chapter 1 introduction to logic

7/21/2019 Chapter 1 Introduction to Logic

http://slidepdf.com/reader/full/chapter-1-introduction-to-logic 1/102

Math 29: Basic Concepts in Mathematics

Reymart S. Lagunero

Departamento ng Matematika at Agham Pangkompyuter

Unibersidad ng Pilipinas Baguio

04 Agosto 2015

Reymart S. Lagunero (UP na, Baguio pa!) Math 29: Introduction to Logic 04 Agosto 2015 1 / 91



Chapter 1: Introduction to Logic

What is Logic?

What is Logic?




Chapter 1: Introduction to Logic

What is Logic?

What is Logic?

Definition

Logic is the study of the methods and principles used to distinguish good(correct) from bad (incorrect) reasoning. There are objective criteria withwhich correct reasoning may be defined. If these criteria are not known,then they cannot be used. The aim of the study of logic is to discover andmake available those criteria that can be used to test arguments, and to

sort good arguments from bad ones.




Chapter 1: Introduction to Logic Statements

Statements

Definition

We define a statement intuitively as a sentence that can be assigned

either to the class of things we would call TRUE or to the class of thingswe would call FALSE, but not both. The TRUTH VALUE of a statementis “true” if the statement is true and “false” if the statement is false.

We often use P or Q to denote statements, or perhaps P 1, P 2, . . . , P n if

there are several statements involved.





Examples

Example

Some examples of statements are:

1 Today is Thursday.

2 I passed the final exam in Mathematics 53.

3 2 is a prime number.

4 There is an integer between 1 and 2.





Examples

Example

The sentence “The real number r is rational.” is a statement provided weknow precisely what real number r is being referred to. Without thisadditional information, however, it is impossible to judge whether thissentenced is true or false. Such a sentence is often referred to as an opensentence or an indeterminate.


Ch 1 I d i L i S




Examples

Example

We do not want to consider paradoxes as statements. For example “Thissentence is false.” cannot be either true or false. If you think this sentenceis true, then it is false. But, if it is false, then it is true. This sentence iscalled the liar paradox.


Ch t 1 I t d ti t L i L i l C ti



Chapter 1: Introduction to Logic Logical Connectives

Logical Connectives

Complex statements can be constructed from simple ones by means of logical connectives. Five logical connectives will be considered in thissection, namely: ¬, ∨, ∧, =⇒ and ⇐⇒ .






Negation

Definition

Negation The negation of a statement P is the statement ¬P , read as

“not P ”. The negation ¬P is true when P is false, and false when P istrue, that is

P ¬P

T F F T

(1.1)






Example

Example

P ¬P

It is raining today. It is not raining today.

2 is a prime number. 2 is not a prime number.2 is a composite number.

My shirt is black. My shirt is not black.

Observe in the last example that negating the statement “My shirt is

black” into “My shirt is white” is not logically correct because not wearinga black shirt does not necessarily mean that a person is wearing a whiteshirt (perhaps, a pink shirt!).






Conjunction

Definition

Conjunction The conjunction of two statements P and Q is thestatement P ∧ Q, read as “P and Q”. The conjuction P ∧ Q is true only

if both P and Q are true; otherwise, P ∧ Q is false, that is,

P Q P ∧ Q

T T T T F F

F T F F F F

(1.2)






Example

Example

Suppose that

P : John is a sophomore.

andQ : 2 is less than 1.

Then P ∧ Q is the statement “John is a sophomore and 2 is less than 1.”





p g g

Disjunction

Definition

Disjunction The disjunction of two statements P and Q is the statementP ∨ Q, read as “P or Q”. The disjunction P ∨ Q is true if at least one of

P and Q is true; otherwise, P ∨ Q is false, that is,

P Q P ∨ Q

T T T T F T

F T T F F F

(1.3)





g g

Example

Example

From the last example, P ∨ Q is the statement “John is a sophomore or 2is less than 1.”





Exclusive Disjunction

Remark

To address a statement involving P and Q that is true when precisely oneof them is true, we use the exclusive disjunction, denoted by P ∨̇Q, that

is,P Q P ∨̇ Q

T T F T F T F T T F F F

(1.4)





Implication

Definition

Implication For statements P and Q, the implication is the statement: “If P , then Q”, and is denoted by P =⇒ Q. The truth table for P ⇒ Q is

given byP Q P ⇒ Q

T T T T F F F T T F F T

(1.5)





Example

Example

A student is taking the Math 29 class and is currently receiving a B+. Heconsults his instructor a few days before the final exam and asks him, “Is

there any chance that I can get an A in this course?” His instructor looksthrough his grade book and says, “If you earn an A in the final exam, thenyou will receive a A for your final grade.”





Example

Example

A student is taking the Math 29 class and is currently receiving a B+. Heconsults his instructor a few days before the final exam and asks him, “Is

there any chance that I can get an A in this course?” His instructor looksthrough his grade book and says, “If you earn an A in the final exam, thenyou will receive a A for your final grade.” We now check the truth orfalseness of this implication based on the various combinations of truthtables of the statements P : You earn an A on the final exam and Q: Youreceive a A for your final grade, which make up the implication.





...continuation: Example

Suppose first that P and Q are both true, that is, the student receives an

A on his final exam and later learns that he got an A for his final grade inthe course. Did his instructor tell the truth?






Suppose first that P and Q are both true, that is, the student receives an

A on his final exam and later learns that he got an A for his final grade inthe course. Did his instructor tell the truth? I think we all agree thatshe did! So if P and Q are true, then so is P ⇒ Q, which agrees with thefirst row of the truth table.






Suppose that P is true and Q is false. So the student got a A on his final

exam but did not receive a A as final grade, say he received B. Certainlyhis instructor did not do as she promised. What he said was false, whichagrees with the second row of the truth table.






Suppose that P is false and Q is true. In this case, the student did not getan A on his final exam, say he earned B, but when he received his final

grades, he learned that his final grade was an A. How could thishappen?






Suppose that P is false and Q is true. In this case, the student did not getan A on his final exam, say he earned B, but when he received his final

grades, he learned that his final grade was an A. How could thishappen? Perhaps his instructor made a mistake, or perhaps the finalexam was unusually difficult, and a grade of B on it was exceptionallygood performance. In either case, the instructor did not lie; he told thetruth. This agrees with the third row of the table.






Suppose that P and Q are both false. That is, suppose the student didnot get an A on his final exam, and he also did not get an A for a finalgrade. The instructor did not lie here either. He promised nothing if thestudent did not get an A on the final exam. So the instructor told thetruth, and this agrees with the fourth row of the table.





Indeterminates

We have concerned ourselves only with statements. In mathematics,however, we are often interested in sentences containing variables andwhose truthfulness or falseness is only known once we have assigned values

to the variables, previously known as indeterminate. Just as newstatements can be formed from two statements P and Q by taking theirnegation, conjunction, or disjunction, new indeterminate can be formedfrom two open sentences P and Q by taking their negation, conjunction,or disjunction. Surprisingly, P ⇒ Q could be a statement even though P

and Q are not statements.





Example

Example

Consider the indeterminate p : x = −3 and q : |x|= 3, where x ∈ R.Indeed, it would be more appropriate to write

p(x) : x = −3 and q (x) : |x|= 3.

In this case, we have p(x) ⇒ q (x) can be stated as

If x = −3, then |x|= 3.

This implication is, in fact, a true statement.





Biconditional

Definition

Biconditional The biconditional of P and Q, denoted by P ⇐⇒ Q, istrue precisely when both P and Q have the same truth values, that is,

P Q P ⇔ Q

T T T T F F F T F

F F T

(1.6)





Example

Suppose thatP : An integer is divisible by 6.

andQ : An integer is divisible by 2 and 3.

Then P ⇔ Q is the statement “An integer is divisible by 6 if and only if itis divisible by 2 and 3.”





Rules of Precedence

For statements expressed symbolically, punctuation is accomplished byparentheses. To keep statements from looking cluttered, we will usecertain conventions for leaving out parentheses.





Rules of Precedence

For statements expressed symbolically, punctuation is accomplished byparentheses. To keep statements from looking cluttered, we will usecertain conventions for leaving out parentheses. The hierarchial order of the logical connectives is as follows: ¬, then ∨ or ∧ have the sameprecedence, followed by =⇒ , and then ⇐⇒ .





Rules of Precedence

The following is a list of ambiguous statements and thus, groupingsymbols have to be present when writing them.

1 ¬P ∧ Q does not stand for ¬(P ∧ Q); it stands for (¬P ) ∧ Q.

2 P ∨ Q ⇒ Q stands for (P ∨ Q) ⇒ R, not for P ∨ (Q ⇒ R).3 P ∨ R ∧ T is ambiguous, since it is not clear whether to apply ∨ or ∧

first.

4 P ⇒ Q ⇒ R is ambiguous, since either occurrence of ⇒ can be

applied first.


Tautologies, Contradictions and Logical Equivalence



Prime, Composite, and Component Statements

Statements without logical connectives are called prime statements while

statements with connectives are called composite statements. Thestatements which formed a composite statement are called its componentstatements.


Tautologies, Contradictions and Logical Equivalence Tautology



Definition (Tautology)

A statement S is called a tautology if it is true for all possiblecombinations of truth values of the component statements that composeS .

Remark

The symbol I will denote a statement that always has truth value T .





Example

Prove that the following compound statements is a tautology.

1 P ∨ (¬P ).

2 (¬Q) ∨ (P ⇒ Q).

3 P ∧ (P ⇒ Q) ⇒ Q (Modus Ponens)



E i 5 Mi



Exercise: 5 Minutes

1 Construct the truth table of the following complex statement:

¬(P ∨ Q) ∧ (P ∧ (¬Q ∨ (¬P ∨ Q))) .

When is the whole statement true?

2 When will the whole statement be true given

¬(P ∧ Q) ∨ R =⇒ ¬P ∨ R.


Tautologies, Contradictions and Logical Equivalence Contradictions



Definition (Contradiction)A statement S is called a contradiction if it is false for all possiblecombinations of truth values of the component statements that are usedto form S .

Remark

In fact, if a compound statement S is a tautology, then the compoundstatement ¬S is a contradiction. The symbol O will denote a statementthat always has truth value F .


Tautologies, Contradictions and Logical Equivalence Contradictions



Example

Prove that the following compound statements is a contradiction.

1

P ∧ (¬P ).2 (P ∧ Q) ∧ (Q ⇒ (¬P )).


Tautologies, Contradictions and Logical Equivalence Logical Equivalence



Definition (Logical equivalence)

We say that statements P and Q are logically equivalent, denoted by

P ≡ Q, if they have the same truth values for all combinations of truthvalues of their component statements.



Theorem



Theorem

It is clear that logical equivalence is reflexive, commutative and transitive.

We list some of the well-known logical equivalence in the followingtheorem.

Theorem

For statements P , Q and R, we have

Commutative Laws

P ∨ Q ≡ Q ∨ P P ∧ Q ≡ Q ∧ P

Associative Laws

P ∨ (Q ∨ R) ≡ (P ∨ Q) ∨ R P ∧ (Q ∧ R) ≡ (P ∧ Q) ∧ R)

Distributive Laws

P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R) P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)



continuation: Theorem



...continuation: Theorem

DeMorgan’s Laws

¬(P ∨ Q) ≡ (¬P ) ∧ (¬Q) ¬(P ∧ Q) ≡ (¬P ) ∨ (¬Q)

Rule of Double Negation

P ≡ ¬(¬P )

Or-form of an ImplicationP ⇒ Q ≡ (¬P ) ∨ Q

Contrapositive of an Implication

P ⇒ Q ≡ (¬Q) ⇒ (¬P )

Rule for Direct Proof

(P ∧ R) ⇒ Q ≡ P ⇒ (R ⇒ Q)

Biconditional

P ⇔ Q ≡ (P ⇒ Q) ∧ (Q ⇒ P )Reymart S. Lagunero (UP na, Baguio pa!) Math 29: Introduction to Logic 04 Agosto 2015 35 / 91


continuation: Theorem



...continuation: Theorem

Adjunction

(P ⇒ R) ∧ (P ⇒ Q) ≡ (P ⇒ (R ∧ Q))

Rule for proof by contradiction

(P ∧ ¬Q) ⇒ O ≡ (P ⇒ Q)

The symbol ≡ has the least precedence among the logical connectivesintroduced above.


Tautologies, Contradictions and Logical Equivalence Quantified Statements



Definition (Universal quantifier)

The universal quantifier ∀x means “for all x” or “for any x.”





Example

If x is a real number, then x2 ≥ 0. This statement is an implication, of course, and can be rephrased by

The square of every real number is nonnegative

orFor every real number x, we have x2 ≥ 0.

If we define the open sentence P (x) by P (x) : x2 ≥ 0, then we can rewritethe above statement in terms of the universal quantifier as

(∀x ∈ R)(P (x)).





Notice that if the set of complex numbers were under consideration, thenthe statement in the previous example would be false, since i2 = −1. Sothe truth value of a quantified statement may depend on the priorspecification of a set from which all elements are understood to come.

Such a set is called the universal set. In general, the statement(∀x ∈ U )(P (x)) is true if P (a) is true for all substitutions a ∈ U . If auniversal set U is not explicitly stated, the student can assume thatU = R.





Definition (Existential quantifier)The existential quantifier ∃x means “there exists an x such that” or“there is an x such that.”

Example

There exists a real number x such that x2 = 3. If we let P (x) : x2 = 3then this statement can be rewritten in terms of the existential quantifieras

(∃x ∈ R)(P (x)).





In general, the statement (∃x ∈ U )(P (x)) is true if there is at least onesubstitution a ∈ U for which P (a) is true.Some statements contain more than one quantifiers. As you will see, theorder of quantifiers is important. For instance, the statement

(∃u)(∀x)(P (x, u)) is read as “There exists an element u such that for allx, we have P (x, u)” while the statement (∀x)(∃u)(P (x, u)) represent thestatement “For every x there exists an element u such that P (x, u).”





Example

The statement “Every person has a mother” is of the form

(∀x)(∃y)(y is the mother of x)

while the statement “Someone is the mother of everyone” is of the form(∃y)(∀x)(y is the mother of x).

If the universal set is the set of all people who have ever been alive, thefirst statement is true, but the second is false.





Assignment

For each of the following statements, use your creativity and constructseveral statements that are logically equivalent to the given statement.

1 No elements of the set A exceeds m.

2 Some element of the set A exceeds m.

3 A contains an element that is greater than every element of B.

4 Every element of A is greater than every element of B.


Tautologies, Contradictions and Logical Equivalence Negating Statements



Using DeMorgan’s laws, we saw that

¬(P ∨ Q) ≡ (¬P ) ∧ (¬Q) and ¬(P ∧ Q) ≡ (¬P ) ∨ (¬Q).

Using the or-form of an implication, we see that

¬(P ⇒ Q) ≡ ¬(¬P ∨ Q) ≡ P ∧ (¬Q),

while using the fact that P ⇔ Q ≡ (P ⇒ Q) ∧ (Q ⇒ P ), we have

¬(P ⇔ Q) ≡ ¬[(P ⇒ Q) ∧ (Q → P )] ≡ (P ∧ (¬Q)) ∨ (Q ∧ (¬P )).



Now, we consider negating statements with quantifiers. Suppose someonek th f ll i l i



makes the following claim

Every student in the class passed the first exam.

For you to deny this claim, what sort of statement would you make? Youwould probably say something like

At least one person in the class failed the first exam.

If we let U be the set of students in the class and let Q be the set of students who passed the first exam. We can write the original statementin terms of the universal quantifier as

(∀x ∈ U )(x ∈ Q)

and its negation would be

(∃x ∈ U )(x /∈ Q)





In general, we can say that

¬[(∀x ∈ U )(P (x))] ≡ (∃x)(¬P (x)).

So the trick to negating a universal statement is that the ¬ symbol crawlsover the ∀x ∈ U and converts it to ∃x as it goes.





Example

Write in terms of the quantifiers the negation of the following statements.

1 For all x ∈ Z, x2 ≥ 0.

2 For all x ∈ Z, if x is divisible by 6 then x is divisible by 3.3 For all x ∈ R and x > 1, we have x3 − x > 0.





Now we negate statements involving the existential quantifier. Considerthe following statement

There is a passenger traveling in the jeepney who didn’t pay his fare.

If we let U be the set of all passengers riding in the jeepney and R be the

set of passengers who didn’t pay his fare, in terms of the existentialquantifier, the statement becomes

(∃x ∈ U )(x ∈ R).





To negate this, we could say

Every passenger in the jeepney paid his fare

which is equivalent to(∀x ∈ U )(x /∈ R).

In general, we can say that

(∃x ∈ U )(P (x)) ≡ (∀x ∈ U )(¬P (x)).





Example

Write in terms of the quantifiers the negation of the following statements.

1 Someone in this class cheated on the final exam.

2 There exists a natural number x such that x ≤ y for all y ∈ N .

3 For every > 0 there exists a δ > 0 such that,

if 0 < |x − a|< δ , then |f (x) − L|< .





Sometimes it is important to know not only that something exists, butalso that exactly one such thing exists. If there exists exactly one thingwith a certain property, we say that it exists uniquely. The mathematicalstatement

(∃! x ∈ U )(P (x))is read “There exists a unique x such that P (x).” The standard way of defining existence is the following.





Definition (Unique existence)We say that there exists a unique x with property P provided that

1 there exists x ∈ U with property P , and

2 for all x1 ∈ U and x2 ∈ U , if x1 and x2 both have property P , then

x1 = x2.In other words

[(∃x ∈ U )(P (x))] ∧ [(∀x1 ∈ U )(∀x2 ∈ U )(P (x1) ∧ P (x2)) ⇒ (x1 = x2)].


Tautologies, Contradictions and Logical Equivalence Disproving a Statement



Sometimes we are presented with a mathematical statement and we areasked to verify whether it is a tautology. Of course, you can create a truthtable for the given statement and check if it is indeed a tautology, that is,if all entries in the main column are T . The combination of truth valuesassigned to the component statements in any row that produces a false inthe main column of the truth table is called a counterexample.





Example

Show that (P ⇒ Q) ∧ Q ⇒ P is not a tautology.





Example


P Q (P ⇒ Q) ∧ Q ⇒ P

T T T T F T

F T F F F T





Example


P Q (P ⇒ Q) ∧ Q ⇒ P

T T T T F T

F T F F F T

The counterexample is P false and Q is true. Claiming the above exampleto be a tautology is a well-known fallacy of logic. It is called the fallacy of

asserting the conclusion, since the conclusion Q has been asserted aspart of the hypothesis.



Counterexamples will become important to us in disproving a givenmathematical statement. To disprove a given mathematical statement, we



mathematical statement. To disprove a given mathematical statement, weprove that its negation is a tautology. This is where our techniques of negation come in handy.






p g ,prove that its negation is a tautology. This is where our techniques of negation come in handy.

Example

Consider the statement

For all sets A, B, and C , if A ∪ C = B ∪ C , then A = B.






p g ,prove that its negation is a tautology. This is where our techniques of negation come in handy.

Example



We can disprove the statement by proving its negation to be true, that is,we would want to show that the statement






p gprove that its negation is a tautology. This is where our techniques of negation come in handy.

Example




There exists sets A, B, and C such that A ∪ C = B ∪ C and A = B

is true.






prove that its negation is a tautology. This is where our techniques of negation come in handy.

Example




There exists sets A, B, and C such that A ∪ C = B ∪ C and A = B

is true. For instance, let A = {1}, B = {2}, and C = {1, 2}. ThenA ∪ C = B ∪ C , but A = B . Again, such an example is also called acounterexample.


Tautologies, Contradictions and Logical Equivalence Translating English Sentences to Logical Expressions



Translating English Sentences to Logical Expressions


Tautologies, Contradictions and Logical Equivalence Translating English Sentences to Logical Expressions

Example



Translate the following English sentences into logical expressions.1 You can only access the Internet from campus only if you are a

computer science major or you are not a freshman.

Solution:

We let a, c, and f represent “You can access the Internet from campus”,“You are a computer science major”, and “You are a freshman”,

respectively. Noting that “only if” is one way a conditional statement canbe expressed, this sentence can be represented as a =⇒ (c ∨ ¬f ).

2 You cannot ride the roller coaster if you are under 4 feet tall unlessyou are older than 16 years old.

Solution:

Let q , r, and s represent “You can ride the roller coaster”, “You are under4 feet tall” and “You are older than 16 years old”, respectively. Then thesentence can be translated to (r ∧ ¬s) =⇒ ¬q .


Tautologies, Contradictions and Logical Equivalence Logic Puzzles



Logic Puzzles



Example



Smullyan posed many puzzles about an island that has two kinds of inhabitants, knights, who always tell the truth, and their opposites, knaves,who always lie. You encounter two people A and B. What are A and B if A says “B is a knight” and B says “The two of us are opposite types?”



Example



A father tells his two children, a boy and a girl, to play in their backyardwithout getting dirty. However, while playing, both children get mud ontheir foreheads. When the children stop playing, the father says “At leastone of you has a muddy forehead,” and then asks the children to answer

“Yes” or “No” to the question: “Do you know whether you have a muddyforehead?” The father asks this question twice. What will the childrenanswer each time this question is asked, assuming that a child can seewhether his or her sibling has a muddy forehead, but cannot see his or herown forehead? Assume that both children are honest and that the children

answer each question simultaneously.



Example



Four friends have been identified as suspects for an unauthorized accessinto a computer system. They have made statements to the investigatingauthorities. Alice said “Carlos did it.” John said “I did not do it.” Carlossaid “Diana did it.” Diana said “Carlos lied when he said that I did it.”

If the authorities also know that exactly one of the four suspects istelling the truth, who did it? Explain your reasoning.

If the authorities also know that exactly one is lying, who did it?Explain your reasoning.



Example



Vhong, Jinggy, and Juanito are all enrolled in Math 29. If Vhong comeslate then so is Jinggy. Either Jinggy or Juanito comes late but never both

at the same time. Either Vhong or Juanito or both are always late. If Vhong is on time then Juanito is also on time. Who is always on time?Justify your answer using truth table.


Rules of Inference for Propositional Logic



In this section, we introduce proof techniques and apply them to provingstatements from a collection of premises.

Definition

A proof is a step-by-step demonstration that a statement can be derivedfrom a collection of premises. A premise is a statement that is assumed inthe context of a proof.


Rules of Inference for Propositional Logic Modus Ponens

Rule of Inference: Modus Ponens



From P and P implies Q, infer Q. In other words,

P ∧ (P =⇒ Q) =⇒ Q






Example

1 Premises: T , T =⇒ S . Prove S .

2 Premises: T =⇒ R, R =⇒ S , T . Prove S .

3 Premises:x is even or x is odd.If x is even or x is odd, then x is not even implies that x is odd.x is not even.

Prove that x is odd.






Exercise

Premises:

¬(R ∧ T ) =⇒ ¬R ∨ ¬P

¬(R ∧ T )

¬R

¬R ∨ ¬T =⇒ (¬R =⇒ T )

Prove: T .


Rules of Inference for Propositional Logic Adjoining Premises

Rule of Inference: Adjoining Premises



Any statement can be adjoined to the premises in a proof if it can beproved from the premises.

Example

Premises:

P ∨ Q =⇒ (¬P =⇒ Q)P ∨ Q

¬P

Q =⇒ P

Q =⇒ RProve: R.



Exercise



Example

Premises:

P ∨ Q =⇒ (¬P =⇒ Q)

P ∨ Q¬P

Q =⇒ R

R =⇒ S ∧ T

Prove: S ∧ T .



Remark



We say that P is provable from a set of premises if there is a proof of P

from the set of premises.


Rules of Inference for Propositional Logic Direct Proof of an Implication

Rule of Inference: Direct Proof of an Implication



To prove an implication P =⇒ Q from a set of premises R, it is sufficientto assert the hypothesis (Hyp) P as an additional premises and show thatthe conclusion Q is provable from the augmented set of premises. DPI is

based on the tautology

(R =⇒ (P =⇒ Q)) ⇐⇒ (R ∧ P =⇒ Q) .



Example



Example

Premises:

x is even or x is odd.

If x is odd, then x2 is not even.

If x is even or x is odd, then x is not even implies that x is odd.

Prove: If x is not even, then x2 is not even.

Reymart S Lagunero (UP na Baguio pa!) Math 29: Introduction to Logic 04 Agosto 2015 72 / 91


Exercise



Premises:

P

Q =⇒ (P =⇒ ¬R)

Prove: Q =⇒ ¬R.


Rules of Inference for Propositional Logic Adjunction

Rule of Inference: Adjunction



If P and Q are provable from the same set of premises, then P ∧ Q is

provable from that set of premises. Adjunction is based on the tautology

(R =⇒ P ) ∧ (R =⇒ Q) ⇐⇒ (R =⇒ P ∨ R)

Example

Premises:

P =⇒ Q

Q ∧ (P ∨ R) =⇒ S ∧ T

P P =⇒ P ∨ R

Prove: S ∧ T .


Rules of Inference for Propositional Logic Adjunction

Exercise



Premises:

Q =⇒ P

QP ∧ Q =⇒ R ∨ T

Prove: R ∨ T .


Rules of Inference for Propositional Logic Substitution

Rule of Inference: Substitution



Assume that P 2 is obtained from P 1 by substituting R for any occurrence

of S in P 1. Then we can derive P 2 from S ⇐⇒ R and P 1. In otherwords, we may substitute R for S using the premise S ⇐⇒ R. We mayalso substitute S for R whenever S ⇐⇒ R is a premise.

Example

Premises:

R ∨ Q ⇐⇒ P ∨ Q

R ∨ Q =⇒ S

P =⇒ P ∨ Q

P

Prove: S .


Rules of Inference for Propositional Logic Substitution

Exercise



Premises:

Q ∨ P ⇐⇒ (¬Q =⇒ P )

¬Q

R =⇒ (Q ∨ P )

R

Prove: P .


Rules of Inference for Propositional Logic Contradiction

Rule of Inference: Contradiction

To prove Q from a set of premises, it is sufficient to use ¬Q as an



p Q p , Qadditional premise to prove any contradiction, for example, a statement of the form R ∧ ¬R. Contradiction is based on the tautology

(P ∧ ¬Q) =⇒ O ⇐⇒ (P =⇒ Q)

ExamplePremises:

If Kathy is not the murderer, then Mary is the murderer.

If Kathy is the murderer, then Jane is an accomplice.

If Mary is the murderer, then Noreen does not have an alibi.Noreen has an alibi.

Prove: Kathy is the murderer.



Exercise



Premises:

¬(P ∨ Q) =⇒ ¬P ∧ ¬Q

¬P =⇒ R

¬Q =⇒ ¬R

¬P ∧ ¬Q =⇒ ¬P

¬P ∧ ¬Q =⇒ ¬Q

Prove: P ∨ Q.





Remark

If we can prove P from a set of premises, then we say that P logicallyfollows, or simply follows, from the set of premises. An argument is asequence of steps proposed as a proof. If an argument is, in fact, a proof,

then it is called a valid argument; if it is not a proof, then it is called aninvalid argument.



Additional Examples



State which rule of inference is the basis of the following arguments:

It is below freezing now. Therefore, it is either below freezing orraining now.

It is below freezing and raining now. Therefore, it is below freezingnow.

If it rains today, then we will not have a barbecue today. If we do nothave a barbecue today, then we will have a barbecue tomorrow.Therefore, if it rains today, then we will have a barbecue tomorrow.



Additional Examples



Show that the premises “It is not sunny this afternoon and it is colderthan yesterday”, “We will go swimming only if it is sunny”, “If we donot go swimming, then we will take a canoe trip”, and “If we take acanoe trip, then we will be home by sunset” lead to the conclusion

“We will be home by sunset”.Show that the premises “If you send me an e-mail message, then Iwill finish writing the program,” “If you do not send me an e-mailmessage, then I will go to sleep early,” and “If I go to sleep early, thenI will wake up feeling refreshed” lead to the conclusion “If I do not

finish writing the program, then I will wake up feeling refreshed.”



Additional Examples



Is the following argument valid?

“If you do every problem in this book, then you will learn discretemathematics.You learned discrete mathematics. Therefore, you did every

problem in this book.”

Is it correct to assume that you did not learn discrete mathematics if youdid not do every problem in the book, assuming that if you do everyproblem in this book, then you will learn discrete mathematics?


Rules of Inference for Quantified Statements Universal Instantiation

Universal Instantiation



Universal instantiation is the rule of inference used to conclude thatP (c) is true, where c is a particular member of the domain, given the

premise ∀xP (x). Universal instantiation is used when we conclude fromthe statement “All women are wise” that “Lisa is wise,” where Lisa is amember of the domain of all women.


Rules of Inference for Quantified Statements Universal Generalization

Universal Generalization



Universal generalization is the rule of inference that states that ∀xP (x)is true, given the premise that P (c) is true for all elements c in thedomain. Universal generalization is used when we show that ∀xP (x) istrue by taking an arbitrary element c from the domain and showing that

P (c) is true. The element c that we select must be an arbitrary, and not aspecific, element of the domain. That is, when we assert from ∀xP (x) theexistence of an element c in the domain, we have no control over c andcannot make any other assumptions about c other than it comes from thedomain.


Rules of Inference for Quantified Statements Existential Instantiation

Existential Instantiation



Existential instantiation is the rule that allows us to conclude that thereis an element c in the domain for which P (c) is true if we know that∃xP (x) is true. We cannot select an arbitrary value of c here, but rather it

must be a c for which P (c) is true. Usually we have no knowledge of whatc is, only that it exists. Because it exists, we may give it a name (c) andcontinue our argument.

R t S L (UP B i !) M th 29 I t d ti t L i 04 A st 2015 86 / 91

Rules of Inference for Quantified Statements Existential Generalization

Existential Generalization



Existential generalization is the rule of inference that is used to concludethat ∃xP (x) is true when a particular element c with P (c) true is known.

That is, if we know one element c in the domain for which P (c) is true,then we know that ∃xP (x) is true.

R t S L (UP B i !) M th 29 I t d ti t L i 04 A t 2015 87 / 91

Combining Rules of Inferences Universal Modus Ponens

Universal Modus Ponens



This rule tells us that if ∀x(P (x) =⇒ Q(x)) is true, and if P (a) is truefor a particular element a in the domain of the universal quantifier, then

Q(a) must also be true. To see this, note that by universal instantiation,P (a) =⇒ Q(a) is true. Then, by modus ponens, Q(a) must also be true.


Combining Rules of Inferences Universal Modus Tollens

Universal Modus Tollens



Universal modus tollens combines universal instantiation and modus

tollens.





Salamat sa pakikinig!

♥

R S L (UP B i !) M h 29 I d i L i 04 A 2015 90 / 91


Math 29: Basic Concepts in Mathematics



Reymart S. Lagunero

Departamento ng Matematika at Agham Pangkompyuter

Unibersidad ng Pilipinas Baguio

04 Agosto 2015

R S L (UP B i !) M h 29 I d i L i 0 A 201 91 / 91

chapter 1 introduction to logic

Documents