logic: learning objectives

Discrete Mathematical Structures: Theory and Applications 1

Logic: Learning Objectives

Learn about statements (propositions)

Learn how to use logical connectives to combine statements

Explore how to draw conclusions using various argument forms

Become familiar with quantifiers and predicates

CS

Boolean data type

If statement

Impact of negations

Implementation of quantifiers


Mathematical Logic

Definition: Methods of reasoning, provides rules and techniques to determine whether an argument is valid

Theorem: a statement that can be shown to be true (under certain conditions)

Example: If x is an even integer, then x + 1 is an odd integer

This statement is true under the condition that x is an integer is true


Mathematical Logic

A statement, or a proposition, is a declarative sentence that is either true or false, but not both

Lowercase letters denote propositionsExamples:

p: 2 is an even number (true)

q: 3 is an odd number (true)

r: A is a consonant (false)

The following are not propositions:p: My cat is beautiful

q: Are you in charge?


Mathematical Logic Truth value

One of the values “truth” or “falsity” assigned to a statement

True is abbreviated to T or 1False is abbreviated to F or 0

NegationThe negation of p, written ∼p, is the statement obtained

by negating statement p Truth values of p and ∼p are oppositeSymbol ~ is called “not” ~p is read as as “not p”Example:

p: A is a consonant~p: it is the case that A is not a consonant

q: Are you in charge?


Mathematical Logic

Truth Table

ConjunctionLet p and q be statements.The conjunction of p and

q, written p ^ q , is the statement formed by joining statements p and q using the word “and”

The statement p∧q is true if both p and q are true; otherwise p∧q is false


Mathematical Logic

ConjunctionTruth Table for Conjunction:


Mathematical Logic

Disjunction

Let p and q be statements. The disjunction of p and q, written p v q , is the statement formed by joining statements p and q using the word “or”

The statement p v q is true if at least one of the statements p and q is true; otherwise p v q is false

The symbol v is read “or”


Mathematical Logic

DisjunctionTruth Table for

Disjunction:


Mathematical Logic

Implication

Let p and q be statements.The statement “if p then q” is called an implication or condition.

The implication “if p then q” is written p q

p q is read:

“If p, then q”

“p is sufficient for q”

q if p

q whenever p


Mathematical Logic

ImplicationTruth Table for Implication:

p is called the hypothesis, q is called the conclusion


Mathematical Logic

ImplicationLet p: Today is Sunday and q: I will wash the car.

The conjunction p q is the statement:p q : If today is Sunday, then I will wash the car

The converse of this implication is written q pIf I wash the car, then today is Sunday

The inverse of this implication is ~p ~qIf today is not Sunday, then I will not wash the car

The contrapositive of this implication is ~q ~pIf I do not wash the car, then today is not Sunday


Mathematical Logic

BiimplicationLet p and q be statements. The statement “p if and

only if q” is called the biimplication or biconditional of p and q

The biconditional “p if and only if q” is written p q

p q is read:“p if and only if q”“p is necessary and sufficient for q”“q if and only if p”“q when and only when p”


Mathematical Logic

BiconditionalTruth Table for the Biconditional:


Mathematical Logic

Statement Formulas Definitions

Symbols p ,q ,r ,...,called statement variables

Symbols ~, ^, v, →,and ↔ are called logical

connectives1) A statement variable is a statement formula2) If A and B are statement formulas, then the

expressions (~A ), (A ^ B) , (A v B ), (A → B )

and (A ↔ B ) are statement formulas Expressions are statement formulas that are

constructed only by using 1) and 2) above


Mathematical Logic

Precedence of logical connectives is:

~ highest

^ second highest

v third highest

→ fourth highest

↔ fifth highest


Mathematical LogicExample:

Let A be the statement formula (~(p v q )) → (q ^

p )Truth Table for A is:


Mathematical Logic

Tautology

A statement formula A is said to be a tautology if the truth value of A is T for any assignment of the truth values T and F to the statement variables occurring in A

Contradiction

A statement formula A is said to be a contradiction if the truth value of A is F for any assignment of the truth values T and F to the statement variables occurring in A


Mathematical Logic

Logically ImpliesA statement formula A is said to logically imply a

statement formula B if the statement formula A → B is a tautology. If A logically implies B, then symbolically we write A → B

Logically EquivalentA statement formula A is said to be logically

equivalent to a statement formula B if the statement formula A ↔ B is a tautology. If A is logically equivalent to B , then symbolically we write A ≡ B


Mathematical Logic


Mathematical Logic

Proof of (~p ^ q ) → (~(q →p ))


Mathematical LogicProof of (~p ^ q ) → (~(q →p )) [continued]


Validity of Arguments

Proof: an argument or a proof of a theorem consists of a finite sequence of statements ending in a conclusion

Argument: a finite sequence of statements.

The final statement, , is the conclusion, and the statements are the premises of the argument.

An argument is logically valid if the statement formula

is a tautology.

AAAAA nn,...,,,,

1321

An

AAAA n 1321...,,,,

AAAAA nn

1321...


Validity of Arguments - Example RPRQQP

P Q R Premises Valid

T T T T T T T T

T T F T F F F T

T F T F T F T T

T F F F T F F T

F T T T T T T T

F T F T F F T T

F F T T T T T T

F F F T T T T T

QP RQ RP



Valid Argument FormsModus Ponens (Method of Affirming)

P Q Premises Conclusion

Q

Valid

T T T T T T

T F F F F T

F T T F T T

F F T F F T

QP



Valid Argument Forms

Modus Tollens (Method of Denying)

QPP Q Premises Conclusion Valid

T T T F F F T

T F F T F F T

F T T F F T T

F F T T T T T

Q P



Valid Argument FormsDisjunctive Syllogisms

Disjunctive Syllogisms


Validity of Arguments Valid Argument Forms

Hypothetical Syllogism (proven earlier)

Dilemma



Valid Argument FormsConjunctive Simplification

Conjunctive Simplification



Valid Argument FormsDisjunctive Addition

Disjunctive Addition



Valid Argument FormsConjunctive Addition


Validity of Arguments – Formal Derivation

Prove Formal Derivation Rule Comment

1. P Q Premise2. Q R Premise

3. P Assumption Assume P4. Q 1,3, MP5. R 2,4, MP R is now proved

6. P R DT Discharge P, ie, P is no longer to be used, and conclude that P R

Uses Deduction Theorem (DT)

RPRQQP


Quantifiers and First Order Logic

Have dealt with Propositional Logic (Calculus) so far

Propositional variables, constants, expressions

Dealt with truth or falsity of expressions as a whole

Consider:1. All cats have tails2. Tom is a cat3. Tom has a tail

Cannot conclude 3, given 1 and 2 using propositional logic

Predicate Calculus – allows us to identify individuals such as Tom together with properties and predicates.



Predicate or Propositional Function

Let x be a variable and D be a set; P(x) is a sentence

Then P(x) is called a predicate or propositional function with respect to the set D if for each value of x in D, P(x) is a statement; i.e., P(x) is true or false

Moreover, D is called the domain of the discourse and x is called the free variable


Quantifiers and First Order LogicPropositional function example #1

Let P(x) be the statement: x is an odd integer

Let D be the set of all positive integers.

Then P is a propositional function with domain of discourse D.

• For each x in D , P(x) is a proposition, i.e. a sentence which is either true or false.

• P(1): 1 is an odd integer – True

• P(14): 14 is an odd integer - False


Quantifiers and First Order LogicPropositional function example #2

Let P(x) be the statement: the baseball player hit over .300 in 2003

Let D be the set of all baseball players.

Then P is a propositional function with domain of discourse D.

• For each x in D , P(x) is a proposition, i.e. a sentence which is either true or false.

• P(Barry Bonds): Barry Bonds hit over .300 in 2003 - True

• P(Alex Rodriguez): Alex Rodriguez hit over .300 in 2003 - False



Predicate or Propositional Function

Example: Q(x,y) : x > y, where the Domain is the set

of integers Q is a 2-place predicate Q is T for Q(4,3) and Q is F for Q (3,4)



Universal Quantifier

Let P(x) be a predicate and let D be the domain of the discourse. The universal quantification of P(x) is the statement:

For all x, P(x) or

For every x, P(x)

The symbol is read as “for all and every”

Two-place predicate:

)( xPx),( yxPyx



Universal Quantifier Examples

Consider the statement

It is true if P(x) is true for every x in D

It is false if P(x) is false for at least one x in D

Consider with D being the set of all real numbers.

The statement is true because for every real number x, it is true that the square of x is positive or zero.

Consider that with D being the set of

real numbers is false. Why?

xxP

02 xx

012 xx



Existential Quantifier

Let P(x) be a predicate and let D be the domain of the discourse. The existential quantification of P(x) is the statement:

There exists x, P(x)

The symbol is read as “there exists”

Bound VariableThe variable appearing in: or

)( xPx

)( xPx )( xPx



Existential Quantifier Example

Consider

It is true since there is at least one real number x for which the proposition is true. Try x=2

Suppose that P is a propositional function whose domain of discourse consists of the elements d1,…,dn. The following pseudocode determines whether

is true.

5

2

12x

xx

xxP



Negation of Predicates (DeMorgan’s Laws) Example:

If P(x) is the statement “x has won a race” where the domain of discourse is all runners, then the universal quantification of P(x) is , i.e., every runner has won a race. The negation of this statement is “it is not the case that every runner has won a race. Therefore there exists at least one runner who has not won a race. Therefore:

and so,

)(~ )( ~ xPxxPx

)( xPx

)(~ xPx

)(~ )( ~ xPxxPx



Negation of Predicates (DeMorgan’s Laws)

)(~ )( ~ xPxxPx



Formulas in Predicate LogicAll statement formulas are considered formulasEach n, n =1,2,...,n-place predicate P( )

containing the variables is a formula. If A and B are formulas, then the expressions

~A, (A∧B), (A∨B) , A →B and A↔B are statement formulas, where ~, ∧, ∨, → and ↔ are logical connectives

If A is a formula and x is a variable, then ∀x A(x) and ∃x A(x) are formulas

All formulas constructed using only above rules are considered formulas in predicate logic

xxx n,...,,

21

xxx n,...,,

21



Additional Rules of InferenceIf the statement ∀x P(x) is assumed to be true,

then P(a) is also true,where a is an arbitrary member of the domain of the discourse. This rule is called the universal specification (US)

If P(a) is true, where a is an arbitrary member of the domain of the discourse, then ∀x P(x) is true. This rule is called the universal generalization (UG)

If the statement ∃x P (x) is true, then P(a) is true, for some member of the domain of the discourse. This rule is called the existential specification (ES)

If P(a) is true for some member a of the domain of the discourse, then ∃x P(x) is also true. This rule is called the existential generalization (EG)



CounterexampleAn argument has the form ∀x (P(x ) → Q(x )),

where the domain of discourse is DTo show that this implication is not true in the

domain D, it must be shown that there exists some x in D such that (P(x ) → Q(x )) is not true

This means that there exists some x in D such that P(x) is true but Q(x) is not true. Such an x is called a counterexample of the above implication

To show that ∀x (P(x) → Q(x)) is false by finding an x in D such that P(x) → Q(x) is false is called the disproof of the given statement by counterexample


Logic and CS

Logic is basis of ALULogic is crucial to IF statements

ANDORNOT

Implementation of quantifiersLooping

Database Query LanguagesRelational AlgebraRelational CalculusSQL

logic: learning objectives

Documents