# 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 |

â€¦â€¦_{>} | If. Than implies | _{P}â€¦._{>q} | If p then q P implies q |

_{<}â€¦…_{>} | Is equivalent to, if and only if | P_{<}â€¦.._{>}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 pâ€¦â€¦ _{> }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 | pâ€¦â€¦._{>}q |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

## What is biconditional?

*The proposition pâ€¦â€¦â€¦ _{>}q^qâ€¦â€¦.._{>}p is written as p_{<}â€¦â€¦â€¦_{>}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 | pâ€¦â€¦â€¦_{>}q | qâ€¦â€¦._{.>}p | p_{<}â€¦â€¦â€¦_{>}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 pâ€¦â€¦._{>}q?

*let pâ€¦â€¦. _{>}q be a given conditional .than qâ€¦â€¦.._{>}p is called the converse of pâ€¦â€¦_{>}q .*

i**ts truth table is**:

p | q | pâ€¦â€¦._{>}q | qâ€¦â€¦.._{>}p |

T | T | T | T |

T | F | F | T |

F | T | T | F |

F | F | T | T |

## What is the inverse of the conditional pâ€¦â€¦*â€¦*_{>}q?

_{>}q

*Let pâ€¦â€¦. _{>}q be is a given conditional,~pâ€¦â€¦_{>}~q is called the inverse of pâ€¦._{.>}q,*

**Its truth table is:**

p | q | pâ€¦._{>}q | ~p | ~q | ~pâ€¦_{>}~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 pâ€¦.._{>}q?

*let pâ€¦.. _{>}q be a given conditional,~qâ€¦..>~p is called the contrapositive of pâ€¦â€¦_{>}q,*

**it truth table is:**

p | q | pâ€¦.._{>}q | ~p | ~q | ~qâ€¦.._{>}~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:**

pâ€¦â€¦_{>}~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

- Universal quantifiers.
- Existential quantifier.

### What are universal quantifiers?

The symbol âˆ€ used, for all is a universal quantifier.

### What is an **Existential quantifier**?

The symbol! âˆƒ 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 âˆ€ xâˆˆU,xâˆˆÏ†â€¦â€¦_{>}xâˆˆA

The antecedent or hypothesis of this conditional is false because no xâˆˆU is a member of Ï†.

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

## Frequently Asked Question-FAQs

### What is induction?

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

### What is the deduction?

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

### 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.

### Types of the quantifiers:

There are two types of quantifier

Universal quantifiers.

Existential quantifier.

### What is the quantifier?

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

### 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.

### What is Aristotelian logic?

Deductive logic, which is the kind of logic that regards every statement as either true or false and nothing else, is called Aristotelian logic.

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