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Classical predicate logic

T: uU [0,1]

U: universe of all propositions.

All elements u U are true for proposition P are called the truth set of P: T(P).

Those elements u U are false for P are called falsity set of P: F(P).

T(Y) = 1 T(Ø) = 0

Logic connectives

Disjunction

Conjunction

Negation –

Implication

Equivalence

If xA, T(P) =1 otherwise T(P) = 0

Or

xA(x)={ 1 if x A, otherwise it is 0 }

If T(p)T()=0 implies P true, false, or true P false. P and are mutually exclusive propositions.

Given a proposition P: xA, P: xA, we have the following logical connectives:

Disjunction

PQ: x A or x B

hence, T(PQ) = max(T(P),T(Q))

Conjunction

PQ: xA and xB

hence T(P Q)= min(T(P),T(Q))

Negation

If T(P) =1, then T(P) = 0 then T(P) =1

Implication

(P Q): xA or xB

Hence , T(P Q)= T(P Q)

Equivalence

1, for T(P) = T(Q)

(P Q): T(PQ)=

0, for T(P) T(Q)

The logical connective implication, i.e.,P Q (P implies Q) presented here is also known as the classical implication.

P is referred to as hypothesis or antecedent

Q is referred to as conclusion or consequent.

T(PQ)=(T(P)T(Q))

Or PQ= (AB is true)

T(PQ) = T(PQ is true) = max (T(P),T(Q))

(AB)= (AB)= AB

So (AB)= AB

Or AB false AB

Truth table for various compound propositions

PQ: If x A, Then y B, or PQ AB

The shaded regions of the compound Venn diagram in the following figure represent the truth domain of the implication, If A, then B(PQ).

X

A

B Y

IF A, THEN B, or IF A , THEN C

PREDICATE LOGIC

(PQ)(PS)

Where P: xA, AX

Q: yB, BY

S: yC, CY

SET THEORETIC EQUIVALENT

(A X B)(A X C) = R = relation ON X Y

Truth domain for the above compound proposition.

Some common tautologies follow:

BB X

AX; A X X

AB (A(AB))B (modeus ponens)

(B(AB))A (modus tollens)

Proof:

(A(AB)) B

(A(AB)) B Implication

((AA) (AB))B Distributivity

((AB))B Excluded middle laws

(AB)B Identity

(AB)B Implication

(AB)B Demorgans law

A(BB) Associativity

AX Excluded middle laws

X T(X) =1 Identity; QED

Proof

(B(AB))A

(B(AB))A

((BA)(BB)) A

((BA))A

(BA)A

(BA)A

(BA)A

B(AA)

BX = X T(X) =1

Truth table (modeus ponens)

Contradictions

BB

A; A

Equivalence

PQ is true only when both P and Q are true or when both P and q are false.

Example

Suppose we consider the universe positive integers X={1 n8}. Let P = “n is an even number “ and let Q =“(3n7)(n6).” then T(P)={2,4,6,8} and T(Q) ={3,4,5,7}. The equivalence PQ has the truth set T(P Q)=(T(P)T(Q)) (T(P) (T(Q)) ={4} {1} ={1,4}

T(A)

Venn diagram for equivalence

T(B)

Exclusive or

Exclusive Nor

Exclusive or P “” Q

(AB) (AB)

Exclusive Nor

(P “” Q)(PQ)

Logical proofs

Logic involves the use of inference in everyday life.

In natural language if we are given some hypothesis it is often useful to make certain conclusions from them the so called process of inference (P1P2….Pn) Q is true.

Hypothesis : Engineers are mathematicians. Logical thinkers do not believe in magic. Mathematicians are logical thinkers.

Conclusion : Engineers do not believe in magic.

Let us decompose this information into individual propositions

P: a person is an engineer

Q: a person is a mathematician

R: a person is a logical thinker

S: a person believes in magic

The statements can now be expressed as algebraic propositions as

((PQ)(RS)(QR))(PS)

It can be shown that the proposition is a tautology.

ALTERNATIVE: proof by contradiction.

Deductive inferences

The modus ponens deduction is used as a tool for making inferences in rule based systems. This rule can be translated into a relation between sets A and B.

R = (AB)(AY)

Now suppose a new antecedent say A’ is known, since A implies B is defined on the cartesian space X Y, B can be found through the following set theoretic formulation B= AR= A((AB)(AY))

Denotes the composition operation. Modus ponens deduction can also be used for compound rule.

Whether A is contained only in the complement of A or whether A’ and A overlap to some extent as described next:

IF AA, THEN y=B

IF AA THEN y =C

IF AA , AA, THEN y= BC

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