Logic

Logic is the study of correct reasoning or good arguments. It is often defined in a more narrow sense as the science of deductively valid inferences or of logical truths. In this sense, it is equivalent to formal logic and constitutes a formal science investigating how conclusions follow from premises in a topic-neutral way or which propositions are true only in virtue of the logical vocabulary they contain. When used as a countable noun, the term "a logic" refers to a logical formal system. Formal logic contrasts with informal logic, which is also part of logic when understood in the widest sense. There is no general agreement on how the two are to be distinguished. One prominent approach associates their difference with the study of arguments expressed in formal or informal languages. Another characterizes informal logic as the study of ampliative inferences, in contrast to the deductive inferences studied by formal logic. But it is also common to link their difference to the distinction between formal and informal fallacies.

Logic is based on various fundamental concepts. It studies arguments, which are made up of a set of premises together with a conclusion. Premises and conclusions are usually understood either as sentences or as propositions and are characterized by their internal structure. Complex propositions are made up of other propositions linked to each other by propositional connectives. Simple propositions have subpropositional parts, like singular terms and predicates. In either case, the truth of a proposition usually depends on the denotations of its constituents. Logically true propositions constitute a special case since their truth depends only on the logical vocabulary used in them.

The arguments or inferences made up of these propositions can be either correct or incorrect. An argument is correct if its premises support its conclusion. The strongest form of support is found in deductive arguments: it is impossible for their premises to be true and their conclusion to be false. This is the case if they follow a rule of inference, which ensures the truth of the conclusion if the premises are true. A consequence of this is that deductive arguments cannot arrive at any substantive new information not already found in their premises. They contrast in this respect with ampliative arguments, which may provide genuinely new information. This comes with an important drawback: it is possible for all their premises to be true while their conclusion is still false. Many arguments found in everyday discourse and the sciences are ampliative arguments. They are sometimes divided into inductive and abductive arguments. Inductive arguments usually take the form of statistical generalizations while abductive arguments are inferences to the best explanation. Arguments that fall short of the standards of correct reasoning are called fallacies. For formal fallacies, the source of the error is found in the form of the argument while informal fallacies usually contain errors on the level of the content or the context. Besides the definitory rules of logic, which determine whether an argument is correct or not, there are also strategic rules, which describe how a chain of correct arguments can be used to arrive at one's intended conclusion. In formal logic, formal systems are often used to give a precise definition of correct reasoning using a formal language.

Systems of logic are theoretical frameworks for assessing the correctness of reasoning and arguments. Aristotelian logic focuses on reasoning in the form of syllogisms. Its traditional dominance was replaced by classical logic in the modern era. Classical logic is "classical" in the sense that it is based on various fundamental logical intuitions shared by most logicians. It consists of propositional logic and first-order logic. Propositional logic ignores the internal structure of simple propositions and only considers the logical relations on the level of propositions. First-order logic, on the other hand, articulates this internal structure using various linguistic devices, such as predicates and quantifiers. Extended logics accept the basic intuitions behind classical logic and extend it to other fields, such as metaphysics, ethics, and epistemology. This happens usually by introducing new logical symbols, such as modal operators. Deviant logics, on the other hand, reject certain classical intuitions and provide alternative accounts of the fundamental laws of logic. While most systems of logic belong to formal logic, some systems of informal logic have also been proposed. One prominent approach understands reasoning as a dialogical game of persuasion while another focuses on the epistemic role of arguments. Logic is studied in and applied to various fields, such as philosophy, mathematics, computer science, and linguistics. Logic has been studied since Antiquity, early approaches including Aristotelian logic, Stoic logic, Anviksiki, and the mohists. Modern formal logic has its roots in the work of late 19th-century mathematicians such as Gottlob Frege.


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