3.1 Logic[1YS]
Propositions
A proposition may depend on some variables. Examples:
”the person x by trade is a baker”,
”the number x is greater than 9”.
We write
to say that \(P(x)\) is the symbol that summarizes the proposition written on the right.
Propositional logic
it will also be a ”well-formed formula”.
For convenience, in this Section, we also add to the language the constants \(V\) and \(F\) which are respectively always true and always false, in every evaluation. 1 In the construction of well-formed formulas they are treated as variables. Note that we have not introduced the equality connective ”\(=\)”. When all variables can only take true/false values, the equality \(a=b\) can be interpreted as \(a\iff b\). In more general contexts (as in the case of set theory) instead, ”equality” needs a precise definition.
First-order logic
In the first order logic we add the connectives \(∀\), which reads ”for each” and \(∃\), which reads ”exists”. We must therefore enlarge the family of well-formed formulas.
Note that, in many examples, quantified variables are assumed to be elements of a ”set”.
An example of a logical proposition would be: ”\(n\) is an even number”. We can use logical propositions as atoms in the construction of well-formed formulas.