logic-systems
Formal logic
is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule i.e. a rule that is not about any particular thing or property. In many definitions of logic, logical inference and inference with purely formal content are the same. Formal logic has started to take shape from the second half of the XIX century, with the use of formal and rigorous mathematical methods in the study of logic, which has introduced a shift towards a symbolic notation and language. The works of Aristotle contain the earliest known formal study of logic; modern formal logic follows and expands on Aristotle.
Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property.
The works of Aristotle contain the earliest known formal study of logic. Modern formal logic follows and expands on Aristotle. In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language.
Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic.
Informal logic
Informal logic is the study of natural language arguments. The study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, informal logic is not logic at all.
Symbolic logic
is the study of symbolic abstractions that capture the formal features of logical inference. It is often divided into two main branches: propositional and predicate logic.
Mathematical logic
is a subfield of mathematics exploring the applications of formal logic to mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Math logic is often divided into the fields of set theory, model theory, computability theory, proof theory, constructive mathematics. These areas share basic results on logic, particularly first-order logic, and definability.
Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.
However, agreement on what logic is has remained elusive, and although the field of universal logic has studied the common structure of logics, in 2007 Mossakowski et al. commented that "it is embarrassing that there is no widely acceptable formal definition of logic".
Description logics
Description logics (DL) are a family of formal knowledge representation languages, more expressive than propositional, but less expressive than FOL.
In contrast to FOL, the core reasoning problems for DLs are usually decidable, and efficient decision procedures have been designed and implemented for these problems.
There are general, spatial, temporal, spatiotemporal, and fuzzy descriptions logics, and each DL features a different balance between expressivity and reasoning complexity by supporting different sets of mathematical constructors.
DLs are used in AI to describe and reason about the relevant concepts of an application domain, known as terminological knowledge.
A DL models concepts, roles and individuals, and their relationships. The fundamental modeling concept of a DL is the axiom - a logical statement relating roles and/or concepts.
https://en.wikipedia.org/wiki/Metatheorem https://en.wikipedia.org/wiki/Extensionality https://en.wikipedia.org/wiki/Extension_(semantics) https://en.wikipedia.org/wiki/Intension https://en.wikipedia.org/wiki/Comprehension_(logic) https://en.wikipedia.org/wiki/Outline_of_logic https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Logic https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Philosophy
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