^^Logica matematica.

Logica.

what is logic still remains one of the main subjects of research and debates in the field of philosophy of logic.

Anche l'attuale (2020) descrizione su wikipedia e' un po' sparpagliata.

Un buon punto di partenza e'

• wp/Logic  per il discorso piu' generale
• wp/Mathematical_logic  per la logica in forma matematica.

Esistono diversi tipi di logica

1. Philosophical logic  wp
2. Informal logic
3. Formal logic
4. Mathematical logic wp

Logical form

History. "Form" is important to logic !

Aristotle employed variable letters to represent valid inferences, one of Aristotle's greatest inventions.

According to the followers of Aristotle

only the logical principles stated in schematic terms belong to logic, and not those given in concrete terms.

ref: wp/Logical_form

Mathematical logic

≡ logistic  ≡  symbolic logic  ≡  algebra of logic  ≡  formal logic (more recently)

emerged in the mid-19th century as a subfield of mathematics, reflecting the confluence of two traditions: formal philosophical logic and mathematics.

It is the set of logical theories elaborated in the course of the last [nineteenth] century with the aid of an artificial notation and a rigorously deductive method. Before this emergence, logic was studied with rhetoric, with calculationes, through the syllogism, and with philosophy. The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics.

Formal logic

• syllogistic logic_wp
• Logica delle proposizioni. E' la piu' semplice.
• Logica dei predicati.

senso1:  creare un sistema formale nel quale fare logica. Def possibile post1900, dato che e' allora che si definisce cosa sia un sistema formale.

senso2: una loro formalita' l'avevano pero' anche i predecessori, per cui lo si puo' dire anche della loro logica.

wp/Formal_system | wp/Formal logical systems

Reason compared to related concepts

1. Compared to logic_wp
2. Reason compared to cause-and-effect thinking, and symbolic thinking
3. Reason, imagination, mimesis, and memory
4. Logical reasoning methods and argumentation
   1. Deductive reasoning
   2. Inductive reasoning
   3. Analogical reasoning
   4. Abductive reasoning
   5. Fallacious reasoning

ref: wp/Reason

Articoli piu' generici

Inference

Conseguenza logica.

en: Logical consequence, entailment;  fr: Déduction logique; de: Implikation;  es: Consecuencia lógica

wp/Logical_consequence

argomento

premessa  inferenza  conclusione.
Premessa e conclusione sono proposizioni

wp/Argument

In logic and philosophy, an argument is a series of statements, called the premises, intended to determine the degree of truth of another statement, the conclusion.

Deductive reasoning ≡ deductive logic ≡  "top-down logic"

goes from premises to conclusions. If all premises are true, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.

wp/Deductive_reasoning

inductive reasoning  ≡ Inductive inference  ≡  "bottom-up logic"

1. a method of reasoning in which the premises are viewed as supplying some evidence, but not full assurance, for the truth of the conclusion.
2. a method where one's experiences and observations, including what are learned from others, are synthesized to come up with a general truth.
3. Many dictionaries define inductive reasoning as the derivation of general principles from specific observations (arguing from specific to general), although there are many inductive arguments that do not have that form.
4. Inductive reasoning is distinct from deductive reasoning. While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument is probable, based upon the evidence given.

wp/Inductive_reasoning

wp/Natural_deduction

Interpretation of a formal language

An interpretation is an assignment of meaning to the symbols of a formal language.

Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation.

formal semantics (=def) the general study of interpretations of formal languages

ref: Interpretation_(logic) | Semantics_of_logic formal semantics

Sentence VS proposition

the logician traditionally is not interested in the sentence as uttered but in the proposition, an idealised sentence suitable for logical manipulation.

proposition_(philosophy)  the meaning of a declarative sentence, where "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the non-linguistic bearer of truth or falsity which makes any sentence that expresses it either true or false.

proposition_(mathematics) a statement that may or may not be true;

axiom a statement that is taken to be true within a domain of discourse.

ref: Proposition

Inferenza VS implicazione

wp/Logic#Inference

Occa: ho sempre detto  per "se X allora Y"  "X implica Y", ma secondo l'articolo la relazione tra i 2 e' piu' complessa.

Frase dichiarativa, interrogativa

Links wp

www.cap-lore.com/Turnstile.html  ⊢

The logician is concerned with

• what follows from what
• what follows from some particular set of axioms.

So an handy notation

“Γ ⊢ A”   means:  Γ is a set of axioms,  A is a particular proposition

• according to some set of deduction rules under discussion,
  A may be deduced from statements in Γ.

Often the Γ is omitted when the discussion is in context of a particular set of axioms and then “⊢ A” means that A may be deduced from those axioms, i.e. that A is a theorem in the formal system. Sometimes more than one set of deduction rules are under discussion and then there will be a subscript to the turnstile.

The simplest and most common deduction rule is merely modus ponens.

philosophical logic ≡  philosophy of logic