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Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.

Arguments can be either correct or incorrect. An argument is correct if its premises support its conclusion. Deductive arguments have the strongest form of support: if their premises are true then their conclusion must also be true. This is not the case for ampliative arguments, which arrive at genuinely new information not found in the premises. Many arguments in everyday discourse and the sciences are ampliative arguments. They are divided into inductive and abductive arguments. Inductive arguments are statistical generalizations, such as inferring that all ravens are black based on many individual observations of black ravens.[2] Abductive arguments are inferences to the best explanation, for example, when a doctor concludes that a patient has a certain disease which explains the symptoms they suffer.[3] Arguments that fall short of the standards of correct reasoning often embody fallacies. Systems of logic are theoretical frameworks for assessing the correctness of arguments.

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Logic has been studied since antiquity. Early approaches include Aristotelian logic, Stoic logic, Nyaya, and Mohism. Aristotelian logic focuses on reasoning in the form of syllogisms. It was considered the main system of logic in the Western world until it was replaced by modern formal logic, which has its roots in the work of late 19th-century mathematicians such as Gottlob Frege. Today, the most used system is classical logic. It consists of propositional logic and first-order logic. Propositional logic only considers logical relations between full propositions. First-order logic also takes the internal parts of propositions into account, like predicates and quantifiers. Extended logics accept the basic intuitions behind classical logic and extend it to other fields, such as metaphysics, ethics, and epistemology. Deviant logics, on the other hand, reject certain classical intuitions and provide alternative explanations of the basic laws of logic.

The word "logic" originates from the Greek word "logos", which has a variety of translations, such as reason, discourse, or language.[4] Logic is traditionally defined as the study of the laws of thought or correct reasoning,[5] and is usually understood in terms of inferences or arguments. Reasoning is the activity of drawing inferences. Arguments are the outward expression of inferences.[6] An argument is a set of premises together with a conclusion. Logic is interested in whether arguments are correct, i.e. whether their premises support the conclusion.[7] These general characterizations apply to logic in the widest sense, i.e., to both formal and informal logic since they are both concerned with assessing the correctness of arguments.[8] Formal logic is the traditionally dominant field, and some logicians restrict logic to formal logic.[9]

Formal logic is also known as symbolic logic and is widely used in mathematical logic. It uses a formal approach to study reasoning: it replaces concrete expressions with abstract symbols to examine the logical form of arguments independent of their concrete content. In this sense, it is topic-neutral since it is only concerned with the abstract structure of arguments and not with their concrete content.[10]

Formal logic is interested in deductively valid arguments, for which the truth of their premises ensures the truth of their conclusion. This means that it is impossible for the premises to be true and the conclusion to be false.[11] For valid arguments, the logical structure of the premises and the conclusion follows a pattern called a rule of inference.[12] For example, modus ponens is a rule of inference according to which all arguments of the form "(1) p, (2) if p then q, (3) therefore q" are valid, independent of what the terms p and q stand for.[13] In this sense, formal logic can be defined as the science of valid inferences. An alternative definition sees logic as the study of logical truths.[14] A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true in all possible worlds and under all interpretations of its non-logical terms, like the claim "either it is raining, or it is not".[15] These two definitions of formal logic are not identical, but they are closely related. For example, if the inference from p to q is deductively valid then the claim "if p then q" is a logical truth.[16]

Formal logic uses formal languages to express and analyze arguments.[17] They normally have a very limited vocabulary and exact syntactic rules. These rules specify how their symbols can be combined to construct sentences, so-called well-formed formulas.[18] This simplicity and exactness of formal logic make it capable of formulating precise rules of inference. They determine whether a given argument is valid.[19] Because of the reliance on formal language, natural language arguments cannot be studied directly. Instead, they need to be translated into formal language before their validity can be assessed.[20]

The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, a logic is a logical formal system. Distinct logics differ from each other concerning the rules of inference they accept as valid and the formal languages used to express them.[21] Starting in the late 19th century, many new formal systems have been proposed. There are disagreements about what makes a formal system a logic.[22] For example, it has been suggested that only logically complete systems, like first-order logic, qualify as logics. For such reasons, some theorists deny that higher-order logics are logics in the strict sense.[23]

When understood in a wide sense, logic encompasses both formal and informal logic.[24] Informal logic uses non-formal criteria and standards to analyze and assess the correctness of arguments. Its main focus is on everyday discourse.[25] Its development was prompted by difficulties in applying the insights of formal logic to natural language arguments.[26] In this regard, it considers problems that formal logic on its own is unable to address.[27] Both provide criteria for assessing the correctness of arguments and distinguishing them from fallacies.[28]

Another characterization identifies informal logic with the study of non-deductive arguments. In this way, it contrasts with deductive reasoning examined by formal logic.[35] Non-deductive arguments make their conclusion probable but do not ensure that it is true. An example is the inductive argument from the empirical observation that "all ravens I have seen so far are black" to the conclusion "all ravens are black".[36]

A further approach is to define informal logic as the study of informal fallacies.[37] Informal fallacies are incorrect arguments in which errors are present in the content and the context of the argument.[38] A false dilemma, for example, involves an error of content by excluding viable options. This is the case in the fallacy "you are either with us or against us; you are not with us; therefore, you are against us".[39] Some theorists state that formal logic studies the general form of arguments while informal logic studies particular instances of arguments. Another approach is to hold that formal logic only considers the role of logical constants for correct inferences while informal logic also takes the meaning of substantive concepts into account. Further approaches focus on the discussion of logical topics with or without formal devices and on the role of epistemology for the assessment of arguments.[40]

Premises and conclusions are the basic parts of inferences or arguments and therefore play a central role in logic. In the case of a valid inference or a correct argument, the conclusion follows from the premises, or in other words, the premises support the conclusion.[41] For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars is a red planet". For most types of logic, it is accepted that premises and conclusions have to be truth-bearers.[41][a] This means that they have a truth value: they are either true or false. Contemporary philosophy generally sees them either as propositions or as sentences.[43] Propositions are the denotations of sentences and are usually seen as abstract objects.[44] For example, the English sentence "the tree is green" is different from the German sentence "der Baum ist grn" but both express the same proposition.[45]

Propositional theories of premises and conclusions are often criticized because they rely on abstract objects. For instance, philosophical naturalists usually reject the existence of abstract objects. Other arguments concern the challenges involved in specifying the identity criteria of propositions.[43] These objections are avoided by seeing premises and conclusions not as propositions but as sentences, i.e. as concrete linguistic objects like the symbols displayed on a page of a book. But this approach comes with new problems of its own: sentences are often context-dependent and ambiguous, meaning an argument's validity would not only depend on its parts but also on its context and on how it is interpreted.[46] Another approach is to understand premises and conclusions in psychological terms as thoughts or judgments. This position is known as psychologism. It was discussed at length around the turn of the 20th century but it is not widely accepted today.[47] 17dc91bb1f