Lecture 09

We have previously learned about inductive reasoning being a "bottom-up" logic. Deductive reasoning is the exact reverse of this process because it is a "top-down" logic. Top-down means, it uses higher or more general justifications to back a specific conclusion.

Deductive reasoning is a logical approach which uses general ideas in order to draw specific conclusions [1]. Basically, we can describe deductive reasoning by saying something like, "since this is true for all, this must be true for this specific case."

Unlike inductive reasoning, in which the premises only support the conclusion, in deductive reasoning, the premises validate the conclusion. Also, in inductive reasoning, the conclusion could only be valued as probably true while in deductive reasoning, the conclusion is guaranteed to be true (provided, of course, that the premises are true).

For example:

Premise:
All forms of life are made of cells.
A jellyfish is a form of life.

Conclusion:
Therefore, a jellyfish is made of cells.

An important note anyone should remember is that:

"NOT all conclusions drawn from a deductive reasoning are true! "

Conclusions from a deductive reason are valid but not necessarily true. There are statements which are valid but are actually false. Being valid is different from being true.

A conclusion is valid if it was drawn from deductive reasoning without violating any rules of logic (as in Lecture 09). In other words, a conclusion is valid if it is logically an implication of the premises.

But what if the premise is false?

A conclusion from a false premise, no matter if it is valid, still remains to be false. Take the following as an example.

Premise:
Everything that can swim is a fish.
Pedro can swim.

Conclusion:
Therefore, Pedro is a fish.

The above is an example of a conclusion which is logically valid, but is false. This is because the premise, "everything that can swim is a fish" is false. There are other things which could swim but are not actually fishes like turtles, humans, etc. So before validating the conclusion, never forget to also examine the premises if they are true.

Legendary German mathematician David Hilbert dreamt of a "flawless mathematics" [2]. That is, mathematics which is perfect in every way: everything built on a complete system of logical foundations and every statement validly deduced from universal premises. Hilbert believed that mathematics is a complete truth, valid, and deterministic. (Then and now, his views have always been debated.)

In the quest of reaching "validity and truth", mathematicians love deductive reasoning. In fact, it may be what divides mathematics from other sciences like statistics, physics, and others. Mathematicians prefer "valid truth" rather than "inferred truth". Deductive reasoning is a standard in mathematics.

So in mathematics, theorems, formulas, algorithms, and other mathematical statements, are always proven using deductive reasoning.

Mathematical statements which are proven true are called theorems. For example, the statement

"the sum of two odd numbers is even"

(which we shall prove later) is a theorem. Theorems also have different kinds like propositions, corollaries, lemmas, etc. but we will not study those for now. We will just collectively call all of them as theorems.

Not all mathematical statements are theorems. For example, statements which are true for a few known cases but not proven in general are called conjectures. Some of the most famous examples are Goldbach's conjecture (see here) and the even perfect number conjecture (see here).

Axioms are statements which are assumed to be true. They are like rules or assumptions in a mathematical system. Postulates, sometimes confused with axioms, are statements which are believed to be obviously true that they do not anymore require to be proven. Definitions are biconditional statements which specifies a mathematical term.

We go back to theorems.

Theorems generally have two parts: the hypothesis and the conclusion. The hypothesis serves as the premise while the conclusion is its implication. If P is the hypothesis and Q is the conclusion, then a theorem is basically in the form:

P ⇒ Q

or "if P, then Q". Consider the following theorem:

If a is even, then a + 1 is odd.

The hypothesis is "a is even" (note: "if" is not part of the hypothesis). The conclusion is "a + 1 is odd (note: "then" is not part of the conclusion).

Not all the time that theorems are expressed in the basic form P ⇒ Q. For example, the following theorem is not directly written in this form.

The sum of two odd numbers is even.

If this is the case, one may have to convert it into its basic form. For example, the above can also be written as:

If a and b are odd numbers, then a + b is even.

A mathematical proof is a sequence of arguments, all logically valid, and each one either being implied by the former or is a theorem which has already been proven, all to deductively validate a certain mathematical statement [3]. There are many overlapping strategies in proving. We will classify the most basic two: direct and indirect proofs.

A proof has three parts:

• assumptions/hypotheses

• body

• conclusion

Assumptions are the arguments which starts the proof. They are called assumptions because they are presumed to be true. They could be the hypothesis of the theorem or some other mathematical statements that are generally known to be true. The body is the main part of the theorem where a sequence of logically valid arguments are presented one after another ultimately leading to the conclusion. When the proof reaches the conclusion, then the theorem is considered proven.

Given a theorem P ⇒ Q, a direct proof is a method of proving which starts with the hypothesis P as the assumption; followed by the body which then ultimately leads to the conclusion Q.

Step 1. Assume the hypothesis (assumption).
Step 2. Present a series of arguments (body).
Step 3. Arrive at the conclusion (conclusion).

For example, let us prove the following theorem:

Theorem. If a is odd, then a + 1 is even.

Proof. Let a be an odd number. Then,

a = 2n + 1

for some integer n. Now,

a + 1 = (2n + 1) + 1
a + 1 = 2n + 2
a + 1 = 2(n + 1)

Therefore, a + 1 is even.

Next, let us prove:

Theorem. The sum of two even numbers is even.

To make this easier, let us first convert the above theorem into its standard form:

Theorem. If a and b are even numbers, then a + b is even.

Proof. Let a and b be even numbers. Then,

a = 2n
b = 2m

for some integers m and n.

Now,

a + b = 2n + 2m
a + b = 2(n + m)

Therefore, a + b is even.

Given an implication P ⇒ Q, the statement ¬Q ⇒ ¬P is called its contrapositive. The most important property of a contrapositive is that it is logically equivalent to the original implication:

P ⇒ Q ≡ ¬Q ⇒ ¬P

In other words, they have the same truth value. When P ⇒ Q is true, then ¬Q ⇒ ¬P is also true and when P ⇒ Q is false, ¬Q ⇒ ¬P is also false. This gives us the idea that if we could prove ¬Q ⇒ ¬P, that is equivalent to proving the theorem P ⇒ Q itself.

Step 1. Assume the negation of the conclusion (assumption).
Step 2. Present a series of arguments (body).
Step 3. Arrive at the negation of the hypothesis (conclusion).

Let us prove the theorem:

Theorem. If n² is even, then n is even.

Proof. Suppose n is not even. Then, n is odd. So,

n = 2k + 1

for some integer k.

Now,

n² = (2k + 1)²
= 2k² + 4k + 1
= 2(k² + 2k) + 1

which means that n² is odd. Hence, n² is not even. Therefore, by contraposition, if n² is even, then n is even.

Theorem. If 7n + 9 is even, then n is odd.

Proof. Suppose n is not odd. Then, n is even. So,

n = 2k

for some integer k.

Now,

7n + 9 = 7(2k) + 9
= 14k + 9
= 14k + 8 + 1
= 2(7k + 4) + 1

Thus, 7n + 9 is odd. This means that it is not even. Therefore, by contraposition, if 7n + 9 is even, then n is odd.

Proof by contradiction, in Latin, is "reductio ad absurdum" (trans. "reducing to absurdity"). It is an argument which, basically, is saying, "this should be true because otherwise, it would be absurd or false".

Proof by contradiction capitalizes in the fact that in order to prove a theorem to be true, we can also show that the contrary is false. If we have a theorem P ⇒ Q, an exercise in Lecture 07 tells us that this is logically equivalent to ¬P ∨ Q. By De Morgan's Law, the negation would be

¬(¬P ∨ Q) ≡ ¬(¬P) ∧ ¬Q ≡ P ∧ ¬Q

So if we can show that P ∧ ¬Q is false, then it would prove the theorem to be true.

Step 1. Assume the hypothesis to be true and the conclusion to be false (assumption).
Step 2. Present a series of arguments (body).
Step 3. Arrive at a contradiction or a statement that is always false (conclusion).

Let us prove, for example, the following theorem.

Theorem. If 3n + 2 is odd, then n is odd.

Proof. Let 3n + 2 be odd and n be not odd. So, n is even. This implies that

n = 2k

for some integer k. Substituting this into 3n + 2 gives us

3n + 2 = 3(2k) + 2
= 6k + 2
= 2(3k + 1)

which means 3n + 2 is even. We arrived at a contradiction since we assumed that 3n + 2 is odd. Therefore, if 3n + 2 is odd, then n is odd.

Theorem. If n² is even, then n + 1 is odd.

Proof. Suppose n² is even and n + 1 is even. Then,

n + 1 = 2k

for some integer k. Subtracting 1 from both sides gives

n = 2k - 1

Substituting this into n² yields

n² = (2k - 1)²
= 4k² - 4k + 1
= 2(2k² - 2k) + 1

Hence, n² is odd. We arrived at a contradiction since we assumed that n² is even. Therefore, if n² is even, then n + 1 is odd.

[1] Bhandari, P. (2022). What Is Deductive Reasoning? | Explanation & Examples. From Scribbr [Click here]
[2] Veritasium (22 May 2021). Math's Fundamental Flaw [Video]. YouTube [Click here.]
[3] Stefanowicz, A. (2014). Proofs and Mathematical Reasoning. University of Birmingham. [Click here]