The Foundations: Logic and Proofs

The Foundations: Logic and Proofs

Predicates and Quantifiers Section 1.4 Section Summary Predicates Variables Quantifiers Universal Quantifier Existential Quantifier Negating Quantifiers De Morgans Laws for Quantifiers Translating English to Logic 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(3) P(3) P(3)

P(-1) P(-1) P(-1) P(-1) 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 (optional) !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 1: 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)))

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