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Date: Mon, 8 Apr 2002 21:14:00 EDT
Subject: Re: [lojban] why is if ganai..gi?
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In a message dated 4/8/2002 1:59:25 PM Central Daylight Time, 
gordon.dyke@bluewin.ch writes:


> I'm still following this logic course.
> 
> we are still doing propositional logic, using the following axioms (among
> others and using G for Gamma (hypothesis), - for not, | to separate
> judgments from hypothesesesesesse ahemm and F for false):
> 
> G, A | F || G | -A J red absurd.
> G,-A | F || G | A K red absurd.
> 

Well, I'm not sure what the "J red absurd" and "K red absurd" are, but the 
rules are perfectly standard ones for negation -- in classical bivalent 
logic.

<The lecturer explained that "intuitive" logicians preferred not to accept
the K red absurd. as it is used to demonstrate:

|| G | A or -A exclusion of a third case>

"Intuitive" is a bad choice of words here; the logicians concerned are not 
more touchy-feely than most and so on but rather are Intuitionists, 
supporters of a praticular school (or cluster of schools) in the Philosophy 
of Mathematics and, to a certain extent, of a mathematical methodology 
sometimes called rigorous constructivism. The idea of the latter (which is 
where the denial of tertium non datur, the theoremhood of P v ~P, comes in) 
is that you can claim that a mathematical object exists only if you can 
actually show how to construct it from the mathhematical givens (usually, the 
natural numbers and the basic operations on them -- some types of recursion 
may even not be allowed). In particular, you cannot go from the fact that 
assuming there is no such object leads to a contradiction to the claim that 
there is such an object: the second rule in your list or ~~P => P is rejected 
as a general principle (though usually P => ~~P).

<I seem to be getting cause and consequence of being "intuitive" mixed up
here, but anyways:>

Intuitionism is not intuitive (well, it is to some people, but not to most).

<From the refgramm, it seems to be accepted that a proposition is either true
or false.

But why force this into lojban:

A => B <=> -B or A

when this judgment can only be made given K red. or third case exclusion>

The Refgramm does say this sort of thing from time to time, but also says 
other things as well, including allowing that {jei P} could range over real 
[0,1] as well as {0,1}. The point is that the syntax of Lojban does not 
determine this issue: however many truth values there are (Intuitionist logic 
has, in fact, nondenumerably many, but cannot actually be described in terms 
of truth values), there will be an "or" and an "and" and a "if... then" 
defined as "not... or " and an "iff" and so on. In fact, often several of 
them. Lojban has not theorems nor axioms -- it's a langauge not a formal 
system -- so there is no guarantee that {P anai P} is necessarily true in 
Lojban (whatever that means). And, the fact that "if ... then..." is defined 
in a certain way (or that we often use a Lojban expression that means "not... 
or", regardless of whether it is "if... then" or not) does not affect whether 
P v ~P is necessarily true or even true at all. (Note -- in case this is the 
problem -- that "A or not B" is very different from "A or not A": the second 
says pretty close to that A has only two possible truth values, the first one 
says only either A has a True value or B doesn't, without noting how many 
other values each might have taken on. Indeed, it doesn't even say how many 
values are counted as True for the present purpose.)





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<HTML><FONT FACE=arial,helvetica><BODY BGCOLOR="#ffffff"><FONT style="BACKGROUND-COLOR: #ffffff" SIZE=2>In a message dated 4/8/2002 1:59:25 PM Central Daylight Time, gordon.dyke@bluewin.ch writes:<BR>
<BR>
<BR>
<BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">I'm still following this logic course.<BR>
<BR>
we are still doing propositional logic, using the following axioms (among<BR>
others and using G for Gamma (hypothesis), - for not, | to separate<BR>
judgments from hypothesesesesesse ahemm and F for false):<BR>
<BR>
G, A | F&nbsp; ||&nbsp; G | -A&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; J red absurd.<BR>
G,-A | F&nbsp; ||&nbsp; G | A&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; K red absurd.<BR>
</BLOCKQUOTE><BR>
<BR>
Well, I'm not sure what the "J red absurd" and "K red absurd" are, but the rules are perfectly standard ones for negation -- in classical bivalent logic.<BR>
<BR>
&lt;The lecturer explained that "intuitive" logicians preferred not to accept<BR>
the K red absurd. as it is used to demonstrate:<BR>
<BR>
|| G | A or -A&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; exclusion of a third case&gt;<BR>
<BR>
"Intuitive" is a bad choice of words here; the logicians concerned are not more touchy-feely than most and so on but rather are Intuitionists, supporters of a praticular school (or cluster of schools) in the Philosophy of Mathematics and, to a certain extent, of a mathematical methodology sometimes called rigorous constructivism.&nbsp; The idea of the latter (which is where the denial of tertium non datur, the theoremhood of P v ~P, comes in) is that you can claim that a mathematical object exists only if you can actually show how to construct it from the mathhematical givens (usually, the natural numbers and the basic operations on them -- some types of recursion may even not be allowed).&nbsp; In particular, you cannot go from the fact that assuming there is no such object leads to a contradiction to the claim that there is such an object: the second rule in your list or ~~P =&gt; P is rejected as a general principle (though usually P =&gt; ~~P).<BR>
<BR>
&lt;I seem to be getting cause and consequence of being "intuitive" mixed up<BR>
here, but anyways:&gt;<BR>
<BR>
Intuitionism is not intuitive (well, it is to some people, but not to most).<BR>
<BR>
&lt;From the refgramm, it seems to be accepted that a proposition is either true<BR>
or false.<BR>
<BR>
But why force this into lojban:<BR>
<BR>
A =&gt; B &lt;=&gt; -B or A<BR>
<BR>
when this judgment can only be made given K red. or third case exclusion&gt;<BR>
<BR>
The Refgramm does say this sort of thing from time to time, but also says other things as well, including allowing that {jei P} could range over real [0,1] as well as {0,1}.&nbsp; The point is that the syntax of Lojban does not determine this issue: however many truth values there are (Intuitionist logic has, in fact, nondenumerably many, but cannot actually be described in terms of truth values), there will be an "or" and an "and" and a "if... then" defined as "not... or "&nbsp; and an "iff" and so on.&nbsp; In fact, often several of them.&nbsp; Lojban has not theorems nor axioms -- it's a langauge not a formal system -- so there is no guarantee that {P anai P} is necessarily true in Lojban (whatever that means).&nbsp; And, the fact that "if ... then..." is defined in a certain way (or that we often use a Lojban expression that means "not... or", regardless of whether it is "if... then" or not) does not affect whether P v ~P is necessarily true or even true at all.&nbsp; (Note -- in case this is the problem -- that "A or not B" is very different from "A or not A": the second says pretty close to that A has only two possible truth values, the first one says only either A has a True value or B doesn't, without noting how many other values each might have taken on.&nbsp; Indeed, it doesn't even say how many values are counted as True for the present purpose.)<BR>
<BR>
<BR>
<BR>
<BR>
</FONT></HTML>
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