Philosophy of logic

Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application. This involves questions about how logic is to be defined and how different logical systems are connected to each other. It includes the study of the nature of the fundamental concepts used by logic and the relation of logic to other disciplines. According to a common characterisation, philosophical logic is the part of the philosophy of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. But other theorists draw the distinction between the philosophy of logic and philosophical logic differently or not at all. Metalogic is closely related to the philosophy of logic as the discipline investigating the properties of formal logical systems, like consistency and completeness.

Various characterizations of the nature of logic are found in the academic literature. Logic is often seen as the study of the laws of thought, correct reasoning, valid inference, or logical truth. It is a formal science that investigates how conclusions follow from premises in a topic-neutral manner, i.e. independent of the specific subject matter discussed. One form of inquiring into the nature of logic focuses on the commonalities between various logical formal systems and on how they differ from non-logical formal systems. Important considerations in this respect are whether the formal system in question is compatible with fundamental logical intuitions and whether it is complete. Different conceptions of logic can be distinguished according to whether they define logic as the study of valid inference or logical truth. A further distinction among conceptions of logic is based on whether the criteria of valid inference and logical truth are specified in terms of syntax or semantics.

Different types of logic are often distinguished. Logic is usually understood as formal logic and is treated as such for most of this article. Formal logic is only interested in the form of arguments, expressed in a formal language, and focuses on deductive inferences. Informal logic, on the other hand, addresses a much wider range of arguments found also in natural language, which include non-deductive arguments. The correctness of arguments may depend on other factors than their form, like their content or their context. Various logical formal systems or logics have been developed in the 20th century and it is the task of the philosophy of logic to classify them, to show how they are related to each other, and to address the problem of how there can be a manifold of logics in contrast to one universally true logic. These logics can be divided into classical logic, usually identified with first-order logic, extended logics, and deviant logics. Extended logics accept the basic formalism and the axioms of classical logic but extend them with new logical vocabulary. Deviant logics, on the other hand, reject certain core assumptions of classical logic and are therefore incompatible with it.

The philosophy of logic also investigates the nature and philosophical implications of the fundamental concepts of logic. This includes the problem of truth, especially of logical truth, which may be defined as truth depending only on the meanings of the logical terms used. Another question concerns the nature of premises and conclusions, i.e. whether to understand them as thoughts, propositions, or sentences, and how they are composed of simpler constituents. Together, premises and a conclusion constitute an inference, which can be either deductive and ampliative depending on whether it is necessarily truth-preserving or introduces new and possibly false information. A central concern in logic is whether a deductive inference is valid or not. Validity is often defined in terms of necessity, i.e. an inference is valid if and only if it is impossible for the premises to be true and the conclusion to be false. Incorrect inferences and arguments, on the other hand, fail to support their conclusion. They can be categorized as formal or informal fallacies depending on whether they belong to formal or informal logic. Logic has mostly been concerned with definitory rules, i.e. with the question of which rules of inference determine whether an argument is valid or not. A separate topic of inquiry concerns the strategic rules of logic: the rules governing how to reach an intended conclusion given a certain set of premises, i.e. which inferences need to be drawn to arrive there.

The metaphysics of logic is concerned with the metaphysical status of the laws and objects of logic. An important dispute in this field is between realists, who hold that logic is based on facts that have mind-independent existence, and anti-realists like conventionalists, who hold that the laws of logic are based on the conventions governing the use of language. Logic is closely related to various disciplines. A central issue in regard to ontology concerns the ontological commitments associated with the use of logic, for example, with singular terms and existential quantifiers. An important question in mathematics is whether all mathematical truths can be grounded in the axioms of logic together with set theory. Other related fields include computer science and psychology.