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Subject: [lojban] lojban and propositional logic
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=20

*coi rodo*


 I've just finished a beginner's class of logic at the swedish university=
=20
(what don't you do to achieve lo ka jbocre someday? zo'o), and now I just=
=20
want to test if I'm able to express some different logical propositions bau=
=20
la lojban. Would you please correct me if you see any errors? mi ba ckire

Yes, it's a bit long, but if you don't find it interesting, don't read it.=
=20


 The text hereunder is just about propositional logic. Maybe I continue=20
with quantifiers (e.g. predicate logic) later in the already existing=20
threads about =E2=80=9Dexact quantifiers=E2=80=9D and =E2=80=9Dfree variabl=
es=E2=80=9D, respectively. But=20
as far as I understand it, the text hereunder show at least how clumsy the=
=20
scope of {zo'u} is to use, when expressing compound propositions. Please=20
correct me if I'm wrong about something, so I don't learn wrongly.


 1. *Atomic sentence* ({slebri} ? {stodzabri} ?)


 *FOL:* SameShape(a, b)


 *lojban:* [abuboi by zo'u] abu by tairmi'u=20


 but as =E2=80=9Dabu=E2=80=9D and =E2=80=9Dby=E2=80=9D could be interpreted=
 as variables rather than=20
individual constants, maybe the following sentence is a better translation?


 *lojban:* [la abus la bys zo'u] la abus la bys tairmi'u=20


 2. *Atomic sentence with complex terms*


 *FOL*: Taller (father(max), max)


 *lojban*: [lo patfu be la maks la maks zo'u] lo patfu be la maks la maks=
=20
rajyclamau [zo'e]


 In FOL the complex term =E2=80=9Dfather(max)=E2=80=9D is interpreted as a =
function, a=20
=E2=80=9Dname-like=E2=80=9D term.

In lojban {lo patfu be la maks} is interpreted as a description with an=20
inner predicate/selbri, and according to the xorlo gadri proposal =E2=80=9D=
any term=20
without an explicit outer quantifier is a *constant*, i.e. not a quantified=
=20
term.=E2=80=9D.


 Probably I should add the inner quantifier {lo *pa *patfu be la maks}.=20
Otherwise, it would mean =E2=80=9Dsomething whatever which has something to=
 do with=20
Max' father=E2=80=9D, right?


 3. *Negations of atomic sentence: literals* ({nafcumslebri} ?)


 *FOL*: =C2=ACHome(max)


 *lojban:* [la maks zo'u] la maks na zdazva [default: his own home]


 *question**:* Is di'u logical equivalent to the following three sentences?


 *lojban:* naku la maks zo'u la maks ku zdazva


 *lojban:* la maks naku zo'u la maks ku zdazva


 *lojban:* [la maks zo'u] la maks ku naku zdazva


 4. *Boolean connectives* (of logical sentences/bridi) ijek *and negations*


 *FOL:* =C2=AC(Home(seb) =E2=88=A7 Home(max))


 *lojban:* naku zo'u la seb zdazva ije la maks zdazva


 di'u negates both sentences, ki'u according to CLL =E2=80=9DIn general, th=
e scope=20
of a prenex that precedes a sentence extends to following sentences that=20
are joined by ijeks=E2=80=9D


 *FOL:* =C2=ACHome(seb) =E2=88=A7 Home(max)


 *lojban:* naku zo'u tu'e la seb zdazva tu'u ije la maks zdazva


 So here I use {tu'e...tu'u} to terminate the scope of zo'u. A bit clumsy?=
=20
Wouldn't it have been better if {zo'u} got it's own terminator?


 Or


 *lojban:* la seb zdazva na.ije la maks zdazva


 5. *DeMorgan's First Law*


 *FOL:* =C2=AC(P =E2=88=A7 Q) =E2=87=94 ( =C2=ACP =E2=88=A8 =C2=ACQ)


 lojban: bu'a bu'e zo'u tu'e naku zo'u bu'a ije bu'e ti'u idu'ibo tu'e na=
=20
bu'a ija na bu'e tu'u tu'u


 6. *A tautology** *({?})*: **Law of excluded middle, and conditionals*


 *FOL:* Cube(a) =E2=88=A8 =C2=ACCube(a)


 *lojban:* [la abus zo'u] tu'e la abus kubli tu'u ija tu'e naku zo'u la=20
abus kubli


 or without prenex:


 *lojban:* la abus kubli ija la abus na kubli


 which is logical equivalent to the material conditional:


 *FOL:* Cube(a) =E2=86=92 Cube(a)


 lojban: la abus kubli ijanai la abus kubli


 7. *=E2=80=9DUnless=E2=80=9D and biconditional*


 *glico:* Seb is at school unless Max is home


 *lojban:* la seb ku zvati le ckule se.ijanai (?) naku zo'u la maks ku=20
zdazva


 the english proposition is equivalent to and the lojban proposition should=
=20
be equivalent to:


 *glico:* Unless Max is home, then Seb is at school


 *lojban:* naku zo'u tu'e la maks ku zdazva tu'u ijanai la seb ku zvati le=
=20
ckule


 *FOL:* =C2=ACHome(max) =E2=86=92 School(seb)


 and the biconditional:


 *FOL:* Home(max) =E2=86=94 School(seb)


 *lojban:* la maks ku zdazva ijo la seb ku zvati le ckule


 *glico:* Max is at home if and only if Seb is at school


 or


 *glico:* Max is at home just in case Seb is at school


=20

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<p style=3D"margin-bottom: 0cm"><i>coi rodo</i></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">I've just finished a beginner's class
of logic at the swedish university (what don't you do to achieve lo
ka jbocre someday? zo'o), and now I just want to test if I'm able to
express some different logical propositions bau la lojban. Would you
please correct me if you see any errors? mi ba ckire</p>
<p style=3D"margin-bottom: 0cm">Yes, it's a bit long, but if you don't
find it interesting, don't read it.=20
</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">The text hereunder is just about
propositional logic. Maybe I continue with quantifiers (e.g.
predicate logic) later in the already existing threads about =E2=80=9Dexact
quantifiers=E2=80=9D and =E2=80=9Dfree variables=E2=80=9D, respectively. Bu=
t as far as
I understand it, the text hereunder show at least how clumsy the
scope of {zo'u} is to use, when expressing compound propositions.
Please correct me if I'm wrong about something, so I don't learn
wrongly.</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">1. <font size=3D"4"><b>Atomic sentence</b><=
/font>
({slebri} ? {stodzabri} ?)</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i>FOL:</i> <font face=3D"Verdana, sans-ser=
if"><font size=3D"3">SameShape(a,
b)</font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i>lojban:</i> [abuboi by zo'u] <font face=
=3D"Verdana, sans-serif"><font size=3D"3">abu
by tairmi'u </font></font>
</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">but as =E2=80=9Dabu=E2=80=9D and =E2=80=9Db=
y=E2=80=9D could be
interpreted as variables rather than individual constants, maybe the
following sentence is a better translation?</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i>lojban:</i> [la abus la bys zo'u] <font =
face=3D"Verdana, sans-serif"><font size=3D"3">la
abus la bys tairmi'u</font></font>=20
</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">2.  <font size=3D"4"><b>Atomic sentence
with complex terms</b></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i>FOL</i>: <font face=3D"Verdana, sans-ser=
if"><font size=3D"3">Taller
(father(max), max)</font></font></p>
<p style=3D"margin-bottom: 0cm; font-weight: normal"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an</span></i>:
[lo patfu be la maks la maks zo'u] <font face=3D"Verdana, sans-serif"><font=
 size=3D"3">lo
patfu be la maks la maks rajyclamau</font></font> [zo'e]</p>
<p style=3D"margin-bottom: 0cm; font-weight: normal"><br>
</p>
<p style=3D"margin-bottom: 0cm; font-weight: normal">In FOL the complex
term =E2=80=9Dfather(max)=E2=80=9D is interpreted as a function, a =E2=80=
=9Dname-like=E2=80=9D
term.</p>
<p style=3D"margin-bottom: 0cm"><span style=3D"font-weight: normal">In
lojban {lo patfu be la maks} is interpreted as a description with an
inner predicate/selbri, and according to the xorlo gadri proposal
=E2=80=9Da</span>ny term without an explicit outer quantifier is a
<strong>constant</strong>, i.e. not a quantified term.=E2=80=9D.</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">Probably I should add the inner
quantifier <font face=3D"Verdana, sans-serif"><font size=3D"3">{lo <i>pa </=
i>patfu
be la maks}</font></font>. Otherwise, it would mean =E2=80=9Dsomething
whatever which has something to do with Max' father=E2=80=9D, right?</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">3. <font size=3D"4"><b>Negations of atomic
sentence: literals</b></font> ({nafcumslebri} ?)</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">FOL<=
/span></i>:
 <font face=3D"Verdana, sans-serif"><font size=3D"3">=C2=ACHome(max)</font>=
</font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i>
[la maks zo'u] <font face=3D"Verdana, sans-serif"><font size=3D"3">la maks
na zdazva</font></font> [default: his own home]</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><span style=3D"font-style: normal"><u><span=
 style=3D"font-weight: normal">question</span></u></span><i><span style=3D"=
font-weight: normal">:</span></i>
Is di'u logical equivalent to the following three sentences?</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i>
<font face=3D"Verdana, sans-serif"><font size=3D"3">naku la maks zo'u la
maks ku zdazva</font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i>
<font face=3D"Verdana, sans-serif"><font size=3D"3">la maks naku zo'u la
maks ku zdazva</font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i>
[la maks zo'u] <font face=3D"Verdana, sans-serif"><font size=3D"3">la maks
ku naku zdazva</font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">4. <font size=3D"4"><b>Boolean connectives<=
/b></font>
(of logical sentences/bridi) ijek <font size=3D"4"><b>and negations</b></fo=
nt></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">FOL:=
</span></i>
 <font face=3D"Verdana, sans-serif"><font size=3D"3">=C2=AC(Home(seb) =E2=
=88=A7
Home(max))</font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i>
<font face=3D"Verdana, sans-serif"><font size=3D"3"><span style=3D"font-wei=
ght: normal">naku
zo'u la seb zdazva ije la maks zdazva</span></font></font></p>
<p style=3D"margin-bottom: 0cm; font-weight: normal"><br>
</p>
<p style=3D"margin-bottom: 0cm">di'u negates both sentences, ki'u
according to CLL =E2=80=9DIn general, the scope of a prenex that precedes a
sentence extends to following sentences that are joined by ijeks=E2=80=9D</=
p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">FOL:=
</span></i>
 <font face=3D"Verdana, sans-serif"><font size=3D"3">=C2=ACHome(seb) =E2=88=
=A7
Home(max)</font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i>
<font face=3D"Verdana, sans-serif"><font size=3D"3">naku zo'u tu'e la seb
zdazva tu'u ije la maks zdazva</font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">So here I use {tu'e...tu'u} to
terminate the scope of zo'u. A bit clumsy? Wouldn't it have been
better if {zo'u} got it's own terminator?</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">Or</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i>
<font face=3D"Verdana, sans-serif"><font size=3D"3"><span style=3D"font-wei=
ght: normal">la
seb zdazva na.ije la maks zdazva</span></font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">5. <font size=3D"4"><b>DeMorgan's First Law=
</b></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">FOL:=
</span></i><span style=3D"font-weight: normal">
</span><font face=3D"Verdana, sans-serif"><font size=3D"3"><span style=3D"f=
ont-weight: normal">=C2=AC(P
=E2=88=A7 Q) </span></font></font><font face=3D"Verdana, sans-serif"><font =
size=3D"3">=E2=87=94
</font></font><font face=3D"Verdana, sans-serif"><font size=3D"3"><span sty=
le=3D"font-weight: normal">(
=C2=ACP =E2=88=A8 =C2=ACQ)</span></font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><span style=3D"font-weight: normal">lojban:
</span><font face=3D"Verdana, sans-serif"><font size=3D"3"><span style=3D"f=
ont-weight: normal">bu'a
bu'e zo'u tu'e naku zo'u bu'a ije bu'e ti'u idu'ibo tu'e na bu'a ija
na bu'e tu'u tu'u</span></font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><span style=3D"font-weight: normal">6. </sp=
an><font size=3D"4"><b>A
tautology</b></font><b> </b><span style=3D"font-weight: normal">({?})</span=
><b>:
</b><font size=3D"4"><b>Law of excluded middle, and conditionals</b></font>=
</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">FOL:=
</span></i><span style=3D"font-weight: normal">
</span><font face=3D"Verdana, sans-serif"><font size=3D"3"><span style=3D"f=
ont-weight: normal">Cube(a)
=E2=88=A8 =C2=ACCube(a)</span></font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i><span style=3D"font-weight: normal">
[la abus zo'u] </span><font face=3D"Verdana, sans-serif"><font size=3D"3"><=
span style=3D"font-weight: normal">tu'e
la abus kubli tu'u ija tu'e naku zo'u la abus kubli</span></font></font></p=
>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><span style=3D"font-weight: normal">or
without prenex:</span></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i><span style=3D"font-weight: normal">
</span><font face=3D"Verdana, sans-serif"><font size=3D"3"><span style=3D"f=
ont-weight: normal">la
abus kubli ija la abus na kubli</span></font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><span style=3D"font-weight: normal">which
is logical equivalent to the material conditional:</span></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">FOL:=
</span></i><span style=3D"font-weight: normal">
</span><font face=3D"Verdana, sans-serif"><font size=3D"3"><span style=3D"f=
ont-weight: normal">Cube(a)
 =E2=86=92 Cube(a)</span></font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><span style=3D"font-weight: normal">lojban:
</span><font face=3D"Verdana, sans-serif"><font size=3D"3"><span style=3D"f=
ont-weight: normal">la
abus kubli ijanai la abus kubli</span></font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">7. <font size=3D"4"><b>=E2=80=9DUnless=E2=
=80=9D and
biconditional</b></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">glic=
o:</span></i><span style=3D"font-weight: normal">
</span><font face=3D"Verdana, sans-serif"><font size=3D"3"><span style=3D"f=
ont-weight: normal">S</span></font></font><span style=3D"font-weight: norma=
l">eb
is at school unless Max is home</span></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i><span style=3D"font-weight: normal">
</span><font face=3D"Verdana, sans-serif"><font size=3D"3"><span style=3D"f=
ont-weight: normal">la
seb ku zvati le ckule se.ijanai <font size=3D"2">(?)</font> naku zo'u la ma=
ks ku zdazva</span></font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><span style=3D"font-weight: normal">the eng=
lish proposition is equivalent to and the lojban proposition should be equi=
valent to:</span></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">glic=
o:</span></i><span style=3D"font-weight: normal">
Unless Max is home, then Seb is at school</span></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i><span style=3D"font-weight: normal">
</span><font face=3D"Verdana, sans-serif"><font size=3D"3"><span style=3D"f=
ont-weight: normal">naku
zo'u tu'e la maks ku zdazva tu'u ijanai la seb ku zvati le ckule</span></fo=
nt></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">FOL:=
</span></i>
<font face=3D"Verdana, sans-serif"><font size=3D"3">=C2=ACHome(max) =E2=86=
=92
School(seb)</font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">and the biconditional:</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">FOL:=
</span></i>
<font face=3D"Verdana, sans-serif"><font size=3D"3">Home(max)  =E2=86=94=20
School(seb)</font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">lojb=
an:</span></i>
<font face=3D"Verdana, sans-serif"><font size=3D"3">la maks ku zdazva ijo
la seb ku zvati le ckule</font></font></p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">glic=
o:</span></i>
<font face=3D"Verdana, sans-serif"><font size=3D"3">M</font></font>ax is at
home if and only if Seb is at school</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm">or</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><i><span style=3D"font-weight: normal">glic=
o:</span></i>
Max is at home just in case Seb is at school</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>
<p style=3D"margin-bottom: 0cm"><br>
</p>

<p></p>

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