^^Connettivi logici.  AND OR NOT IF

Connettivi logici in natural language

not, or, and, if, ...  (no, o, e, se, ...)

behave in quite complex ways which generate ambiguities.

The classical interpretations of mathematical logic are similar to the meanings of natural language expressions, but not identical.

Difference between the mathematical use and the everyday use.

Scopo(estensione) (di un connettivo logico). Ambiguita' dello scopo.

e' il pezzo di frase a cui si riferisce.

Puo' produrre ambiguita' in lingua comune, poiche' non e' univoco il pezzo a cui si riferisce

es: "non_è_vero_che Giulio esce di casa o esce Giulia" non si capisce se il connettivo si riferisce a tutta la frase che segue o solo al primo pezzo.

Logica astratta

in astratto come fatto dalla logica astratta, un connettivo linguistico e' un operatore su variabili logiche, che produce una nuova variabile logica.

Non ci sono piu' le frasi di partenza che si riferiscono a un qualcosa e che possono essere vere o false, ma solo variabili logiche rappresentate da lettere, e che possono assumere 2 valori di verità: vero o falso.

AND

Proposizione logica composta, col connettivo logico:  AND   "e"      ∧

In generale:

la proposizione composta "P AND Q" è fatta da

• "P, Q" proposizioni componenti
• e dalla composizione "AND". 

La proposizione composta AND, e' vera se e solo se

entrambe le proposizioni componenti sono vere.

Mathematical use  VS  everyday use.

Definizione circolare apparente

"P and Q" is true if and only if P is true and Q is true.

It appear to have defined "AND" using the word "AND".

If I substitute actual sentences for P and Q, I get something like this:

"London is in England and Paris is in France" if and only if

"London is in England" and "Paris is in France".

In logic the word "and" connects statements, not objects.

"Jack and Jill went up the hill"

Logic rewrite: "Jack went up the hill and Jill went up the hill".

Be aware: a sentence like "x and y are true" is a different use of the "and" and should think of it as a useful shorthand for "x is true and y is true".

OR (inclusive)

OR  used in mathematics is always the inclusive or rather than the exclusive or.

NOT

"not P" is the statement that

That is, if P is true, then "not P" is false, and if P is false, then "not P" is true.

If you want to understand the statement "not P"

ask yourself "What are all the exact circumstances that need to hold for P to be false?"

Truth table

NOT e' un connettivo unario, non e' il solo, in tutto sono 4.

negation operator  negates the truth value of the propositional variable. That is, if a proposition is true its negation is false and vice-versa.

identity operator doesn't change the truth value of the propositional variable. It acts exactly the same way as to a number being multiplied by one.
⊤ tautology       propositional variable whose valuation is always true.

⊥ contradiction  propositional variable whose valuation is always false.

That is,

the tautology operator ⊤ assigns all values to be true

the contradiction operator ⊥ assigns all values to be false.

distinction, between mathematics and metamathematics.

The former consists of statements like

"31 is a prime number" or

"The angles of a triangle add up to 180".

The latter consists of statements about mathematics rather than of mathematical statements themselves. For example, if I say, "The theorem that the angles of a triangle add up to 180 was known to the Greeks," then I’m not talking about triangles (except indirectly) but about theorems to do with triangles.

The sort of metamathematics that concerns mathematicians is the sort that discusses properties of mathematical statements (notably whether they are true) and relationships between them (such as whether one implies another).

Here are a few metamathematical statements.

1. "There are infinitely many prime numbers" is true.
2. The continuum hypothesis cannot be proved using the standard axioms of set theory.
3. "There are infinitely many prime numbers" implies "There are infinitely many odd numbers".
4. The least upper bound axiom implies that every Cauchy sequence converges.

In each of these four sentences I didn't make mathematical statements. Rather, I referred to mathematical statements. The grammatical reason for this is that the word "implies", in the English language, is supposed to link two noun phrases. You say that one thing implies another.

Links librosito

1. ^^Logica delle proposizioni. Calcolo delle proposizioni.
2. Congiunzioni linguistiche, e significato.
3. AND, OR; circuito AND, circuito OR.
4. Parsing.
5. Logic alphabet.
6. Conjunctive & disgiuntive normal form.
7. Arithmetic. Operators. Python.

Links inet

1. wp/Logical_connective
2. proofwiki/Count_of_Truth_Functions ho messo questo link per ri-conoscere proofwiki, e per apprezzare che ci sia una pagina dedicata al Count_of_Truth_Functions
3. mathigon.org/logical-connectives-and-unary-operators sembrerebbe un bel sito

Approfond

lg: and  or  not  if ...