Predicate Logic, Universal and Existential Quantifiers

(述語論理、全称限量子、存在限量子)

Discrete Mathematics I

6th lecture, Oct. 21, 2016

http://www.sw.it.aoyama.ac.jp/2016/Math1/lecture6.html

Martin J. Dürst

AGU

© 2005-16 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

 

Schedule for the Next Few Weeks

 

Summary of Last Lecture

 

Last Week's Homework: Submission

Please read the submission instructions carefully!

 

Tautology and Contradiction

 

Types of Symbolic Logic

 

Limitations of Propositions

With propositions, related statements have to be made separately

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

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.".

 

Predicates

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.

 

Examples of Predicates

sunny(today), sunny(tomorrow), sunny(yesterday), even(2), even(5), ...

Generalization: sunny(x), even(y), ...

Using predicates, we can express new things:

Similar to propositions, predicates can be true or false.

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

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

 

First Order Predicate Logic

 

Universal Quantifier

Example: ∀n∈ℕ: even (n) → even(n+2)

Readings:

General form: ∀x: P (x)

∀ is the A of "for All", inverted.

Readings in Japanese:

 

Existential Quantifier

Example: ∃y∈ℕ: odd (y)

Readings:

General form: ∃y: P (y)

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

Readings in Japanese:

 

Example: 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∈ℕ, b∈ℕ: abs(a) ≠ s(b)
  5. P(1) ∧ (∀a∈ℕ: (P(a)→P(s(a)))) ⇒ ∀a∈ℕ: P(a)

 

Laws for Quantifiers

  1. ¬∀x: P(x) = ∃x: ¬P(x)
  2. ¬∃x: P(x) = ∀x: ¬P(x)
  3. (∀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

 

This Week's Homework

Deadline: October 27, 2016 (Thursday), 19:00.

Format: A4 single page (using both sides is okay; NO cover page), easily readable handwriting (NO printouts), name (kanji and kana) and student number at the top right

Where to submit: Box in front of room O-529 (building O, 5th floor)

Problem 1: Prove/check the following laws using truth tables:

  1. Reductio ad absurdum (A→¬A = ¬A)
  2. Contraposition
  3. The associative law for disjunction
  4. One of De Morgan's laws

Problem 2: Prove transitivity of implication (((AB) ∧ (BC)) ⇒ (AC)) by formula manipulation.
Hint: Show that ((AB) ∧ (BC)) → (AC) is a tautology by simplifying it to T.

For each simplification step, indicate which law you used.

 

Additional Homework

Deadline: November 10, 2016 (Thursday), 19:00.

Format: A4 single page (using both sides is okay; NO cover page; an additional page is okay if really necessary, but staple the pages together at the top left corner), easily readable handwriting (NO printouts), name (kanji and kana) and student number at the top right

Where to submit: Box in front of room O-529 (building O, 5th floor)

Problem 1: Show that the Wolfram axiom of Boolean logic is a tautology (you can use either a truth table or formula manipulation).

Problem 2: For ternary (three-valued) logic, create truth tables for conjunction, disjunction, and negation. The three values are T, F, and ?, where ? stands for "unknown" (in more words: "maybe true, maybe false, we don't know").

Hint: What's the result of "?∨T"? ? can be T or F, but in both cases the result will be T, so ?∨T=T.

Problem 3: For each of the laws 1, 5, 8, 11, and 12 of "Laws for Quantifiers", imagine a concrete example and explain it. For laws 11 and 12, give examples for both why the implication works one way and why the implication does not work the other way.

 

Glossary

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
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)