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Tuesday 26 April, Logical Inference Patterns
Aim: show you basic logical styles of analyzing valid inference, and their abstraction level.
The first is the logic of sentence combinations analyzed via truth tables, a form of algebra
over binary numbers, and historically the origin of computing machines. This system is
wellunderstood, though the precise computational complexity of its valid principles is still
unknown: an issue on the Cray Millennium List of major open problems in mathematics.
The second topic is quantifiers over objects. We start from simple syllogistic inferences
that are easy to check via Venn diagrams, and generalize this to the ubiquitous theme of
‘monotonicity reasoning’. Next, we look at logics of relations with stacked quantifiers.
Such languages can express all of mathematics, and hence analyze sophisticated scientific
reasoning. The price of this power: determining validity in the full logic of quantifiers is
‘undecidable’, there is no algorithm that can do it (let alone truth tables or some fast visual
method), an insight related to the classical limitation results on computability of last week.
So, there is a range of logics, from simple to complex, dealing with a spectrum from
domestic inferences to scientific reasoning. Is this then a unique ‘yardstick of validity’?
Our third topic shows this is not quite true, since logic today also analyzes many different
reasoning styles. Many of these have come up in linguistics and artificial intelligence. We
show how a simple enrichment of models for propositional logic, adding a binary relation
of ‘more plausible’ between worlds, describes forms of default reasoning whose laws differ
from standard logics, but can otherwise be studied with the same mathematical rigor. The
resulting picture today is that logic analyzes ‘natural reasoning styles’, of which we know
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 Spring '09

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