

Why: This is a valid Barbara syllogism (AAA-1). Since every whale is a mammal and every mammal is warm-blooded, every whale must be warm-blooded. Option A commits the fallacy of the converse, since not every warm-blooded animal need be a mammal.

• A. All snakes are mammals.
• B. Some snakes are mammals.
• C. No snakes are mammals. ✓
• D. Some mammals are snakes.

Why: This is a valid Celarent syllogism (EAE-1). If no reptiles are mammals and all snakes are reptiles, then no snake can be a mammal. The universal negative premise carries through to the conclusion.

• A. All doctors are wealthy.
• B. Some doctors are wealthy.
• C. No doctors are wealthy.
• D. It cannot be determined whether any doctors are wealthy. ✓

Why: The premises do not guarantee that the wealthy educated people include any doctors. The 'some' in the second premise may refer entirely to non-doctors, so no conclusion about doctors' wealth follows necessarily.

• A. All penguins are flightless.
• B. Some penguins are flightless.
• C. No penguins are flightless.
• D. All flightless animals are penguins. ✓

Why: Option D is not valid either, so the intended answer is B: the premises establish only that some birds are flightless, and since all penguins are birds, it cannot be guaranteed that the flightless birds include penguins. Strictly, no option is fully guaranteed here, but B is the closest valid reading only if penguins are assumed to be among the flightless birds. Option B is therefore the best answer under the standard reading of the premises as given.

• A. Some A are C.
• B. No A are C. ✓
• C. All C are A.
• D. Some C are not A.

Why: This is a valid Camestres-type inference (AEE-2). If every A is a B and no C is a B, then no A can be a C, because any A would have to be a B, which no C is. The conclusion No A are C follows necessarily.

• A. The ground is dry.
• B. The ground is wet. ✓
• C. It is not raining.
• D. Nothing follows.

Why: Modus ponens states that from 'If P, then Q' and P, we can conclude Q. Here P is 'it is raining' and Q is 'the ground is wet,' so the ground is wet.

• A. 12 is odd.
• B. 12 is not divisible by 2.
• C. 12 is even. ✓
• D. 12 is divisible by 8.

Why: Applying modus ponens with P = 'divisible by 4' and Q = 'even,' and given P for 12, we conclude Q: 12 is even.

• A. Yes, by modus ponens.
• B. Yes, because evacuation always follows the alarm.
• C. No, this is the fallacy of affirming the consequent. ✓
• D. No, because the alarm is never sounded.

Why: From 'If P, then Q' and Q, we cannot conclude P. Employees might evacuate for other reasons, so this is the fallacy of affirming the consequent, not modus ponens.

• A. Shape X has three sides.
• B. Shape X is a triangle.
• C. Shape X has four sides. ✓
• D. Shape X is not a square.

Why: Modus ponens: since Shape X is a square (P), it follows that Shape X has four sides (Q).

• A. The student studied hard.
• B. The student did not study hard. ✓
• C. The student passed the exam.
• D. The student will pass next time.

Why: This is modus tollens: from 'If P, then Q' and not-Q, we conclude not-P. Since the student did not pass, the student did not study hard.

• A. If it is raining, the ground is wet. The ground is wet. Therefore, it is raining. ✓
• B. If it is raining, the ground is wet. It is raining. Therefore, the ground is wet.
• C. If it is raining, the ground is wet. It is not raining. Therefore, the ground is not wet.
• D. All birds have feathers. A robin is a bird. Therefore, a robin has feathers.

Why: Affirming the consequent takes the form 'If P then Q; Q; therefore P.' Option A reasons from the wet ground back to rain, which is invalid because the ground could be wet for other reasons, such as a sprinkler. Option B is valid modus ponens, option C is denying the antecedent (also invalid, but a different fallacy), and option D is a valid syllogism.

• A. Affirming the consequent
• B. Denying the antecedent ✓
• C. Undistributed middle
• D. Affirming the disjunct

Why: The argument has the form 'If P then Q; not P; therefore not Q.' This is denying the antecedent. Passing could still occur through other means, so the conclusion does not follow necessarily from the premises.

• A. All cats are mammals. All dogs are mammals. Therefore, all cats are dogs. ✓
• B. All roses are flowers. Some flowers fade quickly. Therefore, some roses fade quickly.
• C. No fish are mammals. Whales are mammals. Therefore, whales are not fish.
• D. All squares are rectangles. This shape is a square. Therefore, this shape is a rectangle.

Why: The middle term 'mammals' is never distributed (it is not referring to all mammals in either premise), so the premises do not link cats and dogs to each other. Both are simply subsets of mammals, so the conclusion does not follow.

• A. Valid, because the disjunction excludes the gardener
• B. Invalid, because both could have done it together ✓
• C. Valid, because the butler is mentioned first
• D. Invalid, because it commits the fallacy of composition

Why: The premise 'either...or' is inclusive in ordinary reasoning, so both the butler and the gardener could have been involved. Concluding that the gardener did not do it from the butler's involvement is an invalid inference (affirming a disjunct) unless the disjunction is stated as exclusive.

• A. Drawing a general conclusion from a few unrepresentative cases
• B. Presenting only two options when more possibilities exist ✓
• C. Attacking the person instead of the argument
• D. Assuming that what is true of the parts is true of the whole

Why: A false dilemma (false dichotomy) restricts a situation to two alternatives when other options are available. It is a common informal fallacy that can make a deductive argument appear forced even when its logical structure is sound.

• A. Affirming the consequent: If it rains, the ground is wet. The ground is wet. Therefore, it rained.
• B. Modus ponens: If it rains, the ground is wet. It rains. Therefore, the ground is wet. ✓
• C. Denying the antecedent: If it rains, the ground is wet. It does not rain. Therefore, the ground is not wet.
• D. Undistributed middle: All cats are animals. All dogs are animals. Therefore, all cats are dogs.

Why: Modus ponens (P → Q, P, therefore Q) is a valid form: whenever the premises are true, the conclusion must be true. Options A and C are invalid forms (affirming the consequent and denying the antecedent), and D commits the fallacy of the undistributed middle.

• A. If n is not odd, then n is not a prime greater than 2. ✓
• B. If n is odd, then n is a prime greater than 2.
• C. If n is not a prime greater than 2, then n is not odd.
• D. If n is even, then n is a prime greater than 2.

Why: The contrapositive of P → Q is ¬Q → ¬P, and it is logically equivalent to the original statement. Here P is 'n is a prime greater than 2' and Q is 'n is odd', so ¬Q → ¬P is 'if n is not odd, then n is not a prime greater than 2'.

• A. Direct proof
• B. Proof by contradiction ✓
• C. Proof by induction
• D. Proof by cases

Why: Proof by contradiction assumes the statement is false and shows that this assumption leads to a logical impossibility, so the original statement must be true. Direct proof starts from the hypotheses, induction uses a base case and inductive step, and case analysis splits the problem into exhaustive subcases.

• A. Invalid, because the conclusion is false.
• B. Valid and sound, because the premises are true and the conclusion follows necessarily. ✓
• C. Valid but unsound, because one premise is false.
• D. Invalid, because it uses the word 'all'.

Why: The argument is valid: its form (All M are W; x is M; therefore x is W) guarantees the conclusion if the premises hold. Both premises are true, so the argument is also sound. Validity depends on form, not on the truth of the conclusion alone, and universal quantifiers are perfectly acceptable in valid arguments.

• A. Using a previously proven theorem to establish a new result.
• B. Assuming the conclusion you are trying to prove as one of the premises. ✓
• C. Splitting a proof into separate cases that cover all possibilities.
• D. Providing a counterexample to disprove a universal claim.

Why: Begging the question occurs when the conclusion is assumed among the premises, so the argument proves nothing new. Using earlier proven theorems, case analysis, and counterexamples are all legitimate proof strategies.