Logica fuzzy

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A.A. passati

• (Marra, Aguzzoli), A.A. 2007-2008

Information

Course's website
Times and classrooms:
Monday 15:30 - 17:30 - Room 5 (Ground floor - via comelico 39)
Friday 10:30 - 12:30 - Room 5

Lessons' notes

This notes are written in english to help foreign students to follow this course.

Classical Propositional Logic

Lesson 1 - 5th october

In this lesson we are going to describe the classical propositional logic (L.P.C) language.

Sintax

Let's ${\mathbb {N}}=\{1,2,...\}$ be the set of the natural numbers and ${\mathbb {A}}=\{(,),X,|,\$,\rightarrow ,\land ,\lor ,\neg ,\top ,\bot \}$ an alphabet of symbols.
${\mathbb {A}}^{\star }$ is the set of strings on this alphabet. For example $(\top \land \bot ),(\neg \land ,X((\in {\mathbb {A}}^{\star }$.
We have now to define the set of the "well formed formula" $FORM$, that is the set of the element of the L.P.C.
Definition (well formed formulas). $FORM\subset {\mathbb {A}}^{\star }$ is defined with some conditions:

1. $\top ,\bot \in FORM$
2. $\forall n\in {\mathbb {N}}$
   $\qquad X||...|\$\in FORM$, where | is taken n times. For example: $X|\$,X||\$,X|||\$,....\in FORM$
3. if $\alpha ,\beta \in form$, then $(\neg \alpha )\in FORM$.
4. if $\alpha ,\beta \in form$, then $(\alpha \land \beta )\in FORM$.
5. if $\alpha ,\beta \in form$, then $(\alpha \lor \beta )\in FORM$.
6. if $\alpha ,\beta \in form$, then $(\alpha \rightarrow \beta )\in FORM$.

Well formed formulas is often abbreviated with f.b.f. The strings in condition (2) are called "propositional variable" or "atomic formula" or "propositional letter" or simply "variable". They are abbreviated with this notation: $X_{1},X_{2},X_{3},...,X_{n}$(In other books p, q, r, .., are used to refer to variables).
The set of variables is $VAR$.
$VAR\subset FORM$
Example of well formed formulas:
$(((X_{1}\land X_{2})\lor \top )\rightarrow (X_{1}\lor \bot ))\in FORM$
Instead
$(X_{1}\land \land )\notin FORM$
In this notes we omit (, ) where it is clear the precedence of the connectives. (Connectives are $\neg ,\land ,\lor ,\rightarrow ,\bot ,\top$)
Why don't we use directly $X_{1},X_{2},...$ or $p,q,r,s,...$?
Because it is important that the alphabet is a finited set. Although, in this notes, we use $X_{1},X_{2},...,$notation. It is essential, even, that the L.P.C. language is decidability, that is it must exist an algorithm that tell us if a string is $\in FORM$ or not. For this reason it is crucial the
Unique readability of well formed formulas. For all $\alpha \in FORM$one (and ony one) of the following sentences must be true:

1. $\alpha =\bot$ or $\alpha =\top$ but not both.
2. Exists an unique $n\in {\mathbb {N}}$ such that $\alpha =X_{n}$.
3. Exists an unique $\beta \in FORM$ such that $\alpha =(\neg \beta )$.
4. Exist unique $\beta ,\gamma \in FORM$ such that $\alpha =(\beta \land \gamma )$.
5. Exist unique $\beta ,\gamma \in FORM$ such that $\alpha =(\beta \lor \gamma )$.
6. Exist unique $\beta ,\gamma \in FORM$ such that $\alpha =(\beta \rightarrow \gamma )$.

Let's, infact, consider the parser of the algorithm we talked above: When it find, for example, $\neg (\alpha \land \beta )$, to decide if the string belong to FORM, it have to decide (for the unique readability of well formed formulas) if $\alpha \land \beta \in FORM$. But how can it do this? Deciding if $\alpha \in FORM$ and $\beta \in FORM$. This is what the above theorem tell us. Try to draw the parser tree. It is unique thanks to, again, that theorem.

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