Re: Model vs. World

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Date: Thu, 17 Jun 93 12:45:48 -0400
From: schubert@cs.rochester.edu
Message-id: <9306171645.AA23911@ash.cs.rochester.edu>
To: cg@cs.umn.edu, fritz@rodin.wustl.edu, interlingua@ISI.EDU
Subject: Re:  Model vs. World

> ...  Since when is a Tarskian model itself 
> a CORRESPONDENCE?  It is a domain set with a set of predicates and 
> relations (sets of tuples) on the domain set.  (This is an 
> abstract combinatorial structure, namely a relational structure or 
> directed hypergraph.)  Nichtwahr? 

Nicht wahr. Sure, a model assumes a domain of individuals, and sometimes
additional sets (of functions, relations) and structures (such as
algebras or semi-lattices), but the key component of a model is a
correspondence between "word" and "object" (a value assignment). I refer
you, for instance, to Hughes and Cresswell, Modal Logic, p. 135, on 
models for the Lower predicate Calculus. The term "model" does get 
used somewhat variably even in logic, and maybe you're used to a 
different convention.

> >Relative to such a correspondence, there is assuredly no discrepancy
> >between truth in the model and actual truth. For instance, if part of
> >the assumed correspondence is that the predicate constant `US-president'
> >corresponds to a set which includes the actual person, Pat Hayes, AND
> >the individual constant `Pat-Hayes' corresponds to Pat Hayes, then
> >  US-president(Pat-Hayes)
> >is true in the model and true in actuality (i.e., the individual denoted
> >by `Pat-Hayes' really IS a member of the set denoted by `US-president').
> >Of course, this involves a use of the symbol `US-president' in a way
> >rather far removed from the way speakers of English might prefer to use
> >it -- but logic is very permissive in that regard. 

> Yes, and if a further "part of the assumed correspondence" is 
> that the predicate-constant `US-President' corresponds to being a 
> U.S. President, the sentence is true in the model and in fact false.  

In a standard (extensional) model for the LPC, there IS no such further 
assumption to be made. Truth of closed sentences is fully determined by 
the extensions assigned to the individual, predicate, and function 
constants. On the other hand, if you have in mind some sort of 
intensional model that assigns a property as well as an extension
(in any given world) to each predicate constant, then making this 
further assumption does not lead to a model at all, since any reasonable
notion of a model of this sort will insist on having the individual 
denoted by constant "Pat-Hayes" in the EXTENSION of "US-president" 
just in case that same individual is assigned the PROPERTY denoted by 
"US-president" (in a given world). In other words, an individual can't 
"happen to be" in the set of US presidents without also having the 
property of being a US president. So in any model of this sort, once 
you've said that "US-president" shall denote the ACTUAL property of 
being a US president (however you cash this out formally), the only way 
you can rig the model to render 
  US-president(Pat-Hayes)
true in the model is by letting "Pat-Hayes" denote something other
than the ACTUAL Pat Hayes, such as Bill Clinton. Again, all you end up 
with is a use of a symbol -- in this case "Pat-Hayes" -- in a way that 
might annoy some US residents but will leave logicians unfazed.

If you feel so inclined, have the last word. I'm bowing out, as I
think it is of limited value to both of us (not to mention interlingua)
to find out exactly what the other means by "model". -Len