Propositional logic: let´s formalize!

Logic deals with the formalization of natural language and reasoning methods. We will begin with an introduction to the logic of compound statements, which is a basis for many other logical systems and is also called propositional logic.

Propositional logic is a formal system in which the basic units are statements (or propositions)

The basic assumption is that each proposition represents a sentence that is either true or false (but not both)

Simple propositions are denoted by propositional variables (p, q, r, s……), statements can be combined via logical connectives into more complex , compound propositions.

The connectives used to form more complex propositions include:

·         negation (¬, ~,  . , read “not” or every natural language expression that means “not”, like: “It is not the case”, “It´s false that”, and so on)

·         Conjuntion: (˄, &,  read “and” or every natural language expression that means “and”, like “but”, althoug, also, as well as, and so on)

·         Disjuntion (˅, read “or” and every natural language expression that means “or”, like “either”)

·         Conditional (→,  ᴝ, read “if….then….” but it also can be used to translate “if,” “whenever”, “…provided that…”,  “…is sufficient for….”, “….is necessary for…” and so on.

·         Biconditional (↔, read “…if and only if…”, but also: “when and only when”, “is equivalent to “, “is equal to”, “is both necessary and sufficient for”, “just in case of..then..”

Parentheses are also used to mark the way the connectives linked the propositions, they are used when the connectives changed the ordinary logical order which is:

• Whenever a negation affects the whole conjunction or disjunction, conditional or bicontional , for example:

“It´s false that my brother can speak French and my sister too”: ¬(p˄q)

• Whenever the logical dominance (the usual capability that  the connectives have, to connect the propositions) changes  in the natural language formulation. So the order in “logical dominance” is: ↔, →, and ˄,˅ which have the same dominance.

Parentheses are often omitted to increase readability but one has to be careful to avoid  ambiguous expressións.

Here you can practice what you has just learn. you will find exercises at the bottom of the page.

On this page there are more ejercises, solutions are at the bottom. 

Anyway, if you are still in a mess, you can watch  this videos (in Spanish, ufff!!) that show you how solving certain  sort of  formalizacion exercises.