The Foundations: Logic and Proofs - Valdosta State University

The Foundations: Logic and Proofs - Valdosta State University

The Foundations: Logic and Proofs Chapter 1, Part I: Propositional Logic With Question/Answer Animations Propositional Logic Section 1.1 Propositions A proposition is a declarative sentence that is either true or false. Examples of propositions: a) b) c) d) e) The Moon is made of green cheese.

Trenton is the capital of New Jersey. Toronto is the capital of Canada. 1+0=1 0+0=2 Examples that are not propositions. a) Sit down! b) What time is it? c) x + 1 = 2 d) x + y = z Propositional Logic Constructing Propositions Propositional Variables: p, q, r, s, The proposition that is always true is denoted by T and the proposition that is always false is denoted by F. Compound Propositions; constructed from logical connectives and other propositions Negation Conjunction

Disjunction Implication Biconditional Compound Propositions: Negation The negation of a proposition p is denoted by p and has this truth table: p p T

F F T Example: If p denotes The earth is round., then p denotes It is not the case that the earth is round, or more simply The earth is not round. Conjunction The conjunction of propositions p and q is denoted by p q and has this truth table: p T

q T pq T T F F F T F F F

F Example: If p denotes I am at home. and q denotes It is raining. then p q denotes I am at home and it is raining. Disjunction The disjunction of propositions p and q is denoted by p q and has this truth table: p q p q T

T T T F T F T T F F

F Example: If p denotes I am at home. and q denotes It is raining. then p q denotes I am at home or it is raining. The Connective Or in English In English or has two distinct meanings. Inclusive Or - In the sentence Students who have taken CS202 or Math120 may take this class, we assume that students need to have taken one of the prerequisites, but may have taken both. This is the meaning of disjunction. For p q to be true, either one or both of p and q must be true. Exclusive Or - When reading the sentence Soup or salad comes with this entre, we do not expect to be able to get both soup and salad. This is the meaning of Exclusive Or (Xor). In p q , one of p and q must be true, but not both. The truth q p q

table forp is: T T F T F T F T T F F F

Implication If p and q are propositions, then p q is a conditional statement or implication which is read as if p, then q and has this truth table: q p q p T T T T F F

F T T F F T Example: If p denotes I am at home. and q denotes It is raining. then p q denotes If I am at home then it is raining. In p q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). Understanding Implication In p q there does not need to be any connection

between the antecedent or the consequent. The meaning of p q depends only on the truth values of p and q. These implications are perfectly fine, but would not be used in ordinary English. If the moon is made of green cheese, then I have more money than Bill Gates. If the moon is made of green cheese then Im on welfare. If 1 + 1 = 3, then your grandma wears combat boots. Understanding Implication (cont) One way to view the logical conditional is to think of an obligation or contract. If I am elected, then I will lower taxes. If you get 100% on the final, then you will get an A.

If the politician is elected and does not lower taxes, then the voters can say that he or she has broken the campaign pledge. Something similar holds for the professor. This corresponds to the case where p is true and q is false. Different Ways of Expressing p q if p, then q if p, q q unless p q if p q whenever p q follows from p p implies q p only if q q when p q when p p is sufficient for q q is necessary for p

a necessary condition for p is q a sufficient condition for q is p Converse, Contrapositive, and Inverse From p q we can form new conditional statements . q p is the converse of p q q p is the contrapositive of p q p q is the inverse of p q Example: Find the converse, inverse, and contrapositive of It raining is a sufficient condition for my not going to town. Solution: converse: If I do not go to town, then it is raining. inverse: If it is not raining, then I will go to town. contrapositive: If I go to town, then it is not raining.

Biconditional If p and q are propositions, then we can form the biconditional proposition p q , read as p if and only if q . The biconditional p q denotes the proposition with this truth table: p q p q T T T T

F F F T F F F T If p denotes I am at home. and q denotes It is raining. then p q denotes I am at home if and only if it is raining.

Expressing the Biconditional Some alternative ways p if and only if q is expressed in English: p is necessary and sufficient for q if p then q , and conversely p iff q Truth Tables For Compound Propositions Construction of a truth table: Rows Need a row for every possible combination of values for the atomic propositions. Columns Need a column for the compound proposition (usually at far right) Need a column for the truth value of each

expression that occurs in the compound proposition as it is built up. This includes the atomic propositions Example Truth Table Construct a truth table for p q r r pq pq r

T T T F T F T T F T T

T T F T F T F T F F

T T T F T T F T F F T

F T T T F F T F F T

F F F T F T Equivalent Propositions Two propositions are equivalent if they always have the same truth value. Example: Show using a truth table that the biconditional is equivalent to the contrapositive. Solution: p

q p q q p q p T T F F T T T

F F T F F F T T F T

T F F T T T T Using a Truth Table to Show NonEquivalence Example: Show using truth tables that neither the converse nor inverse of an implication are not equivalent to the implication. p Solution: q

p q p p q T T F F T q T qp T T

F F T F T T F T T F

T F F F F T T T T T Problem

How many rows are there in a truth table with n propositional variables? Solution: 2n We will see how to do this in Chapter 6. Note that this means that with n propositional variables, we can construct 2n distinct (i.e., not equivalent) propositions. Precedence of Logical Operators Operator Precedence 1 2

3 4 5 p q r is equivalent to (p q) r If the intended meaning is p (q r ) then parentheses must be used. Applications of Propositional Logic Section 1.2 Translating English Sentences Steps to convert an English sentence to a

statement in propositional logic Identify atomic propositions and represent using propositional variables. Determine appropriate logical connectives If I go to Harrys or to the country, I will not go shopping. p: I go to Harrys q: I go to the country. r: I will go shopping. If p or q then not r. Example Problem: Translate the following sentence into propositional logic: You can access the Internet from campus only if you are a computer science major or you

are not a freshman. One Solution: Let a, c, and f represent respectively You can access the internet from campus, You are a computer science major, and You are a freshman. a (c f ) System Specifications System and Software engineers take requirements in English and express them in a precise specification language based on logic. Example: Express in propositional logic: The automated reply cannot be sent when the file system is full Solution: One possible solution: Let p denote The automated reply can be sent and q denote The file system is full. q p

Consistent System Specifications Definition: A list of propositions is consistent if it is possible to assign truth values to the proposition variables so that each proposition is true. Exercise: Are these specifications consistent? The diagnostic message is stored in the buffer or it is retransmitted. The diagnostic message is not stored in the buffer. If the diagnostic message is stored in the buffer, then it is retransmitted. Solution: Let p denote The diagnostic message is stored in the buffer. Let q denote The diagnostic message is retransmitted The specification can be written as: p q, p, p q. When p is false and q is true all three statements are true. So the specification is consistent. What if The diagnostic message is not retransmitted is added. Solution: Now we are adding q and there is no satisfying assignment. So the specification is not consistent. Raymond Smullyan (Born

An island has two kinds of inhabitants, knights, who always 1919) Logic Puzzles tell the truth, and knaves, who always lie. You go to the island and meet A and B. A says B is a knight. B says The two of us are of opposite types. Example: What are the types of A and B? Solution: Let p and q be the statements that A is a knight and B is a knight, respectively. So, then p represents the proposition that A is a knave and q that B is a knave. If A is a knight, then p is true. Since knights tell the truth, q must also be true. Then (p q) ( p q) would have to be true, but it is not. So, A is not a knight and therefore p must be true. If A is a knave, then B must not be a knight since knaves always lie. So, then both p and q hold since both are knaves.

Logic Circuits (Studied in depth in Chapter 12) Electronic circuits; each input/output signal can be viewed as a 0 or 1. represents False represents True Complicated circuits are constructed from three basic circuits called gates. 0 1 The inverter (NOT gate)takes an input bit and produces the negation of that bit. The OR gate takes two input bits and produces the value equivalent to the disjunction of the two bits. The AND gate takes two input bits and produces the value equivalent to the conjunction of the two bits. More complicated digital circuits can be constructed by combining these basic circuits to produce the desired output given the input signals by

building a circuit for each piece of the output expression and then combining them. For example: Propositional Equivalences Section 1.3 Tautologies, Contradictions, and Contingencies A tautology is a proposition which is always true. Example: p p A contradiction is a proposition which is always false. Example: p p A contingency is a proposition which is P a tautology p

p p p p neither nor a contradiction, such F T F as p T F T T F Logically Equivalent

Two compound propositions p and q are logically equivalent if pq is a tautology. We write this as pq or as pq where p and q are compound propositions. Two compound propositions p and q are equivalent if and only if the columns in a truth table giving their truth values agree. This truth table show p q is equivalent to p q. p q p p q

p q T T F T T T F F F F

F T T T T F F T T T

De Morgans Laws Augustus De Morgan 18061871 This truth table shows that De Morgans Second Law holds. p q p q (pq) (pq) pq T

T F F T F F T F F T

T F F F T T F T F F F

F T T F T T Key Logical Equivalences Identity Laws: Domination Laws: Idempotent laws: , , ,

Double Negation Law: Negation Laws: , Key Logical Equivalences (cont) Commutative Laws: Associative Laws: Distributive Laws: Absorption Laws: , More Logical Equivalences Constructing New Logical Equivalences We can show that two expressions are logically equivalent by developing a series of logically

equivalent statements. To prove that we produce a series of equivalences beginning with A and ending with B. Keep in mind that whenever a proposition (represented by a propositional variable) occurs in the equivalences listed earlier, it may be replaced by an arbitrarily complex compound proposition. Equivalence Proofs Example: Show that is logically equivalent to Solution: Equivalence Proofs Example: Show that is a tautology. Solution:

Propositional Satisfiability A compound proposition is satisfiable if there is an assignment of truth values to its variables that make it true. When no such assignments exist, the compound proposition is unsatisfiable. A compound proposition is unsatisfiable if and only if its negation is a tautology. Questions on Propositional Satisfiability Example: Determine the satisfiability of the following compound propositions: Solution: Satisfiable. Assign T to p, q, and r. Solution: Satisfiable. Assign T to p and F to q. Solution: Not satisfiable. Check each possible assignment of truth values to the propositional variables and none will make the proposition true.

Notation Needed for the next example. Sudoku A Sudoku puzzle is represented by a 99 grid made up of nine 33 subgrids, known as blocks. Some of the 81 cells of the puzzle are assigned one of the numbers 1,2, , 9. The puzzle is solved by assigning numbers to each blank cell so that every row, column and block contains each of the nine possible numbers. Example Encoding as a Satisfiability Problem Let p(i,j,n) denote the proposition that is true when the number n is in the cell in the ith

row and the jth column. There are 99 9 = 729 such propositions. In the sample puzzle p(5,1,6) is true, but p(5,j,6) is false for j = 2,3,9 Encoding (cont) For each cell with a given value, assert p(d,j,n), when the cell in row i and column j has the given value. Assert that every row contains every number. Assert that every column contains every number. Encoding (cont) Assert that each of the 3 x 3 blocks contain every number. (this is tricky - ideas from chapter 4 help)

Assert that no cell contains more than one number. Take the conjunction over all values of n, n, i, and j, where each variable ranges from 1 to 9 and , of Solving Satisfiability Problems To solve a Sudoku puzzle, we need to find an assignment of truth values to the 729 variables of the form p(i,j,n) that makes the conjunction of the assertions true. Those variables that are assigned T yield a solution to the puzzle. A truth table can always be used to determine the satisfiability of a compound proposition. But this is too complex even for modern computers for large problems. There has been much work on developing efficient methods for solving satisfiability problems as many practical problems can be translated into satisfiability problems.

Predicates and Quantifiers Section 1.4 Propositional Logic Not Enough If we have: All men are mortal. Socrates is a man. Does it follow that Socrates is mortal? Cant be represented in propositional logic. Need a language that talks about objects, their properties, and their relations. Later well see how to draw inferences. Introducing Predicate Logic Predicate logic uses the following new features: Variables:

x, y, z Predicates: P(x), M(x) Quantifiers (to be covered in a few slides): Propositional functions are a generalization of propositions. They contain variables and a predicate, e.g., P(x) Variables can be replaced by elements from their domain. Propositional Functions Propositional functions become propositions (and have truth values) when their variables are each replaced by a value from the domain (or bound by a quantifier, as we will see later). The statement P(x) is said to be the value of the propositional function P at x. For example, let P(x) denote x > 0 and the domain be

the integers. Then: P(-3) is false. P(0) is false. P(3) is true. Often the domain is denoted by U. So in this example U is the integers. Examples of Propositional Functions Let x + y = z be denoted by R(x, y, z) and U (for all three variables) be the integers. Find these truth values: R(2,-1,5) Solution: F R(3,4,7) Solution: T R(x, 3, z) Solution: Not a Proposition

Now let x - y = z be denoted by Q(x, y, z), with U as the integers. Find these truth values: Q(2,-1,3) Solution: T Q(3,4,7) Solution: F Q(x, 3, z) Solution: Not a Proposition Compound Expressions Connectives from propositional logic carry over to predicate logic. If P(x) denotes x > 0, find these truth values: P(3) P(-1) P(3) P(-1) P(3) P(-1)

P(-1) P(3) Solution: T Solution: F Solution: F Solution: T Expressions with variables are not propositions and therefore do not have truth values. For example, P(3) P(y) P(x) P(y) When used with quantifiers (to be introduced next), these expressions (propositional functions) become propositions. Quantifiers Charles Peirce (18391914) We need quantifiers to express the meaning of English

words including all and some: All men are Mortal. Some cats do not have fur. The two most important quantifiers are: Universal Quantifier, For all, symbol: Existential Quantifier, There exists, symbol: We write as in x P(x) and x P(x). x P(x) asserts P(x) is true for every x in the domain. x P(x) asserts P(x) is true for some x in the domain. The quantifiers are said to bind the variable x in these expressions. Universal Quantifier

x P(x) is read as For all x, P(x) or For every x, P(x) Examples: 1) 2) 3) If P(x) denotes x > 0 and U is the integers, then x P(x) is false. If P(x) denotes x > 0 and U is the positive integers, then x P(x) is true. If P(x) denotes x is even and U is the integers, then x P(x) is false. Existential Quantifier x P(x) is read as For some x, P(x), or as

There is an x such that P(x), or For at least one x, P(x). Examples: 1. 2. 3. If P(x) denotes x > 0 and U is the integers, then x P(x) is true. It is also true if U is the positive integers. If P(x) denotes x < 0 and U is the positive integers, then x P(x) is false. If P(x) denotes x is even and U is the integers, then x P(x) is true. Uniqueness Quantifier !x P(x) means that P(x) is true for one and only one x in the

universe of discourse. This is commonly expressed in English in the following equivalent ways: There is a unique x such that P(x). There is one and only one x such that P(x) Examples: 1. If P(x) denotes x + 1 = 0 and U is the integers, then !x P(x) is true. 2. But if P(x) denotes x > 0, then !x P(x) is false. The uniqueness quantifier is not really needed as the restriction that there is a unique x such that P(x) can be expressed as: x (P(x) y (P(y) y =x)) Thinking about Quantifiers When the domain of discourse is finite, we can think of

quantification as looping through the elements of the domain. To evaluate x P(x) loop through all x in the domain. If at every step P(x) is true, then x P(x) is true. If at a step P(x) is false, then x P(x) is false and the loop terminates. To evaluate x P(x) loop through all x in the domain. If at some step, P(x) is true, then x P(x) is true and the loop terminates. If the loop ends without finding an x for which P(x) is true, then x P(x) is false. Even if the domains are infinite, we can still think of the quantifiers this fashion, but the loops will not terminate in some cases. Properties of Quantifiers The truth value of x P(x) and x P(x) depend on both the propositional function P(x) and on the

domain U. Examples: 1. If U is the positive integers and P(x) is the statement x < 2, then x P(x) is true, but x P(x) is false. 2. If U is the negative integers and P(x) is the statement x < 2, then both x P(x) and x P(x) are true. 3. If U consists of 3, 4, and 5, and P(x) is the statement x > 2, then both x P(x) and x P(x) are true. But if P(x) is the statement x < 2, then both x P(x) and x P(x) are false. Precedence of Quantifiers The quantifiers and have higher precedence than all the logical operators. For example, x P(x) Q(x) means (x P(x)) Q(x) x (P(x) Q(x)) means something different. Unfortunately, often people write x P(x)

Q(x) when they mean x (P(x) Q(x)). Translating from English to Logic Example 1: Translate the following sentence into predicate logic: Every student in this class has taken a course in Java. Solution: First decide on the domain U. Solution 1: If U is all students in this class, define a propositional function J(x) denoting x has taken a course in Java and translate as x J(x). Solution 2: But if U is all people, also define a propositional function S(x) denoting x is a student in this class and translate as x (S(x) J(x)). x (S(x) J(x)) is not correct. What does it mean? Translating from English to Logic Example 2: Translate the following sentence into predicate logic: Some student in this class has taken a course in Java.

Solution: First decide on the domain U. Solution 1: If U is all students in this class, translate as x J(x) Solution 2: But if U is all people, then translate as x (S(x) J(x)) x (S(x) J(x)) is not correct. What does it mean? Returning to the Socrates Example Introduce the propositional functions Man(x) denoting x is a man and Mortal(x) denoting x is mortal. Specify the domain as all people. The two premises are: The conclusion is: Later we will show how to prove that the conclusion follows from the premises.

Equivalences in Predicate Logic Statements involving predicates and quantifiers are logically equivalent if and only if they have the same truth value for every predicate substituted into these statements and for every domain of discourse used for the variables in the expressions. The notation S T indicates that S and T are logically equivalent. Example: x S(x) x S(x) Thinking about Quantifiers as Conjunctions and Disjunctions If the domain is finite, a universally quantified proposition is equivalent to a conjunction of propositions without quantifiers

and an existentially quantified proposition is equivalent to a disjunction of propositions without quantifiers. If U consists of the integers 1,2, and 3: Even if the domains are infinite, you can still think of the quantifiers in this fashion, but the equivalent expressions without quantifiers will be infinitely long. Negating Quantified Expressions Consider x J(x) Every student in your class has taken a course in Java. Here J(x) is x has taken a course in Java and the domain is students in your class. Negating the original statement gives It is not the case that every student in your class has taken Java. This implies that There is a student in your class who has not taken Java. Symbolically x J(x) and x J(x) are

equivalent Negating Quantified Expressions (continued) Now Consider x J(x) There is a student in this class who has taken a course in Java. Where J(x) is x has taken a course in Java. Negating the original statement gives It is not the case that there is a student in this class who has taken Java. This implies that Every student in this class has not taken Java Symbolically x J(x) and x J(x) are equivalent De Morgans Laws for Quantifiers The rules for negating quantifiers are: The reasoning in the table shows that: These are important. You will use these.

Translation from English to Logic Examples: 1. Some student in this class has visited Mexico. Solution: Let M(x) denote x has visited Mexico and S(x) denote x is a student in this class, and U be all people. x (S(x) M(x)) 2. Every student in this class has visited Canada or Mexico. Solution: Add C(x) denoting x has visited Canada. x (S(x) (M(x)C(x))) Some Fun with Translating from English into Logical Expressions U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd

T(x): x is a thingamabob Translate Everything is a fleegle Solution: x F(x) Translation (cont) U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd T(x): x is a thingamabob Nothing is a snurd. Solution: x S(x) What is this equivalent to? Solution: x S(x) Translation (cont) U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd T(x): x is a thingamabob

All fleegles are snurds. Solution: x (F(x) S(x)) Translation (cont) U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd T(x): x is a thingamabob Some fleegles are thingamabobs. Solution: x (F(x) T(x)) Translation (cont) U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd T(x): x is a thingamabob No snurd is a thingamabob. Solution: x (S(x) T(x)) What is this

equivalent to? Solution: x (S(x) T(x)) Translation (cont) U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd T(x): x is a thingamabob If any fleegle is a snurd then it is also a thingamabob. Solution: x ((F(x) S(x)) T(x)) System Specification Example Predicate logic is used for specifying properties that systems must satisfy. For example, translate into predicate logic: Every mail message larger than one megabyte will be compressed. If a user is active, at least one network link will be

available. Decide on predicates and domains (left implicit here) for the variables: Let L(m, y) be Mail message m is larger than y megabytes. Let C(m) denote Mail message m will be compressed. Let A(u) represent User u is active. Let S(n, x) represent Network link n is state x. Now we have: Charles Lutwidge Dodgson (AKA Lewis Caroll) (1832-1898) The first two are called premises and the third is called the Lewis Carroll Example conclusion. 1. All lions are fierce. 2. Some lions do not drink coffee.

3. Some fierce creatures do not drink coffee. Here is one way to translate these statements to predicate logic. Let P(x), Q(x), and R(x) be the propositional functions x is a lion, x is fierce, and x drinks coffee, respectively. 1. x (P(x) Q(x)) 2. x (P(x) R(x)) 3. x (Q(x) R(x)) Later we will see how to prove that the conclusion follows from the premises.

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