Natural deduction is a method for establishing the validity of propositional arguments that is simpler than the truth table method.

Through a series of steps the conclusion is derived from the premises. Each step in the derivation or proof must depend on a rule of inference. Rules of inference permit us to glean new information from information given, either by "breaking up" or combining information already obtained.

We begin with four rules of inference.

Modus Ponens (MP)

Given a conditional statement and its antecedent, you may conclude that the consequent is true.

Given: p > q, p

Conclude: q

Modus Tollens (MT)

Given a conditional statement and the negation of its consequent, you may conclude that the antecedent is false.

Given: p > q, ~q

Conclude: ~p

Hypothetical Syllogism (HS)

Given two conditionals where the consequent of one is the identical with the antecedent of the other, you may conclude with a third conditional that combines the antecedent of the one with the matching consequent with the consequent of the one with the matching antecedent.

Given: p > q, q > r

Conclude: p > r

Disjunctive Syllogism (DS)

Given a disjunction and the negation of the first disjunct you may conclude that the second disjunct is true.

Given: p v q, ~p

Conclude: q

Notice that rules of inference are presented with statement variables, which means that either simple or compound statements can stand in for the variables. The following inference is an instance of the disjunctive syllogism.

In this example let (A * B) stand for p and ~(B > C) stand for q in the argument form.

In a natural deduction proof you are given a series of premises that may be assumed to be true and a conclusion that you are asked to prove is true. The premises should be listed and numbers. and the conclusion is set beside the last premise and separated with a slash. Use the rules of inference to deduce pieces of information from what is given to derive the conclusion. New information that is derived along the way is then available for further derivations. The conclusion to be proved must be the last line of the proof.

Example 1:

1. A > B

2. A ___/B

3. B ___1, 2, MP

Lines 1 and 2 and given as premises and you are asked to prove that the conclusion B is true. Applying the rule Modus Ponens to lines 1 and 2 you derive line 3. The justification for the move is listed next to line 3.

Example 2:

1. A > B

2. D v ~B

3. ~D ___/~A

You need to show that ~A is true. Looking at premise 1 you see that if you knew that ~B you could use MT to derive ~A. Can you show that ~B? Looking at premise 2 you see that if you could show that ~D you could use DS to derive ~B. Can you show that ~D? Yes, in fact premise 3 indicates that ~D. Now list the steps in order.

4. ~B ___2, 3, DS

5. ~A ___1, 4, MT

In detail.

Example 3:

1. M > N

2. A

3. A > (N > O) ___/M > O

The conclusion you are asked to prove is not found in the premises as stated. But since it is a conditional you might be able to use HS to derive it should you find the right lines of information, viz., two conditionals, one with M as an antecedent and a consequent that is identical with the antecedent of another conditional whose consequent is O.

M > Something

Something > O

Line 1 is a conditional whose antecedent is M. Line 3 contains a conditional whose consequent is a conditional whose consequent is O. But first you must "detach" this consequent (N > O) from the larger conditional. To do that you need to show that A is true so you can use MP. In fact, line 2 tells you that A is true.

4. N > O ___2, 3, MP

5. M > O ___1, 4, HS

In detail.

Tips:

1. Work backwards by trying to "find" the conclusion in the premises or by considering what you would need to do last to derive the conclusion, and then considering what you would need to know to do that and how you would show those additional pieces of information are true.

2. Use MP to "detach" a consequent. Remember that conditionals do not assert both the antecedent and consequent; they state only that the consequent is true IF the antecedent is true.

Suppose you have the two premises:

1. A > (B > C)

2. B

With these two premises alone you cannot derive C using MP. First you must detach the conditional B > C from the larger conditional. So you will need to know that A is true.

3. Do not commit the fallacy of affirming the consequent. Even if the consequent of a conditional is true, the antecedent could be false while the whole conditional is true.

4. Use DS to "detach" the second disjunct of a disjunction. Remember that disjunctions do not assert both disjuncts; they state only that one or the other (or both) is true.

5. Use MT to show that the negation of the antecedent of a conditional is correct. You must also know that the negation of the consequent is true.

6. Do not commit the fallacy of denying the antecedent. If the antecedent of a conditional is false, the consequent might still be true.

7. Use HS to create a "chain" argument that links the antecedent of one conditional to the consequent of another. The "middle" statement drops out.