Maharashtra Board Class 12 Logic Chapter 2 Deductive Proof PDF Download

Deductive Proof

If someone offers you a ticket to Europe tour or Asia tour then Logic is on your side. If you accept the ticket for Europe but not Asia, you can prove the Conclusion by showing that its denial is impossible.

When an individual says 6 + 4 is same as 4 + 6 then that individual is using the rule of Logic.

Teacher's Note

Logic helps us prove things are true or false. Just like in a math class, if your teacher says 2 + 3 = 5 and 3 + 2 = 5, you know both are the same thing in Logic too.

Exam Trick

Remember: Deductive Proof means proving something is true by using clear steps. Like climbing stairs one by one to reach the top.

Points to Remember

Deductive Proof uses clear steps from what we know to what we want to prove.
There are three types of Deductive Proofs.
Logic rules help us prove things are right or wrong.
We can use different methods like Direct Proof, Conditional Proof, and Indirect Proof.

2.1 Formal Proof of Validity

There are two types of methods used by logicians for deciding or proving the validity of arguments.

1) Decision Procedure such as Truth Table Method, Shorter truth table method, Truth tree etc. are used to decide validity of arguments.

2) Methods that are not Decision procedure such as Deductive proof, Conditional proof, Indirect proof are used to prove validity of arguments.

Truth-table is a purely mechanical method for deciding whether an argument is valid or invalid. However it is not a convenient method when an argument contains many different truth-functional statements. In such cases there are other methods in Logic for establishing the validity of arguments and one of the method is the Method of Deductive Proof.

The Deductive Proof is of three types. They are:

(1) The Direct Deductive Proof

(2) Conditional Proof

(3) Indirect Proof

In the Method of Direct Deductive Proof, the conclusion is deduced directly from the premises by a sequence of Elementary valid argument forms. The Elementary valid argument forms, used for this purpose are called the Rules of Inference. We have already dealt with direct deductive proof and we know that the Direct Deductive proof is based on nine rules of inference and ten rules based on rule of replacement as follows.

Teacher's Note

Direct Deductive Proof means proving your answer step by step using rules. Like solving a math problem by showing each calculation clearly on paper.

Exam Trick

Remember: There are 9 rules of inference and 10 rules of replacement. Total 19 rules in direct proof. Write this number if you forget.

Points to Remember

Direct Deductive Proof uses step-by-step reasoning.
Rules of Inference are Elementary valid argument forms.
There are 9 Rules of Inference.
There are 10 Rules based on Replacement.
These rules help us prove if an argument is true.

Rules of Inference

(i) Rule of Modus Ponens (M.P.)

\(p \supset q\)

\(p\)

\(\therefore q\)

(ii) Rule of Modus Tollens (M.T.)

\(p \supset q\)

\(\sim q\)

\(\therefore \sim p\)

(iii) Rule of Hypothetical syllogism (H.S.)

\(p \supset q\)

\(q \supset r\)

\(\therefore p \supset r\)

(iv) Rule of Disjunctive syllogism (D.S.)

\(p \vee q\)

\(\sim p\)

\(\therefore q\)

(v) Rule of Constructive Dilemma (D.D.)

\((p \supset q) . (r \supset s)\)

\(p \vee r\)

\(\therefore q \vee s\)

(vi) Rule of Destructive Dilemma (D.D.)

\((p \supset q) . (r \supset s)\)

\(\sim q \vee \sim s\)

\(\therefore \sim p \vee \sim r\)

Teacher's Note

These are the main rules we use in logic. Modus Ponens means if P is true and P leads to Q, then Q is true. Like if your teacher says get good marks then you can go to the movie, and you get good marks, so you can go.

Exam Trick

Remember: M.P. (Modus Ponens) is the most common rule. If you see an if-then statement and the first part is true, the second part is true.

Points to Remember

Modus Ponens: If P then Q, and P is true, so Q is true.
Modus Tollens: If P then Q, and Q is false, so P is false.
Disjunctive Syllogism: P or Q, not P, so Q is true.
Hypothetical Syllogism: If P then Q, if Q then R, so if P then R.
Dilemma rules work with two or more choices.

(vii) Rule of Conjunction (Conj.)

\(p\)

\(q\)

\(\therefore p . q\)

(viii) Rule of Simplification (Simp.)

\(p . q\)

\(\therefore p\)

(ix) Rule of Addition (Add.)

\(p\)

\(\therefore p \vee q\)

Teacher's Note

Conjunction means putting two things together. If you have math homework AND you have science homework, then you have both homeworks to do.

Exam Trick

Remember: Conjunction uses a dot (.), Simplification takes it apart, Addition adds or to something.

Points to Remember

Conjunction joins two statements together with AND.
Simplification takes out one part from a joined statement.
Addition adds OR to any statement.
These three rules are very easy to use.
They help make proofs shorter and simpler.

(i) Rule of Double Negation (D.N.)

\(\sim \sim p \equiv p\)

(ii) De-Morgan's Law (De. M.)

\(\sim (p . q) \equiv (\sim p \vee \sim q)\)

\(\sim (p \vee q) \equiv (\sim p . \sim q)\)

(iii) Associative Laws (Assoc.)

\([(p . q) . r)] \equiv [p . (q . r)]\)

\([(p \vee q) \vee r)] \equiv [p \vee (q \vee r)]\)

(iv) Distributive Laws (Dist.)

\([p . (q \vee r) \equiv [(p . q) \vee (p . r)]\)

\([p \vee (q . r) \equiv [(p \vee q) . (p \vee r)]\)

(v) Commutative Law (Comm.)

\((p . q) \equiv (q . p)\)

\((p \vee q) \equiv (q \vee p)\)

(vi) Rule of Transposition (Trans.)

\((p \supset q) \equiv (\sim q \supset \sim p)\)

(vii) Rule of Material Implication (M. Imp.)

\((p \supset q) \equiv (\sim p \vee q)\)

(viii) Rule of Material Equivalence (M. Equi)

\((p \equiv q) \equiv [(p \supset q) . (q \supset p)]\)

\((p \equiv q) \equiv [(p . q) \vee (\sim p . \sim q)]\)

(ix) Rule of Exportation (Export.)

\([(p . q) \supset r] \equiv [p \supset (q \supset r)]\)

(x) Rule of Tautology (Taut.)

\(p \equiv (p . p)\)

\(p \equiv (p \vee p)\)

Teacher's Note

De Morgan's Law is very useful. It says if you have NOT (A AND B), it is same as (NOT A OR NOT B). Like if you don't have both a pen and pencil, you are missing the pen or the pencil or both.

Exam Trick

Remember: Double Negation means two NOTs cancel each other. Like NOT (NOT happy) means you are happy. Commutative means order does not matter, like 2+3 = 3+2.

Points to Remember

Double Negation: Two NOT signs cancel each other out.
De Morgan's Law: Break down the NOT with AND and OR.
Commutative Law: Order does not matter in AND and OR.
Transposition: Switch the parts and add NOT to both.
Distributive Law: Spread AND or OR across brackets.

2.2 Conditional Proof

The method of Conditional Proof is used to establish the validity of arguments, when the conclusion of an argument is an implicative (conditional) proposition. The method of Conditional Proof is based upon the Rule of Conditional Proof.

The Rule of Conditional Proof enables us to construct shorter proofs of validity for some arguments. Further by using it, we can prove the validity of some arguments which cannot be proved by using the above nineteen rules.

The Rule of Conditional Proof may be expressed in a simple way: By assuming the antecedent of the conclusion as an additional premise, when its consequent is deduced as the conclusion, the original conclusion will be taken to have been proved.

While using Conditional Proof, it should be noted that the conclusion can be any statement equivalent to a conditional statement. In such a case, first the equivalent conditional statement is derived and then the Rule of Conditional Proof is used. However, in this chapter, we will use Conditional Proof only when the conclusion is a conditional statement.

To illustrate let us construct a Conditional Proof of Validity for the following argument:

Teacher's Note

Conditional Proof is used when you want to prove an if-then statement. You assume the if part is true, then try to prove the then part is true. Like saying if you study hard, you will pass the exam.

Exam Trick

Remember: In Conditional Proof, assume the first part (antecedent) and try to prove the second part (consequent). Use a bent arrow to show where the assumption starts and ends.

Points to Remember

Conditional Proof is used for if-then conclusions only.
You assume the first part of the if-then statement.
Then you try to prove the second part using rules.
The assumption starts and ends with a bent arrow.
This method makes some proofs shorter and easier.

Example: 1

\(\sim M \supset N\)

\(\therefore \sim N \supset M\)

The proof may be written as follows:

1. \(\sim M \supset N\) \(/ \therefore \sim N \supset M\)

2. \(\sim N\) Assumption

3. \(\sim \sim M\) 1, 2 . M.T.

4. \(M\) 3 . D.N.

Here the step 2 is the antecedent of the conclusion. It is used as an assumption. The assumption should be indicated by bent arrow. From the premise 1 and the assumption, one has deduced the consequent of the conclusion by the Rule of M.T.

However the proof is not complete. One has yet to arrive at the conclusion. To do so one more step remains to be taken, i.e. to write down the conclusion, \(\sim N \supset M\).

The proof is now written by adding step 5 thus:

1. \(\sim M \supset N\) \(/ \therefore \sim N \supset M\)

2. \(\sim N\) Assumption

3. \(\sim \sim M\) 1, 2 . M.T.

4. \(M\) 3. D.N.

5. \(\sim N \supset M\) 2 - 4, C.P.

The conclusion step 5 has not been deduced from the assumption. So the conclusion lies outside the scope of the assumption. i.e. the scope of the assumption ends up with the last step which follows from step 4. To mark this out clearly the device of a bent arrow is used. The head of the arrow points at the assumption and its shaft runs down till it reaches the last statement which is deduced on its basis. Then the arrow bends inwards and discharges (closes) the assumption. The last step i.e. step 5, where the conclusion is written, will lie outside the scope of assumption.

The proof may now be written down as:

1. \(\sim M \supset N\) \(/ \therefore \sim N \supset M\)

2. \(\sim N\)

3. \(\sim \sim M\) 1, 2 . M.T.

4. \(M\) 5 . D.N.

5. \(\sim N \supset M\) 2 - 4, C.P.

The head of the arrow indicates that step 2 is an assumption. So the word assumption need not be written as the justification.

If the conclusion has a compound proposition with more than one conditional statement as its components, then the antecedents of all the conditional statements can be assumed as additional premises.

Let us take an example of this type:

Example: 2

1. \((X \supset Y) \supset Z\)

2. \(A \supset (B \cdot C)\) \(/ \therefore (X \supset Z) \cdot (A \supset B