zantufa jonma'o smuni/ja: Difference between revisions

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3項の場合: {'''ga A gi B gi C'''}=(A∨B)∨C
3項の場合: {'''ga A gi B gi C'''}=(A∨B)∨C


[[Image:venn-ga.png|thumb|ga zei venn]]
[[Image:venn-ga.png|thumb|ga zei ven]]
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3項の場合: {'''ga nai A gi B gi C'''}=(¬A∨B)∨C
3項の場合: {'''ga nai A gi B gi C'''}=(¬A∨B)∨C


[[Image:venn-ganai.png|thumb|ga zei nai zei venn]]
[[Image:venn-ganai.png|thumb|ga zei nai zei ven]]
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[[Image:venn-ganaibo.png|thumb|ga zei nai zei bo zei venn]]
[[Image:venn-ganaibo.png|thumb|ga zei nai zei bo zei ven]]
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[[Image:venn-inaja.png|thumb|i zei na zei ja zei venn]]
[[Image:venn-inaja.png|thumb|i zei na zei ja zei ven]]
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3項の場合: {'''ge A gi B gi C'''}=(A∧B)∧C
3項の場合: {'''ge A gi B gi C'''}=(A∧B)∧C


[[Image:venn-ge.png|thumb|ge zei venn]]
[[Image:venn-ge.png|thumb|ge zei ven]]
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3項の場合: {'''ge nai A gi B gi C'''}=(¬A∧B)∧C
3項の場合: {'''ge nai A gi B gi C'''}=(¬A∧B)∧C


[[Image:venn-genai.png|thumb|ge zei nai zei venn]]
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[[Image:venn-genaibo.png|thumb|ge zei nai zei bo zei venn]]
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3項の場合: {'''go A gi B gi C'''}=(A↔B)↔C
3項の場合: {'''go A gi B gi C'''}=(A↔B)↔C


[[Image:venn-go.png|thumb|go zei venn]]
[[Image:venn-go.png|thumb|go zei ven]]
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[[Image:venn-gobo.png|thumb|go zei bo zei venn]]
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3項の場合: {'''go nai A gi B gi C'''}=(¬A↔B)↔C
3項の場合: {'''go nai A gi B gi C'''}=(¬A↔B)↔C


[[Image:venn-gonai.png|thumb|go zei nai zei venn]]
[[Image:venn-gonai.png|thumb|go zei nai zei ven]]
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[[Image:venn-gonaibo.png|thumb|go zei nai zei bo zei venn]]
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[[Image:venn-ijonai.png|thumb|i zei jo zei nai zei venn]]
[[Image:venn-ijonai.png|thumb|i zei jo zei nai zei ven]]
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[[Image:venn-gu.png|thumb|gu zei venn]]
[[Image:venn-gu.png|thumb|gu zei ven]]


===gu bo A gi B gi C gi ...===
===gu bo A gi B gi C gi ...===
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[[Image:venn-gunai.png|thumb|gu zei nai zei venn]]
[[Image:venn-gunai.png|thumb|gu zei nai zei ven]]


===gu nai bo A gi B gi C gi ...===
===gu nai bo A gi B gi C gi ...===

Latest revision as of 00:25, 11 July 2016

banjubu'o / français / 日本語 / English

前置型論理接続詞の意味論

非公式のロジバン構文解析器 zantufa の前置型接続は、公式文法に比べて非常に豊かな表現力を持つ。

zantufa 自体は構文解析器であり意味論を規定しないが、 zantufa 文法の豊かな表現力を有効に活用するものとして、以下のような前置型論理接続詞の意味論を提案する。

以下のロジバン文は zantufa_1.16 の文法に従う。

このページで使われる記号のリスト

記号 意味
¬P na ku P
P ∨ Q P i ja Q
"so'i zo ja" ((((P0 ∨ P1) ∨ P2) ∨ ...) ∨ Pn)
P ∧ Q P i je Q
"so'i zo je" ((((P0 ∧ P1) ∧ P2) ∧ ...) ∧ Pn)
P ↔ Q P i jo Q
"so'i zo jo" ((((P0 ↔ P1) ↔ P2) ↔ ...) ↔ Pn)
P ⊕ Q P i jonai Q
"so'i lu jonai li'u" ((((P0 ⊕ P1) ⊕ P2) ⊕ ...) ⊕ Pn)
P ⊏ Q P i ju Q
"so'i zo ju" ((((P0 ⊏ P1) ⊏ P2) ⊏ ...) ⊏ Pn)
= P0
P ⊐ Q P i se ju Q
"so'i lu seju li'u" ((((P0 ⊐ P1) ⊐ P2) ⊐ ...) ⊐ Pn)
= Pn

以下の説明中、3項や4項の接続例について真理値表を与えることがある。 偽を 0、 真を 1 で表す。

3項の接続例について Venn 図を与えることがある。 白色は偽、オレンジ色は真を表す。

ga

ga A gi B gi C gi ...

定義

ga P0 gi P1 gi ... gi Pn (gi'i)
= "so'i zo ja"

例

3項の場合: {ga A gi B gi C}=(A∨B)∨C

ga zei ven
A B C A∨B (A∨B)∨C
1 1 1 1 1
0 1 1 1 1
1 0 1 1 1
0 0 1 0 1
1 1 0 1 1
0 1 0 1 1
1 0 0 1 1
0 0 0 0 0

ga bo A gi B gi C gi ...

定義

ga bo P0 gi P1 gi ... gi Pn (gi'i)
= ga P0 gi P1 gi ... gi Pn (gi'i)
= "so'i zo ja"

ga nai A gi B gi C gi ...

定義

ga nai P0 gi P1 gi ... gi Pn (gi'i)
= ga ¬P0 gi P1 gi ... gi Pn (gi'i)

例

3項の場合: {ga nai A gi B gi C}=(¬A∨B)∨C

ga zei nai zei ven
A B C ¬A∨B (¬A∨B)∨C
1 1 1 1 1
0 1 1 1 1
1 0 1 0 1
0 0 1 1 1
1 1 0 1 1
0 1 0 1 1
1 0 0 0 0
0 0 0 1 1

ga nai bo A gi B gi C gi ...

定義

ga nai bo P0 gi P1 gi ... gi Pn (gi'i)
"ganaibo"

例

3項の場合:

{ga nai bo A gi B gi C}
= ((A→B)∧(B→C))
= ((¬A∨B)∧(¬B∨C))
ga zei nai zei bo zei ven
A B C ¬A∨B ¬B∨C (¬A∨B)∧(¬B∨C)
1 1 1 1 1 1
0 1 1 1 1 1
1 0 1 0 1 0
0 0 1 1 1 1
1 1 0 1 0 0
0 1 0 1 0 0
1 0 0 0 1 0
0 0 0 1 1 1
相違点

{ga nai bo A gi B gi C} ≠ {A i na ja B i na ja C}

A i na ja B i na ja C
= (A → B) → C
= ¬(¬A ∨ B) ∨ C
i zei na zei ja zei ven
A B C ¬A∨B ¬(¬A∨B)∨C
1 1 1 1 1
0 1 1 1 1
1 0 1 0 1
0 0 1 1 1
1 1 0 1 0
0 1 0 1 0
1 0 0 0 1
0 0 0 1 0

ge

ge A gi B gi C gi ...

定義

ge P0 gi P1 gi ... gi Pn (gi'i)
= "so'i zo je"

例

3項の場合: {ge A gi B gi C}=(A∧B)∧C

ge zei ven
A B C A∧B (A∧B)∧C
1 1 1 1 1
0 1 1 0 0
1 0 1 0 0
0 0 1 0 0
1 1 0 1 0
0 1 0 0 0
1 0 0 0 0
0 0 0 0 0

ge bo A gi B gi C gi ...

定義

ge bo P0 gi P1 gi ... gi Pn (gi'i)
= ge P0 gi P1 gi ... gi Pn (gi'i)
= "so'i zo je"

ge nai A gi B gi C gi ...

定義

ge nai P0 gi P1 gi ... gi Pn (gi'i)
= ge ¬P0 gi P1 gi ... gi Pn (gi'i)

例

3項の場合: {ge nai A gi B gi C}=(¬A∧B)∧C

ge zei nai zei ven
A B C ¬A∧B (¬A∧B)∧C
1 1 1 0 0
0 1 1 1 1
1 0 1 0 0
0 0 1 0 0
1 1 0 0 0
0 1 0 1 0
1 0 0 0 0
0 0 0 0 0

ge nai bo A gi B gi C gi ...

定義

ge nai bo P0 gi P1 gi ... gi Pn (gi'i)
"genaibo"

例

3項の場合:

{ge nai bo A gi B gi C}
= {ge nai A gi nai B gi nai C}
= {A i na je nai B i je nai C}
= (¬A∧¬B)∧¬C
ge zei nai zei bo zei ven
A B C ¬A∧¬B (¬A∧¬B)∧¬C
1 1 1 0 0
0 1 1 0 0
1 0 1 0 0
0 0 1 1 0
1 1 0 0 0
0 1 0 0 0
1 0 0 0 0
0 0 0 1 1

go

go A gi B gi C gi ...

定義

go P0 gi P1 gi ... gi Pn (gi'i)
= "so'i zo jo"

例

3項の場合: {go A gi B gi C}=(A↔B)↔C

go zei ven
A B C A↔B (A↔B)↔C
1 1 1 1 1
0 1 1 0 0
1 0 1 0 0
0 0 1 1 1
1 1 0 1 0
0 1 0 0 1
1 0 0 0 1
0 0 0 1 0

go bo A gi B gi C gi ...

定義

go bo P0 gi P1 gi ... gi Pn (gi'i)
"gobo"

例

3項の場合:

{go bo A gi B gi C}
= ((A↔B)∧(B↔C))
go zei bo zei ven
A B C A↔B B↔C (A↔B)∧(B↔C)
1 1 1 1 1 1
0 1 1 0 1 0
1 0 1 0 0 0
0 0 1 1 0
1 1 0 1 0 0
0 1 0 0 0 0
1 0 0 0 1 0
0 0 0 1 1 1

go nai A gi B gi C gi ...

定義

go nai P0 gi P1 gi ... gi Pn (gi'i)
= go ¬P0 gi P1 gi ... gi Pn (gi'i)

例

3項の場合: {go nai A gi B gi C}=(¬A↔B)↔C

go zei nai zei ven
A B C ¬A↔B (¬A↔B)↔C
1 1 1 0 0
0 1 1 1 1
1 0 1 1 1
0 0 1 0 0
1 1 0 0 1
0 1 0 1 0
1 0 0 1 0
0 0 0 0 1

go nai bo A gi B gi C gi ...

定義

以下のように再帰的に定義される。

2項 P0, P1 の場合:
go nai bo P0 gi P1 (gi'i)
= go nai P0 gi P1 (gi'i)
= P0 ⊕ P1
= ¬P0 ↔ P1
3項以上 P0, ... , Pn (n≥2) の場合:
go nai bo P0 gi ... gi Pn (gi'i)
= ((go nai bo P0 gi ... gi Pn-1 (gi'i)) ⊕ Pn) ∧ "gonaibo_na"

例

3項の場合:
{go nai bo A gi B gi C}
= ((go nai A gi B) ⊕ C) ∧ ¬(A∧B)
= ((A⊕B)⊕C) ∧ ¬(A∧B)
go zei nai zei bo zei ven
A B C A⊕B (A⊕B)⊕C ¬(A∧B) ((A⊕B)⊕C) ∧ ¬(A∧B)
1 1 1 0 1 0 0
0 1 1 1 0 1 0
1 0 1 1 0 1 0
0 0 1 0 1 1 1
1 1 0 0 0 0 0
0 1 0 1 1 1 1
1 0 0 1 1 1 1
0 0 0 0 0 1 0
4項の場合:
{go nai bo A gi B gi C gi D}
= ((go nai bo A gi B gi C) ⊕ D) ∧ ¬(A∧B) ∧ ¬(B∧C) ∧ ¬(C∧A)
A B C D go nai bo A gi B gi C (go nai bo A gi B gi C) ⊕ D ¬(A∧B) ¬(B∧C) ¬(C∧A) ((go nai bo A gi B gi C) ⊕ D) ∧ ¬(A∧B) ∧ ¬(B∧C) ∧ ¬(C∧A)
1 1 1 1 0 1 0 0 0 0
0 1 1 1 0 1 1 0 1 0
1 0 1 1 0 1 1 1 0 0
0 0 1 1 1 0 1 1 1 0
1 1 0 1 0 1 0 1 1 0
0 1 0 1 1 0 1 1 1 0
1 0 0 1 1 0 1 1 1 0
0 0 0 1 0 1 1 1 1 1
1 1 1 0 0 0 0 0 0 0
0 1 1 0 0 0 1 0 1 0
1 0 1 0 0 0 1 1 0 0
0 0 1 0 1 1 1 1 1 1
1 1 0 0 0 0 0 1 1 0
0 1 0 0 1 1 1 1 1 1
1 0 0 0 1 1 1 1 1 1
0 0 0 0 0 0 1 1 1 0
相違点

{go nai bo P0 gi P1 gi ... gi Pn (gi'i)} ≠ {P0 i jo nai P1 i jo nai ... i jo nai Pn}

例えば4項まで {i jo nai} で接続すると真理値表は以下のようになり、 {go nai bo A gi B gi C gi D} と異なる。

{A i jo nai B i jo nai C i jo nai D}
= {A i na jo B i na jo C i na jo D}
= ((A ⊕ B) ⊕ C) ⊕ D
i zei jo zei nai zei ven
A B C D A ⊕ B (A ⊕ B) ⊕ C ((A ⊕ B) ⊕ C) ⊕ D
1 1 1 1 0 1 0
0 1 1 1 1 0 1
1 0 1 1 1 0 1
0 0 1 1 0 1 0
1 1 0 1 0 0 1
0 1 0 1 1 1 0
1 0 0 1 1 1 0
0 0 0 1 0 0 1
1 1 1 0 0 1 1
0 1 1 0 1 0 0
1 0 1 0 1 0 0
0 0 1 0 0 1 1
1 1 0 0 0 0 0
0 1 0 0 1 1 1
1 0 0 0 1 1 1
0 0 0 0 0 0 0

gu

gu A gi B gi C gi ...

定義

gu P0 gi P1 gi ... gi Pn (gi'i)
= "so'i zo ju"
= P0

例

3項の場合:

{gu A gi B gi C}
= (A ⊏ B) ⊏ C
= A
gu zei ven

gu bo A gi B gi C gi ...

定義

gu bo P0 gi P1 gi ... gi Pn (gi'i)
= gu P0 gi P1 gi ... gi Pn (gi'i)
= "so'i zo ju"
= P0

gu nai A gi B gi C gi ...

定義

gu nai P0 gi P1 gi ... gi Pn (gi'i)
= gu ¬P0 gi P1 gi ... gi Pn (gi'i)
= ¬P0

例

3項の場合:

{gu nai A gi B gi C}
= {gu ¬A gi B gi C}
= ¬A
gu zei nai zei ven

gu nai bo A gi B gi C gi ...

定義

gu nai bo P0 gi P1 gi ... gi Pn (gi'i)
gu ¬P0 gi ¬P1 gi ... gi ¬Pn (gi'i)
= ¬P0
= gu nai P0 gi P1 gi ... gi Pn (gi'i)

segu

se xi ky gu A gi B gi C gi ... (se gu, te gu, ve gu, xe gu, ..., se xi ro gu)

se xi ky gu P0 gi P1 gi ... gi Pn (gi'i)
= P0 i se ju P1 i se ju ... i se ju Pk i ju ... i ju Pn
= "ky lu seju li'u" i ju ... i ju Pn
= Pk i ju ... i ju Pn
= "ny vu'u ky zo ju"
= Pk
{se xi no gu} = {gu}
{se xi pa gu} = {se gu}
{se xi re gu} = {te gu}
{se xi ci gu} = {ve gu}
{se xi vo gu} = {xe gu}
...
{se xi ro gu P0 gi P1 gi ... gi Pn (gi'i)} = Pn

se xi ky gu bo A gi B gi C gi ...

se xi ky gu bo P0 gi P1 gi ... gi Pn (gi'i)
= se xi ky gu P0 gi P1 gi ... gi Pn (gi'i)
= Pk

se xi ky gu nai A gi B gi C gi ...

k=0 のとき
se xi no gu nai P0 gi P1 gi ... gi Pn (gi'i)
= gu ¬P0 gi P1 gi ... gi Pn (gi'i)
= ¬P0
k≠0 のとき
se xi ky gu nai P0 gi P1 gi ... gi Pn (gi'i)
= se xi ky gu ¬P0 gi P1 gi ... gi Pn (gi'i)
= Pk

se xi ky gu nai bo A gi B gi C gi ...

se xi ky gu nai bo P0 gi P1 gi ... gi Pn (gi'i)
= se xi ky gu ¬P0 gi ¬P1 gi ... gi ¬Pk gi ... gi ¬Pn (gi'i)
= ¬Pk
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