vBenthem3 - 1 Tuesday 26 April, Logical Inference Patterns...

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1 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 well-understood, 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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vBenthem3 - 1 Tuesday 26 April, Logical Inference Patterns...

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