incoming information.[154] Correct reasoning and the arguments it is based on follow the laws of
probability, for example, the principle of conditionalization. Bad or irrational reasoning, on the other
hand, violates these laws.[155]
Logic is studied in various fields. In many cases, this is done by applying its formal method to specific
topics outside its scope, like to ethics or computer science.[156] In other cases, logic itself is made the
subject of research in another discipline. This can happen in diverse ways. For instance, it can involve
investigating the philosophical assumptions linked to the basic concepts used by logicians. Other ways
include interpreting and analyzing logic through mathematical structures as well as studying and
comparing abstract properties of formal logical systems.[157]
Philosophy of logic is the philosophical discipline studying the scope and nature of logic.[59] It examines
many presuppositions implicit in logic, like how to define its basic concepts or the metaphysical
assumptions associated with them.[158] It is also concerned with how to classify logical systems and
considers the ontological commitments they incur.[159] Philosophical logic is one of the areas within the
philosophy of logic. It studies the application of logical methods to philosophical problems in fields like
metaphysics, ethics, and epistemology.[160] This application usually happens in the form of extended or
deviant logical systems.[161]
Metalogic is the field of inquiry studying the properties of formal logical systems. For example, when a
new formal system is developed, metalogicians may study it to determine which formulas can be proven
in it. They may also study whether an algorithm could be developed to find a proof for each formula and
whether every provable formula in it is a tautology. Finally, they may compare it to other logical systems
to understand its distinctive features. A key issue in metalogic concerns the relation between syntax and
semantics. The syntactic rules of a formal system determine how to deduce conclusions from premises,
i.e. how to formulate proofs. The semantics of a formal system governs which sentences are true and
which ones are false. This determines the validity of arguments since, for valid arguments, it is
impossible for the premises to be true and the conclusion to be false. The relation between syntax and
semantics concerns issues like whether every valid argument is provable and whether every provable
argument is valid. Metalogicians also study whether logical systems are complete, sound, and consistent.
They are interested in whether the systems are decidable and what expressive power they have.
Metalogicians usually rely heavily on abstract mathematical reasoning when examining and formulating
metalogical proofs. This way, they aim to arrive at precise and general conclusions on these topics.[162]
The term "mathematical logic" is sometimes used as a synonym of "formal logic". But in a more
restricted sense, it refers to the study of logic within mathematics. Major subareas include model theory,
proof theory, set theory, and computability theory.[164] Research in mathematical logic commonly
addresses the mathematical properties of formal systems of logic. However, it can also include attempts
Areas of research
Philosophy of logic and philosophical logic
Metalogic
Mathematical logic