Lógica

A Lógica é um ramo tanto da Filosofia quanto da Matemática. O sistema lógico (ou simplesmente a lógica) é um conjunto de regras para raciocínio sobre um determinado assunto. Muitos sistemas diferentes de lógica foram construídos ao longo do tempo. Esses sistemas artificiais de raciocínio têm encontrado atualmente muitas aplicações práticas na computação, como por exemplo nas aplicações de Inteligência artificial.

De forma superficial, lógica é o estudo de sistemas prescritivos de raciocínio, ou seja, sistemas que definem como se "deveria" realmente pensar para não errar, usando a razão, dedutivamente e indutivamente. A forma como as pessoas realmente raciocinam é estudado noutras áreas, como na psicologia cognitiva.

Como ciência, a lógica define a estrutura de declaração e argumento e elabora fórmulas através das quais estes podem ser codificados. Implícita no estudo da lógica está a compreensão do que gera um bom argumento e de quais os argumentos que são falaciosos.

A lógica filosófica lida com descrições formais da linguagem natural. A maior parte dos filósofos assumem que a maior parte do raciocínio "normal" pode ser capturado pela lógica, desde que se seja capaz de encontrar o método certo para traduzir a linguagem corrente para essa lógica.

Abaixo estão discussões mais específicas sobre alguns sistemas lógicos. Veja também: lista de tópicos em lógica.

Lógica Aristotelica

A Lógica aristotélica foi iniciada por Aristóteles. Embora seja possível que Aristóteles tenha aprendido de alguém anterior, o primeiro estudo do raciocínio foi atribuído a ele. Aristóteles e seus discípulos concluíram que dois dos mais importantes princípios da lógica são a lei da não-contradição e a lei do terceiro excluído. Esta lógica é atualmente conhecida por vários nomes, que a distinguem de sistemas lógicos mais recentes, e.g., Lógica Aristotélica ou Lógica bivalente clássica.

A lei da não-contradição diz que nenhuma afirmação pode ser verdadeira e falsa ao mesmo tempo e a lei do terceiro excluído diz que uma afirmação deve ser ou verdadeira ou falsa. Combinadas, estas duas leis requerem dois valores de verdade que são mutuamente exclusivos. Uma afirmação pode ser falsa ou verdadeira, mas, de modo algum, pode ser verdadeira e falsa ao mesmo tempo.

Lógica formal

Formal logic, also called symbolic logic, is concerned primarily with the structure of reasoning. Formal logic deals with the relationships between concepts and provides a way to compose proofs of statements. In formal logic, concepts are rigorously defined, and sentences are translated into a precise, compact, and unambiguous symbolic notation.

Some examples of symbolic notation are:

Lowercase letter p, q and r with italic font are conventionally used to denote propositions:

p: 1 + 2 = 3

This statement defines p is 1 + 2 = 3 and that is true.

Two propositions can be combined using conjunction, disjunction or conditional. They are called binary logical operators. Such combined propositions are called compound propositions. For example,

p: 1 + 1 = 2 and "logic is the study of reasoning."

In this case, and is a conjunction. The two propositions can differ totally from each other.

In mathematics and computer science, one may want to state a proposition depending on some variables:

p: n is an odd integer.

This proposition can be either true or false according to the variable n.

A proposition with free variables is called propositional function with domain of discourse D. To form an actual proposition, one uses quantifiers. For every n, or for some n, can be specified by quantifiers: either the universal quantifier or the existential quantifier. For example,

for all n in D, P(n).

This can be written also as:

$\forall n\in D,P(n)$

When there are several free variables free, the standard situation in mathematical analysis since Weierstrass, the quantifications for all ... there exists or there exists ... such that for all (and more complex analogues) can be expressed.

Lógica matemática

Lógica Matemática é o uso de lógica formal para estudar o raciocínio matemático. No início do século XX, lógicos filosóficos incluindo (Frege, Russell) tentaram provar que matemática poderia ser inteiramente reduzida à lógica. Eles diziam que descobrir a forma lógica de uma sentença era na verdade revelar a forma "certa" de dizê-la, ou revelar alguma essência previamente escondida. A redução falhou, mas hoje em dia, a lógica é aceita como uma forma precisa de descrever o raciocínio matemático.

Philosophical logic

Philosophical logic is essentially a continuation of the traditional discipline that was called "Logic" before it was supplanted by the invention of Mathematical logic. It is concerned with the elucidation of ideas such as reference, predication, identity, truth, quantification, existence, and others. Philosophical logic has a much greater concern with the connection between natural language and logic. See Philosophical logic.

Predicate logic

Gottlob Frege, in his Begriffsschrift, discovered a way to rearrange many sentences to make their logical form clear, to show how sentences relate to one another in certain respects. Prior to Frege, formal logic had not been successful beyond the level of sentential logic: it could represent the structure of sentences composed of other sentences using such words as "and", "or", and "not," but it could not break sentences down into smaller parts. It could not show how "Cows are animals" entails "Parts of cows are parts of animals."

Sentential logic explains the workings of words such as "and", "but", "or", "not", "if-then", "if and only if", and "neither-nor". Frege expanded logic to include words such as "all", "some", and "none". He showed how we can introduce variables and "quantifiers" to rearrange sentences.

• "All humans are mortal" becomes "All things x are such that, if x is a human then x is mortal." which may be written symbolically

$\forall x(H(x)\Rightarrow M(x))$

• "Some humans are vegetarian" becomes "There exists some (at least one) thing x such that x is human and x is vegetarian" which may be written symbolically

$\exists x(H(x)\wedge V(x))$.

Frege treats simple sentences without subject nouns as predicates and applies them to "dummy objects" (x). The logical structure in discourse about objects can then be operated on according to the rules of sentential logic, with some additional details for adding and removing quantifiers. Frege's work started contemporary formal logic.no,

Frege adds to sentential logic (1) the vocabulary of quantifiers (upside-down A, backward E) and variables, (2) a semantics that explains that the variables denote individual objects and the quantifiers have something like the force of "all" "some" in relation to those objects, and (3) methods for using these in language. To introduce an "All" quantifier, you assume an arbitrary variable, prove something that must hold true of it, and then prove that it didn't matter which variable you chose, that would have held true. An "All" quantifier can be removed by applying the sentence to any particular object at all. A "Some" (exists) quantifier can be added to a sentence true of any object at all; it can be removed in favor of a term about which you are not already presupposing any information.

Multi-valued Logic

Systems which go beyond these two distinctions are known as non-Aristotelian logics, or multi-valued logics.

In the early 20th century Jan Łukasiewicz investigated the extension of the traditional true/false values to include a third value, "possible".

Logics such as fuzzy logic have since been devised with an infinite number of "degrees of truth", e.g., represented by a real number between 0 and 1. Bayesian probability can be interpreted as a system of logic where probability is the subjective truth value.

Lógica e computadores

Lógica é extensivamente usada em áreas como Inteligência Artificial, e Ciências da computação.

Nas décadas de 50 e 60, researchers predicted that when human knowledge could be expressed using logic with mathematical notation, it would be possible to create a machine that reasons, or artificial intelligence. This turned out to be more difficult than expected because of the complexity of human reasoning. Logic programming is an attempt to make computers do logical reasoning and Prolog programming language is commonly used for it.

In symbolic logic and mathematical logic, proofs by humans can be computer-assisted. Using automated theorem proving the machines can find and check proofs, as well as work with proofs too lengthy to be written out by hand.

In computer science, Boolean algebra is the basis of hardware design, as well as much software design.

Quote

Logic, logic, logic. Logic is the beginning of wisdom, Valeris, not the end. From Star Trek VI: The Undiscovered Country

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See also analytic proposition; college logic; argument form; validity; soundness; cogency; deduction and induction; modus ponens; affirming the consequent; modus tollens; disjunctive syllogism, faith, Scientific method; fuzzy logic; history of logic; set theory