Artificial Intelligence/Logic/Representation/Propositional calculus

One of the points of logic is that you can reason about statements even when you don't know what those statements mean. We can replace statements, or "propositions," with variable names.

So, for example, you can say "It's raining and I'm wet," which is a representation as characters describing an utterance in natural language. In logic, we might represent this like so:

$(a\wedge b)$

In this case the proposition "It's raining" is represented by "a" and "I'm wet" is represented by b. The $\wedge$ symbol stands for "and."

Let's talk a bit about propositions. Propositions are statements that can be either true or false, and nothing else. This is called "the law of excluded middle," because there's nothing allowed in the middle of true and false. The law helps us know what kinds of utterances are candidates for propositions and which are not. For example, questions (e.g., What color is he wearing?), exclamations (e.g., Wow!), and commands (e.g., Study harder.) are not propositions.

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 $\alpha$, $\beta$ or $\gamma$ 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.