Philosophical logic
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical logic in a wider sense as the study of the scope and nature of logic in general. In this sense, philosophical logic can be seen as identical to the philosophy of logic, which includes additional topics like how to define logic or a discussion of the fundamental concepts of logic. The current article treats philosophical logic in the narrow sense, in which it forms one field of inquiry within the philosophy of logic.
An important issue for philosophical logic is the question of how to classify the great variety of non-classical logical systems, many of which are of rather recent origin. One form of classification often found in the literature is to distinguish between extended logics and deviant logics. Logic itself can be defined as the study of valid inference. Classical logic is the dominant form of logic and articulates rules of inference in accordance with logical intuitions shared by many, like the law of excluded middle, the double negation elimination, and the bivalence of truth.
Extended logics are logical systems that are based on classical logic and its rules of inference but extend it to new fields by introducing new logical symbols and the corresponding rules of inference governing these symbols. In the case of alethic modal logic, these new symbols are used to express not just what is true simpliciter, but also what is possibly or necessarily true. It is often combined with possible worlds semantics, which holds that a proposition is possibly true if it is true in some possible world while it is necessarily true if it is true in all possible worlds. Deontic logic pertains to ethics and provides a formal treatment of ethical notions, such as obligation and permission. Temporal logic formalizes temporal relations between propositions. This includes ideas like whether something is true at some time or all the time and whether it is true in the future or in the past. Epistemic logic belongs to epistemology. It can be used to express not just what is the case but also what someone believes or knows to be the case. Its rules of inference articulate what follows from the fact that someone has these kinds of mental states. Higher-order logics do not directly apply classical logic to certain new sub-fields within philosophy but generalize it by allowing quantification not just over individuals but also over predicates.
Deviant logics, in contrast to these forms of extended logics, reject some of the fundamental principles of classical logic and are often seen as its rivals. Intuitionistic logic is based on the idea that truth depends on verification through a proof. This leads it to reject certain rules of inference found in classical logic that are not compatible with this assumption. Free logic modifies classical logic in order to avoid existential presuppositions associated with the use of possibly empty singular terms, like names and definite descriptions. Many-valued logics allow additional truth values besides true and false. They thereby reject the principle of bivalence of truth. Paraconsistent logics are logical systems able to deal with contradictions. They do so by avoiding the principle of explosion found in classical logic. Relevance logic is a prominent form of paraconsistent logic. It rejects the purely truth-functional interpretation of the material conditional by introducing the additional requirement of relevance: for the conditional to be true, its antecedent has to be relevant to its consequent.
Definition and related fields
The term "philosophical logic" is used by different theorists in slightly different ways. When understood in a narrow sense, as discussed in this article, philosophical logic is the area of philosophy that studies the application of logical methods to philosophical problems. This usually happens in the form of developing new logical systems to either extend classical logic to new areas or to modify it to include certain logical intuitions not properly addressed by classical logic. In this sense, philosophical logic studies various forms of non-classical logics, like modal logic and deontic logic. This way, various fundamental philosophical concepts, like possibility, necessity, obligation, permission, and time, are treated in a logically precise manner by formally expressing the inferential roles they play in relation to each other. Some theorists understand philosophical logic in a wider sense as the study of the scope and nature of logic in general. On this view, it investigates various philosophical problems raised by logic, including the fundamental concepts of logic. In this wider sense, it can be understood as identical to the philosophy of logic, where these topics are discussed. The current article discusses only the narrow conception of philosophical logic. In this sense, it forms one area of the philosophy of logic.Central to philosophical logic is an understanding of what logic is and what role philosophical logics play in it. Logic can be defined as the study of valid inferences. An inference is the step of reasoning in which it moves from the premises to a conclusion. Often the term "argument" is also used instead. An inference is valid if it is impossible for the premises to be true and the conclusion to be false. In this sense, the truth of the premises ensures the truth of the conclusion. This can be expressed in terms of rules of inference: an inference is valid if its structure, i.e. the way its premises and its conclusion are formed, follows a rule of inference. Different systems of logic provide different accounts for when an inference is valid. This means that they use different rules of inference. The traditionally dominant approach to validity is called classical logic. But philosophical logic is concerned with non-classical logic: it studies alternative systems of inference. The motivations for doing so can roughly be divided into two categories. For some, classical logic is too narrow: it leaves out many philosophically interesting issues. This can be solved by extending classical logic with additional symbols to give a logically strict treatment of further areas. Others see some flaw with classical logic itself and try to give a rival account of inference. This usually leads to the development of deviant logics, each of which modifies the fundamental principles behind classical logic in order to rectify their alleged flaws.
Classification of logics
Modern developments in the area of logic have resulted in a great proliferation of logical systems. This stands in stark contrast to the historical dominance of Aristotelian logic, which was treated as the one canon of logic for over two thousand years. Treatises on modern logic often treat these different systems as a list of separate topics without providing a clear classification of them. However, one classification frequently mentioned in the academic literature is due to Susan Haack and distinguishes between classical logic, extended logics, and deviant logics. This classification is based on the idea that classical logic, i.e. propositional logic and first-order logic, formalizes some of the most common logical intuitions. In this sense, it constitutes a basic account of the axioms governing valid inference. Extended logics accept this basic account and extend it to additional areas. This usually happens by adding new vocabulary, for example, to express necessity, obligation, or time. These new symbols are then integrated into the logical mechanism by specifying which new rules of inference apply to them, like that possibility follows from necessity. Deviant logics, on the other hand, reject some of the basic assumptions of classical logic. In this sense, they are not mere extensions of it but are often formulated as rival systems that offer a different account of the laws of logic.Expressed in a more technical language, the distinction between extended and deviant logics is sometimes drawn in a slightly different manner. On this view, a logic is an extension of classical logic if two conditions are fulfilled: all well-formed formulas of classical logic are also well-formed formulas in it and all valid inferences in classical logic are also valid inferences in it. For a deviant logic, on the other hand, its class of well-formed formulas coincides with that of classical logic, while some valid inferences in classical logic are not valid inferences in it. The term quasi-deviant logic is used if it introduces new vocabulary but all well-formed formulas of classical logic are also well-formed formulas in it and even when it is restricted to inferences using only the vocabulary of classical logic, some valid inferences in classical logic are not valid inferences in it. The term "deviant logic" is often used in a sense that includes quasi-deviant logics as well.
A philosophical problem raised by this plurality of logics concerns the question of whether there can be more than one true logic. Some theorists favor a local approach in which different types of logic are applied to different areas. Early intuitionists, for example, saw intuitionistic logic as the correct logic for mathematics but allowed classical logic in other fields. But others, like Michael Dummett, prefer a global approach by holding that intuitionistic logic should replace classical logic in every area. Monism is the thesis that there is only one true logic. This can be understood in different ways, for example, that only one of all the suggested logical systems is correct or that the correct logical system is yet to be found as a system underlying and unifying all the different logics. Pluralists, on the other hand, hold that a variety of different logical systems can all be correct at the same time.
A closely related problem concerns the question of whether all of these formal systems actually constitute logical systems. This is especially relevant for deviant logics that stray very far from the common logical intuitions associated with classical logic. In this sense, it has been argued, for example, that fuzzy logic is a logic only in name but should be considered a non-logical formal system instead since the idea of degrees of truth is too far removed from the most fundamental logical intuitions. So not everyone agrees that all the formal systems discussed in this article actually constitute logics, when understood in a strict sense.