Previously, we've seen the power of quantified statements- its ability to prove theorems through contrapositive. In this post we will deal with the idea of inductive and deductive reasoning. In addition, we will encounter the concept of syllogism and dilemma.

NOTE:

_{Disclaimer: this is a summary of section 2.6 from the book A Gentle Introduction to the Art of Mathematics: by Joe Fields, the content apart from rephrasing is identical, all equations are screenshots of the book and the same examples are treated.}

Mathematical Arguments

An argument in mathematics is different from the layman’s connotation of the word. For mathematics and other scholarly areas, it does not involve discord. A mathematical argument is a sequence of logically connected statements designed to produce an agreement as to the validity of a proposition. [J. Fields]

This design generally follows one of two possibilities - inductive reasoning or deductive reasoning.

Inductive Reasoning

In inductive reasoning, a long list of premises is presented whose truths are considered to be apparent to all, each of which provides evidence that the desired conclusion is true. So an inductive argument represents a kind of statistical thing, you have all these statements that are true each of which indicates that the conclusion is most likely true. A strong inductive argument amounts to what attorneys call a “ preponderance of the evidence.” [1]

Occasionally a person who has been convicted of a crime based on a preponderance of the evidence is later found to be innocent. This usually happens when new evidence is discovered that incontrovertibly proves that he or she cannot be guilty. In a nutshell: inductive arguments can be wrong.

In contrast, a deductive argument can only turn out to be wrong under certain well-understood circumstances.

Deductive Reasoning

Deduction is the process by which we determine new truths from old. Nothing new is actually generated in a deduction; the hidden truth was lying within the hypotheses.

The statements that can sometimes be deduced from others can be remarkably surprising. A good argument against deduction is that it is not effective to discover new truths. Inductive reasoning is far superior for the majority of us.

A deductive argument is essentially just a long sequence of statements, but there is some additional structure. The last statement in the list is the conclusion – the statement to be proved. The statements before a conclusion are known as premises.

Premises may be further subdivided into five sorts:

1. Axioms
2. Definitions
3. Previously proved theorem
4. Hypotheses
5. Deductions

In any mathematical fields, you just need to completely memorize the axioms and standard definitions of the fields. You can build the theorems from these two foundations.

It has long been an axiom of mine that the little things are infinitely the most important ~ Arthur Conan Doyle

One should note that a conclusion is also a deduction: a deduction obeys a strict rule, every deduction follows from the premises that have already been written down by one of the so-called rules of inference.

Rules of Inference

Each of the rules of inference actually amounts to a logical tautology that has been re-expressed as a sort of re-writing rule. Each rule of inference will be expressed as a list of logical sentences that are assumed to be among the premises of the argument, a horizontal bar, followed by the symbol (voiced as “therefore”) and then a new statement that can be placed among the deductions.

Conjunctive Simplification

equivalent to the tautology

Rules of Inference	Tautology Equivalence	English Translation
		"if A and B, therefore B"

Modus Ponens (Latin for “method of affirming”)

related to the tautology

Rules of Inference	Tautology Equivalence	English Translation
		"if A and, if A then B, therefore B"

Modus Tollens (Latin for “method of denying”)

the contrapositive version of modus ponens

Rules of Inference	Tautology Equivalence	English Translation
		"If not B and if A then B, therefore it is not A"

Conjunctive Simplification, Modus Ponens, and Modus Tollens are some of the commonly used rules of inference. There are other ten most common listed in the book of J. Fields. [1]

Syllogisms

Modus Ponens and Modus Tollens are known as syllogisms. A syllogism is an argument form wherein a deduction follows from two premises.

Two Common Syllogism:

1. Hypothetical syllogism: basically asserts a transitive property for implications

Hypothetical syllogism	English Translation
	"if A then B, if B then C, therefore if A then C"

1. Disjunctive syllogism: can be thought of as a statement about alternative

Disjunctive syllogism	English Translation
	"either A or B, if not B, therefore it's A"

Dilemma

A situation in which an individual is faced with an impossible choice, a cute example is the Crocodile’s dilemma:

A crocodile captures a little boy who has strayed too near the river. The child's father appears and the crocodile tells him "Don't worry, I shall either release your son or I shall eat him. If you can say, in advance, which I will do, then I shall release him." The father responds, "You will eat my son." What should the crocodile do?

Types of Dilemma:

Dilemmas in Mathematics:

There are many available problems in mathematics that are classified as dilemma.

1. Prisoner's dilemma
2. Innocent prisoner's dilemma
3. Optional prisoner's dilemma
4. Unscrupulous diner's dilemma

Reference:

1. A Gentle Introduction to the Art of Mathematics by Joe Field

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