Predicate Logic and Quantifiers

(述語論理、全称限量子、存在限量子)

Discrete Mathematics I

7th lecture, November 6, 2020

https://www.sw.it.aoyama.ac.jp/2020/Math1/lecture7.html

Martin J. Dürst

© 2005-20 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Leftovers/summary for last lecture
• Homework due November 5
• Homework due October 29
• Predicates and predicate logic
• Quantifiers
• Laws for Quantifiers
• This week's homework

Leftovers of Last Lecture

Limits of sets

Summary of Last Lecture

• Sets are one of the most basic kinds of objects in Mathematics
• Sets are unordered, and do not contain any repetitions
• There are two notations for sets: Connotation (e.g. {1, 2, 3, 4}) and denotation (e.g. {n|n ∈ ℕ, n>0, n<5})
• Operations on sets include union, intersection, difference, and complement
• The empty set is a subset of any set
• The powerset of a set is the set of all subsets
• The laws for sets are very similar to the laws for predicate logic

Homework Due November 5, Problem 1

Prove/check the following laws using truth tables (important: always add a sentence stating the conclusion):

1. Reductio ad absurdum: A→¬A = ¬A

   [solution removed]

2. Contraposition: A→B = ¬B→¬A

   [solution removed]

3. The associative law for conjunction: (A∧B) ∧ C = A ∧  (B∧C)

   [solution removed]

4. One of De Morgan's laws: ¬(A∧B) = ¬A ∨ ¬B

   [solution removed]

Homework Due November 5, Problem 2

Prove transitivity of implication (((A→B) ∧ (B→C)) ⇒ (A→C)) by formula manipulation.

For each simplification step, indicate which law you used.

Hint: Show that ((A→B) ∧ (B→C)) → (A→C) is a tautology by simplifying it to T.

[solution removed]

Homework Comment

This homework is a good example of a general strategy to simplify formulæ:

• Remove implications and equivalences
• Push negation inside using DeMorgan's laws
• Remove double negations
• Regroup formula using associative/commutative/distributive laws
• Reduce opposites to T/F with law of excluded middle/law of noncontradiction
• Remove T/F or variable terms with properties of T/F

Homework Due October 29

[solution removed]

Types of Symbolic Logic

• Binary (Boolean) logic (using only true and false)
• Multi-valued logic (using e.g. true, false, and unknown)
• Fuzzy logic (including calculation of ambiguity)

• Propositional logic (using only propositions)
• Predicate logic (first order predicate logic,...)
• Temporal logic (integrating temporal relationships)

Limitations of Propositions

With propositions, related statements have to be made separately

Examples:
2 is even. 5 is even.
Today it is sunny. Tomorrow it is sunny. The day after tomorrow, it is sunny.

We can express "If today is sunny, then tomorrow will also be sunny." or "If 2 is even, then 3 is not even".

But we cannot express "If it's sunny on a given day, it's also sunny on the next day." or "If x is even, then x+2 is also even.".

⇒ This problem can be solved using predicates

Examples of Predicates

• even(4): 4 is even
• even(27): 27 is even
• odd(4): 4 is odd
• sunny(November 15, 2019): It is sunny on November 15, 2019
• parent(Ieyasu, Hidetada): Ieyasu is the parent of Hidetada
  
• smaller(3, 7) (or: 3 < 7)

Predicate Overview

• The problem with propositions can be solved by introducing predicates.
• In the same way as propositions, predicates are objectively true or false.
• A predicate is a function (with 0 or more arguments) that returns true or false.
• If the value of an argument is undefined, the result (value) of the predicate is unknown.
• A predicate with 0 arguments is a proposition.

How to Write Predicates

There are two ways to write predicates:

1. Functional notation:
   • The name of the predicate is the name of the function
   • Arguments are enclosed in parentheses after the function name
   • Each predicate has a fixed number of arguments
   • Arguments in different positions have different meanings
   • Reading of predicates depends on their meaning
2. Operator notation:
   • Operators that return true or false are predicates
   • Examples: 3 < 7, 5 ≧ 2, a ∈ B, even(x) ∨ odd(y)

Formulas Containing Predicates

Using predicates, we can express new things:

• sunny(x) → sunny(day after x)
• even(y) → even(y+2)
• even(z) → odd(z+1)

Similar to propositions, predicates can be true or false.

But predicates can also be unknown/undefined, for example if they contain variables.

Even if a predicate is undefined (e.g. even( x)),
a formula containing this predicate can have a defined value (true or false)
(e.g. even(y) → even(y+2), or odd(z) → even(z+24))

First Order Predicate Logic

• The arguments of predicates can be constants, functions (e.g. sin), formulæ,...

  Examples:
  

  • even(2), say(Romeo, 'I love you'), parent(Ieyasu, Hidetada)
  • even(sin(0)), even(2+3×7)
• However, it is not possible to use predicates within predicates
  Counterexample: say(z, parent(y, x))
  (z says "y is the parent of x")
• Higher-order logic allows predicates within predicates

Universal Quantifier

Example: ∀n∈ℕ: even(n) → even(n+2)

Readings:

• For all n, elements of ℕ, if n is even, then n+2 is even.
• For all natural numbers n, if n is even, then n+2 is even.

General form: ∀x: P (x)

∀ is the A of "for All", inverted.

Readings in Japanese:

• 全ての自然数 n において、n が偶数ならば n+2 も偶数である
• 任意の x において、P(x)

Examples of Universal Quantifiers

∀n∈ℕ: n > -1

∀n∈ℕ: ∀m∈ℕ: n+m = m+n

∀a∈ℚ: ∀b∈ℕ: a+b = b+a

∀a∈{T, F}: ∀b∈{T, F}: a∨b = b∨a

Let S be the set of all students, B the set of all books, and let read(s, b) denote the fact that student s reads book b.

Then ∀s∈S: ∀b∈B: read(s, b) means that all students read all books.

Remark: ∀s∈S: ∀b∈B: read(s, b) is interpreted as ∀s∈S: (∀b∈B: read(s, b))

Knowledge about Field of Application

• Propositional logic does not need application knowledge except for the truth value of each proposition.
• Predicate logic combines axioms/theorems/knowledge of logic with axioms/theorems/knowledge of one or more application areas.
• Example: Predicate logic on natural numbers: Peano axioms,...
• Example: Predicate logic for sets: Laws for operations on sets,...
• Example: Size of sets: Knowledge about set operations and arithmetic with natural numbers
• Concrete example:

  ∀s: (male(s) ∨ female(s)) [all students are either male or female]

  ∀s: ¬(male(s) ∧ female(s)) [no student is both male and female]

Existential Quantifier

Example: ∃n∈ℕ: odd (n)

Readings:

• There exists a n, element of ℕ, for which n is odd.
• There is a natural number n so that n is odd.
• There exists a natural number n which is odd.
• There exists an odd natural number.

General form: ∃y: P (y)

∃ is the mirrored form of the E in "there Exists".

Readings in Japanese:

• P(y) が成立する y が存在する
• ある y について、P(y)

Structure of Quantifier Expressions

Example: ∀m, n∈ℕ: m > n → m² ≧ n²

• ∀: Quantifier
• m, n: Variable(s), separated by commas
• ∈ℕ: Set membership (applies to all previous variables connected by commas; unnecessary if there is a single obvious universal set)
• ":": Colon
• m > n → m² ≧ n²: Quantified predicate

More Quantifier Examples

∀n∈ℕ: n + n + n = 3n

∃n∈ℕ: n² = n³

∃n∈ℝ: n² < 50n < n³

∀m, n∈ℕ: 7m + 2n = 2n + 7m

Applied Quantifier Examples

S: Set of students

F: Set of foods

like(p, f): Person p likes food f

1. All students like all foods:
2. Some students like all foods:
3. There is a food that all students like:
4. There is no food that all students like:
5. Each student dislikes a food:

Peano Axioms in Predicate Logic

Peano Axioms (Guiseppe Peano, 1858-1932)

1. 1∈ℕ
2. ∀a∈ℕ: s(a)∈ℕ
3. ¬∃x∈ℕ: s(x) = 1
4. ∀a, b∈ℕ: a ≠ b → s(a) ≠ s(b)
5. P(1) ∧ (∀k∈ℕ: (P(k)→P(s(k)))) ⇒ ∀a∈ℕ: P(a)
   

The Use of Variables with Quantifiers

Bound variable:

Variable quantified by a quantifier
Example: the x in: ∀x: (P(x)∧Q(y))

Free variable:

Variable not quantified by a quantifier
Example: the y in: ∀x: (P(x)∧Q(y))

Closed formula:

A formula without free variables.

Scope:

The part of a formula where a bound variable (or a quantifier) is active.
All occurrences of a bound variable within its scope can be exchanged by another variable.
Example: ∀s∈S: (age(s)≤30 ∧ college(s)=CSE) ⇔ ∀u∈S: (age(u)≤30 ∧ college(u)=CSE)
Using a bound variable outside its scope is an error.
Example: (∀x: P(x))∧Q(x)

Manipulation of Bound Variables

(∀s∈S: age(s)≤30) ∧ (∀t∈S: college(t)=CSE) = ∀u∈S: (age(u)≤30∧college(u)=CSE)

is the same as

(∀s∈S: age(s)≤30) ∧ (∀s∈S: college(s)=CSE) = ∀s∈S: (age(s)≤30∧college(s)=CSE)

There are three different variables s in the last statement.

Advice:

• It is better to use different variable names with each quantifier, but
• You have to understand formuæ that repeatedly used the same variable name.
• Bound variables are similar to local variables in programming languages.

Important Points for Quantifiers

• What is the universal set?
  (Example: ∀i ∈ℕ⁺: i>0)
• Notation (colons, parentheses)
• Definition of used predicates
• Free vs. bound variables

Laws for Quantifiers

1. ¬∀x: P(x) = ∃x: ¬P(x)
2. ¬∃x: P(x) = ∀x: ¬P(x)
3. (X≠{}∧∀x∈X: P(x)) → (∃x: P(x))
4. (∀x: P(x)) ∧ Q(y) = ∀x: P(x)∧Q(y)
5. (∃x: P(x)) ∧ Q(y) = ∃x: P(x)∧Q(y)
6. (∀x: P(x)) ∨ Q(y) = ∀x: P(x)∨Q(y)
7. (∃x: P(x)) ∨ Q(y) = ∃x: P(x)∨Q(y)
8. (∀x: P(x)) ∧ (∀x: R(x)) = ∀x: P(x)∧R(x)
9. (∀x: P(x)) ∨ (∀x: R(x)) ⇒ ∀x: P(x)∨R(x)
10. (∃x: P(x)) ∨ (∃x: R(x)) = ∃x: P(x)∨R(x)
11. (∃x: P(x)) ∧ (∃x: R(x)) ⇐ ∃x: P(x)∧R(x)
12. (∃y: ∀x: P(x, y)) ⇒ (∀x: ∃y: P(x, y))
13. P(x) is a tautology ⇔∀x: P(x) is a tautology

Combination of Quantifiers

(∃y: ∀x: P(x, y)) ⇒ (∀x: ∃y: P(x, y))

(∀x: ∃y: P(x, y)) ⇏ (∃y: ∀x: P(x, y))

The number of prime numbers is infinite.

(This means that whatever big number x we choose, there will always be a bigger prime number y.)

∀x: ∃y: (y > x ∧ prime(y))

Reversing the order of the quantifiers changes the meaning:

∃y: ∀x: (y > x ∧ prime(y))

(There is a prime number y that is bigger than any (natural number) x. This statement is obviously false.)

Proof that the Number of Primes is Infinite

• Assume there is a largest prime number z:
  ∃z: prime(z) ∧ ∀y: prime(y) → y≤z
• Calculate a new number t as 1 plus the product of all prime numbers up to z:
  t = 1 + ∏^z_p=2(prime(p)) p
  Example: z=7; t = 1 + 2·3·5·7 = 211
• ∀x≤z: prime(x)→t mod x = 1 ⇒

  ∃s: (z<s≤t ∧ prime(s)) ⇒

  ∃y: (y > z ∧ prime(y))

This proves that there is no largest prime number, and therefore, the number of primes is infinite.

This is a proof by contradiction.

Summary

• Predicates are limited because they cannot express complex facts
• Propositions are functions or operators that return truth values (true/false)
• First order propositional logic allows formal reasoning for some application domain (e.g. arithmetic, set theory,...)
• The universal quantifier (∀) expresses that a predicate is true for all members of some set(s)
• The existential quantifier (∃) expresses that a predicate is true for some member(s) of some set(s)
• Variables not bound by a quantifier are free variables

This Week's Homework

Deadline: Thursday November 12, 2020, 22:00.

Problems: See handout

If you have a printer, print out and fill in the handout.

If you do not have a printer, use a page of white A4 paper, and number the solutions carefully.

Format: A4 scanned PDF (more than one page is okay, but single file; NO cover page), easily readable handwriting (NO printouts), name (kanji and kana) and student number at the top right

Where to submit: Moodle assignment

Glossary

Mathematical induction

数学的帰納法

Pascal's triangle

パスカルの三角形

predicate logic

述語論理

quantifier

限量子

evaluate

評価する

evaluation

評価

array

配列

tautology

恒真 (式)、トートロジー

contradiction

恒偽 (式)

symbolic logic

記号論理

multi-valued logic

多値論理

fuzzy logic

ファジィ論理

ambiguity

曖昧さ

first-order predicate logic

一階述語論理

temporal logic

時相論理

binary logic

二値論理

generalization

一般化

argument

引数

undefined

未定

higher-order logic

高階述語論理

universal quantifier

全称限量子 (全称記号)

existential quantifier

存在限量子 (存在記号)

inference

推論

College of Science and Engineering

理工学部

native of ...

...出身

bound variable

束縛変数

free variable

自由変数

local variable

局所変数

closed formula

閉論理式

scope

作用領域、スコープ

prime (number)

素数