Information Systems - Applied Logic

Logic: the product of our consideration, our thinking, reasoning.

Five key areas of logic are:

1 Basic Propositional Logic

One would think that logicians would be clear, but often they are not. For a start they tend to use a vocabulary that most people rarely use. Secondly, logicians don't always agree among themselves, so if most people are going to understand what logicians are talking about it would be a good idea if logical concepts could be clearly defined.

Two of the key figures in the philosophy of logic are Aristotle and Frege. In the 19th century, logicians like George Boole and Venn tried to develop methods of describing and applying logic in simple ways. Today logical study includes looking at the proof of compactness as a property of mathematical systems, how agents operate in use language.

A key concept in dealing with reasoning, is that of inference. Traditionally there are two kinds of inference: deductive inference and inductive inference. One characterisation of the difference is that deductive inference goes from the general to the particular, and inductive inference goes from the particular to the general.

An example of deductive inference would be as follows: All apples are coloured red. I have an apple in my hand. THEREFORE I have something coloured red in my hand.

An example of inductive inference would be as follows: All the apples I have seen are coloured red. THEREFORE all apples are red.

The basis of propositional logical languages are atoms and connectives, which are put together to define well-formed formulas. Aristotle uses the subject - predicate construction to describe knowledge, a construction that remains to today.

Propositional languages also contains the notion of classical entailment.

2 First Order Languages

First order languages are basic propositional languages with the addition of quantification. Essentially FOL can be seen as a variety of modal language.

Quantification is described in two ways: FORALL and THEREEXISTS.

FORALL (X) ...

THEREEXISTS (X) ...

One particular subset of the first order languages that has aroused a lot of interest is the situation calculi. These languages create state descriptions, called "situations", out of the truth propositions, as well as "Actions", "Results" and "Facts". Situation Calculus is a formalism which is very popular as a foundation for Artificial Intelligence.




Languages to be considered which are more complex than FOL, are Second-order languages in which functions can operate on relations and functions, and higher-order languages than that. In addition Cantor's set theory needs to be considered in this as well.

3 Non-Monotonic Reasoning

In attempting to model common sense reasoning, the property of monotonicity needs to be considered. A monotonic language is where truths, once established, cannot be revised - such is the case with mathematical systems. Defeasible reasoning, is non-monotonic and models our own thought processes much more accurately. One way of understanding this "heuristic reasoning" is through the use of ranked interpretations. The study of heuristics is a curious area: what is the way we reason, and how do we determine our conclusions.

John McCarthy worked on research into Artificial Intelligence and nonmonotonic reasoning using an approach called minimal model semantics.

You can do your own experiments with this: go to the page of logic problems and solve the problems yourself. Write careful detailed notes of how you actually solved the problems: where did you start? How did you come to the conclusion? Did you go down any blind alleys? When did you recognise you went down a blind alley.

4 Automated Reasoning

Automated reasoning programs like OTTER have been developed to mimic basic logical deduction. These rely of logic statements using the horn clause.

5 Modal Logic

The classification of propositions using possibility, impossibility, contingency, or necessity.

CTL is an example.

SDNF - Strong Disjunctive Normal Form "State Description Normal Form"

Axiomatic System
The family of logics appears now to be determined, to be fixed. No doubt this family will develop, but the question is how, and in what way will prove useful. It is probably fair to say the logic has no "basis in reality", the only reality being that which suits the purposes of the human processing requirements.

Logic is really a language, like any other, just looks a little strange, and because of its simpleness, can produce some startling results, which would not be possible with a conventionally spoken language.

"Logic is the study of prescriptive systems of reasoning" ... whatever that means!

At this point in time we are saying that set theory encompasses all known logics.
RDF/URI the diagrams for a book on LOGIC - now if the future is pictures - remember BUZAN Mindmapping and the principles of memory with pictures - how about we describe logic with a series of pictures and publish that in a book, in addition, we add a little bit of spice and RDF/URI the diagrams - COLOUR and CAPITALS - BUZAN Mentions CAPITALS and COLOUR as good ways to jog the memory - these need to be incorporated in such logic diagrams



COSC470 Course Notes - Otago University - Hans van Ditmarsch


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Artificial Intelligence Research Laboratory Iowa State University http://www.cs.iastate.edu/~honavar/aigroup.html