FOM: logical vs mathematical

Martin Davis martin at
Sun Sep 10 14:29:04 EDT 2000

The issue of in what sense higher order logic can reasonably be considered 
as "logical" while set theory must be regarded as "mathematical" has come 
up in exchanges between Harvey Friedman and Joe Shipman. Quine addressed 
this (I think in a conference paper in 1960). He pointed out that the 
theorems of what he called SOL (by which he meant the standard at that time 
deductive systems studied by Hilbert-Ackermann and Church) divide neatly 
into two classes: those that depend on comprehension and those that follow 
just from the first order deductive rules. He proposed that only the 
theorems of the second class should be considered logical since those of 
the first, in their intended interpretation, have ontological presuppositions.

I was convinced and have taken that position ever since.


                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at
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