Picturing negation

Continuing his musings on negation Ed Ockham has a post on Wittgenstein's picture theory of language.  He reproduces a drawing from the Notebooks which illustrates W's picturing of negation.
This smacks of complementation.  But for this to work a proposition must be a set within some universe.  Can we reconcile this with something like a picture theory?

Let's suppose our language has names for N objects, a, b, c,... and M predicates F, G, H,... with which to describe them.   A table such as the one below, with objects labelling the rows and predicates labelling the columns, and a tick or cross in each cell, characterises a possible way the world might be.  There are 2**(NM) such possible worlds: each cell can be filled in two ways and there are NM cells in all.

F G H ...

The above table stands for the possible world in which Fa, Ga, Gb, Fc, and Hc are all true and all other predications are false.  We can associate the atomic proposition Fa with a set of tables, namely those for which there is a tick in the row labelled 'a' and the column labelled 'F'.  In effect we associate the proposition Fa with the set of tables standing for the possible worlds in which it is true.  More generally, if proposition p is associated to table set Sp then affirming p amounts to the claim that the actual world is represented by a member of Sp.  Starting with the atomic propositions of the form Fa we can recursively define the table set associated to any proposition built from atomic propositions and logical connectives by the following rules:  if p --> Sp and q-->Sq then
  • p and q --> Sp ∧ Sq
  • p or q --> Sp ∨ Sq
  • ~p --> U ∖ Sp  where U denotes the set of all tables.  
Quantified propositions can also be associated to table sets.  ∀x.Fx maps to the set of tables with a tick in every cell in the F column.  ∃x.Fx maps to the set of tables with at least one tick in the F column.  More general, non-atomic predicate functions can be thought of as labelling further 'virtual' columns added to the right of the table.  But the ticks and crosses in each such column are determined completely by the distribution of ticks and crosses in the 'core' table.

To what extent can we see this as a picture theory?   Each table can be thought of as a picture, complete in every detail down to the level of resolution of the language, not of the world, but of a possible world.  And each table constitutes a point in a logical space of possibilities.  But to capture W's insight that negation is complementation we have to see propositions as sets of pictures.

Now, this view is quite close to the established option of taking propositions to be sets of possible worlds.  But I'm not taking a position in the metaphysics of propositions.   I'm offering a semantics for a tiny formalised fragment of English.  A rather absurd semantics, perhaps, but a semantics has to be given in other than linguistic terms, else we get nowhere.  Consider this:  When someone asserts proposition p to me, being a cautious chap, I ask myself Is p true?  We think of this as somehow 'offering up p to the world' to see if it 'fits'.  But can this be anything other than metaphorical?  Suppose, in asserting p, my interlocutor is offering me a set of tables.  I have my own set of tables associated to the conjunction of the propositions I believe.  The process of 'offering for fit' involves checking if the two sets of tables have an element in common.  If they do then my interlocutor's proposition and my beliefs are compatible.  If not, not.

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