This page contains discussions of experimental/scientific/philosophical/logical aspects of Lojban that are non-official and not for everyday usage. You've been warned.
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.
List of symbols used on this page
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 P0gi P1gi ... 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 P0gi P1gi ... gi Pn (gi'i)
=
ga P0gi P1gi ... gi Pn (gi'i)
=
ga nai A gi B gi C gi ...
Definition
ga nai P0gi P1gi ... gi Pn (gi'i)
=
ga ¬P0gi P1gi ... 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 P0gi P1gi ... 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 P0gi P1gi ... 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 P0gi P1gi ... gi Pn (gi'i)
=
ge P0gi P1gi ... gi Pn (gi'i)
=
ge nai A gi B gi C gi ...
Definition
ge nai P0gi P1gi ... gi Pn (gi'i)
=
ge ¬P0gi P1gi ... 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 P0gi P1gi ... 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 P0gi P1gi ... 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 P0gi P1gi ... 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 P0gi P1gi ... gi Pn (gi'i)
=
go ¬P0gi P1gi ... 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.
2-ary case of P0 and P1:
go nai bo P0gi P1 (gi'i)
=
go nai P0gi P1 (gi'i)
=
P0 ⊕ P1
=
¬P0 ↔ P1
3-ary case of P0, ... , Pn (n≥2):
go nai bo P0gi ... gi Pn (gi'i)
=
((go nai bo P0gi ... 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 P0gi P1gi ... gi Pn (gi'i)} ≠ {P0i jo nai P1i 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 P0gi P1gi ... 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 P0gi P1gi ... gi Pn (gi'i)
=
gu P0gi P1gi ... gi Pn (gi'i)
=
=
P0
gu nai A gi B gi C gi ...
Definition
gu nai P0gi P1gi ... gi Pn (gi'i)
=
gu ¬P0gi P1gi ... 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 P0gi P1gi ... gi Pn (gi'i)
gu ¬P0gi ¬P1gi ... gi ¬Pn (gi'i)
=
¬P0
=
gu nai P0gi P1gi ... 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 P0gi P1gi ... gi Pn (gi'i)
=
P0i se ju P1i se ju ... i se ju Pki ju ... i ju Pn
=
i ju ... i ju Pn
=
Pki 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 P0gi P1gi ... gi Pn (gi'i)}
=
Pn
se xi ky gu bo A gi B gi C gi ...
se xi ky gu bo P0gi P1gi ... gi Pn (gi'i)
=
se xi ky gu P0gi P1gi ... gi Pn (gi'i)
=
Pk
se xi ky gu nai A gi B gi C gi ...
In the case of k=0
se xi no gu nai P0gi P1gi ... gi Pn (gi'i)
=
gu ¬P0gi P1gi ... gi Pn (gi'i)
=
¬P0
In the case of k≠0
se xi ky gu nai P0gi P1gi ... gi Pn (gi'i)
=
se xi ky gu ¬P0gi P1gi ... gi Pn (gi'i)
=
Pk
se xi ky gu nai bo A gi B gi C gi ...
se xi ky gu nai bo P0gi P1gi ... gi Pn (gi'i)
=
se xi ky gu ¬P0gi ¬P1gi ... gi ¬Pkgi ... gi ¬Pn (gi'i)
i ju ... i ju Pn
