(述語論理の応用)

http://www.sw.it.aoyama.ac.jp/2015/Math1/lecture7.html

© 2005-15 Martin J. Dürst Aoyama Gakuin University

- Summary and homework from last lecture
- Applications of predicate logic
- Examples for various laws
- Quantifiers and variables
- Quantifiers and sums/products

- There are many different kinds of symbolic logic: Propositional logic, predicate logic,...
- Predicates take arguments (propositions don't take arguments)
- Predicate logic allows more general inferences than propositional logic
- Predicate logic uses universal quantifiers (∀) and existential quantifiers (∃)

The examples all are about the students taking Discrete Mathematics I.

Predicates and functions used:

age(`s`): A student's age (fully completed years)

college(`s`): A student's faculty or college (example: College of
Science and Engineering)

female(`s`), male(`s`): Predicates for a student's
gender.

native(`s`, `k`): True if student `s` is a native
of prefecture `k` (using "abroad" for students from outside Japan)

Law 5:

(∃`x`: P(`x`)) ∧ Q(`y`) = ∃`x`:
(P(`x`)∧Q(`y`))

Concrete example:

There is a student from Shizuoka, and student `y` is older than 20
means that there is a case where there is a student from Saitama and student
`y` is older than 20, and the other way round.

(∃`s`: native(`s`, Shizuoka)) ∧
age(`y`)>20 = ∃`s`: (native(`s`,
Shizuoka)∧age(`y`)>20)

Law 8:

(∀`x`: P(`x`)) ∧ (∀`x`: R(`x`)) =
∀`x`: P(`x`)∧R(`x`)

Concrete example:

All students are less that 30 years old, and all students belong to the College
of Science and Engineering. = All students are less than 30 years old and
belong to the College of Science and Engineering.

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

Law 11:

(∃`x`: P(`x`)) ∧ (∃`x`: R(`x`)) ←
∃`x`: P(`x`) ∧ R(`x`)

Example of how the left side follows from the right:

There is a student who is a native of Hiroshima and who is male. → There is a
student who is a native of Hiroshima, and there is a student who is male.

(∃`s`: native(`s`, Hiroshima)) ∧ (∃`s`:
male(`s`)) ← ∃`s`: (native(`s`, Hiroshima) ∧
male(`s`))

Example of how the right side does not follow from the left side:

There is a student who is a native from Hokkaido, and there is a student who is
female. However, this does not imply that there is a student who is a native
from Hokkaido and is female.

(∃`s`: native(`s`, Hokkaido)) ∧ (∃`s`:
female(`s`)) ↛ ∃`s`: (native(`s`, Hokkaido) ∧
female(`s`))

Law 12:

(∃`y`: ∀`x`: P(`x`, `y`)) →
(∀`x`: ∃`y`: P(`x`, `y`))

Example of how the left side follows from the right:

There is an age `y` (e.g. 30) so that for `all` students,
their age is below `y`. From this follows that for `each`
student, there is an age for which the student's age is lower.

(∃`y`: ∀`s`:
age(`s``)<``y`) → (∀`s`:
∃`y`: age(`s``)<``y`)

Example of how the right side doesn't follow from the left side:

All students are native of some prefecture (or abroad). But this does not mean
that there is a single prefecture of which all students are native.

(∀`s`: ∃`y`: native(`s`, `y`)) ↛
(∃`y`: ∀`s`: native(`s`, `y`))

(∃`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.)

- Assume there is a largest prime number
`z`:

∃`z`: prime(`z`) ∧ ∀`y`: prime(`y`) →`y`≤`z` - Calculate a new number
`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 ⇒prime(

`t`) ∨ ∃`s`: (`z``<``s`<`t`∧ prime(`s`)) ⇒∃

`y`: (`y`>`x`∧ prime(`y`))

- 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`: (age(`s`)≤30 ∧ college(`s`)=CSE) ⇔ ∀`u`: (age(`u`)≤30 ∧ college(`u`)=CSE)

Using a bound variable outside its scope is an error.

Example: (∀`x`: P(`x`))∧Q(`x`)

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

is the same as

∀`s`: age(`s`)≤30) ∧ (∀`s`:
college(`s`)=CSE) = ∀`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 be able to handle formuæ that repeatedly used the same variable name, too.

Sum: ∑^{∞}_{i}_{=1}
1/`i`^{2} = 1 + 1/4 + 1/9 + 1/16 + 1/25 + ...

Product: ∏^{∞}_{i}_{=1}
1+1/(-2)^{i} = ...

Universal quantification: (∀`i` ∈ℕ^{+}:
`i`>0) = **⋀**^{∞}_{i}_{=1}
`i`>0 = 1>0 ∧2>0 ∧3>0 ∧...

Existential quantification: (∃`i` ∈ℕ^{+}:
odd(`i`)) = **⋁**^{∞}_{i}_{=1}
odd(`i`) = odd(1)∨odd(2)∨odd(3)∨...

Quantification is a generalization of conjunction/disjunction to more than two operands in the same way that sum and product are a generalization of addition/multiplication to more than two operands.

∀`i` (`i`<0⋀`i`>5): odd(`i`) =
T

∃`i` (`i`<0⋀`i`>5): odd(`i`) =
F

Laws 1 and 2 introduced in the last lecture are generalizations of DeMorgan's laws:

- ¬∀
`x`: P(`x`) = ∃`x`: ¬P(`x`)

¬(P(x_{1})∧P(x_{2})∧P(x_{3})∧P(x_{4})∧...) =

= (¬P(x_{1})∨¬P(x_{2})∨¬P(x_{3})∨¬P(x_{4})∨...) - ¬∃
`x`: P(`x`) = ∀`x`: ¬P(`x`)

¬(P(x_{1})∨P(x_{2})∨P(x_{3})∨P(x_{4})∨...) =

= (¬P(x_{1})∧¬P(x_{2})∧¬P(x_{3})∧¬P(x_{4})∧...)

Simplify ¬(∃x: P(x) → ∀y: ¬Q(y))

¬(∃x: P(x) → ∀y: ¬Q(y)) [removing implication]

= ¬(¬∃x: P(x) ∨ ∀y: ¬Q(y)) [deMorgan's law]

= ¬¬∃x: P(x) ∧ ¬∀y: ¬Q(y) [law 1 of last lecture]

= ¬¬∃x: P(x) ∧ ∃y: ¬¬Q(y) [double negation]

= ∃x: P(x) ∧ ∃y: Q(y)

Actual example:

Let P(`x`) mean "it is raining in `x`", and
Q(`y`) "it is snowing `y`"

Then the original formula says "It's wrong that if it rains somewhere, then it snows nowhere". The final formula says "There is a place where it rains and there is a place where it snows".

- Propositional logic does not nead application knowledge except for the truth value of each proposition.
- Predicate logic combines axioms/theorems/knowledge of logic with the 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]∀

`s`∈`S`: (∃`k`∈`K`: native(`s`,`k`) ∧(∀`h`∈`K`:`h`=`k`∨¬native(`s`,`h`))) [all students are native of exactly one prefecture]

Deadline: November 12, 2015 (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)

Using your high school books/materials or other sources, research the following terms related to sets, and write a definition and short explanation for each of them:

- Set
- Element
- Set union
- Set intersection
- Set difference
- Subset
- Proper subset
- Empty set
- Power set

(高校の本・資料や他の情報を活用して、上記の集合に関連する概念を調査し、定義と簡単な説明を書きなさい。)

- 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
- closed formula
- 閉論理式
- scope
- 作用領域、スコープ
- sum
- 総和
- product
- 総積
- prime number
- 素数
- infinite
- 無限 (な)
- set
- 集合
- element
- 元、要素
- (set) union
- 和集合
- (set) intersection
- 積集合
- (set) difference
- 差集合
- subset
- 部分集合
- proper subset
- 真 (しん) の部分集合
- empty set
- 空 (くう) 集合
- power set
- べき (冪) 集合