# [FOM] Concerning definition of formulas

Richard Heck rgheck at brown.edu
Tue Oct 2 15:02:38 EDT 2007

```Alex Blum wrote:
> Arnon Avron wrote:
>
>> So the real answer to the question (in the spirit of Poincare)
>> is that there are basic concepts which  cannot
>> really be defined, and can only be explained in terms of themselves
>> (or some equivalent notions). There is no way to explain the quantifiers
>> "forall" and "exists" without using at least one of these quantifiers,
>> and the same applies to other logical notions (I believe that
>> anybody who has taught a basic course in logic, and explained
>> Tarski's semantics,  has faced some student claiming: "but
>> you use "forall" to define the meaning of "forall"!").
>>
> I wonder if giving the truth conditions of say the universal quantifier
> is intended to do more than make precise the use of the formal
> counterpart of  ''for all ', rather than define it.  And thus at least
> in this case circularity does not enter.
>
There's no circularity in a definition of \forall in terms of the
natural language "for all". If there were a circularity, it would have
to be in an attempt to define \forall in terms of \forall, or "for all"
in terms of "for all". Of course, one there are familiar theories that
characterize (say) the truth-conditions of \forall using \forall, but
there is no circularity there, either. Then, indeed, you are not
defining it, but rather theorizing about it. Tarski doesn't help matters
by running together `definitions' of truth, on the one hand, with
`theories' of truth, on the other.

What Arnon was talking about was something much less concrete, which is
the question what sorts of "concepts" can be defined, and in what sorts
of terms. And in this case, one doesn't have to sympathize with
Poincare's specific take on all of this to recognize that, in some
sense, not all concepts can be defined. Frege was fond of making this
point. (Pinches of salt and all that.) Still, I'd be reluctant myself to
rest very much on this point. It's simply not clear what concepts are,
or what it would be define one, or for that matter exactly what the
purpose of such definitions are supposed to be. Obviously, it's circular
to say, "The concept of universal quantification is the concept of
universal quantification". But that's not what we're considering. And
other sorts of circularity, if such there are, may be problems if the
definition is supposed to accomplish one thing but not if it's supposed
to accomplish another, or may not be problems at all. Much more would
need to be said before we could even begin to debate that question. Or
so it seems to me.

Richard

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Richard G Heck, Jr
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Brown University
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