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Can a classical ML contain OL?
I have a question...

A classical, Tarskian system standardly contains homophonic definitions as the base clauses, e.g.,

'G' denotes Gs.

This is understood to be a sentence of the metalanguage (ML) which defines a predicate of the object language (OL). And when ML contains OL, the standard assumption is that the expression 'G' used on the right-hand side is identical to the expression mentioned on the that numerically one expression, 'G', is an expression of both ML and of OL.

However, if homophonic definitions are admissible, then (given the compositionality of negation) the following sentence also seems admissible into the system as a definition:

'G' denotes x iff x is ~G.

But such an interpretation suffices for paradox in OL. For instance,  'Socrates is G' will be true in OL iff it is not true in OL (contra PNC). So apparently the second definition is not admissible in a classical system. But (again by the compositionality of negation) that means the former cannot be admitted into the system either. And more broadly, it looks like ML cannot contain OL...otherwise ML would rule that these definitions are well-formed.

Gupta and Belnap 1993 take note of something like the second definition on p. 273--but their concerns at that stage of the book are not with classical logic. When I asked Anil what bearing it has for classical logic (about a year ago) he denied it showed anything noteworthy (though I wasn't clear why).

The plot now thickens. About 3 months ago, Jeremy Avigad raised the very same definition to me when commenting on one of my papers. His claim was that it reflect that a classical system cannot contain "directly self-referential" expressions. Here too, I wasn't entirely clear on the reasons given. (After all, a homophonic definition like ' 'dogs' denote dogs' seems importantly different from a self-referring definition like ' 'this very sentence' denotes this very sentence.') More broadly, however, Avigad gave the impression that the second definition indeed shows something substantive about a classical system...but also, his attitude was that all this was old news. (I didn't get any references from him however.)

So my question to you is: What is going on here? Does the latter definition show that ML cannot contain OL in a classical system? If not, what does it show? Or, can someone provide references for further reading?


Can a classical ML contain OL?
Reply to T. Parent
Yes, it can...

The essential point is that languages in the Tarskian setting must be interpreted languages.

Hence, if you change

    'G' denotes x iff x is G
    'G' denotes x iff x is ~G,

you have a different OL. They have different truth definitions. Only in the former case ML contains OL.

Read my,%20meaning,%20translation.pdf



Panu Raatikainen

Ph.D., University Lecturer
Docent in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki


Can a classical ML contain OL?
Thanks very much Panu! I will read your paper in the next few days. I will post again if further questions arise.

Can a classical ML contain OL?
Reply to T. Parent
Dear Panu (and anyone else who may be reading),

I enjoyed your paper very much. Your most helpful thought was the one you expressed above, though your paper helped me appreciate that thought better. However, I'm not quite sure how to apply that thought to a different, related paradox. (For what it's worth, I presented the paradox below to Keith Simmons several years ago, and he seemed to be convinced that there was an issue here. Though he was not convinced by the other paradox, presented at the beginning of this thread.)

To get situated, it will help to review a similar paradox, though this one is clearly due to an ill-behaved semantic term. The semantic term in question is 'empty'; it denotes predicates that fail to denote anything (and is defined on predicates of its own language). More precisely, for any predicate F:

                        (1) F is empty if and only if ~(#y) F denotes y.

(In this, '#' is the existential quantifier and 'denotes' is defined on its own language). Naïvely, if we were allowed an unrestricted use of this predicate, then we could define the following paradoxical predicate 'E.

                        (2) 'E' denotes x iff x = 'E' and 'E' is empty.

We can show that 'E' results in paradox by asking: Is the term 'E' itself E? Suppose first that it is. Then, 'E' denotes 'E', and that means that 'E' is nonempty. But by the second conjunct of the definiens, 'E' denotes 'E' only on the condition where 'E' is empty. Hence, since 'E' is nonempty, it must be that it not denote 'E' after all. So 'E' is not E.

However, if 'E' is not E, then the predicate 'E' does not denote 'E'. But nor could it denote anything else, since nothing else could satisfy the first conjunct of the definiens. Hence, in this case 'E' is empty. And if 'E' is empty, then by definition, 'E' denotes 'E'. So it turns out that 'E' is E.

                       Hence, 'E' is E iff 'E' is not E; contradiction.

            Again, it is no surprise that 'empty' as defined by (1) ends up generating paradox, since it is defined by an unrestricted 'denotes' predicate (which generates paradoxes of its own). But once semantic terms are stratified in the Tarskian way, the predicate 'empty,' like 'denotes,' will no longer generate paradox.

            What is bizarre, however, is that it seems this semantic paradox can be recreated using non-semantic terminology--assuming ML contains OL. Specifically, if 'E' is numerically the same expression in the definiens as in the definiendum, it seems possible to re-introduce the above paradoxical predicate 'E' just by using quantification and negation:

                        (3) 'E' denotes x iff x = 'E' and ~(#y)Ey.

 A bit of reflection should reveal that (3) is equivalent to (2). Or at least, they both can be used to show that 'E' is E iff 'E' is not E, a contradiction.

So my question is: What goes wrong here? Note that Panu's earlier thought seems inapplicable here, since 'E' is assumed to have a constant interpretation throughout (articulated at (3)).