Inductive and Deductive Logic

In daily life, we often draw general conclusions from a limited number of observations or experiences. These are inductive and deductive logic. A person gets a penicillin injection once or twice and experiences a reaction soon often words. He generalizes that he is allergic to penicillin.

What is induction?

The method of drawing conclusions on the basis of a limited number of observations or experiences is called induction.

For example:

A person gets a penicillin injection once or twice and experiences a reaction soon afterward. He generalizes he is allergic to penicillin.

What is the deduction?

To draw conclusions from accepted or well-known facts is called deduction.

For example:

All men are mortal. We are men so we are mortal.

What is the postulate?

Some statements which are accepted as true without proof are called postulates.

What is a proposition?

A declarative statement that may be true or false but not both is called a proposition.

What is Aristotelian logic?

Deductive logic in which every statement is regarded as true or false and there is no other possibility is called the Aristotelian logic.

What is Non-Aristotelian logic?

Deductive logic in which there is scope for a third or fourth possibility is called Non-Aristotelian logic.

Symbolic logic.

symbol	How to be read	Symbolic expression	How to be read
~	not	~p	Not p negation of p
^	and	P^q	P and q
v	or	pvq	P or q
……>	If. Than implies	P….>q	If p then q P implies q
<……>	Is equivalent to, if and only if	P<…..>q	P if and only if q

What is Negation logic?

If p is any proposition, its negation is denoted by ~p. Thus if a statement p is true, ~p is false and if a statement p is false ~p is true.

The truth table is given as:

P	~P
T	F
F	T

What is the conjunction of two statements?

The conjunction of two statements P and q is denoted by p^q and it is true and only if both of its components are true .in all other cases, it is false.

The truth table of p^q is given as:

p	q	P^q
T	T	T
T	F	F
F	T	F
F	F	F

What is the disjunction?

Disjunction of two statements p and q is denoted by pvq and is true if at least one of the statements is true.

It is false when both of them are false.

The truth table of pvq is:

p	q	pvq
T	T	T
T	F	T
F	T	T
F	F	F

What is conditional /implication?

A compound statement of the form if p then q .written as p……_> q.is called a conditional or an implication .P is called the antecedent or hypothesis and q is called the consequent or conclusion.

A conditional is regarded as false only when the antecedent is true and the consequent is false .in all other cases. It is considered to be true.

The truth table is

p	q	p…….>q
T	T	T
T	F	F
F	T	T
F	F	T

What is biconditional?  

The proposition p………_>q^q…….._>p is written as p_<………_>q and is called the biconditional or equivalence.

The bio conditional is considered true when both p and q are true or both p and q are false.

The truth table of p iff q is:

p	q	p………>q	q……..>p	p<………>q
T	T	T	T	T
T	F	F	T	F
F	T	T	F	F
F	F	T	T	T

What is the converse of p……._>q?

let p……._>q be a given conditional .than q…….._>p is called the converse of p……_>q .

its truth table is:

p	q	p…….>q	q……..>p
T	T	T	T
T	F	F	T
F	T	T	F
F	F	T	T

What is the inverse of the conditional p………_>q?

Let p……._>q be is a given conditional,~p……_>~q is called the inverse of p…._.>q,

Its truth table is:

p	q	p….>q	~p	~q	~p…>~q
T	T	T	F	F	T
T	F	F	F	T	T
F	T	T	T	F	F
F	F	T	T	T	T

What is the contrapositive of p….._>q?

let p….._>q be a given conditional,~q…..>~p is called the  contrapositive  of p……_>q,

it truth table is:

p	q	p…..>q	~p	~q	~q…..>~p
T	T	T	F	F	T
T	F	F	F	T	F
F	T	T	T	F	T
F	F	T	T	T	T

What is a tautology?

A statement that is true for all the possible values of the variable involved in it is called tautology.

What is the absurdity?

A statement that is always false is called an absurdity or a contradiction.

For example:

p……_>~q is an absurdity.

What is a contingency?

A statement that can be true or false depending upon the truth values of the variables involved in it is called a contingency.

What is the quantifier?

The words or symbols which convey the idea of quantity or number are called quantifiers.

Types of the quantifiers:

There are two types of quantifier

1. Universal quantifiers.
2. Existential quantifier.

What are universal quantifiers?

The symbol  ∀   used, for all is a universal quantifier.

What is an Existential quantifier?

The symbol! ∃ is used for there exist is an existential quantifier.

 Prove that in any universe the empty set {} is a subset of any set A.

Proof:

Let U be a universal set.

Considered the conditional that ∀  x∈U,x∈φ……_>x∈A

The antecedent or hypothesis of this conditional is false because no x∈U is a member of φ.

Hence the conditional is true because a conditional is false only when the antecedent is true and the conclusion is false.

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