2. Deductive Reasoning - the process of making specific conclusions based on general 

principles. It is the method of coming to a conclusion using logical stages and a set of premises or assumptions. It is a methodical approach to problem-solving that starts with broad concepts and builds specific conclusions utilizing logical reasoning.

Example:

1. All women are mortal. I am a woman. Therefore, I am mortal. (Basic rule: If p implies q and p is true, q must follow.)

2. Given two extra angles, one of which is 120 degrees in depth, the other is 180 degrees in depth. (Basic Rule: extra angles 300 total  degrees in depth.)

Rules of Inference - are rules that make it possible to derive logical inferences from given premises. Deductive reasoning employs these criteria to create sound arguments that logically flow from the premises. 

The following are a few of the most widely applied inference rules:

1. Addition - if p is true, then p or q is true.

Example: 

It is raining. Therefore, it is either raining or drizzling.

Solution:

p

----------

p v q

2. Simplification - if p and q are true, then p is true.

Example:

John is small and John is smart. Therefore, John is small

Solution: 

p ^ q

----------

∴ p

3. Conjunction - if p and q are true, then p and q together are true.

Example:

Mari is pretty. Mari is lovely. Therefore, Mari is pretty and lovely.

Solution:

p

q

----------

∴ p ^ q

4. Modus Ponens - if p implies q, and p is true, then q must be true.

Example:

If it is raining, then the streets are wet. It is raining. Therefore, the streets are wet.

Solution:

p ⇒ q

p

----------

∴ q

5. Modus Tollens - If p implies q, and q is false, then p must be false.

Example:

If it is raining, then the streets are wet. The streets are not wet. Therefore, it is 

not raining.

Solution:

p ⇒ q

ㄱq

----------

∴ㄱp

6. Hypothetical Syllogism - if p implies q, and q implies r, then p implies r.

Example: 

If it rains today, the picnic will be canceled. If the picnic is canceled, we won't need to buy groceries. Therefore, if it rains today,  we won’t need to buy groceries.

Solution:

p ⇒ q

q ⇒ r

----------

∴ p ⇒ r

7. Disjunctive Syllogism - If p or q is true, and p is false, then q must be true.

Example:

Either I will go to the party tonight or I will stay at home and study. I will not go to the party tonight. Therefore, I will stay at home and study.

Solution:

p v q

ㄱp

----------

∴ q

Fallacies - are incorrect reasoning which appears to follow the rules of inference but are based on contingencies rather than tautologies.

1. Fallacy of Affirming the Conclusion - based on the compound proposition [(p ⇒ q)  ^ q] ⇒ p.

Example:

If you make brownies all day, then you will be a baker. You made brownies all 

day. Therefore, you will be a baker.

Solution:

p ⇒ q

p

----------

∴ q

2. Fallacy of Denying the Hypothesis - based on the compound proposition  [(p ⇒ q) ^ㄱp] ⇒ㄱp.

Example:

If you make brownies all day, then you will be a baker. You did not make brownies all day. Therefore, you will not be a baker.

Solution:

p ⇒ q

ㄱp

----------

∴ ㄱq

Direct proof - Using logical deductions, the conclusion is directly demonstrated from the 

stated premises or axioms. For instance, we can assume that two numbers are even and then use the definition of even numbers to demonstrate that their addition is also even in order to demonstrate that the sum of two even numbers is even.

Indirect Proof - By assuming the negative of the statement to be proved and demonstrating that this results in a contradiction, the conclusion is demonstrated. For instance, to demonstrate that the square root of 2 is irrational, we can first assume that it is rational and then explain that this leads to a contradiction with the arithmetic fundamental theorem.

Proof by Contradiction - We begin by supposing that the conclusion is untrue and then demonstrate how this contradicts the stated premises or axioms. For instance, in order to demonstrate that there are infinitely many prime numbers, we can first assume that there are only a finite amount of prime numbers. Then, we can demonstrate how this assumption contradicts the basic theorem of mathematics.

Proof by Mathematical Induction - The claim is established for a simple situation before it is demonstrated that if the claim holds true for any integer n, it must also hold true for n+1. For instance, we can demonstrate that it holds for n=1 and then demonstrate that if it holds for n=k, it also holds for n=k+1 in order to demonstrate that the sum of the first n natural numbers is n(n+1)/2.

Proof by Counterexample - A claim is refuted by citing a concrete instance that conflicts with it. For instance, we can use the counterexample of 2, which is a prime number but is even, to refute the claim that "all prime numbers are odd."

Proof Construction - The statement has an explicit solution that is created. The Lagrange four-square theorem can be used to prove that each positive integer can be written as the sum of four squares.