Applied Logic

Unit 5: Conditional Logic

Learning Objectives

  • Identify the antecedent and consequent of a conditional statement
  • Construct the converse, inverse, and contrapositive of a conditional
  • Recognise which transformations preserve truth and which do not
  • Apply five standard valid propositional argument forms
  • Identify invalid argument forms that mimic valid ones
1

The Conditional Statement

The conditional statement — If P, then Q — is the most important logical connective in this course. It captures the idea that one thing follows from another: if the condition P is satisfied, the outcome Q must occur.

If P, then Q
P = the antecedent (the condition, the "if" part)
Q = the consequent (the result, the "then" part)

The conditional is false only when P is true and Q is false — that is, when the condition is met but the promised outcome fails to materialise. In all other cases, the conditional is true.

PQIf P, then Q
TTT
TFF
FTT
FFT

The last two rows may seem counterintuitive. If P is false (it is not raining), the conditional "If it rains, the ground is wet" is not violated — there is no claim about what happens when it does not rain. A false antecedent makes the whole conditional trivially true. Truth tables are covered in detail in Unit 6.

2

Converse, Inverse, and Contrapositive

Three transformations can be applied to a conditional. Only one of them always preserves the truth of the original statement.

NameFormExampleLogically equivalent?
Original If P, then Q If it rains, the ground is wet. —
Converse If Q, then P If the ground is wet, it has rained. No — the ground could be wet for other reasons
Inverse If not P, then not Q If it does not rain, the ground is not wet. No — same problem as the converse
Contrapositive If not Q, then not P If the ground is not wet, it has not rained. Yes — always logically equivalent to the original

The contrapositive is always logically equivalent to the original conditional. This is the basis of modus tollens from Unit 3: denying Q (the ground is not wet) allows us to validly conclude not P (it did not rain).

Confusing the conditional with its converse is one of the most common reasoning errors — and underlies the fallacy of affirming the consequent (Unit 3).

Practice

For each conditional, write the contrapositive and decide whether the converse is logically equivalent to the original.

  1. If a number is divisible by 4, it is divisible by 2.
  2. If someone is a bachelor, they are unmarried.
  3. If a patient has COVID-19, they have a fever.
  4. If a letter is sealed, it has a first-class stamp.
3

Five Valid Propositional Argument Forms

Five argument forms using conditionals are always valid. Learn to recognise them — they appear constantly in academic arguments, legal reasoning, and scientific writing.

1. Modus Ponens (MP)

Affirming the antecedent

If P, then Q.
P.
Therefore, Q.

If it rains, the ground is wet. It is raining. Therefore, the ground is wet.

2. Modus Tollens (MT)

Denying the consequent

If P, then Q.
Not Q.
Therefore, not P.

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

3. Hypothetical Syllogism (HS)

Chaining conditionals

If P, then Q.
If Q, then R.
Therefore, if P, then R.

If study hours increase, grades improve. If grades improve, scholarships are awarded. Therefore, if study hours increase, scholarships are awarded.

4. Disjunctive Syllogism (DS)

Elimination of a disjunct

P or Q.
Not P.
Therefore, Q.

The fault is in the hardware or the software. It is not in the hardware. Therefore, it is in the software.

5. Constructive Dilemma (CD)

Either way, an outcome follows

P or Q.
If P, then R.
If Q, then S.
Therefore, R or S.

The project will run over budget or over schedule. If over budget, funding is cut. If over schedule, the client complains. Therefore, funding is cut or the client complains.

4

Identify the Argument Form

Identify which of the five valid forms (MP, MT, HS, DS, CD) each argument uses, or state whether it is invalid.

  1. If a patient tests positive for COVID-19, they must self-isolate. This patient tested positive. Therefore, this patient must self-isolate.
  2. Either the experiment failed because of equipment error or because of human error. There was no equipment error. Therefore, it failed because of human error.
  3. If the server is down, users cannot log in. If users cannot log in, orders are lost. Therefore, if the server is down, orders are lost.
  4. If it is raining, she carries an umbrella. She is carrying an umbrella. Therefore, it is raining.
  5. If the alarm sounds, the building is evacuated. The building is not being evacuated. Therefore, the alarm is not sounding.

  1. Modus Ponens (MP) — valid. P → Q; P; therefore Q.
  2. Disjunctive Syllogism (DS) — valid. P or Q; not P; therefore Q.
  3. Hypothetical Syllogism (HS) — valid. P → Q; Q → R; therefore P → R.
  4. Invalid — Affirming the Consequent. P → Q; Q; therefore P. She could be carrying an umbrella for any reason (sun, habit, etc.).
  5. Modus Tollens (MT) — valid. P → Q; not Q; therefore not P.
5

Check Your Understanding

Which transformation of 'If P, then Q' is always logically equivalent to the original?

Correct! The contrapositive (If not Q, then not P) is always logically equivalent to the original conditional. The converse (If Q, then P) and the inverse (If not P, then not Q) are logically equivalent to each other but not to the original.
Not quite — review the material and try again. The contrapositive (If not Q, then not P) is always logically equivalent to the original conditional. The converse (If Q, then P) and the inverse (If not P, then not Q) are logically equivalent to each other but not to the original.

A conditional statement 'If P, then Q' is false only when:

Correct! A conditional is false exactly when the antecedent is true and the consequent is false — the condition was met but the promised outcome did not occur. If the antecedent is false, the conditional is trivially true (there is no claim about what happens when P is false).
Not quite — review the material and try again. A conditional is false exactly when the antecedent is true and the consequent is false — the condition was met but the promised outcome did not occur. If the antecedent is false, the conditional is trivially true (there is no claim about what happens when P is false).

'P or Q. Not P. Therefore, Q.' is which valid argument form?

Correct! This is the Disjunctive Syllogism: given that at least one of P or Q is true, and P is false, Q must be true. It is valid regardless of the content of P and Q.
Not quite — review the material and try again. This is the Disjunctive Syllogism: given that at least one of P or Q is true, and P is false, Q must be true. It is valid regardless of the content of P and Q.

Review

Expand each concept to check your understanding before moving on.

A conditional has two parts: the antecedent (P — the "if" part, the condition) and the consequent (Q — the "then" part, the result). The conditional is false only when P is true and Q is false. In all other cases it is true.

The contrapositive (If not Q, then not P) is logically equivalent to the original conditional — they always have the same truth value. The converse and inverse are equivalent to each other but not to the original. Confusing the original with its converse is the source of the fallacy of affirming the consequent.

1. Modus Ponens (MP): If P then Q; P; therefore Q.
2. Modus Tollens (MT): If P then Q; not Q; therefore not P.
3. Hypothetical Syllogism (HS): If P then Q; if Q then R; therefore if P then R.
4. Disjunctive Syllogism (DS): P or Q; not P; therefore Q.
5. Constructive Dilemma (CD): P or Q; if P then R; if Q then S; therefore R or S.

Affirming the consequent: If P then Q; Q; therefore P — invalid because Q can be true for reasons other than P. Denying the antecedent: If P then Q; not P; therefore not Q — invalid for the same reason. Both sound plausible but fail because they confuse sufficiency with necessity.

Key concepts covered in this unit: conditional statement, antecedent, consequent, converse, inverse, contrapositive, logical equivalence, modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, constructive dilemma, affirming the consequent (invalid), denying the antecedent (invalid).

Proceed to Unit 6: Truth Tables when ready.

Applied Logic
Overview 01 Propositions and Truth 02 Arguments and Inference 03 Deductive Reasoning 04 Inductive Reasoning 05 Conditional Logic 06 Truth Tables 07 Informal Fallacies I 08 Informal Fallacies II 09 Argument Mapping 10 Applied Reasoning