[FOM] First Order Logic

[Richard Heck]

On 09/08/2013 01:20 AM, Panu Raatikainen wrote:
> Let me ask a clarificatory question:
>
> Is it Harvey's (or anyone's) view that compactness in itself is a
> desirable property of a logic? And if so, why more exactly?

My understanding is that Harvey thinks *completeness* is a desirable, 
and indeed non-negotiable, property of anything rightly called "logic" 
(or, perhaps more weakly, anything usable as "logic" for foundational 
purposes). If so, then the view is one also found in Quine.

And if, indeed, we are talking about *certain* foundational purposes, 
then I don't see that there is much room for argument. If those purposes 
require us to be able to tell whether something counts as a correct 
proof, then we need (i) to be able to decide whether something is an 
axiom and (ii) to be able to decide whether a given step in a proof is 
correct, in the sense that it is permissible given the previous steps in 
the proof. Given various assumptions about the finitude of proofs, it 
will follow easily that consequence has to be r.e. Since such 
assumptions might not be universally shared, I won't make any stronger 
claim. But, as I said, it's easy enough to see why one might think 
completeness important, for *certain* purposes.

Richard


-- 
-----------------------
Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University

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