[lit-ideas] The General's Box
- From: Donal McEvoy <donalmcevoyuk@xxxxxxxxxxx>
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Wed, 19 May 2010 19:27:24 +0000 (GMT)
In "Popper Selections" in the "Notes to p.35" [at 383], P discusses the
paradoxical character of the status of the props that constitute the TLP - an
issue still alive within Wittgensteinian "exegesis" if the difference of
opinion between Monk and McGinn in the BBC "In Our Time" programme on
Wittgenstein is any indication [this programme, with others, is now archived
on-line: http://www.bbc.co.uk/programmes/p0054945]. ;
P notes first that W's "The totality of true propositions is...the totality of
the natural sciences" "asserts its own untruth, and is therefore
contradictory." However, this leaves open whether it is a false or senseless
_p_. P then goes on to argue that any such _p_ "which implies its own
meaningless is not meaningless but false, since the predicate 'meaningless', as
opposed to 'false', does not give rise to paradoxes". By "false" P means that
such a _p_ "_will be self-contradictory and neither meaningless nor genuinely
paradoxical_; it will be a meaningful proposition merely because it asserts of
every expression of a certain kind that it is not a proposition (i.e. not a
well-formed formula); and such an assertion will be true or false, but not
meaningless, simply because to be (or not to be) a well-formed proposition is a
property of expressions......Modifying an idea of J.N.Findlay's we can write:
_The expression obtained by substituting for the variable in the following
expression 'The expression obtained by substituting for the variable in the
following expression _x_ the quotation name of this expression, is not a
statement' the quotation name of this expression is not a statement_.
And what we have just written turns out to be a self-contradictory
statement. (If we write twice 'is a false statement' instead of 'is not a
statement', we obtain a paradox of the liar; if we write 'is a non-demonstrable
statement', we obtain a Godelian statement in Findlay's writing.)"
In the next "Notes to p.35" [p.385], P uses the following "The General's Box"
argument to attack as insufficient the tripartite division of expressions into
true, false and senseless [or not well-formed]:-
"The General's Chief Counter-espionage Officer is provided with three boxes,
labelled (i) 'General's Box, (ii) 'Enemy's Box' (to be made accessible to the
enemy's spies), and (iii) 'Waste Paper', and is instructed to distribute all
information arriving before 12 o'clock among these three boxes, according to
whether this information is (i) true, (ii) false, or (iii) meaningless.
"For a time he receives information which he can easily distribute....The
last message _M_ ...disturbs him a little [however], for _M_ reads:
'From the set of all statements placed, or to be placed, within the box
labelled "General's Box", the statement "0=1" cannot validly be derived.'
At first, the [Officer] hesitates whether he should not put _M_ into box (ii).
But since he realises that, if put into (ii), _M_ would supply the enemy with
valuable true information, he ultimately decides to put _M_ into (i).
"But this turns out to be a big mistake. For the symbolic logicians (experts
in logistic?) on the General's staff, after formalizing (and 'arithmetizing')
the contents of the General's box, discover that they obtain a set of
statements which contains an assertion of its own consistency; and this,
according to Godel's second theorem on decidability, leads to a contradiction,
so that '0=1' can actually be deduced from the presumably true information
supplied to the General.
"The solution...[is recognising] that the tripartition claim is unwarranted,
at least for ordinary languages; and we can see from Tarski's theory of truth
that no definite number of boxes will suffice."
Donal
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