Predicate Logic, Quantifiers

(述語論理、限量子)

Discrete Mathematics I

6th lecture, Oct. 24, 2014

http://www.sw.it.aoyama.ac.jp/2014/Math1/lecture6.html

Martin J. Dürst

© 2005-14 Martin J. Dürst 青山学院大学

Today's Schedule

• Schedule for the next few weeks
• Leftovers/summary/homework for last lecture
• Predicates
• Quantifiers
• Laws for Quantifiers
• This week's homework

Schedule for the Next Few Weeks

• October 24 (today): Predicate Logic, Quantifiers
• October 31: No lecture (Aoyama Festival)
• November 7: No lecture (trip abroad)
• November 14: Next lecture: Application of Predicate Logic

Leftovers from Last Lecture

Laws for equivalence and implication.

Summary of Last Lecture

• All Boolean formulæ can be expressed using only NAND or only NOR.
• Logic circuits can be built from gates to implement Boolean functions.
• Boolean logic can be axiomatized, there are many different ways to do this (learn at least one set of axioms).
• Logical operations important for symbolic logic are implication and equivalence.

Last Week's Homework: Problem One

For each of the 16 Boolean functions of two Boolean variables A and B (same as problem 1 of last week), find the shortest formula using only NAND.
You can use NAND with an arbitrary number of arguments, but you cannot use T or F.

Hint: Start with simple formulæ using NAND and check which function they represent.

[都合により削除]  

Problem Two

Draw logic circuits of the following three Boolean formulæ:

[都合により削除]

Tautology and Contradiction

• A Boolean formula that is always true is called a tautology.
• A Boolean formula that is always false is called a contradiction.
• All laws are tautologies, but there are also tautologies that we don't call laws.
  (Example: T→A = ¬¬A)

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 prepositions)
• Predicated logic (first order predicate logic,...)
• Temporal logic (integrating temporal relationships)

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 predicates, propositions 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:

• sunny(x) → sunny(day after x)
• even(x) → even(x+2)

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 or false; e.g. even(x) → even(x+2))

First Order Predicate Logic

• The arguments of predicates can be constants, functions, formulæ,...

  Examples:
  

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

Universal Quantifier

Example: ∀n∈ℕ: even (n) → even(n+2)

Readings:

• For all n element 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 x において、P(x)
• 任意の x において、P(x)

Existential Quantifier

Example: ∃y∈ℕ: odd (y)

Readings:

• There is a y element ℕ for which y is odd.
• There exists a natural number y 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)

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 30, 2014 (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
2. Contraposition
3. The associative law for disjunction
4. One of De Morgan's laws

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

For each simplification step, indicate which law you used.

Additional Homework

Deadline: November 13, 2014 (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 "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 8-11 of "Laws for Quantifiers", imagine concrete examples and explain it. For laws 9 and 11, 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)