Unit 3: Deductive Reasoning

Learning Objectives

Explain deductive reasoning and distinguish it from inductive reasoning
Identify valid and invalid deductive argument forms
Recognise modus ponens and modus tollens
Understand categorical syllogisms
Apply the concepts of validity and soundness to evaluate deductive arguments

What is Deductive Reasoning?

Deductive reasoning moves from the general to the specific. It starts with one or more general premises and draws a specific conclusion that must be true if the premises are true. Deductive arguments are truth-preserving: if the premises are true and the argument form is valid, the conclusion cannot be false.

P1: All mammals are warm-blooded.
P2: Whales are mammals.
C: Therefore, whales are warm-blooded.

Notice that if P1 and P2 are true, C must be true. There is no possible world in which the premises hold and the conclusion fails. This is what makes deductive reasoning so powerful — and why it is the foundation of mathematics, formal logic, and proof-based disciplines.

Deductive vs inductive — the key distinction

Deductive reasoning

Moves from general premises to a specific conclusion. If valid and premises are true, the conclusion is certain. A valid deductive argument cannot have true premises and a false conclusion.

Inductive reasoning

Moves from specific observations to a general conclusion. Even a strong inductive argument only makes the conclusion probable. New evidence can always overturn an inductive conclusion.

Validity

A deductive argument is valid if it is impossible for the premises to be true and the conclusion to be false. Validity is a structural property: it is about the logical relationship between premises and conclusion, not about whether the premises are actually true.

This means a valid argument can have false premises — and even a false conclusion:

Valid

All cats can fly.
Whiskers is a cat.
Therefore, Whiskers can fly.

The first premise is false — but the argument is valid because the conclusion follows necessarily from the premises.

Invalid

All dogs are mammals.
All cats are mammals.
Therefore, all dogs are cats.

Both premises are true, but the conclusion does not follow. The argument is invalid.

Soundness

An argument is sound if it is (1) valid and (2) all its premises are actually true. A sound argument guarantees a true conclusion.

Valid?	Premises true?	Result
Yes	Yes	Sound — conclusion is guaranteed true
Yes	No	Valid but unsound — conclusion may be false
No	Yes	Invalid — conclusion does not follow
No	No	Invalid and unsound

Standard Argument Forms

Certain argument forms appear so frequently in deductive reasoning that they have been named. Recognising these forms helps you evaluate arguments quickly.

Categorical syllogism

A syllogism has two premises and a conclusion. The classic form involves three terms — the major, minor, and middle term — distributed across the premises and conclusion.

  All M are P.   (major premise)

  All S are M.   (minor premise)

  All S are P.   (conclusion)

Modus Ponens — affirming the antecedent

  If P, then Q.   (conditional premise)

  P is true.     (affirm the antecedent)

  Therefore, Q.  (conclusion)

Example: If it is raining, the ground is wet. It is raining. Therefore, the ground is wet.

Modus Tollens — denying the consequent

  If P, then Q.    (conditional premise)

  Q is false.     (deny the consequent)

  Therefore, not P. (conclusion)

Example: If it is raining, the ground is wet. The ground is not wet. Therefore, it is not raining.

Two common invalid forms

Affirming the consequent (invalid)

If P, then Q.
Q is true.
Therefore, P.

The ground is wet does not prove it is raining — there are many other possible causes.

Denying the antecedent (invalid)

If P, then Q.
P is false.
Therefore, not Q.

It is not raining does not prove the ground is dry — it could be wet for other reasons.

Four Types of Reasoning — Video

Watch this explanation of different types of reasoning. Pay particular attention to how deductive reasoning is contrasted with inductive and abductive reasoning.

Four types of reasoning

Evaluate These Arguments

For each argument below, determine: (a) Is it valid? (b) Are the premises true? (c) Is it sound? Where possible, identify the argument form (syllogism, modus ponens, modus tollens, etc.).

All viruses can infect organisms. COVID-19 is a virus. Therefore, COVID-19 can infect organisms.
If a person is infected with COVID-19, they will develop a fever. This person has a fever. Therefore, this person is infected with COVID-19.
If a letter is sealed, it must have a first-class stamp. This letter does not have a first-class stamp. Therefore, this letter is not sealed.
COVID-19 can kill people. Something that kills people is dangerous. Therefore, COVID-19 is dangerous.
All people in that room two weeks ago developed COVID-19. She was in that room two weeks ago. Therefore, she developed COVID-19.

Valid — categorical syllogism. Both premises are true. Sound.
Invalid — affirming the consequent. Fever has many causes; COVID-19 is only one. Having a fever does not prove COVID-19 infection.
Valid — modus tollens. If sealed → first class; not first class → not sealed. Both premises true. Sound.
Valid — categorical syllogism. Both premises are true. Sound.
Valid — categorical syllogism. However, the first premise is likely false (not everyone in the room necessarily developed COVID-19). Unsound.

Review

Expand each concept to check your understanding before moving on.

Deductive reasoning moves from general premises to a specific conclusion. It is truth-preserving: if the premises are true and the argument is valid, the conclusion must be true. This certainty distinguishes deduction from induction, where conclusions are only probable.

Validity is a structural property: the conclusion follows necessarily from the premises. A valid argument can have false premises. Soundness adds a factual requirement: all premises must be true. A sound argument is valid with true premises, guaranteeing a true conclusion.

If P then Q. P is true. Therefore, Q. This is the most basic valid deductive form — affirming the antecedent. Example: If it rains, the ground is wet. It is raining. Therefore, the ground is wet.

If P then Q. Q is false. Therefore, P is false. This is denying the consequent — equally valid. Example: If it rains, the ground is wet. The ground is not wet. Therefore, it is not raining.

Affirming the consequent: If P then Q; Q; therefore P — invalid. Denying the antecedent: If P then Q; not P; therefore not Q — invalid. Both look superficially like valid forms but do not preserve truth.

Key concepts covered in this unit: deductive reasoning, general-to-specific inference, validity, soundness, categorical syllogism, modus ponens, modus tollens, affirming the consequent (invalid), denying the antecedent (invalid), universal case, particular case.

Proceed to Unit 4: Inductive Reasoning when ready.