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Course 13 · Lesson 3.3

Deductive arguments and validity

11 minute read

Lessons in this course

Course 13 outline

Module 1: What Apologetics Is and Is Not

  1. The word apologia and 1 Peter 3:15 in context
  2. Paul and the apostles giving reasons: Acts 17 and Acts 26
  3. What apologetics is for: defense, clarification and invitation
  4. What apologetics cannot do: conversion, certainty and the work of the Spirit

Module 2: Four Methods of Apologetics

  1. Classical apologetics: first God, then Christ
  2. Evidential apologetics: the weight of historical evidence
  3. Presuppositional apologetics: starting from Christian premises
  4. Cumulative-case apologetics: many lines of support
  5. Comparing the methods and choosing among them

Module 3: Logic and Argument Construction

  1. What an argument is and is not
  2. Finding premises and conclusions in ordinary writing
  3. Deductive arguments and validity
  4. Soundness, inductive strength and abductive reasoning
  5. Mapping an argument step by step

Module 4: Evidence, Inference and the Burden of Proof

  1. Who carries the burden of proof, and when
  2. Types of evidence: testimony, documents, experience and background knowledge
  3. From evidence to conclusion: inference and its pitfalls
  4. Common fallacies and how to spot them fairly

Module 5: Fair Argument: Steelmanning, Rebuttal and Humility

  1. Steelmanning: restating an opposing argument at its strongest
  2. Rebuttal, counterargument and defeaters
  3. Intellectual humility and the limits of what you know
  4. Speaking with gentleness and respect: tone, listening and honesty

Module 6: Faith and Reason

  1. Faith and reason in the Fathers: Tertullian, Augustine and Anselm
  2. Aquinas and the harmony of faith and reason
  3. Pascal, Kierkegaard and the limits of argument
  4. Reformed epistemology and properly basic belief

All courses

Program / Course 13: Foundations of Apologetics / Lesson 3.3

What you will learn

  • Define a deductive argument and explain what it means for one to be valid.
  • Recognize three common valid forms: modus ponens, modus tollens and the categorical syllogism.
  • Recognize two common invalid forms that look similar: affirming the consequent and denying the antecedent.
  • Test an argument for validity by trying to describe a case where the premises are true and the conclusion false.

The big idea

A deductive argument is one that aims to guarantee its conclusion. It is valid when the premises, if true, would make it impossible for the conclusion to be false. Validity concerns the connection between premises and conclusion, not the truth of the premises. A valid argument can have false premises, and an invalid argument can have true ones. Learning a few common forms, and a simple test using counterexamples, makes it possible to check many arguments quickly and to avoid being fooled by ones that only sound right.

Key terms

Deductive argument
An argument intended to show that the conclusion must be true if the premises are.
Valid
Describing a deductive argument in which it is impossible for all the premises to be true and the conclusion false.
Invalid
Describing a deductive argument that is not valid, so that the premises could all be true while the conclusion is false.
Conditional statement
A statement of the form, if A then B. A is called the antecedent and B the consequent.
Modus ponens
A valid form: if A then B; A; therefore B. The Latin means the mode that affirms.
Modus tollens
A valid form: if A then B; not B; therefore not A. The Latin means the mode that denies.
Counterexample
A case, real or imagined, in which the premises of an argument form are true and the conclusion false, which shows the form to be invalid.
Syllogism
An argument with two premises and a conclusion, classically about categories, such as all A are B; all B are C; so all A are C.

Background

The study of deductive reasoning goes back to Aristotle, who described the syllogism, and to the Stoics, who studied the logic of if-then statements. Medieval writers gave the forms their Latin names, which are still used. The ideas are not difficult, but they require attention to wording.

This lesson uses small, everyday examples to show the forms. They are not about religion, on purpose. It is easier to see the structure when the content does not stir strong feelings. The next lesson takes up the question of when a good argument is also sound.

Step-by-step explanation

Step 1: What makes an argument deductive

A deductive argument claims more than that its conclusion is likely. It claims that the conclusion follows with necessity. If all the premises are true, the conclusion cannot be false. An argument that fails to meet this standard may still be a good argument of another kind, but it is not a good deductive one.

Whether an argument is meant to be deductive depends on the writer’s aim and on its wording. Phrases such as it must be that, it follows necessarily and certainly suggest deduction. Words such as probably, likely and suggests point to another kind of reasoning, treated in the next lesson.

Step 2: What validity means

A deductive argument is valid when it is impossible for the premises to be true and the conclusion false. This is a statement about connection. It does not say that the premises are true. Consider: All dogs are reptiles. All reptiles can fly. Therefore all dogs can fly. The premises are false and the conclusion is false, but the argument is valid, because if the premises were true, the conclusion would have to be.

Validity depends on form, and form is the pattern left when the particular subject matter is replaced by letters. The argument above has the form: all A are B; all B are C; so all A are C. Every argument with that form is valid, whatever A, B and C stand for.

Step 3: Learn the valid forms

Three forms are enough to begin. Modus ponens: if it rained, the street is wet; it rained; so the street is wet. Modus tollens: if it rained, the street is wet; the street is not wet; so it did not rain. The syllogism: all monks live under a rule; Basil was a monk; so Basil lived under a rule. In each case, the conclusion cannot be false if the premises are true.

Modus tollens is worth special attention because it is used in many debates. It lets a person argue from the falsity of a consequence back to the falsity of the claim that implied it. The reverse reasoning, discussed below, does not work.

Step 4: Watch for look-alike invalid forms

Two invalid forms resemble the valid ones. Affirming the consequent has the form: if A then B; B; so A. For example: if it rained, the street is wet; the street is wet; so it rained. This does not follow, since a street cleaner may have wet the street. Denying the antecedent has the form: if A then B; not A; so not B. For example: if it rained, the street is wet; it did not rain; so the street is not wet. This also does not follow, for the same reason.

These mistakes are common in religious debate. If God exists, the world has order; the world has order; so God exists, is an example of the first. The conclusion may or may not be true, but this argument does not establish it. A careful reader will notice the pattern and ask for more.

Step 5: Test with a counterexample

A simple test for validity is to try to imagine a case in which the premises are true and the conclusion false. If one can be found, the argument is invalid. If, after honest effort, none can be found, the argument is probably valid. The method works because the definition of validity says that no such case can exist.

To use the test, replace the content with something easy to picture. For the form, all A are B; all C are B; so all A are C, replace the letters with dogs, animals and cats: all dogs are animals; all cats are animals; so all dogs are cats. The premises are true and the conclusion false, so the form is invalid. This approach is better than relying on a feeling that the argument sounds right.

  1. What is the difference between a valid argument and a true conclusion?
  2. Can an argument be valid and still have false premises? Give an example.
  3. Why is the street-is-wet argument from the wet street to rain invalid?
  4. What is the form of modus tollens, and why does it work?
  5. How would you use a counterexample to show that an argument form is invalid?
  6. Why might it be easier to learn logical forms with examples that do not touch religion?

Examples

A valid argument with false premises

All planets are made of cheese. Mars is a planet. Therefore Mars is made of cheese. If the two premises were true, the conclusion would be true. The argument is valid even though a premise is false.

An invalid argument with true premises and a true conclusion

Some birds are black. Swans are birds. Therefore swans are black. Both premises are true, yet the conclusion is false. The argument is invalid because the premises do not force the conclusion. (An invalid argument can also happen to reach a true conclusion, but the form gives no guarantee.)

Modus tollens applied

If the monastery had a rule of silence, then the monks would not have spoken at supper. The monks did speak at supper. Therefore the monastery did not have a rule of silence. This is valid. The premises may be mistaken, but if they are true, the conclusion cannot be false. This is a constructed example.

Different perspectives

Aristotelian and medieval tradition

Aquinas and other scholastic writers relied on syllogisms to organize theology, arguing that sound reasoning is part of the proper use of the mind God has given. They also held that some truths about God are received through faith and are not the conclusion of any syllogism.

Orthodox theological tradition

Orthodox theologians have used deductive reasoning in defining doctrine at the councils, for example in arguing about the two natures of Christ. They hold that logic serves the Church’s confession of the mystery and does not exhaust it.

Protestant scholastics and modern Protestants

Post-Reformation Protestant theologians continued to teach syllogistic reasoning. In the twentieth century, Gordon Clark argued strongly that Christian theology should be logically consistent and deduced from Scripture, while others warn that paradox may mark the limits of human logic.

Modern logic

Contemporary logicians define validity in the same way but describe it using formal systems that make the testing of forms precise. They also study logics in which the standard rules are altered for special purposes, but the classical rules described here remain the foundation.

Why it matters

Many arguments in apologetics have a deductive form: if God exists then something follows, something is observed, therefore God exists or does not. Recognizing the valid and invalid forms prevents mistakes on both sides. A student who knows modus tollens can see how a single established fact can overturn a theory. A student who knows affirming the consequent can see when a conclusion has been reached by a shortcut that only looks like a proof.

Common misunderstandings

Misunderstanding: A valid argument is one with a true conclusion.

Correction: Validity concerns the link between premises and conclusion. A valid argument can have a false conclusion if a premise is false.

Misunderstanding: If an argument has true premises, it must be valid.

Correction: True premises do not guarantee that the conclusion follows. The form must also be valid.

Misunderstanding: Invalid arguments always have false conclusions.

Correction: An invalid argument may have a true conclusion. It simply fails to show that conclusion.

Misunderstanding: Deductive reasoning is the only good reasoning.

Correction: Much of science, history and everyday life depends on reasoning that is not deductive. The next lesson treats inductive and abductive arguments.

Knowledge check

What does it mean to say that an argument is valid?

It means that it is impossible for all the premises to be true while the conclusion is false. Validity concerns the connection between premises and conclusion. It does not require that the premises are in fact true, and it does not by itself show that the conclusion is true.

Write the form of modus ponens and modus tollens.

Modus ponens: if A then B; A; therefore B. Modus tollens: if A then B; not B; therefore not A. Both are valid. Each uses a conditional statement and a second premise that either affirms the antecedent or denies the consequent.

What is affirming the consequent, and why is it invalid?

It has the form: if A then B; B; therefore A. It is invalid because B might be true for other reasons. If it rained, the street is wet; the street is wet; so it rained, ignores the chance that the street was washed some other way.

How can a counterexample show that an argument is invalid?

If one can describe a case in which every premise is true and the conclusion false, then the premises do not guarantee the conclusion, and the argument is invalid. The case may be imagined, as long as it is genuinely possible.

Can an argument be valid and have a false conclusion?

Yes, if at least one premise is false. A valid argument with a false conclusion must have a false premise, since a valid argument with all true premises cannot have a false conclusion. This is why validity alone does not settle whether to accept a conclusion.

Think it through

  1. Where have you heard affirming the consequent in a religious or political discussion?
  2. Why is it satisfying to find a valid argument, and why might that be misleading?
  3. What do you do when you believe the conclusion but find the argument invalid?
  4. Can you think of a case where modus tollens was used to reject a belief you once held?

Independent assignment

Test five arguments

In your own notebook, write 400 to 600 words. Write five short arguments of your own, on any subject, two valid and three invalid (at least one affirming the consequent and one denying the antecedent). For each, write the form with letters, and either state why it is valid or give a counterexample. Keep this for yourself.

Further study