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 of philosophical logic in a broader 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.

The term "philosophical logic" is used by different theorists in slightly different ways. 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. 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.

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.

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: (1) all well-formed formulas of classical logic are also well-formed formulas in it and (2) all valid inferences in classical logic are also valid inferences in it. For a deviant logic, on the other hand, (a) its class of well-formed formulas coincides with that of classical logic, while (b) some valid inferences in classical logic are not valid inferences in it. The term quasi-deviant logic is used if (i) it introduces new vocabulary but all well-formed formulas of classical logic are also well-formed formulas in it and (ii) 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 refers primarily to propositional logic and first-order logic. For this reason, it neglects many topics of philosophical importance not relevant to mathematics, like the difference between necessity and possibility, between obligation and permission, or between past, present, and future. The concepts pertaining to propositional logic include propositional connectives, like "and", "or", and "if-then". Singular terms refer to objects and predicates express properties of objects and relations between them. Quantifiers constitute a formal treatment of notions like "for some" and "for all". They can be used to express whether predicates have an extension at all or whether their extension includes the whole domain. Quantification is only allowed over individual terms but not over predicates, in contrast to higher-order logics. Whether they are true or false is specified by the formal semantics. Possible worlds semantics specifies the truth conditions of sentences expressed in modal logic in terms of possible worlds. On this view, a sentence modified by the <math>\Diamond</math>-operator is true if it is true in at least one possible world while a sentence modified by the <math>\Box</math>-operator is true if it is true in all possible worlds. Of central importance in ethics are the concepts of obligation and permission, i.e. which actions the agent has to do or is allowed to do. Deontic logic usually expresses these ideas with the operators <math>O</math> and <math>P</math>. Just as in alethic modal logic, there is a discussion in philosophical logic concerning which is the right system of axioms for expressing the common intuitions governing deontic inferences. Expressed formally: . While similar approaches are often seen in physics, logicians usually prefer an autonomous treatment of time in terms of operators. This is also closer to natural languages, which mostly use grammar, e.g. by conjugating verbs, to express the pastness or futurity of events.

Epistemic

Epistemic logic is a form of modal logic applied to the field of epistemology.

Higher-order

Higher-order logics extend first-order logic by including new forms of quantification. In first-order logic, quantification is restricted to singular terms. It can be used to talk about whether a predicate has an extension at all or whether its extension includes the whole domain. This way, propositions like (there are some apples that are sweet) can be expressed. In higher-order logics, quantification is allowed not just over individual terms but also over predicates. This way, it is possible to express, for example, whether certain individuals share some or all of their predicates, as in (there are some qualities that Mary and John share). In first-order logic, this concerns only individuals, which is usually seen as an unproblematic ontological commitment. In higher-order logic, quantification concerns also properties and relations. In classical logic, every singular term has to denote an object in the domain of quantification. This is usually understood as an ontological commitment to the existence of the named entity. But many names are used in everyday discourse that do not refer to existing entities, like "Santa Claus" or "Pegasus". This threatens to preclude such areas of discourse from a strict logical treatment. Free logic avoids these problems by allowing formulas with non-denoting singular terms.

Karel Lambert, who coined the term "free logic", has suggested that free logic can be understood as a generalization of classical predicate logic just as predicate logic is a generalization of Aristotelian logic. On this view, classical predicate logic introduces predicates with an empty extension while free logic introduces singular terms of non-existing things. Formal semantics of classical logic can define the truth of their expressions in terms of their denotation. But this option cannot be applied to all expressions in free logic since not all of them have a denotation.

Many-valued

Many-valued logics are logics that allow for more than two truth values. They reject one of the core assumptions of classical logic: the principle of the bivalence of truth. The most simple versions of many-valued logics are three-valued logics: they contain a third truth value. In Stephen Cole Kleene's three-valued logic, for example, this third truth value is "undefined". So since it is true that "the sun is bigger than the moon", it is possible to infer that "the sun is bigger than the moon or Spain is controlled by space-rabbits". According to the disjunctive syllogism, one can infer that one of these disjuncts is true if the other is false. Without paraconsistent logics, dialetheism would be hopeless since everything would be both true and false.

Relevance

Relevance logic is one type of paraconsistent logic. As such, it also avoids the principle of explosion even though this is usually not the main motivation behind relevance logic. Instead, it is usually formulated with the goal of avoiding certain unintuitive applications of the material conditional found in classical logic. Classical logic defines the material conditional in purely truth-functional terms, i.e. is false if is true and is false, but otherwise true in every case. According to this formal definition, it does not matter whether and are relevant to each other in any way.