^^Logica matematica, formal logic.

≡ formal system of logic  ≡  logic system

≡ mathematical logic  ≡  logistic  ≡  symbolic logic (historically called)  ≡  algebra of logic  ≡  formal logic (more recently)

Consists of a

1. Linguaggio formale.
2. Proof system.
3. Semantica

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

Esistono diversi tipi di logica

1. Philosophical logic  wp
2. Mathematical logic wp
3. wp/Informal logic  emphasizes argumentation;
   wp/Formal logic    emphasizes wp/implication and wp/inference.

Combinatory_logic | stanford/logic-combinatory

wp/Logic#Systems_of_logic

Storia >>>

per comprendere, un minimo di storia.

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

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

wp/Natural_deduction

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

Argomento, argomentazione >>>

Conseguenza logica.

en: Logical_consequence, entailment, implication; 

fr: Déduction logique;

de: Implikation; 

es: Consecuencia lógica

Inferenza VS implicazione

wp/Logic#Inference | wp/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.

proof system >>>

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, the non-linguistic bearer of truth or falsity which makes any sentence that expresses it either true or false.

proposition_(mathematics) a statement that is true or not true;

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

ref: wp/Proposition

Frase dichiarativa (affermazione), interrogativa

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

Approfond

formal system wp

is an abstract structure used for inferring theorems from axioms according to a set of rules.

These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.

A formal system is essentially an "axiomatic system".

In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics.

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.