# zantufa jonma'o smuni

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## Semantics of forethought logical connectives

The forethought connectives of Zantufa that is a series of parsers for unofficial Lojban are much more expressive than those of official grammar.

Zantufa is syntactic parser, and it does not fix rules for sematics by itself, though.

Here is only a suggestion of semantics of the forethought logical connectives in order to gain profit from expressiveness of Zantufa grammar.

The sentences in Lojban here follow the grammar of Zantufa_1.16.

Symbol Definition
¬P na ku P
P ∨ Q P i ja Q
((((P0 ∨ P1) ∨ P2) ∨ ...) ∨ Pn)
P ∧ Q P i je Q
((((P0 ∧ P1) ∧ P2) ∧ ...) ∧ Pn)
P ↔ Q P i jo Q
((((P0 ↔ P1) ↔ P2) ↔ ...) ↔ Pn)
P ⊕ Q P i jonai Q
((((P0 ⊕ P1) ⊕ P2) ⊕ ...) ⊕ Pn)
P ⊏ Q P i ju Q
((((P0 ⊏ P1) ⊏ P2) ⊏ ...) ⊏ Pn)
= P0
P ⊐ Q P i se ju Q
((((P0 ⊐ P1) ⊐ P2) ⊐ ...) ⊐ Pn)
= Pn

In the following explanation, examples for 3-ary or 4-ary connection are sometimes accompanied with a truth value table. The falseness is represented by 0; the truth is represented by 1.

Examples for 3-ary connection are sometimes accompanied with a Venn diagram. White color represents falseness; orange color represents truth.

## ga

### ga A gi B gi C gi ...

#### Definition

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

#### Example

3-ary case: {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 ...

#### Definition

 ga bo P0 gi P1 gi ... gi Pn (gi'i) = ga P0 gi P1 gi ... gi Pn (gi'i) =

### ga nai A gi B gi C gi ...

#### Definition

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

#### Example

3-ary case: {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 ...

#### Definition

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

#### Example

3-ary case:

 {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
##### Difference

{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 ...

#### Definition

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

#### Example

3-ary case: {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 ...

#### Definition

 ge bo P0 gi P1 gi ... gi Pn (gi'i) = ge P0 gi P1 gi ... gi Pn (gi'i) =

### ge nai A gi B gi C gi ...

#### Definition

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

#### Example

3-ary case: {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 ...

#### Definition

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

#### Example

3-ary case:

 {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 ...

#### Definition

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

#### Example

3-ary case: {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 ...

#### Definition

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

#### Example

3-ary case:

 {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 ...

#### Definition

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

#### Example

3-ary case: {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 ...

#### Definition

It is defined recursively as follows.

 go nai bo P0 gi P1 (gi'i) = go nai P0 gi P1 (gi'i) = P0 ⊕ P1 = ¬P0 ↔ P1
 go nai bo P0 gi ... gi Pn (gi'i) = ((go nai bo P0 gi ... gi Pn-1 (gi'i)) ⊕ Pn) ∧

#### Example

##### 3-ary case:
 {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-ary case:
 {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
##### Difference

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

Here is, for example, the table of truth value of connection by {i jo nai}. It is different from that of {go nai bo A gi B gi C gi D}.

 {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 ...

#### Definition

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

#### Example

3-ary case:

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

### gu bo A gi B gi C gi ...

#### Definition

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

### gu nai A gi B gi C gi ...

#### Definition

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

#### Example

3-ary case:

 {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 ...

#### Definition

 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 = i ju ... i ju Pn = Pk i ju ... i ju Pn = = 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 ...

 se xi no gu nai P0 gi P1 gi ... gi Pn (gi'i) = gu ¬P0 gi P1 gi ... gi Pn (gi'i) = ¬P0
 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