Artificial Intelligence/Logic/Representation/Propositional calculus

The propositional calculus is defined in the context of Boolean constants, where two or more values are computed against each other to produce an accurate description of a concept. Each variable used in the calculus holds a value for it, which is either true to the context or false¹.

Propositional logic deals with the determination of the truth of a sentence. An allowable sentence is called the syntax of proposition. A syntax or sentence holds various propositional symbols, where each symbol holds a proposition that can either be true or false. The names of the symbols can be anything from alphabets like a, b or c to symbols like α, β or γ to variable names like IsOld, and may hold meaning relative to their contexts in the concept. Although, two propositions are constant as per the syntax and have a fixed meaning. They are:

• True - proposition that is always-true.
• False - proposition that is always-false².

In mathematical terms, these values would pronounce more like this:

$u \Leftarrow a \wedge b \wedge c$

In simple terms this means that an object is an instance of a concept u if the conditions a, b and c hold true simultaneously. So, for instance, we are given with a concept that says, one is eligible to drive a car if one has a license, a car and a knowledge of driving. To state this in propositional calculus, you'd have to state this in the following mathematical notation:

$CanDriveCar \Leftarrow HasLicense \wedge HasCar \wedge KnowsHowToDrive$

However simple to denote, the zero-order logic is only capable of describing concepts in a limited context and do not hold much of a descriptive power.

Notes

1. Kubat, Bratko, Ivan and Michalski, Ryszard. Machine Learning and Data Mining: Methods and Applications/A Review of Machine Learning Methods. 1998. ISBN 0471971995. John Wiley: New York.
2. Russell, Stuart and Norvig, Peter. Artificial Intelligence: A modern approach. 2003. ISBN 0130803022. Prentice Hall:New Jersey.