unit 1 of DMS

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Discrete Mathematical Structures Language of Logic

Prepared by: Aisha Rafi Page 1

JECRC UDML COLLEGE OF ENGINEERING(DEPARTMENT OF MATHEMATICS)

Notes

Discrete Mathematical Structures(Subject Code: 4CS3)

Prepared By: Aisha Rafi

Class: B. Tech. II Year, IV Semester

Syllabus

UNIT 1: Proposition ,compound proposition, conjunction, disjunction,Implication ,converse ,inverse & contra positive ,bioconditional

Statement ,tautology, contradiction & contingency, logicalEquivalences, quantifiers ,arguments

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Beyond the Syllabus

Applications of Logical connectives

Learning Objectives

Students will learn about the language of logic

Logical connectiveThis article is about connectives in logical systems. For connectors in natural

languages, see discourse connective. For other logical symbols, see table oflogic symbols.

In logic, a logical connective (also called a logical operator or a truthfunction) is a symbol or word used to connect two or more sentences (ofeither a formal or a natural language) in a grammatically valid way, such

that the sense of the compound sentence produced depends only on the

original sentences.

The most common logical connectives are binary connectives (also called

dyadic connectives) which join two sentences which can be thought of asthe function's operands. Also commonly, negation is considered to be a

unary connective.

Logical connectives along with quantifiers are the two main types of logicalconstants used in formal systems such as propositional logic and predicatelogic.

Natural language

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In the grammar of natural languages two sentences may be joined by agrammatical conjunction to form a grammatically compound sentence.

Some but not all such grammatical conjunctions are truth functions. For

example, consider the following sentences:

A: Jack went up the hill.

B: Jill went up the hill.

C: Jack went up the hill andJill went up the hill.

D: Jack went up the hill so Jill went up the hill.

The words andand so are grammaticalconjunctions joining the sentences(A) and (B) to form the compound sentences (C) and (D). The andin (C) isa logicalconnective, since the truth of (C) is completely determined by (A)

and (B): it would make no sense to affirm (A) and (B) but deny (C).Howeverso in (D) is not a logical connective, since it would be quite

reasonable to affirm (A) and (B) but deny (D): perhaps, after all, Jill went upthe hill to fetch a pail of water, not because Jack had gone up the hill at all.

Various English words and word pairs express logical connectives, and

some of them are synonymous. Examples (with the name of the relationshipin parentheses) are:

"and" (conjunction)

"or" (disjunction) "either...or" (exclusive disjunction) "implies" (implication) "if...then" (implication) "if and only if" (equivalence) "only if" (implication) "just in case" (equivalence) "but" (conjunction) "however" (conjunction) "not both" (NAND) "neither...nor" (NOR)

The word "not" (negation) and the phrases "it is false that" (negation) and "itis not the case that" (negation) also express a logical connective even

though they are applied to a single statement, and do not connect two

statements.

Formal languages

In formal languages, truth functions are represented by unambiguous

symbols. These symbols are called "logical connectives", "logical

operators", "propositional operators", or, in classical logic, "truth-functionalconnectives". See well-formed formula for the rules which allow new well-

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Discrete Mathematical Structu

Prepared by: Aisha Rafi

formed formulas to be constructedusing truth-functional connectives

Logical connectives can be used tcan speak about "n-ary logical con

Common logical connectives

Name / SymbolTr

P=

Truth/Tautology

Proposition P

False/Contradiction

Negation

Binary connectives P=

Q

Conjunction

Alternative denial

Disjunction

Joint denial

Material conditional

Exclusive or

Biconditional

Converse implication

Proposition P

Proposition Q

M o r e i n f o r

es Language of Logic

Page 4

by joining other well-formed formulas.

link more than two statements, so onenective".

th table Venndiagram0 1

1 1

0 1

0 0

1 0

0 0 1 1

0 1 0 1

0 0 0 1

1 1 1 0

0 1 1 1

1 0 0 0

1 1 0 1

0 1 1 0

1 0 0 1

1 0 1 1

0 0 1 1

0 1 0 1

a t i o n

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List of common logical con

Commonly used logical connectiv

Negation (not): , Np, ~

Conjunction (and): , Kp Disjunction (or): , Apq Material implication (if...t Biconditional (if and only

Alternative names for bicondition

For example, the meaning of the s

transformed when the two are co

It is raining and I am indo Ifit is raining, then I am i IfI am indoors, then it is r I am indoors if and only if It is not raining (P)

For statement P= It is rainingan

It is also common to consider the

formula to be connective:

True formula (, 1, Vpq, o

False formula (, 0, Opq,

Negation.

In logic and mathematics, negatio unary logical connective. It is an o

semantic values more generally. Itrue when that proposition is false,is normally identified with the trut

vice versa. In intuitionistic logic, aKolmogorov interpretation, the ne

whose proofs are the refutations osemantic values of formulae are s

theoretic complementation.

Definition

Classical negation is an operation

of a proposition, that produces a vvalue offalse when its operand is


Page 5

ectives

es include:

, &,

en): , Cpq, ,

f): , Epq, ,

l are "iff", "xnor" and "bi-implication".

atements it is rainingand I am indoors is

bined with logical connectives:

rs (P Q)doors (P Q)

aining (Q P)it is raining (P Q)

Q = I am indoors.

lways true formula and the always false

r T)

r F)

, also called logical complement, is aperation on propositions, truth values, or

tuitively, the negation of a proposition isand vice versa. In classical logic negation

h function that takes truth to falsity and

ccording to the BrouwerHeytingation of a proposition p is the proposition

p. In Kripke semantics where thets of possible worlds, negation is set-

on one logical value, typically the value

lue oftrue when its operand is false and arue. So, if statement A is true, then A

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(pronounced "not A") would therethen A would be false.

The truth table ofp is as follows

Truth table of p

p pTrue False

False True

Classical negation can be definedexample, p can be defined as p and Fis absolute falsehood. Conv

proposition p, where "&" is logica

contradiction is false. While these

intuitionistic logic, they do not woare not necessarily false. But in cl

q can be defined as p q, where

Algebraically, classical negation c

Boolean algebra, and intuitionistic

Heyting algebra. These algebras p

intuitionistic logic respectively.

Notation

The negation of a proposition p is

contexts of discussion and fields o

the following:

Notation Vocalizationp not p

p not p

~p not p

Np en pprime,

complement

bar,

barp

bang p

not p

In Set Theory \ is also used to indimembers of U that are not membe


Page 6

fore be false; and conversely, ifA is true,

:

n terms of other logical operations. For F, where "" is logical consequence

rsely, one can define Fas p & p for any

conjunction. The idea here is that any

ideas work in both classical and

rk in Brazilian logic, where contradictionsssical logic, we get a further identity: p

"" is logical disjunction: "not p, orq".

orresponds to complementation in a

negation to pseudocomplementation in a

ovide a semantics for classical and

notated in different ways in various

f application. Among these variants are

cate 'not member of': U \ A is the set of alls of A.

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No matter how it is notated or syas "it is not the case that p", "not t

grammatically) as "not p"

Distributivity

De Morgan's laws provide a way

and disjunction:

Linearity

In Boolean algebra, a linear functi

If there exists a0, a1, ..., an {0,1} (an bn), for all b1, ..., bn {0,1

Another way to express this is thain the truth-value of the operationa linear logical operator.

Self dual

In Boolean algebra a self dual fun

f(a1, ..., an) = ~f(~a1, ..., ~an) for allogical operator.

Rules of inference

There are a number of equivalentusual way to formulate classical ntake as primitive rules of inferenc ofp to both q and q, infer p; thiabsurdum), negation elimination (called ex falso quodlibet), and dou

p). One obtains the rules for intuitiexcluding double negation elimina

Negation introduction states that ifrom p then p must not be the case(intuitionistically) or etc.). Negatifrom an absurdity. Sometimes neg

primitive absurdity sign . In thisfollows an absurdity. Together wit


Page 7

bolized, the negation p / p can be readat p", or usually more simply (though not

f distributing negation over conjunction

, and

.

on is one such that:

such that f(b1, ..., bn) = a0 (a1 b1) ...}.

each variable always makes a differenceor it never makes a difference. Negation is

tion is one such that:

a1, ..., an {0,1}. Negation is a self dual

ays to formulate rules for negation. Onegation in a natural deduction setting is to

negation introduction (from a derivations rule also being called reductio ad

from p and p infer q; this rule also beingble negation elimination (from p inferonistic negation the same way but bytion.

an absurdity can be drawn as conclusion(i.e. p is false (classically) or refutablen elimination states that anything followsation elimination is formulated using a

case the rule says that from p and ph double negation elimination one may

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infer our originally formulated rulabsurdity.

Typically the intuitionistic negationegation introduction and eliminat

introduction (conditional proof) acase one must also add as a primit

Logical conjunction" " redirects here. For the logic gsee Exterior algebra.

Venn diagram of

Venn diagram of

In logic and mathematics, a two-plogical conjunction[1], results in t otherwise the value offalse.

The analogue of conjunction for auniversal quantification, which is

Notation


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, namely that anything follows from an

n p ofp is defined as p . Thenion are just special cases of implication

d elimination (modus ponens). In thisve rule ex falso quodlibet.

te, see AND gate. For exterior product,

ace logical operatorand, also known as ue if both of its operands are true,

(possibly infinite) family of statements isart of predicate logic.

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And is usually expressed with themathematics and logic, the infix o in programming languages, & orrelated control structure, the short-

Definition

Logical conjunction is an operativalues of two propositions, that prof its operands are true.

The conjunctive identity is 1, whiwith 1 will never change the valueconcept of vacuous truth, when cofunction of arbitrary arity, the emset of operands) is often defined a

The truth table of :

INPUT OUTPUT

A B A B

T T T

T F F

F T F

F F F

Propertiescommutativity: yes

associativity: yes


Page 9

prefix operatorK, or an infix operator. Inerator is usually ; in electronics ; and

nd. Some programming languages have acircuit and, written &&, and then, etc.

n on two logical values, typically theoduces a value oftrue if and only if both

h is to say that AND-ing an expressionof the expression. In keeping with the

njunction is defined as an operator orty conjunction (AND-ing over an emptyhaving the result 1.

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distributivity: with various opera

[s

idempotency: yes

monotonicity: yes

truth-preserving: yesWhen all inputs are true, the outp

(to be tested)

falsehood-preserving: yesWhen all inputs are false, the outp

(to be tested)

Walsh spectrum: (1,-1,-1,1)

Nonlinearity: 1 (the function is b


Page 10

ions, especially with or

ow]others

t is true.

t is false.

nt)

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If using binary values for true (1)works exactly like normal arithme

Logical disjunction

"Disjunction" redirects here. For tof chromosomes, see Meiosis. Fordistribution.

Venn diagram of

Venn diagram of

In logic and mathematics, or is a t(inclusive) disjunction and alter represents this operator is also kn

. The "or" operator produces a roperands are true. For example, inor ifB is true, or if both A and B aconjunction.

The "or" operator differs from thewhen both of its inputs are true, wlanguage, outside of contexts such


Page 11

nd false (0), then logical conjunctiontic multiplication.

e logic gate, see OR gate. For separationdisjunctions in distribution, see Disjunct

uth-functional operator also known asation. The logical connective thatwn as "or", and typically written as orsult of true wheneverone or more of itsthis context, "A orB" is true ifA is true,e true. In grammar, or is a coordinating

exclusive or in that the latter returns Falseile "or" returns True. In ordinaryas formal logic, mathematics and

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programming, "or" sometimes hasexample, "Please ring me or sendother, but not both". On the othereither very bright or studies hard"both bright and works hard. In oth

mean inclusive or exclusive or. Usthe context.

Notation

Or is usually expressed with the pIn mathematics and logic, the infi and in programming languages, | a related control structure, the sho

Definition

Logical disjunction is an operativalues of two propositions, that prof its operands are false. More gethat can have one or more literalsoften considered to be a degenerat

The disjunctive identity is 0, whic0 will never change the value of tof vacuous truth, when disjunctio

arbitrary arity, the empty disjunctioperands) is often defined as havi

Truth table

Disjunctions of the arguments on tThe false bits form a Sierpinski tri

The truth table of :

INPUT OUTPUT

A B A OR B

0 0 0

0 1 1

1 0 1

1 1 1


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the meaning of "exclusive or". Forn email" likely means "do one or theand, "Her grades are so good that she's

allows for the possibility that the person iser words, in ordinary language "or" can

ually the intended meaning is clear from

refix operatorA, or with an infix operator.operator is usually ; in electronics, +;

ror. Some programming languages havet-circuit or, written ||, or else, etc.

n on two logical values, typically theoduces a value offalse if and only if botherally a disjunction is a logical formulaeparated only by ORs. A single literal is

e disjunction.

is to say that OR-ing an expression withe expression. In keeping with the conceptis defined as an operator or function of

on (OR-ing over an empty set ofg the result 0.

he leftangle

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Properties

commutativity: yes

associativity: yes

distributivity: with various opera

[s

idempotency: yes

monotonicity: yes



Page 13

ions, especially with and

ow]others

t is true.

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(to be tested)

falsehood-preserving: yesWhen all inputs are false, the outp

(to be tested)

Walsh spectrum: (3,-1,-1,-1)

Nonlinearity: 1 (the function is b

If using binary values for true (1)

works almost like binary addition.while .

Symbol

The mathematical symbol for logiaddition to the word "or", and thefrom the Latin word velfor "or", iexample: "A B " is read as "A orand B are false. In all other cases i

All of the following are disjunctio

The corresponding operation in se

Material conditional


Page 14

t is false.

nt)

nd false (0), then logical disjunction

The only difference is that ,

al disjunction varies in the literature. Inormula "Apq", the symbol " ", derivingcommonly used for disjunction. For

B ". Such a disjunction is false if both Ais true.

s:

theory is the set-theoretic union.

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A material conditional (also knoconsequence", or simply "implicasymbolized by a forward arrow "connection of two, for instance "p typically interpreted as "Ifp, then

between two sentences p, q is typi

1. ;2. ;3. (Although this sy

(i.e. logical implication) ra

As placed within the material conantecedent, and q as the conseque compounds as components, for ex

pq (short for "p and q") is the anteconsequent, of the larger conditiocomponents.

Implication is a form of logical cosuch as "Fred is Mike's brother's sstatement "Fred is Mike's nephew.Mike's brother's son," not a formalargument depends on the the mea"brother," "son," and "nephew," n

In classical logic, the compound p compound: not (both p and not q).that ifp then q" is not often takenq is false" but, when used within clogically equivalent. (Other senseslogical forms.)

Definitions of the material condi

Logicians have many different vie

and approaches to explain its sensAs a truth function

In classical logic, the compound p compound: not both p and not q.only if both p is true and q is false.only if eitherp is false orq is trueof truth values of the componentswhose truth value is entirely a funcomponents. Hence, the compoun

compound pq is logically equivboth)), and to q p (if not q thp q, which is equivalent to q


Page 15

n as "material implication", "materialion") is a logical connective often". A single statement formed from the

q" (called a conditional statement) is q" or "q ifp". The material implication

ally symbolized as

bol is often used for logical consequenceher than for material implication.)

itionals above, p is known as the t, of the conditional. One can also use

mple pq (rs). There, the compoundedent, and the compound rs is theal of which those compounds are

sequence. For instance, in an argumentn. Therefore Fred is Mike's nephew" the

" is a material consequence of "Fred isconsequence. The validity of theings of the words "Fred," "Mike,"t the logical form of the argument. [1]

q is equivalent to the negativeIn everyday English, saying "It is false

as flatly equivalent to saying "p is true andlassical logic, those phrasings are taken asof English "if...then..." require other

tional

ws on the nature of material implication

.

[2]

q is logically equivalent to the negativehus the compound pq is false if andBy the same stroke, pq is true if and

(or both). Thus is a function from pairs, q to truth values of the compound pq,tion of the truth values of the

pq is called truth-functional. The

lent also to p q (either not p, orq (oren not p). But it is not equivalent top.

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Truth table

The truth table associated with the(symbolized as p q) and the loas p q, orCpq) is as follows:

p q p q

T T T

T F F

F T T

F F T

As a formal connective

The material conditional can be ctaken as a set of sentences, satisfyi, in particular the following cha

1. Modus ponens;2. Conditional proof;

3. Classical contraposition;4. Classical reductio.

Unlike truth-functional one, this aexamination of structurally identicsystems, where somewhat differenexample, in intuitionistic logic whvalid rules of inference, (pq) the material conditional is used to

Other properties of implication (foany logical values of variables):

distributivity:

transitivity:

reflexivity:

truth preserving: The inter

assigned a truth value of 'tresult of material implicati


Page 16

material conditional not p or qical implication p implies q (symbolized

nsidered as a symbol of a formal theory,ng all the classical inferences involvingacteristic rules:

proach to logical connectives permits theal propositional forms in various logicalt properties may be demonstrated. Forch rejects proofs by contraposition as

p q is not a propositional theorem, butdefine negation.

llowing expressions are always true, for

retation under which all variables are

ue' produces a truth value of 'true' as an.

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commutativity of antecede

Note that is logicaproperty is sometimes called currconvenient to adopt a right-associ

denotes .

Note also that comparison of truth, and it is sometimes conv

proofs. Such a replacement can be

Logical biconditional

In logic and mathematics, the logithe material biconditional) is theasserting "p if and only ifq", whera conclusion (orconsequent).[1] Toperator is denoted using a doubleequality sign (=), an equivalence sto (p q) (q p), or the XNequivalent to "(not p or q) and (no"(p and q) or (not p and not q)", m

The only difference from materialhypothesis is false but the conclusthe result is true, yet in the bicondi

In the conceptual interpretation, a a 's"; in other words, the sets a annot mean that the concepts have th"trilateral", "equiangular triangle"is the subjectand the consequent iproposition.

In the propositional interpretation,implies a; in other words, that theeither true or false at the same timsame meaning. Example: "The tritriangle ABC has two equal anglecause and the consequent is the co translated by a hypothetical(orco called the hypothesis (or the condi thesis.

A common way of demonstratingthe conjunction of two converse cseparately.


Page 17

ts:

lly equivalent to ; thising. Because of these properties, it istive notation for where

table shows that is equivalent toenient to replace one by the other inviewed as a rule of inference.

al biconditional (sometimes known as logical connective of two statements

e q is a hypothesis (orantecedent) and p isis is often abbreviated p iff q. Theheaded arrow (), a prefixed E (Epq), anign (), orEQV. It is logically equivalentR (exclusive nor) boolean operator. It isq or p)". It is also logically equivalent to

eaning "both or neither".

conditional is the case when theon is true. In that case, in the conditional,tional the result is false.

= b means "All a 's are b 's and all b 's are b coincide: they are identical. This does

e same meaning. Examples: "triangle" andand "equilateral triangle". The antecedents the predicate of a universal affirmative

a b means that a implies b and bpropositions are equivalent, that is to say,. This does not mean that they have thengle ABC has two equal sides", and "The". The antecedent is the premise or the

nsequence. When an implication is ditional) judgment the antecedent is

ion) and the consequent is called the

a biconditional is to use its equivalence tonditionals, demonstrating these

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When both members of the biconseparated into two conditionals, oother its reciprocal. Thus whenevhave a biconditional. A simple theantecedent is the hypothesis and w

theorem.

It is often said that the hypothesisthe thesis the necessary condition sufficient that the hypothesis be trnecessary that the thesis be true fotheorem and its reciprocal are trueand sufficient condition of the theboth cause and consequence.

Definition

Logical equality (also known as bivalues, typically the values of twoif and only if both operands are fal

Truth table

The truth table for (alsofollows:

INPUT OUTPUT

A B A B

0 0 1

0 1 0

1 0 0

1 1 1

More than two statements combin


Page 18

itional are propositions, it can bewhich one is called a theorem and ther a theorem and its reciprocal are true we

orem gives rise to an implication whosehose consequent is the thesis of the

s the sufficient condition of the thesis, andof the hypothesis; that is to say, it ise for the thesis to be true; while it is

r the hypothesis to be true also. When awe say that its hypothesis is the necessaryis; that is to say, that it is at the same time

conditional) is an operation on two logicalpropositions, that produces a value oftruese or both operands are true.

ritten as A B, A = B, orA EQ B) is as

d by are ambiguous:

ay be meant as

,

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or may be used to say that all a

Only for zero or two arguments th

The following truth tables show thargument and in the lines with tw

meant as equivalent to

The central Venn diagram below,and line (ABC ) in this matrixrepresent the same operation.

meant as shorthand for

Properties

commutativity: yes

associativity: yes

distributivity: Biconditional doeseven itself),but logical disjunction (see there)


Page 19

e together true or together false:

s is the same.

e same bit pattern only in the line with noarguments:

n't distribute over any binary function (not

istributes over biconditional.

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idempotency: no

monotonicity: no


falsehood-preserving: noWhen all inputs are false, the outp

Rules of Inference

Like all connectives in first-orderinference that govern its use in for

Biconditional Introduction

Biconditional introduction allowsA follows from B, then A if and o

For example, from the statementsI'm alive, then I'm breathing", it ca


Page 20

t is true.

t is not false.

ogic, the biconditional has rules ofmal proofs.

ou to infer that, if B follows from A, andly if B.

"if I'm breathing, then I'm alive" and "ifn be inferred that "I'm breathing if and

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only if I'm alive" or, equally inferrable, "I'm alive if and only if I'mbreathing."

B AA B

A B

B AA B B A

Tautology

The word tautology was used by the ancient Greeks to describe a statement

that was true merely by virtue of saying the same thing twice, a pejorativemeaning that is still used for rhetorical tautologies. Between 1800 and 1940,the word gained new meaning in logic, and is currently used in mathematicallogic to denote a certain type of propositional formula, without thepejorative connotations it originally possessed.

In 1800, Immanuel Kant wrote in his bookLogic:

"The identity of concepts in analytical judgments can be eitherexplicit(explicita) ornon-explicit(implicita). In the former case

analytic propositions are tautological."

Here analytic proposition refers to an analytic truth, a statement in naturallanguage that is true solely because of the terms involved.

In 1884, Gottlob Frege proposed in his Grundlagen that a truth is analyticexactly if it can be derived using logic. But he maintained a distinctionbetween analytic truths (those true based only on the meanings of theirterms) and tautologies (statements devoid of content).

In 1921, in his Tractatus Logico-Philosophicus, Ludwig Wittgensteinproposed that statements that can be deduced by logical deduction aretautological (empty of meaning) as well as being analytic truths. HenriPoincar had made similar remarks in Science and Hypothesis in 1905.Although Bertrand Russell at first argued against these remarks byWittgenstein and Poincar, claiming that mathematical truths were not onlynon-tautologous but were synthetic, he later spoke in favor of them in 1918:

"Everything that is a proposition of logic has got to be in some sense

or the other like a tautology. It has got to be something that has somepeculiar quality, which I do not know how to define, that belongs tological propositions but not to others."

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Here logical proposition refers tolaws of logic.

During the 1930s, the formalizatioin terms of truth assignments was

applied to those propositional forfalsity of their propositional variaSymbolic Logicby C. I. Lewis anproposition (in any formal logic) tpresentations after

this (such as Stephen Kleene 1967tautology to refer to a logically vaa distinction between tautology an order logic (see below).

Definition and examples

A formula of propositional logic itrue regardless of which valuation

There are infinitely many tautolog

("A or not A"), the law of

only one propositional variable, A.definition, assign A one of the trutother truth value.

("and vice versa), which exp

negation not-B, then not-A which is the principle kno

("if

B", and vice versa), whichthen A implies C"), which

true, and each implies C, tprinciple known as proof b

A minimal tautology is a tautologtautology.

is a tautolan instantiation of .

Verifying tautologies


Page 22

proposition that is provable using the

n of the semantics of propositional logic developed. The term tautologybegan to be

ulas that are true regardless of the truth orles. Some early books on logic (such as Langford, 1932) used the term for any

at is universally valid. It is common in

and Herbert Enderton 2002) to useid propositional formula, but to maintaind logically validin the context of first-

a tautology if the formula itself is alwaysis used for the propositional variables.

ies. Examples include:

the excluded middle. This formula has

Any valuation for this formula must, byvalues true orfalse, and assign A the

ifA implies B then not-B implies not-A",esses the law of contraposition.

("if not-A implies both B and itsmust be false, then A must be true"),n as reductio ad absurdum.

not both A and B, then either not-A or not-

is known as de Morgan's law.("ifA implies B and B implies C,s the principle known as syllogism.

(if at least one ofA orB isen Cmust be true as well), which is they cases.

that is not the instance of a shorter

ogy, but not a minimal one, because it is

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The problem of determining whetin propositional logic. If there arethere are 2n distinct valuations fordetermining whether or not the forone: one need only evaluate the tr

possible valuations. One algorithvaluation causes this sentence to bevery possible valuation.

For example, consider the formula

There are 8 possible valuations fo

represented by the first three colucolumns show the truth of subfora column showing the truth valuevaluation.

A B C

T T T T T T

T T F T F F

T F T F T T

T F F F T T

F T T F T T

F T F F T F

F F T F T T

F F F F T T

Because each row of the final coluverified to be a tautology.

ContradictionsThere are other propositional form

no matterwhat propositions you substitute falways get a


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er a formula is a tautology is fundamentalvariables occurring in a formula then

the formula. Therefore the task ofmula is a tautology is a finite, mechanicalth value of the formula under each of its

ic method for verifying that everye true is to make a truth table that includes

the propositional variables A, B, C,

ns of the following table. The remainingulas of the formula above, culminating inf the original formula under each

T T

F T

T T

T T

T T

T T

T T

T T

mn shows T, the sentence in question is

s with only false substitution instances

r the propositional variables, you will

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false proposition as a result. These propositional forms are calledcontradictions.Again recall the example of such a proposition:

p p

Again, it does not matter whether you substitute a true proposition or a false

proposition forp, the resulting proposition will be false

DefinitionContradictions are propositional forms all of whose substitution instancesare false.

ContingenciesFinally, there are propositional forms that do not determine the truth-value

of theirsubstitution instances they are called contingencies. When a propositionthat has thepropositional form of a contingency is true, the proposition is said to becontingentlytrue, when such a proposition is false, it is said to be contingently false.Consider thefollowing propositional form:~pDepending on whether you substitute a false or a true proposition forp, theresulting proposition will be true or false, respectively

DefinitionContingencies are those propositional forms some of whose substitutioninstances aretrue and some of whose substitution instances are false

4. Truth-Table Definitions of a Tautology, a Contradiction, aContingency

Logicians have invented the truth-table method for checking whether a

propositionalform is tautologous, contradictory or contingent. Truth tables forpropositional formsallow to determine all the possible truth-values that the substitution instancesof thoseforms can have.

A propositional form that is true in all rows of its truth table is a tautology.

A propositional form that is false in all rows of its truth table is acontradiction.

A propositional form that is true in at least one row of its truth table andfalse in at

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least one row of its truth table is a contingency.

Lets consider an example of a contingency: p (p q). Now note,however,that there are two propositional variables in that propositional form, viz. p

and q. Wethus need to consider more possibilities in fact, four of truth-valueassignmentsfor variables p and q:

The propositional form p (p q) is a contingency because there are atleast one row(in fact there are two such rows rows 1 and 2) where it is true, and there isat leastone row in fact there are two such rows rows 3 and 4) where it is false

Logically equivalent

Two sentences are Logically Equivalent if and only if they are true inprecisely the same situations. Another way to think of a pair of logicallyequivalent sentences is as two sentences that say the exact same thing. Ifone of them is true, the other is true; if one of them is false, the other is false.

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The notion of logical equivalence can be broadened to apply to collectionsof sentences of any size (not just pairs).

[edit] Truth Table Test for Logical EquivalenceThe easiest way to test whether two sentences are logically equivalent is to

make a truth-table for the sentences in question. Once you fill out the truth-table for the sentences, you look at thecolumn under the main connective of each sentence. The two sentences arelogically equivalent exactly when the column under the main connective ofthe first sentence is identical to the column under the main connective of thesecond sentence. Notice that this captures in a formal way the informal ideathat two sentences are logically equivalent just in case they are either bothtrue or both false in any given situation.

[edit] ExamplesThe sentences (A B) and (A B) are logically equivalent, as thefollowing truth-table shows:

A B (A B) (A B)

T T F F

T F F F

F T F F

F F T T

The sentences (A B) and (A B) are also logically equivalent, as thefollowing truth-table shows:

A B (A B) (A B)

T T F F

T F T T

F T T T

F F T T

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The sentences (PQ) and (QP)following truth-table demonstrate

P Q (PQ) (QP)

T T T T

T F F T

F T T F

F F T T

Quantifiers

The words in italics are quantifierreformulating any one of these exsentences, each a simple predicatewas chipped. These examples alsoexpressions in natural language caFortunately, for mathematical assesyntactically more straightforward

There are two quantifiers, the univquantifier. The traditional symbolinverted letter "A", which stands fsymbol for the existential quantififor "there exists" or "exists".

An example of quantifying an Enthe statement, "All of Peters frien

beach", we can identify key aspecquantifiers. So, let x be any one paPeter's friends, P(x) be the predicaand lastly Q(x) the predicate "x li

. Which is reor Q of x."

Some other quantified expressions

for a formula P. Variant notations


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are not logically equivalent, as the:

. There exists no simple way ofressions as a conjunction or disjunction ofof an individual such as That wine glass

suggest that the construction of quantifiedn be syntactically very complicated.rtions, the quantification process is

ersal quantifier and the existentialfor the universal quantifier is " ", anr "for all" or "all". The correspondingr is " ", a rotated letter "E", which stands

lish statement would be as follows. Givens either like to dance or like to go to the

s and rewrite using symbols includingrticular friend of Peter, X the set of allte (mathematical logic) "x likes to dance",es to go to the beach". Then we have, :d, "for all x that's a member of X, P of x

are constructed as follows,

include, for set

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Xand set members x

All of these variations also apply tvariations for the universal quanti

Equivalent Expressions

IfXis a domain ofx and P(x) is auniversal proposition is expressed

which equivalently reads "ifx is iP(x) is indeterminate. Note that thxbe in X, so it can be any x in X, ithe expression, or the truth of

additionally requires that xbe sucreason behind calling x a "boundread as "for some x in X, P(x) is fa

P(x) is false." So, we now have thexistential proposition:

Thus, together with negation, onlyquantifier is needed to perform bo

which shows that to disprove a "fthan to find an x for which the pre

to disprove a "there exists an x" prpredicate is false for all x.

Various Normal Forms

Disjunctive Normal FormsConjunctive Normal Forms


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o universal quantification. Otherier are

redicate dependent on x, then thein Boolean algebra terms as

X, then P(x) is true." Ifx is not in X, thentruth of the expression requires only thatdependent ofP(x), whereas the falsity of

that P(x) evaluates to false; this is theariable." This last expression can thus belse," or "there exists an x in Xsuch that

equivalent Boolean expression for the

one of either the universal or existentialh tasks:

r all x" proposition, one needs no moreicate is false. Similarly,

oposition, one needs to show that the

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Principal Disjunctive Normal FormsPrincipal Conjunctive Normal Forms

VARIOUS NORMAL FORMS

In this section you will know about four kinds of normal forms i.e.Disjunctive, Conjunctive, principal Disjunctive & Principal Conjunctivenormal forms.

Disjunctive Normal Forms

We shall use the words 'product' and 'sum' in place of the logical connectives'conjunction' and 'disjuncdon' respectively. Before defining what is called'disjunctive normal forms' we first introduce some other concepts needed inthe sequel. In a formula, a product of the variables and their negations is

called an elementary product. Similarly a sum of the variables and theirnegations is called an elementary sum.

Let P and Q be any two variables. Then P., P Q, Q P P, P P,Q P are some examples of elementary products; and P, P Q, Q P P, P P , Q P are examples of elementary sums of twovariables. A part of the elementary sum or product which is itself anelementary sum Or product is called a factor of the Original sum Or ProductThe elementary sums or products satisfy the following properties. We onlystate them without proof.

1. An elementary product is identically false if and only if it contains atleast one pair of factors in which one is the negation of the other.

2. An elementary sum is identically true if and only if it contains atleastone pair of factors in which one is the negation of the other.

We now discuss some examples.

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It should be noted that the disjunctive normal form of a given formula is notunique. For example, consider P ( Q R ). This is already in disjunctivenormal form.

We can write

P ( Q R ) ( P Q ) ( P R )

( P P ) ( P Q ) ( P R ) ( Q P )

zaand this is another equivalent normal form. In subsequent sections weintroduce the concepts of Principal normal forms which give unique normalform of a given formula.

Conjunctive Normal Forms

A formula which consists of a product of elementary sums and is equivalentto a given formula is called a conjunctive normal form of the given formula.

Principal Disjunctive Normal Forms

For two Statement variables P and Q, construct all possible formulas whichconsist of Conjunctions of P or its negation and conjunctions of Q or itsnegation such that none of the formulas contain both a variable and itsnegation. Note that any formula which is obtained by commuting theformulas in the conduction should not be included in the list because anysuch formula will be equivalent to one included in the list. For example, anyone of P Q or Q P is included in the list, but not the both. For two

variables P and Q, there are 2

2

= 4 formulas included in the list these aregiven by

P Q, P Q, P Q, P Q.

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These formulas are called minterms. One can construct truth table andobserve that no two minterms are equivalent. The truth table is given below :

PQ P Q P Q P Q P QT T T F F FT F F T F FF T F F T FF F F F F T

Given one formula, an equivalent formula, consisting of disjunctions ofminterrns only is known as its Principal Disjunctive normal, form or sum-of-products canonical form.

Principal Conjunctive Normal Forms

Given a number of variables, maxterm of these variables is a formulawhich consists of disjunctions in which each variable or its negation but notboth appear only once. Observe that the maxterms are the duals of minterms.Note that each of the minterms has the truth value T for exactly onecombination of the truth values of the variables and this fact can be seenfrom truth table. Therefore each of the maxterms has the truth value F forexactly one combination of the truth values of the variables. This fact can bederived from the duality principle (using the properties of minterms) or canbe directly obtained from truth table.

For a given formula, an equivalent formula consisting of conductions of themaxterms only is known as its principal conjunctive normal form or theproduct-of-sums canonical form.

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VALID INFERENCE USING TRUTH TABLES AND DIRECTMETHOD OF PROOF

Let A and B two statement formulas. It is said that 'B follows from A' or 'Bis a conclusion of the premise A' or 'B is a consequence of the premise A' iffA B is a tautology.

Let { A1, A2, ...... An } be a set of formulas. It is said that a conclusion Bfollows from the set of premises { A1, A2, ...... An } iff

A1 A2 .... An B is a tautology.

Given a set of premises and a conclusion, it is possible, by using truth table,to know whether the conclusion follows from the given premises. But thismethods becomes tedious and time consuming when the number of variablespresent in all the formulas representing premises and conclusion are large.Also there are situations where truth table technique is no longer available.Because of these facts other methods have been developed and we discussthem in following sections.

RULES OF INFERENCE ( RULES P AND T)

We shall describe the method by which one derives that a formula is aconsequence of a set of given Premises ( also called hypotheses ). We firstquote two rules of inference called the rules P and T.

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Rule P: A premise may be introduced at any point of the derivation.

Rule T : A formula S may be introduced in a derivation if S istautologically implied by any, one or more of the proceedingformulas in the derivation.

We now, list some important implications and equivalences which will beused in the derivations. All the implications and equivalences which will belisted now, are not independent of each other; some of them can be derived

from some others. So one could start with minimum number of them. but wewill not follow such an axiomatic scheme in this introductory text. Some ofthese were also listed earlier while discussing implications.

Implications

3.4.2 Equivalences

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CONSISTENCY OF PREMISES AND INDIRECT METHOD OFPROOF

If R is any formula, note that R R is a contradiction. A set of formulasA1, A2, ...., Am. is said to be consistent if their conjunction has the truthvalue T for some assignment of the truth values to the atomic variablesappearing in the formulas A1, A2, ...., Am. Otherwise the formulas are calledinconsistent. In otherwords, A1, A2, ...., Am are said to be inconsistent if for

every assignment of the truth values to the atomic variables, atleast one ofthe formulas A1, A2, ...., Am is false, i.e. their conjuction is identically false.

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To be precise, a set of formulas A1, A2, ...., Am is consistent if theirconjuction is a contradiction, i.e. if

A1 A2 .... Am R R

where R is any formula.

The notion of inconsistency is used in a procedure called 'proof bycontradiction orreductio ad absurdum or indirect method of proof'. In orderto prove that a conclusion C follows from a set of premises A1, A2, ...., Amby using the method 'proof by contradiction' the procedure is as follows :

Assume that C is false and consider C as an additional premise. If the newset of premises C, A1, A2, ...., Am is inconsistent, they imply acontradiction, then the assumption that C is true does not holdsimultaneously with A

1 A

2 ..... A

mbeing true. Therefore C is true

whenever A1 A2 ..... Am is true. Thus the conclusion C follows fromthe premises A1, A2, ...., Am.

Proof by contradiction is sometimes very much convenient. But if required,it can be eliminated and replaced by another method, called a conditionalproof (CP) ). To demonstrate this observe that

' P ( Q Q ) ' P ..... (1)

In the proof by contradiction we show

A1, A2, ...., Am C

by showing

A1, A2, ...., Am, C R P

Now this can be converted to the following by using rule CP

A1, A2, ...., Am C ( R R ) ..... (2)

From (1) and (2) by using the rule of double negation one can prove that

A1, A2, ...., Am C

and this is the required derivation.

AUTOMATIC PROVING OF THEOREMS

In this section we reformulate the rules of inference theory for statementcalculus and describe a procedure of derivation which can be conductedmechanically.

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Before describing this procedure, we first discuss some of the shortcomingsof the procedure used in the earlier section. We shall then describe a set of10 rules an axiom schema and rules for sequents.

First of all observe that rule P permits the introduction of a premise at any

point in the derivation, but it does not suggest either the Premises or the stepat which it should introduced. Similarly Rule T allows us to introduce anyformula which follows from the previous steps, but it does not suggest anydefinite choice of such a formula nor there is any guidance for the use of anyparticular equivalence. Similarly Rule CP does not tell anything about thestages at which an antecedent is to be introduced as an additional premise,nor does it indicate the stage at which it is again incorporated into theconditional. At every step, such decisions are taken from a large number ofalternatives, with the ultimate aim of reaching the conclusion.

We now describe our system which is given as follows :

Variables

The capital letters Of English alphabet A, B, C, ..... P, Q, R, ..... will be usedto denote the statement variables. They are also used to represent thestatement formulas, but these notations will be clear from the context

Connectives

The connectives , , , , appear in the formulas with the order of

precedence as given above.

String Of Formulas

A string of formulas is defined as follows :

i) any formula is a string of formulas,

ii) if a and b are string of formulas, then ( a , b and b , a are strings offormulas,

iii) only those strings which are obtained from steps (i) and (ii) arestrings of formulas; with the exception of the empty string which isalso a string of formulas.

The order in which the formulas appear in a string is not important; thestrings A, B, C; B, C, A; A, C, B etc. are the Same.

Sequents

If a and b are strings of formulas then a b is called a sequent in which a isthe antecedent and b the consequent of the sequent. A sequent is trueif either atleast one of the formulas of the antecedent is false or atleast one of

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the formulas of the consequent is tproof and the latter is called the T

Occasionally we have sequents wanticedent or as consequent. The e

logical constant 'true' or T and thelogical constant 'false' or F.

Axiom Scheme

If a , and b strings of formulas sucvariable only, then the sequent avariable common. For example, Avariables is an axiom.

TheoremThe following sequents are theore

i) every axiom, is a theorem.

ii) if a sequent a is a theoremthe use of one of the 10 rules of tha theorem,

iii) sequents obtained from (i)

Beyond the Syllabus

Applications in computer scienc

Operators corresponding to logicalanguages.

Bitwise operation

Disjunction is often used for bitwi

0 or 0 = 0 0 or 1 = 1 1 or 0 = 1 1 or 1 = 1


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rue. The former case is called the vacuousivial proof.

ich have empty strings of formulasmpty antecedent, is interpreted as the

empty consequent is interpreted as the

that every formula in both a and b isb is an axiom if a and b have atleast one

, B, C P, B, R where A, B, C, P, R are

s of our system.

and a sequent b results from cc throughe system which are given below, then b is

and (ii) are the only theorems.

disjunction exist in most programming

se operations. Examples:

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1010 or 1100 = 1110

The or operator can be used to set bits in a bit field to 1, by or-ing the fieldwith a constant field with the relevant bits set to 1. For example, x = x |0b00000001 will force the final bit to 1 while leaving other bits unchanged.

Logical operation

Many languages distinguish between bitwise and logical disjunction byproviding two distinct operators; in languages following C, bitwisedisjunction is performed with the single pipe (|) and logical disjunction withthe double pipe (||) operators.

Logical disjunction is usually short-circuited; that is, if the first (left)operand evaluates to true then the second (right) operand is not evaluated.The logical disjunction operator thus usually constitutes a sequence point.

Although in most languages the type of a logical disjunction expression isboolean and thus can only have the value true or false, in some (such asPython and JavaScript) the logical disjunction operator returns one of itsoperands: the first operand if it evaluates to a true value, and the secondoperand otherwise.

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