Handbook of the History of Logic: - Fordham University Faculty

586 Mikko Yrjönsuuri
is false’ provide an illuminating example. Walter Burley, for example, considers
whether one should grant ‘the positum is false’ if it really is. Thus, if the positum is
‘You are in Rome’ in a disputation held in Oxford, one should concede ‘the positum
is false’ as the first proposition. According to Burley, this has the implication that
then one should deny ‘’you are in Rome’ is the positum’, which may result in a
complicated (and curious) disputation with sentences like ‘”you are in Rome” is
true and irrelevant’ being conceded in Oxford [Burley, 1963, 62]. It seems that
Burley was thinking about the fact that there is nothing inconsistent in supposing
that one is participating an obligational disputation in Rome with a false positum,
and insofar as this was his idea, he was not dealing with self-referential paradoxes
in this context. 6
THOMAS BRADWARDINE
Despite work done by many authors, including such excellent logicians as Walter
Burley and William Ockham, it seems that the medieval discussion concerning the
paradoxes of self-reference did not make much visible progress for more than a century.
The account given in Insolubilia Monacensia appears almost as advanced as
some late thirteenth-century ones. This may be partly due to a modern pretension
that progress would be marked by the offered solutions becoming better in some
respect, while the thirteenth century authors apparently were more interested in
forming a clear picture in how self-reference leads to a paradox and in mapping the
various versions of the paradox that can be built. Ockham, to mention a different
kind of a case, appears to have been simply not very interested in the topic. He
ends the relevant section in his Summa logicae to the comment that it is there just
for the sake of completeness. 7
In any case, the discussion was brought to a turning point by Thomas Bradwardine
in the early 1320s. 8 His discussion of the topic seems to be the first that
gives the feeling of a systematic solution cast in a logically careful format. Bradwardine
was one of the so-called Oxford Calculators, which was a group of scholars
working in Oxford and in effect producing a new mathematical and logical way
of doing philosophy that was to have repercussions even until Galileo Galilei and
early modern science in general. Bradwardine has in recent discussion been best
known for his work in mathematical physics, but it is clear that his solution of the
insolubles was a very remarkable achievement. Also other calculators, including
Richard Kilvington and William Heytesbury, produced treatments of insolubles
that are of a very high logical quality. 9
Bradwardine’s treatise is systematically structured. It starts off with discussions
6 For an edition of Burley’s treatise on the insolubles, see [Roure, 1970].
7 [Ockham, 1974, 746]. In respect to the substance of what Ockham has to say about insolubles,
it seems clear that this part of Summa logicae must have been written without knowledge of
Bradwardine’s work.
8 For Bradwardine’s position in the discussion, see e.g. [Read, 2002, esp. pp. 189–199].
9 For a general discussion of the Oxford calculators, see e.g. [Sylla, 1982].